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ALGORITHMIC MARKET DESIGN
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF ECONOMICS
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Mohammad Akbarpour
June 2015
http://creativecommons.org/licenses/by-nc/3.0/us/
This dissertation is online at: http://purl.stanford.edu/xf731pn2513
© 2015 by Mohammad Akbarpour. All Rights Reserved.
Re-distributed by Stanford University under license with the author.
This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.
ii
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Paul Milgrom, Primary Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Matthew Jackson
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Alvin Roth
Approved for the Stanford University Committee on Graduate Studies.
Patricia J. Gumport, Vice Provost for Graduate Education
This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.
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Abstract
This thesis consists of two essays that exploit algorithmic techniques to solve two
matching market design problems.
The first essay introduces a simple benchmark model of dynamic matching in
networked markets, where agents arrive and depart stochastically and the network of
acceptable transactions among agents forms a random graph. The main insight of our
analysis is that waiting to thicken the market can be substantially more important
than increasing the speed of transactions. We also show that naıve local algorithms
that maintain market thickness by choosing the right time to match agents, but do
not exploit global network structure, can perform very close to optimal algorithms.
Finally, our analysis asserts that having information about agents’ departure times
is highly valuable. To elicit agents’ departure times when it is private, we design an
incentive-compatible continuous-time dynamic mechanism without transfers.
The second essay extends the scope of random allocation mechanisms, in which
the mechanism first identifies a feasible “expected allocation” and then implements
it by randomizing over nearby feasible integer allocations. Previous literature had
shown that the cases in which this is possible are sharply limited. We show that if
some of the feasibility constraints can be treated as goals rather than hard constraints
then, subject to weak conditions that we identify, any expected allocation that satis-
fies all the constraints and goals can be implemented by randomizing among nearby
integer allocations that satisfy all the hard constraints exactly and the goals at least
approximately. We show that by adding ex post utility goals to random serial dicta-
torship, we can construct a strategy-proof mechanism with the same ex ante utility
that is nearly ex post fair.
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To:
Those children who are victims of poverty; children who are born in un-
derprivileged families, who are deprived of nurturing childhood developmental oppor-
tunities, and consequently who will not perform as well as their privileged peers in
school. The challenge of these children is not to thrive, but to survive. In a paral-
lel universe, in which there is less poverty and less inequality, they would have had
incredible scientific contributions. The void of the next page is dedicated to those
“missing contributions.”
v
vi
Acknowledgements
The opportunity to learn from my advisors has been an extraordinary gift. I am
indebted to Paul Milgrom for being the best advisor anyone could ever imagine. His
continuous encouragement to think out of the box, his incredible depth and breadth of
knowledge in economics and computer science, and his confidence in me were essential
in my academic development. To Al Roth for inspiring me on a daily basis. His door
was always open to intellectual conversations. His unique way of thinking about
markets influenced me in an irreversible way. To Matt Jackson for caring about novel
ideas, as opposed to brute force calculations. He changed my way of thinking about
a valuable research in economics. To Mitch Polinsky for his life-changing trust in
me. And to Paul and Eva, Al and Emilie, Matt and Sara, and Mitch and Joan, for
welcoming students as family members.
I am truly thankful to my parents, Mansoureh and Mohammadali; my sisters,
Maryam and Fatemeh; and my grandparents for their unconditional love. They made
my childhood beautiful and they never stopped loving me.
I am deeply grateful to my wonderful coauthors – Shayan, Shengwu, Afshin, and
Sam – for being great friends, as well as passionate scientists. I learned a lot from
them, and I am looking forward to collaborate with them on many more projects to
come. To tens of economists and computer scientists who inspired me in the past few
years (I name a few of them at the beginning of each Chapter). To my classmates
at Stanford economics department for their significant impact on my mental growth
as an economist. To Ali Naghi Mashayekhi and Masoud Nili for encouraging me to
pursue economics, and to Yahya Tabesh and Amin Saberi for supporting me in my
way from engineering to economics. To Caro Lucas for proving that one can fall
vii
in love with knowledge – Rest in peace, Caro. To Vahid Karimipour for making
Maxwell’s equations as beautiful as Hafez’s poems. To Amir Asghari for making high
school mathematics as joyful as The Neverhood. To Farhad Meysami for making me
fall in love with scientific thinking.
I am thankful to the wonderful founding team of KelaseDars and Khan Academy
Farsi – Shima, Reza, Sahar, and Alireza. They taught me that real happiness is in
sharing what you have. To all my friends, who together we watched movies, camped
at Tahoe and Yosemite, played football and squash, talked, walked, debated, smiled,
and shared moments, feelings and stories. To name a few (in a random order): Mo-
hammad, Maryam, Nima, Reza, Arash, Nushin, Mohsen, Parastoo, Babak, Alireza,
Mahnoosh, Saeed, Mohsen, Zeinab, Tahereh, Mohammadreza, Azar, Behnam, Hazhir,
Sara, Rad, Mahmoud, Behrad, Neda, Hadi, Hanieh, Hossein, Homeira, Adel, Shayan,
Farnaz, Hamed, Maryam, Hamed, Marzieh, Mohsen, Narges, Shima, Shahin, Leili,
Ian, Soheil, Amin, Farid, Reza, Masoud, Milad, Ali, Kaveh, Leila, Pouyan, Nazanin,
Mohammad, Salman, Sina, Sanam, Soha, Mohammad, Hessam, Kaveh, Keyvan, Ah-
madali, Amir – Thank you all for making me much happier.
Last but, of course, not the least, I am truly thankful to Shima, my best friend
and the kindest supporter, who was there whenever I needed her, made me smile even
when I was sad, and influenced me more than any other person in my life.
viii
Contents
Abstract iv
Acknowledgements vii
1 Introduction 1
2 Dynamic Matching Markets 6
2.0.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Our Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Timing in Matching Markets . . . . . . . . . . . . . . . . . . . 21
2.2.2 Welfare Under Discounting and Optimal Waiting Time . . . . 26
2.2.3 Information and Incentive-Compatibility . . . . . . . . . . . . 29
2.2.4 Technical Contributions . . . . . . . . . . . . . . . . . . . . . 32
2.3 Performance of the Optimum and Periodic Algorithms . . . . . . . . 33
2.3.1 Loss of the Optimum Online Algorithm . . . . . . . . . . . . . 35
2.3.2 Loss of the Omniscient Algorithm . . . . . . . . . . . . . . . . 36
2.4 Modeling an Online Algorithm as a Markov Chain . . . . . . . . . . . 37
2.4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.2 Markov Chain Characterization . . . . . . . . . . . . . . . . . 39
2.5 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5.1 Loss of the Greedy Algorithm . . . . . . . . . . . . . . . . . . 40
2.5.2 Loss of the Patient Algorithm . . . . . . . . . . . . . . . . . . 45
2.5.3 Loss of the Patient(α) Algorithm . . . . . . . . . . . . . . . . 48
ix
2.6 Welfare and Optimal Waiting Time under Discounting . . . . . . . . 49
2.6.1 Welfare of the Patient Algorithm . . . . . . . . . . . . . . . . 50
2.6.2 Welfare of the Greedy Algorithm . . . . . . . . . . . . . . . . 52
2.7 Incentive-Compatible Mechanisms . . . . . . . . . . . . . . . . . . . . 52
2.8 Concluding Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.8.1 Insights of the Paper . . . . . . . . . . . . . . . . . . . . . . . 57
2.8.2 Discussion of Assumptions . . . . . . . . . . . . . . . . . . . . 58
2.8.3 Further Extensions . . . . . . . . . . . . . . . . . . . . . . . . 59
3 Random Allocation Mechanisms 61
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1.1 Model and Contributions . . . . . . . . . . . . . . . . . . . . . 63
3.1.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2.1 Approximate Implementation . . . . . . . . . . . . . . . . . . 71
3.3 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.1 The Structure of Soft Blocks . . . . . . . . . . . . . . . . . . . 74
3.3.2 Corollary 1: Fully General Soft Structure . . . . . . . . . . . . 76
3.3.3 Corollary 2: Local Structure . . . . . . . . . . . . . . . . . . . 76
3.3.4 Generalized Structures . . . . . . . . . . . . . . . . . . . . . . 78
3.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.4.1 Diversity Requirements in School Choice . . . . . . . . . . . . 80
3.4.2 Distance-based Walk-zone Priorities . . . . . . . . . . . . . . . 82
3.4.3 Ex post Guarantees . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5 Fixing Random Serial Dictatorship . . . . . . . . . . . . . . . . . . . 85
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
A Missing Proofs From Chapter 2 89
A.1 Auxiliary Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
A.2 Proof of Theorem 2.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 90
A.2.1 Stationary Distributions: Existence and Uniqueness . . . . . . 90
A.2.2 Upper bounding the Mixing Times . . . . . . . . . . . . . . . 91
x
A.2.3 Mixing time of the Greedy Algorithm . . . . . . . . . . . . . . 91
A.2.4 Mixing time of the Patient Algorithm . . . . . . . . . . . . . . 93
A.3 Proofs from Section 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 96
A.3.1 Proof of Lemma 2.5.4 . . . . . . . . . . . . . . . . . . . . . . . 96
A.3.2 Proof of Lemma 2.5.7 . . . . . . . . . . . . . . . . . . . . . . . 97
A.3.3 Proof of Lemma 2.5.8 . . . . . . . . . . . . . . . . . . . . . . . 97
A.3.4 Proof of Proposition 2.5.9 . . . . . . . . . . . . . . . . . . . . 98
A.3.5 Proof of Lemma 2.5.10 . . . . . . . . . . . . . . . . . . . . . . 100
A.4 Proofs from Section 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . 102
A.4.1 Proof of Lemma 2.6.3 . . . . . . . . . . . . . . . . . . . . . . . 102
A.5 Proofs from Section 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . 103
A.5.1 Proof of Lemma 2.7.4 . . . . . . . . . . . . . . . . . . . . . . . 103
A.6 Small Market Simulations . . . . . . . . . . . . . . . . . . . . . . . . 107
B Missing Proofs From Chapter 3 110
B.1 Implementation: A Random Mechanism . . . . . . . . . . . . . . . . 110
B.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
B.1.2 Operation X . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
B.1.3 The Implementation Mechanism . . . . . . . . . . . . . . . . . 115
B.1.4 Approximate Satisfaction of Soft Constraints . . . . . . . . . . 120
B.2 Average Performance of the Matching Algorithm . . . . . . . . . . . 124
B.3 Chernoff Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Bibliography 128
xi
List of Figures
2.1 Patient algorithm is not optimal . . . . . . . . . . . . . . . . . . . . . 23
2.2 Optimal trade frequency as a function of discount rate . . . . . . . . 28
2.3 Greedy algorithm Markov Chain . . . . . . . . . . . . . . . . . . . . . 42
2.4 Patient algorithm Markov Chain . . . . . . . . . . . . . . . . . . . . . 46
3.1 Assignment problem framework . . . . . . . . . . . . . . . . . . . . . 69
3.2 Capacity blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3 A hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4 Failure of bihierarchy assumption . . . . . . . . . . . . . . . . . . . . 71
3.5 An illustration of the deepest level condition . . . . . . . . . . . . . . 74
3.6 Implementation mechanism . . . . . . . . . . . . . . . . . . . . . . . 75
3.7 Local structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.8 Depth k condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.9 School choice application . . . . . . . . . . . . . . . . . . . . . . . . . 82
A.1 A Markov Chain to study the mixing time . . . . . . . . . . . . . . . 95
A.2 Small market simulations . . . . . . . . . . . . . . . . . . . . . . . . . 108
B.1 A floating cycle of length 6 . . . . . . . . . . . . . . . . . . . . . . . . 112
B.2 A floating path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
B.3 Average performance simulations I . . . . . . . . . . . . . . . . . . . 125
B.4 Average performance simulations II . . . . . . . . . . . . . . . . . . . 126
xii
Chapter 1
Introduction
Economics is the science of the allocation of scarce resources. Consequently, who gets
what is one of the most fundamental questions of economics. In some marketplaces,
such as the New York Stock Exchange, prices determine who gets what. In such
markets, if you can afford something, you can have it. In some other marketplaces,
such as the allocation of students to public schools, prices play very little or no role.
Matching markets, to start with one definition, are markets in which prices are
not the only facto determining who gets what. In some matching markets, such
as the allocation of organs, monetary transfers are fully precluded. In some other
matching markets, such as the labor market or the college admissions market, prices
play some role but they are not the only facto determining the allocation outcome.
An undergraduate education at Stanford University, for instance, is expensive, but
there are many more people who are willing to pay Stanford’s tuition fee than there
are people who gain admission.
Matching markets surround us: the kidney exchange market, the National Resi-
dency Matching Program (NRMP), the school choice systems, many offline and online
labor markets such as Upwork and Uber, the allocation of courses to students in busi-
ness schools, foster care systems, and the assignment of cadets to military bases are
all examples of markets in which prices do not do all the work.
1
CHAPTER 1. INTRODUCTION 2
Economists (and computer scientists) have widely studied matching markets dur-
ing the past half century.1 Perhaps surprisingly, and despite the fact that hundreds of
papers have studied matching markets, many important aspects of matching markets
are still undertheorized. There are, in my opinion, at least two reasons for this. The
first reason, as simple as it may seem, is that we did not have enough time to study
them. Many matching markets have emerged in recent years; some of them were
formed over the Internet (e.g., online labor markets or Airbnb) and some others were
formed based on recent scientific breakthroughs (e.g., organ transplant technology).
It is not too unrealistic to claim that the arrival rate of new applications motivating
research problems is higher than the arrival rate of researchers who are working to
solve them. The second reason why some aspects of matching markets are underthe-
orized is that the mathematical tools and techniques of the field have not kept up the
pace with the computational complexities of emerging marketplaces.
In my dissertation, I envision Algorithmic Market Design as a conceptual and
technical paradigm that – by exploiting tools from algorithm design and other areas
of theoretical computer science – aims to fill the gap between the theory of matching
market design and the practice of the emerging, complex market design problems,
especially in dynamic and networked environments.
I want to emphasize that in my opinion, the abstractions in simple theoretical
models can be highly valuable because they focus attention on specific aspects of the
dynamics that determine outcomes in a market. Nevertheless, if the requirement of
analytical tractability limits our tools in such a way that essential components or
the interplays between them are overlooked, then we may not achieve the goal of the
modeling exercise. As I will show in this dissertation, dynamical aspects and complex
structures can be crucial parts of optimal designs; therefore the models that abstract
from them for the sake of simplicity can be misleading.
This dissertation contains two essays on matching market design. The first essay
is concerned with an allocation problem in a dynamic, networked environment. The
complexity of the model arises from the fact that the state space of the problem is
the set of all possible networks that stochastically evolve over time. The second essay
1For a discussion of those studies, see Chapter 2.
CHAPTER 1. INTRODUCTION 3
concerns market-making in a static resource allocation problem with complex quotas.
The complexity of the problem arises from the structure of the quotas that must be
satisfied in the allocation.
The first essay2 is concerned with the “option value” of waiting in dynamic, net-
worked markets. In dynamic matching markets (such as the kidney exchange market,
or Uber, or dating platforms), in contrast to static ones, those agents that are not
matched may stay in the market to be matched later. Thus, a matching policy will
not only affect who gets matched today, but will also affect what the composition of
options will be tomorrow. Consequently, designing matching algorithms for such en-
vironments is a dynamic decision problem in which waiting can be valuable. Waiting
expands the planner’s information through at least two different channels. First, it
resolves the uncertainty about the future matching opportunities. Second, waiting
increases the planner’s information about which agents have more urgent needs than
others. On the other hand, waiting can be costly. The central questions of the paper
are: when should the planner wait and how long should the planner wait for?
Because we explicitly model the underlying ‘exchange possibilities network’, the
state space of the planner’s problem is the set of all possible networks, which is com-
putationally complex and not soluble via standard dynamic programming techniques.
This is exactly why we exploit tools and techniques from algorithm design and stochas-
tic processes to analyze this problem. We design some heuristic matching algorithms
and bound their performance. Then, by comparing those bounds to the bounds that
we get for the optimal solutions, we identify key features of matching algorithms in
dynamic environments. In addition, we design an incentive-compatible mechanism
to extract valuable information truthfully and study some interesting comparative
statics results. This is done without explicitly solving the underlying optimization
problems of the model.
The second essay3 of this dissertation is concerned with the allocation of indivisible
goods when cash transfers are prohibited and the final allocation is required to satisfy
2The first essay is based on the paper, Dynamic Matching Market Design, which is a joint workwith Shengwu Li and Shayan Oveis Gharan [6]
3The second essay is based on the paper, Approximate Random Allocation Mechanisms, which isa joint work with Afshin Nikzad [7].
CHAPTER 1. INTRODUCTION 4
multiple quotas. A popular way to solve this problem is the ‘expected assignment’
method. In this mechanism, we first identify a fair and efficient expected assignment,
and then implement it by randomizing over feasible integer allocations. Unfortunately,
the constraint structures for which this is possible are sharply limited. The key
contribution of this paper is to show that by reconceptualizing some constraints from
‘hard’ objectives to ‘goals’, one can accommodate many more constraints into this
allocation problem. The key technical novelty of this result is in designing a unique
matching algorithm that allocates objects to agents in such a way that constraints
are exactly satisfied and goals are satisfied, at least approximately.
Both essays of the thesis are concerned with resource allocation in markets where
prices play very little or no role; where finding the “optimal” solution is not feasible.
In the first essay, finding the optimum is computationally complex. In the second
essay, the optimum satisfaction of constraints is theoretically impossible. To over-
come this issue in both cases, we aim to find “good enough”, rather than optimal,
solutions. In the first essay, by simplifying the design space, we show that a simple
matching algorithm that ignores the network complexity, but chooses the optimal
level of market thickness performs very close to the optimum solution. In the second
essay, we show that by satisfying some constraints in an approximate sense - a “good
enough” solution - one can overcome an impossibility result.
I define Algorithmic Market Design as a subfield of market design, which deals
with market design problems in which finding the optimum solution is either com-
putationally complex or theoretically impossible. In such cases, the approach of an
algorithmic market designer – who knows that “the best” is the greatest enemy of
“the good” – is to design a good enough allocation policy, rather than a perfect one.
It is often the case that an attempt to find an allocation policy that is approximately
optimal will improve our understanding of the key features of the (unachievable) opti-
mum. Furthermore, since we can hardly identify the functional form of the objective
function in most problems, an attempt to understand key features of good allocation
policies in a robust way can be even more valuable than solving for the optimum
policy for a given the functional form.
The two essays of this dissertation are two examples that exhibit the usefulness
CHAPTER 1. INTRODUCTION 5
and applications of Algorithmic Market Design in solving resource allocation problems
of computational complexity. I hope that these two examples, and the tools and
techniques described here, will pave the way for more applications of Algorithmic
Market Design in solving the emerging, complex market design problems.
Chapter 2
Dynamic Matching Markets
The theory of matching has guided the designs of many markets, from school choice,
to kidney exchange, to the allocation of medical residents. In a series of classic
papers, economists have extensively characterized good matching algorithms for static
settings.1 In the canonical set-up, a social planner faces a set of agents who have
preferences over partners, contracts, or combinations thereof. The planner’s goal is
to find a matching algorithm with desirable properties (e.g. stability, efficiency, or
strategy-proofness). The algorithm is run, a match is made, and the problem ends.
Of course, many real-world matching problems are dynamic. In a dynamic match-
ing environment, agents arrive gradually over time. A social planner continually ob-
serves the agents and their preferences, and chooses how to match agents. Matched
agents leave the market. Unmatched agents either persist or depart. Thus, the plan-
ner’s decision today affects the sets of agents and options tomorrow.
Some seasonal markets, such as school choice systems and the National Residency
Matching Program, are well described as static matching problems without intertem-
poral spillovers. However, some markets are better described as dynamic matching
problems. Some examples include:
• Kidney exchange: In paired kidney exchanges, patient-donor pairs arrive over
time. They stay in the market until either they are matched to a compatible
1See [35, 30, 49, 71, 72, 42, 73, 41].
6
CHAPTER 2. DYNAMIC MATCHING MARKETS 7
pair, their condition deteriorates, or they receive a cadaveric kidney from a
waiting list.
• Markets with brokers: Some markets, such as real estate, aircraft, and ship
charters, involve intermediary brokers who receive requests to buy or sell par-
ticular items. A broker facilitates transactions between compatible buyers and
sellers, but does not hold inventory. Agents may withdraw their request if they
find an alternative transaction.
• Allocation of workers to time-sensitive tasks: Both within firms and
online labor markets, such as Uber and oDesk, planners allocate workers to
tasks that are profitable to undertake. Tasks arrive continuously, but may
expire. Workers are suited to different tasks, but may cease to be available.
In dynamic settings, the planner must decide not only which agents to match,
but also when to match them. If the planner waits, new agents may arrive, and a
more socially desirable match may be found. Waiting, in addition, will increase the
planner’s information about which agents’ needs are more urgent than others. On
the other hand, waiting might impose waiting costs on agents.
This paper deals with identifying features of optimal matching algorithms in dy-
namic environments. Our discussion for the benefits and costs of waiting suggests
that static matching models do not capture important features of dynamic match-
ing markets. Obviously, waiting may bring new agents, and thus expand the set of
feasible matchings. More generally, in a static setting, the planner chooses the best
algorithm for an exogenously given set of agents and their preferences. By contrast,
in a dynamic setting, the set of agents and trade options at each point in time depend
endogenously on the matching algorithm.
The optimal timing policy in a dynamic matching problem is not obvious a priori.
In practice, many paired kidney exchanges enact static matching algorithms (‘match-
runs’) at fixed intervals.2 Even then, matching intervals differ substantially between
exchanges: The Alliance for Paired Donation conducts a match-run once a weekday,
2In graph theory, a matching is a set of edges that have no nodes in common.
CHAPTER 2. DYNAMIC MATCHING MARKETS 8
the United Network for Organ Sharing conducts a match-run once a week3, the South
Korean kidney exchange conducts a match-run once a month, and the Dutch kidney
exchange conducts a match-run once a quarter [8]. This shows that policymakers
select different timing policies when faced with seemingly similar dynamic matching
problems. It is therefore useful to identify good timing policies, and to investigate
how policy should depend on the underlying features of the problem.
In this paper, we create and analyze a simple model of dynamic matching on
networks. Agents arrive and depart stochastically. We use binary preferences, where
a pairwise match is either acceptable or unacceptable, generated according to a known
distribution. These preferences are persistent over time, and agents may discount the
future. The set of agents (vertices) and the set of potential matches (edges) form
a random graph. Agents do not observe the set of acceptable transactions, and are
reliant upon the planner to match them to each other. We say that an agent perishes
if she leaves the market unmatched.
The planner’s problem is to design a matching algorithm; that is, at any point
in time, to select a subset of acceptable transactions and broker those trades. The
planner observes the current set of agents and acceptable transactions, but has only
probabilistic knowledge about the future. The planner may have knowledge about
which agents’ needs are urgent, in the sense that he may know which agents will
perish imminently if not matched. The goal of the planner is to maximize the sum
of the discounted utilities of all agents. In the important special case where the cost
of waiting is zero, the planner’s goal is equivalent to minimizing the proportion of
agents who perish. We call this the loss of an algorithm.
In this setting, the planner faces a trade-off between matching agents quickly and
waiting to thicken the market. If the planner matches agents frequently, then matched
agents will not have long to wait, but it will be less likely that any remaining agent
has a potential match (a thin market). On the other hand, if the planner matches
agents infrequently, then there will be more agents available, making it more likely
that any given agent has a potential match (a thick market).
When facing a trade-off between the frequency of matching and the thickness of
3See http://www.unos.org/docs/Update_MarchApril13.pdf
CHAPTER 2. DYNAMIC MATCHING MARKETS 9
the market, what is the optimal timing policy? Because we explicitly model the graph
structure of the planner’s matching problem, the state space of the resulting Markov
Decision Problem is combinatorially complex. Thus, it is not amenable to solution
via standard dynamic programming techniques. Instead, to analyze the model, we
formulate simple matching algorithms with different timing properties, and compare
them to analytic bounds on optimum performance. This will enable us to investigate
whether timing is an important feature of dynamic matching algorithms.
Our algorithms are as follows: The Greedy algorithm attempts to match each agent
upon arrival; it treats each instant as a static matching problem without regard for the
future.4 The Patient algorithm attempts to match patients on the verge of leaving the
market, potentially by matching them to a non-urgent patient. Both these algorithms
are local, in the sense that they look only at the immediate neighbors of the agent
they attempt to match rather than at the global graph structure5. We also study a
family of algorithms that speed up the trade frequency of the Patient algorithm. The
Patient(α) algorithm attempts to match urgent cases, and additionally attempts to
match each non-urgent case at some rate determined by α6.
We now state our main results. First, we analyze the performance of algorithms
with different timing properties, in the benchmark setting where the planner can
identify urgent cases. Second, we relax our informational assumption, and thereby
establish the value of short-horizon information about urgent cases. Third, we exhibit
a dynamic mechanism that truthfully elicits such information from agents.
Our first family of results concerns the problem of timing in dynamic matching
markets. First, we establish that the loss of the Patient algorithm is exponentially
(in the average degree of agents) smaller than the loss of the Greedy algorithm. This
entails that, for even moderately dense markets, the Patient algorithm substantially
outperforms the Greedy algorithm. For example, suppose on average agents perish
4Our analysis of the Greedy Algorithm encompasses waiting list policies where brokers maketransactions as soon as they are available, giving priority to agents who arrived earlier.
5Example 2.2.5 shows a case in which ‘locality’ of the Patient algorithms makes it suboptimal.6More precisely, every non-urgent agent is treated as urgent when an exogenous “exponential
clock” ticks and attempted to be matched either in that instance, or when she becomes truly urgent.
CHAPTER 2. DYNAMIC MATCHING MARKETS 10
after one year. In a market where 1000 agents arrive every year and and the proba-
bility of an acceptable transaction is 1100
, the loss of the Patient algorithm is no more
than 7% of the loss the Greedy algorithm. Thus, varying the timing properties of
simple algorithms has large effects on their performance.
Second, we find that the loss of the Patient algorithm is close to the loss of the
optimum algorithm. Recall that the Patient algorithm is local; it looks only in the
immediate neighborhood of the agents it seeks to match. By contrast, the optimum
algorithm is global and potentially very complex; the matchings it selects depend on
the entire graph structure. Thus, this result suggests that the gains from waiting to
thicken the market are large compared to the total gains from considering the global
network structure.
Third, we find that it is possible to accelerate the Patient algorithm and still
achieve exponentially small loss7. That is, we establish a bound for the tuning pa-
rameter α such that the Patient(α) algorithm has exponentially small loss. Given
the same parameters as in our previous example, under the Patient(α) algorithm, the
planner can promise to match agents in less than 4 months (in expectation) while the
loss is at most 37% of the loss of the Greedy algorithm. Thus, even moderate degrees
of waiting can substantially reduce the proportion of perished agents.
Next, we examine welfare under discounting. We show that for a range of discount
rates, the Patient algorithm delivers higher welfare than the Greedy algorithm, and
for a wider range of discount rates, there exists α such that the Patient(α) algorithm
delivers higher welfare than the Greedy algorithm. Then, in order to capture the
trade-off between the trade frequency and the thickness of the market, we solve for
the optimal waiting time as a function of the market parameters. Our comparative
statics show that the optimal waiting time is increasing in the sparsity of the graph.
Our second family of results relaxes the informational assumptions in the bench-
mark model. Suppose that the planner cannot identify urgent cases; i.e. the planner
has no individual-specific information about departure times. We find that the loss
of the Patient algorithm, which naıvely exploits urgency information, is exponentially
7As before, the exponent is in the average degree of agents.
CHAPTER 2. DYNAMIC MATCHING MARKETS 11
smaller than the loss of the optimum algorithm that lacks such information.8 This
suggests that short-horizon information about departure times is very valuable.
On the other hand, suppose that the planner has more than short-horizon infor-
mation about agent departures. The planner may be able to forecast departures long
in advance, or foresee how many new agents will arrive, or know that certain agents
are more likely than others to have new acceptable transactions. We prove that no ex-
pansion of the planner’s information allows him to achieve a better-than-exponential
loss. Taken as a pair, these results suggest that short-horizon information about de-
parture times is especially valuable to the planner. Lacking this information leads to
large losses, and having more than this information does not yield large gains.
In some settings, however, agents have short-horizon information about their de-
parture times, but the planner does not. Our final result concerns the incentive-
compatible implementation of the Patient(α) algorithm.9 Under private information,
agents may have incentives to mis-report their urgency so as to hasten their match
or to increase their probability of getting matched. We show that if agents are not
too impatient, a dynamic mechanism without transfers can elicit such information.
The mechanism treats agents who report that their need is urgent, but persist, as
though they had left the market. This means that agents trade off the possibility of a
swifter match (by declaring that they are in urgent need now) with the option value
of being matched to another agent who has an urgent need in future. We prove that
it is arbitrarily close to optimal for agents to report the truth in large markets.
The rest of the paper is organized as follows. Section 2.1 introduces our dynamic
matching market model and defines the objective. Section 2.2 presents our main con-
tributions; we recommend that readers consult this section to see a formal statement
of our results without getting into the details of the proofs. Section 2.3 analyzes two
optimal policies as benchmarks and provides analytic bounds on their performance.
8This result has some of the flavor of Bulow and Klemperer’s theorem [25] comparing simpleauctions to optimal negotiations. They show that simple auctions with N + 1 bidders raise morerevenue than optimal mechanisms with N bidders. We show that simple matching algorithms thatthicken the market by exploiting urgency information are better than optimal algorithms that donot.
9Note that the Patient(α) algorithm contains the Patient algorithm as a special case.
CHAPTER 2. DYNAMIC MATCHING MARKETS 12
Section 2.4 models our algorithms as Markov Chains and bounds the mixing times
of the chains. Section 2.5 goes through a deep analysis of the Greedy algorithm, the
Patient algorithm, and the Patient(α) algorithm and bounds their performance. Sec-
tion 2.6 takes waiting costs into account and bounds the social welfare under different
algorithms. Section 2.7 considers the case where the urgency of an agent’s needs is
private information, and exhibits a truthful direct revelation mechanism. Section 2.8
includes the concluding discussions.10
2.0.1 Related Work
There have been several studies on dynamic matching in the literatures of economics,
computer science, and operations research, that each fit a specific marketplace, such
as the real estate market, paired kidney exchange, or online advertising. To the
best of our knowledge, no previous work has offered a general framework for dynamic
matching in networked markets, and no previous work has considered stochastic agent
departures. This paper is also the first to produce analytic results on bilateral dynamic
matching that explicitly account for discounting.11
[54] and [18] study an overlapping generations model of the housing market. In
their models, agents have deterministic arrivals and departures. In addition, the
housing side of the market is infinitely durable and static, and houses do not have
preferences over agents. In the same context, [55] studies a one-sided dynamic housing
allocation problem in which houses arrive stochastically over time. His model is based
on two waiting lists and does not include a network structure. In addition, agents
remain in the waiting list until they are assigned to a house; i.e., they do not perish.
10This paper has benefited from helpful comments of many people. We thank Paul Milgromand Alvin Roth for valuable comments and suggestions. We also thank Itai Ashlagi, Peter Biro,Timothy Bresnahan, Jeremy Bulow, Gabriel Carroll, Ben Golub, Matthew Jackson, Fuhito Kojima,Scott Kominers, Soohyung Lee, Jacob Leshno, Malwina Luczak, Stephen Nei, Muriel Niederle,Afshin Nikzad, Michael Ostrovsky, Takuo Sugaya, Bob Wilson, and Alex Wolitzky for their valuablecomments, as well as several seminar participants for helpful suggestions. All errors remain our own.
11Other analytic results in bilateral dynamic matching concern statistics such as average waitingtimes, which are related to but not identical with discounted utility. See a recent study of dynamicbarter exchange markets by [9], where agents never perish and the main objective is the averagewaiting time. They show that when only pairwise exchanges are allowed, the Greedy algorithm isclose to the optimum, which is similar to Theorem 2.2.11.
CHAPTER 2. DYNAMIC MATCHING MARKETS 13
In the context of live-donor kidney exchanges, [76] studies an interesting model
of the dynamic kidney exchange in which agents have multiple types. In his model,
agents never perish and so, one insight of his model is that waiting to thicken the
market is not helpful when only bilateral exchanges are allowed. This is very different
from the insights of our paper. In the Operations Research and Computer Science
literatures, dynamic kidney matching has been extensively studied, see e.g., [79, 75,
14, 31]. Perhaps most related to our work is that of [11] who construct a discrete-time
finite-horizon model of dynamic kidney exchange. Unlike our model, agents who are
in the pool neither perish, nor bear any waiting cost, and so they do not model agents’
incentives. Their model has two types of agents, one easy to match and one hard
to match, which then creates a specific graph structure that fits well to the kidney
market.
In an independent concurrent study, inspired by online labor markets such as
oDesk, [10] model a dynamic two-sided dynamic matching market and show that
reducing search and screening costs does not necessarily increase welfare. Their main
goal is to analyze congestion in decentralized dynamic markets, as opposed to our goal
which is to study the “when to match” question from a central planning perspective.
The problem of online matching has been extensively studied in the literature
of online advertising. In this setting, advertisements are static, but queries arrive
adversarially or stochastically over time. Unlike our model, queries persist in the
market for exactly one period. [47] introduced the problem and designed a randomized
matching algorithm. Subsequently, the problem has been considered under several
arrival models with pre-specified budgets for the advertisers, [61, 37, 34, 59].
In contrast to dynamic matching, there are numerous investigations of dynamic
auctions and dynamic mechanism design. [67] generalize the VCG mechanism to
a dynamic setting. [12] construct efficient and incentive-compatible dynamic mecha-
nisms for private information settings. [65] and [36] extend Myerson’s optimal auction
result [62] to dynamic environments. We refer interested readers to [66] for a review
of the dynamic mechanism design literature.
CHAPTER 2. DYNAMIC MATCHING MARKETS 14
2.1 The Model
In this section, we provide a stochastic continuous-time model for a bilateral matching
market that runs in the interval [0, T ]. Agents arrive at the market at ratem according
to a Poisson process. Hence, in any interval [t, t+ 1], m new agents enter the market
in expectation. Throughout the paper we assume m ≥ 1. For t ≥ 0, let At be the set
of the agents in our market at time t, and let Zt := |At|. We refer to At as the pool
of the market. We start by describing the evolution of At as a function of t ∈ [0, T ].
Since we are interested in the limit behavior of At, without loss of generality, we may
assume A0 = ∅. We use Ant to denote12 the set of agents who enter the market at
time t. Note that with probability 1, |Ant | ≤ 1. Also, let |Ant0,t0+t1| denote the set of
agents who enter the market in time interval [t0, t1].
Each agent becomes critical according to an independent Poisson process with rate
λ. This implies that, if an agent a enters the market at time t0, then she becomes
critical at some time t0+X where X is an exponential random variable with parameter
λ. Any critical agent leaves the market immediately; so the last point in time that an
agent can get matched is the time that she gets critical. We say an agent a perishes
if a leaves the market unmatched.13
We assume that an agent a ∈ At leaves the market at time t, if any of the following
three events occur at time t:
• a is matched with another agent b ∈ At,
• a becomes critical and gets matched
• a becomes critical and leaves the market unmatched, i.e., a perishes.
Say a enters the market at time t0 and gets critical at time t0 + X where X
is an exponential random variable with parameter λ. By above discussion, for any
matching algorithm, a leaves the market at some time t1 where t0 ≤ t1 ≤ t0 + X
12As a notational guidance, we use subscripts to refer to a point in time or a time interval, whilesuperscripts n, c refer to new agents and critical agents, respectively.
13We intend this as a term of art. In the case of kidney exchange, perishing can be interpreted asa patient’s medical condition deteriorating in such a way as to make transplants infeasible.
CHAPTER 2. DYNAMIC MATCHING MARKETS 15
(note a may leave sooner than t0 +X if she gets matched before getting critical). The
sojourn of a is the length of the interval that a is in the pool, i.e., s(a) := t1 − t0.
We use Act to denote the set of agents that are critical at time t.14 Also, note that
for any t ≥ 0, with probability 1, |Act | ≤ 1.
For any pair of agents, the probability that a bilateral transaction between them is
acceptable is d/m, where 0 ≤ d ≤ m and these probabilities are independent. For the
sake of clarity of notation, we may use q := 1− d/m. For any t ≥ 0, let Et ⊆ At×Atbe the set of acceptable bilateral transactions between the agents in the market (the
set of edges) at time t, and let Gt = (At, Et) be the exchange possibilities graph at
time t. Note that if a, b ∈ At and a, b ∈ At′ , then (a, b) ∈ Et if and only if (a, b) ∈ Et′ ,i.e. the acceptable bilateral transactions are persistent throughout the process. For
an agent a ∈ At we use Nt(a) ⊆ At to denote the set of neighbors of a in Gt. It
follows that, if the planner does not match any agents, then for any fixed t ≥ 0,
Gt is distributed as an Erdos-Reyni graph with parameter d/m and d is the average
degree15 of agents [33].
Let A = ∪t≤TAnt , let E ⊆ A × A be the set of acceptable transactions between
agents in A, and let G = (A,E)16. Observe that any realization of the above stochas-
tic process is uniquely defined given Ant , Act for all t ≥ 0 and the set of acceptable
transactions, E. A vector (m, d, λ) represents a dynamic matching market. With-
out loss of generality, we can scale time so that λ = 1 (by normalizing m and d).
Therefore, throughout the paper, we assume λ = 1, unless otherwise specified17.
Online Matching Algorithms. A set of edges Mt ⊆ Et is a matching if no two
edges share the same endpoints. An online matching algorithm, at any time t ≥ 0,
14In our proofs, we use the fact that Act ⊆ ∪0≤τ≤tAτ . In the example of the text, we havea ∈ Act0+X . Note that even if agent a is matched before getting critical (i.e., t1 < t0 + X), we stillhave that a ∈ Act0+X . Hence, Act is not necessarily a subset of At since it may have agents who arealready matched and left the market. This generalized definition of Act is going to be helpful in ourproofs.
15In an undirected graph, degree of of a node is equal to the total number of edges connected tothat node.
16Note that E ⊇ ∪t≤TEt, and the two sets are not typically equal, since two agents may find itacceptable to transact, even though they are not in the pool at the same time because one of themwas matched earlier.
17See Proposition 2.5.12 for details of why this is without loss of generality.
CHAPTER 2. DYNAMIC MATCHING MARKETS 16
selects a (possibly empty) matching, Mt, in the current acceptable transactions graph
Gt, and the endpoints of the edges in Mt leave the market immediately. We assume
that any online matching algorithm at any time t0 only knows the current graph Gt
for t ≤ t0 and does not know anything about Gt′ for t′ > t0. In the benchmark case
that we consider, the online algorithm can depend on the set of critical agents at time
t; nonetheless, we will extend several of our theorems to the case where the online
algorithm does not have this knowledge. As will become clear, this knowledge has a
significant impact on the performance of any online algorithm.
We emphasize that the random sets At (the set of agents in the pool at time
t), Et (the set of acceptable transactions at time t), Nt(a) (the set of an agent a’s
neighbors), and the random variable Zt (pool size at time t) are all functions of the
underlying matching algorithm. We abuse the notation and do not include the name
of the algorithm when we analyze these variables.
The Goal. The goal of the planner is then to design an online matching algorithm
that maximizes the social welfare, i.e., the sum of the utility of all agents in the
market. Let ALG(T ) be the set of matched agents by time T ,
ALG(T ) := {a ∈ A : a is matched by ALG by time T}.
We may drop the T in the notation ALG(T ) if it is clear from context.
An agent receives zero utility if she leaves the market unmatched. If she is
matched, she receives a utility of 1 discounted at rate δ. More formally, if s(a) is
the sojourn of agent a, then we define the utility of agent a as follows:
u(a) :=
e−δs(a) if a is matched
0 otherwise.
We define the social welfare of an online algorithm to be the expected sum of the
CHAPTER 2. DYNAMIC MATCHING MARKETS 17
utility of all agents in the interval [0, T ], divided by a normalization factor:
W(ALG) := E
1
mT
∑a∈ALG(T )
e−δs(a)
The goal of the planner is to choose an online algorithm that maximizes the welfare
for large values of T (see Theorem 2.5.1, Theorem 2.5.2, and Theorem 2.6.1 for the
dependence of our results on T ).
It is instructive to consider the special case where δ = 0, i.e., the cost of waiting
is negligible compared to the cost of leaving the market unmatched. In this case,
the goal of the planner is to match the maximum number of agents, or equivalently
to minimize the number of perished agents. The loss of an online algorithm ALG is
defined as the ratio of the expected18 number of perished agents to the expected size
of A,
L(ALG) :=E [|A− ALG(T )− AT |]
E [|A|]=
E [|A− ALG(T )− AT |]mT
.
When we assume δ = 0, we will use the L notation for the planner’s loss function.
When we consider δ > 0, we will use the W notation for social welfare.
Each of the above optimization problems can be modeled as a Markov Decision
Problem (MDP)19 that is defined as follows. The state space is the set of pairs (H,B)
where H is any undirected graph of any size, and if the algorithm knows the set of
critical agents, B is a set of at most one vertex of H representing the corresponding
critical agent. The action space for a given state is the set of matchings on the
graph H. Under this conception, an algorithm designer wants to minimize the loss
or maximize the social welfare over a time period T .
Although this MDP has infinite number of states, with small error one can reduce
the state space to graphs of size at most O(m). Even in that case, this MDP has an
exponential number of states in m, since there are at least 2(m2 )/m! distinct graphs of
18We consider the expected value as a modeling choice. One may also be interested in objectivefunctions that depend on the variance of the performance, as well as the expected value. As willbe seen later in the paper, the performance of our algorithms are highly concentrated around theirexpected value, which guarantees that the variance is very small in most of the cases.
19We recommend [16] for background on Markov Decision Processes.
CHAPTER 2. DYNAMIC MATCHING MARKETS 18
size m20, so for even moderately large markets21, we cannot apply tools from Dynamic
Programming literature to find the optimum online matching algorithm.
Optimum Solutions. In many parts of this paper we compare the performance of
an online algorithm to the performance of an optimal omniscient algorithm. Unlike
any online algorithm, the omniscient algorithm has full information about the future,
i.e., it knows the full realization of the graph G.22 Therefore, it can return the
maximum matching in this graph as its output, and thus minimize the fraction of
perished agents. Let OMN(T ) be the set of matched agents in the maximum matching
of G. The loss function under the omnsicient algorithm at time T is
L(OMN) :=E [|A−OMN(T )− AT |]
mT
Observe that for any online algorithm, ALG, and any realization of the probability
space, we have |ALG(T )| ≤ |OMN(T )|.23
It is also instructive to study the optimum online algorithm, an online algorithm
with unlimited computational power. By definition, an optimum online algorithm can
solve the exponential-sized state space Markov Decision Problem and return the best
policy function from states to matchings. We first consider OPTc, the algorithm that
knows the set of critical agents at time t (with associated loss L(OPTc)). We then
relax this assumption and consider OPT, the algorithm that does not know these sets
(with associated loss L(OPT)).
Let ALGc be the loss under any online algorithm that knows the set of critical
20This lower bound is derived as follows: When there are m agents, there are(m2
)possible edges,
each of which may be present or absent. Some of these graphs may have the same structure butdifferent agent indices. A conservative lower bound is to divide by all possible re-labellings of theagents (m!).
21For instance, for m = 30, there are more than 1098 states in the approximated MDP.22In computer science, these are equivalently called offline algorithms.23This follows from a straightforward revealed-preference argument: For any realization, the op-
timum offline policy has the information to replicate any given online policy, so it must do weaklybetter.
CHAPTER 2. DYNAMIC MATCHING MARKETS 19
agents at time t. It follows that
L(ALGc) ≥ L(OPTc) ≥ L(OMN).
Similarly, let ALG be the loss under any online algorithm that does not know the
set of critical agents at time t. It follows that24
L(ALG) ≥ L(OPT) ≥ L(OPTc) ≥ L(OMN).
2.2 Our Contributions
In this section, we present our main contributions and provide intuitions for them.
We first introduce two simple matching algorithms, and then two classes of algorithms
that vary the waiting time. The first algorithm is the Greedy algorithm, which mimics
‘match-as-you-go’ algorithms used in many real marketplaces. It delivers maximal
matchings at any point in time, without regard for the future.
Definition 2.2.1 (Greedy Algorithm:). If any new agent a enters the market at
time t, then match her with an arbitrary agent in Nt(a) whenever Nt(a) 6= ∅. We
use L(Greedy) and W(Greedy) to denote the loss and the social welfare under this
algorithm, respectively.
Note that since |Ant | ≤ 1 almost surely, we do not need to consider the case where
more than one agent enters the market at any point in time. Observe that the graph
Gt in the Greedy algorithm is (almost) always an empty graph. Hence, the Greedy
algorithm cannot use any information about the set of critical agents.
The second algorithm is a simple online algorithm that preserves two essential
characteristics of OPTc when δ = 0 (recall that OPTc is the optimum online algorithm
with knowledge of the set of critical agents):
i) A pair of agents a, b get matched in OPTc only if one of them is critical. This
24Note that |ALG| and |OPT| are generally incomparable, and depending on the realization of Gwe may even have |ALG| > |OPT|.
CHAPTER 2. DYNAMIC MATCHING MARKETS 20
property is called the rule of deferral match: Since δ = 0, if a, b can be matched
and neither of them is critical we can wait and match them later.
ii) If an agent a is critical at time t and Nt(a) 6= ∅ then OPTc matches a. This
property is a corollary of the following simple fact: matching a critical agent does
not increase the number of perished agents in any online algorithm.
Our second algorithm is designed to be the simplest possible online algorithm that
satisfies both of the above properties.
Definition 2.2.2 (Patient Algorithm). If an agent a becomes critical at time t, then
match her uniformly at random with an agent in Nt(a) whenever Nt(a) 6= ∅. We
use L(Patient) and W(Patient) to denote the loss and the social welfare under this
algorithm, respectively.
Observe that unlike the Greedy algorithm, here we need access to the set of critical
agents at time t. We do not intend the timing assumptions about critical agents to
be interpreted literally. An agent’s point of perishing represents the point at which
it ceases to be socially valuable to match that agent. Letting the planner observe the
set of critical agents is a modeling convention that represents high-accuracy short-
horizon information about agents’ departures. An example of such information is the
Model for End-Stage Liver Disease (MELD) score, which accurately predicts 3-month
mortality among patients with chronic liver disease. The US Organ Procurement and
Transplantation Network gives priority to individuals with a higher MELD score,
following a broad medical consensus that liver donor allocation should be based on
urgency of need and not substantially on waiting time. [78] Note that the Patient
algorithm exploits only short-horizon information about urgent cases, as compared
to the Omniscient algorithm which has full information of the future. We discuss
implications of relaxing our informational assumptions in Subsection 2.2.3.
The third algorithm interpolates between the Greedy and the Patient algorithms.
The idea of this algorithm is to assign independent exponential clocks with rates 1/α
where α ∈ [0,∞) to each agent a. If agent a’s exponential clock ticks, the market-
maker attempts to match her. If she has no neighbors, then she remains in the pool
until she gets critical, where the market-maker attempts to match her again.
CHAPTER 2. DYNAMIC MATCHING MARKETS 21
A technical difficulty with the above matching algorithm is that it is not memo-
ryless; that is because when an agent gets critical and has no neighbors, she remains
in the pool. Therefore, instead of the above algorithm, we study a slightly different
matching algorithm (with a worse loss).
Definition 2.2.3 (The Patient(α) algorithm). Assign independent exponential clocks
with rate 1/α where α ∈ [0,∞) to each agent a. If agent a’s exponential clock ticks or
if an agent a becomes critical at time t, match her uniformly at random with an agent
in Nt(a) whenever Nt(a) 6= ∅. In both cases, if Nt(a) = ∅, treat that agent as if she
has perished; i.e., never match her again. We use L(Patient(α)) and W(Patient(α))
to denote the loss and the social welfare under this algorithm, respectively.
It is easy to see that an upper bound on the loss of the Patient(α) algorithm is an
upper bound on the loss of our desired interpolating algorithm. Under this algorithm
each agent’s exponential clock ticks at rate 1α
, so we search their neighbors for a
potential match at rate α := 1 + 1α
. We refer to α as the trade frequency. Note that
the trade frequency is a decreasing function of α25.
In the rest of this section we describe our contributions. To avoid cumbersome
notation, we state our results in the large-market long-horizon regime (i.e., as m→∞and T → ∞). In later sections, we explicitly study the dependency on (m,T ). For
example, we show that the transition of the market to the steady state takes no more
than O(log(m)) time units. In other words, many of the large time effects that we
predict in our model can be seen in poly-logarithmic time in the size of the market.
In addition, we only present an overview of the proofs in this section; the rest of the
paper includes detailed analysis of the model and full proofs.
2.2.1 Timing in Matching Markets
Does timing substantially affect the performance of dynamic matching algorithms?
Our first result establishes that varying the timing properties of simple algorithms
has large effects on their performance. In particular, we show that the number of
25The use of exponential clocks is a modeling convention that enables us to reduce waiting timeswhile retaining analytically tractable Markov properties.
CHAPTER 2. DYNAMIC MATCHING MARKETS 22
perished agents under the Patient algorithm is exponentially (in d) smaller than the
number of perished agents under the Greedy algorithm.
Theorem 2.2.4. For d ≥ 2, as T,m→∞,
L(Greedy) ≥ 1
2d+ 1
L(Patient) ≤ 1
2· e−d/2
As a result,
L(Patient(α)) ≤ (d+ 1) · e−d/2 · L(Greedy)
This theorem shows that the Patient algorithm strongly outperforms the Greedy
algorithm. The intuition behind this finding is that, under the Greedy algorithm,
there is no acceptable transactions among the set of agents in the pool and so all
critical agents imminently perish. On the contrary, the pool is thicker since under
the Patient algorithm is always an Erdos-Reyni random graph (see Proposition 2.4.1
for a proof of this claim), and such market thickness helps the planner to react to
critical cases.
The next question is, are the gains from market thickness large compared to the
total gains from optimization and choosing the right agents to match? First, in the
following example, we show that the Patient algorithm is not optimal because it
ignores the global graph structure.
Example 2.2.5. Let Gt be the graph shown in Figure 2.1, and let a2 ∈ Act , i.e., a2
is critical at time t. Observe that it is strictly better to match a2 to a1 as opposed to
a3. Nevertheless, since the Patient algorithm makes decisions that depend only on the
immediate neighbors of the agent it is trying to match, it cannot differentiate between
a1 and a3 and will choose either of them with equal probability.
To quantify gains of optimization, we compare the Patient algorithm to the om-
niscient algorithm, which obviously bounds the performance of the optimum online
algorithm.
The next theorem shows that no algorithm achieves better than exponentially
small loss. Furthermore, the gains from the right timing decision (moving from the
CHAPTER 2. DYNAMIC MATCHING MARKETS 23
a1 a2 a3 a4
Figure 2.1: If a2 gets critical in the above graph, it is strictly better to match him toa1 as opposed to a3. The Patient algorithm, however, chooses either of a1 or a3 withequal probability.
Greedy algorithm to the Patient algorithm) are much larger than the remaining gains
from optimization (moving from the Patient algorithm to the optimum algorithm).
Theorem 2.2.6. For d ≥ 2, as T,m→∞,
e−d
d+ 1≤ L(OMN) ≤ L(Patient) ≤ 1
2· e−d/2.
This constitutes an answer to the “when to match versus whom to match” ques-
tion. Recall that OMN has perfect foresight, and bounds the performance of any
algorithm, including OPTc, the globally optimal solution. Thus, Theorem 2.2.6 im-
plies that the marginal effect of a globally optimal solution is small relative to the
effect of making the right timing decision. In many settings, optimal solutions may
be computationally demanding and difficult to implement. Thus, this result suggests
that it will often be more worthwhile for policymakers to find ways to thicken the
market, rather than to seek potentially complicated optimal policies.26
A planner, however, may not be willing to implement the Patient algorithm for
various reasons. First, the cost of waiting is usually not zero; in other words, agents
prefer to be matched earlier (we discuss this cost in detail in Subsection 2.2.2). Second,
the planner may be in competition with other exchange platforms and may be able to
attract more agents by advertising reasonably short waiting times. Hence, we study
26It is worth emphasizing that this result (as well as Theorem 2.2.11) proves that local algorithmsare close-to-optimal and since in our model agents are ex ante homogeneous, this shows that “whoto match” is not as important as “when to match”. In settings where agents have multiple types,however, the decision of “who to match” can be an important one even when it is local. For instance,suppose a critical agent has two neighbors, one who is hard-to-match and one who is easy-to-match.Then, everything else constant, the optimal policy should match the critical agent to the hard-to-match neighbor. This result has been formally proved in a new working paper by Akbarpour, Nikzadand Roth, in which they show that breaking the ties in favor of hard-to-match agents reduces theloss.
CHAPTER 2. DYNAMIC MATCHING MARKETS 24
the performance of the Patient(α) algorithm, which introduces a way to speed up
the Patient algorithm. The next result shows that when α is not ‘too small’ (i.e.,
the exponential clocks of the agents do not tick at a very fast rate), then Patient(α)
algorithm still (strongly) outperforms the Greedy algorithm. In other words, even
waiting for a moderate time can substantially reduce perishings.
Theorem 2.2.7. Let α := 1/α + 1. For d ≥ 2, as T,m→∞,
L(Patient(α)) ≤ (d+ 1) · e−d/2α · L(Greedy)
A numerical example clarifies the significance of this result. Consider the case of
a kidney exchange market, where 1000 new patients arrive to the market every year,
their average sojourn is 1 year and they can exchange kidneys with a random patient
with probability 1100
; that is, d = 10. The above result for the Patient(α) algorithm
suggests that the market-maker can promise to match agents in less than 4 months
(in expectation) while the fraction of perished agents is at most 37% of the Greedy
algorithm. Note that if we employ the Patient algorithm, the fraction of perished
agents will be at most 7% of the Greedy algorithm.
We now present a proof sketch for all of the theorems presented in this section.
The details of the proofs are discuss in the rest of the paper.
Proof Overview. [Theorem 2.2.4, Theorem 2.2.6, Theorem 2.2.7, and ??] We first
sketch the proof of Theorem 2.2.4. We show that for large enough values of T and m,
(i) L(Patient) ≤ e−d/2/2 and (ii) L(OPT) ≥ 1/(2d+1). By the fact that L(Greedy) ≥L(OPT) (since the Greedy algorithm does not use information about critical agents),
Theorem 2.2.4 follows immediately. The key idea in proving both parts is to carefully
study the distribution of the pool size, Zt, under any of these algorithms.
For (i), we show that pool size under the Patient algorithm is a Markov chain, it
has a unique stationary distribution and it mixes rapidly to the stationary distribution
(see Theorem 2.4.2). This implies that for t larger than mixing time, Zt is essentially
distributed as the stationary distribution of the Markov Chain. We show that under
the stationary distribution, with high probability, Zt ∈ [m/2,m]. Therefore, any
CHAPTER 2. DYNAMIC MATCHING MARKETS 25
critical agent has no acceptable transactions with probability at most (1−d/m)m/2 ≤e−d/2. This proves (i) (see Subsection 2.5.2 for the exact analysis of the Patient
Markov chain).
For (ii), note that any algorithm which lacks the information of critical agents, the
expected perishing rate is equal to the pool size because any critical agent perishes
with probability one, and agents get critical with rate 1. Therefore, if the expected
pool size is large, the perishing rate is very high. On the other hand, if the expected
pool size is very low, the perishing rate is again very high because there will be many
agents with no acceptable transactions upon their arrival or during their sojourn. We
analyze the trade-off between the above two extreme cases and show that even if the
pool size is optimally chosen, the loss cannot be less than what we claimed in (ii)
(see Theorem 2.3.1).
We prove Theorem 2.2.6 by showing that L(OMN) ≥ e−d
d+1. To do so, we provide
a lower bound for the fraction of agents who arrive the market at some point in
time and have no acceptable transactions during their sojourn (see Theorem 2.3.2).
Note that even with full knowledge about the realization of the process, those agents
cannot be matched.
We now sketch the proof of Theorem 2.2.7. By the additivity of the Poisson
process, the loss of Patient(α) algorithm in a (m, d, 1) matching market is equal to
the loss of the Patient algorithm in a (m, d, α) matching market, where α = 1/α+ 1.
The next step is to show that a matching market (m, d, α) is equivalent to a
matching market (m/α, d/α, 1) in the sense that any quantity in these two markets
is the same up to a time scale (see Definition 2.5.11). By this fact, the loss of the
Patient algorithm on a (m, d, α) matching market at time T is equal to the loss of
Patient algorithm on a (m/α, d/α, 1) market at time αT . But, we have already upper
bounded the latter in Theorem 2.2.4.
Remark 2.2.8. One may interpret the above result as demonstrating the value of
information (i.e. knowledge of the set of critical agents) as opposed to the value of
waiting. This is not our interpretation, since the Greedy algorithm cannot improve
CHAPTER 2. DYNAMIC MATCHING MARKETS 26
its performance even if it has knowledge of the set of critical agents. The graph
Gt is almost surely an empty graph, so there is no possibility of matching a critical
agent in the Greedy algorithm. The Patient algorithm strongly outperforms the Greedy
algorithm because it waits long enough to have a large pool of agents with many
acceptable bilateral transactions.
2.2.2 Welfare Under Discounting and Optimal Waiting Time
In this part we explictly account for the cost of waiting and study online algorithms
that optimize social welfare. It is clear that if agents are very impatient (i.e., they have
very high waiting costs), it is better to implement the Greedy algorithm. On the other
hand, if agents are very patient (i.e., they have very low waiting costs), it is better to
implement the Patient algorithm. Therefore, a natural welfare economics question is:
For which values of δ is the Patient algorithm (or the Patient(α) algorithm) socially
preferred to the Greedy algorithm?
Our next result studies social welfare under the Patient, Patient(α) and Greedy
algorithms. We show that for small enough δ, there exists a value of α such that the
Patient(α) algorithm is socially preferable to the Greedy algorithm.
Theorem 2.2.9. For any 0 ≤ δ ≤ 18 log(d)
, there exists an α ≥ 0 such that as m,T →∞,
W(Patient(α)) ≥W(Greedy).
In particular, for δ ≤ 1/2d and d ≥ 5, we have
W(Patient) ≥W(Greedy).
A numerical example illustrates these magnitudes. Consider a barter market,
where 100 new traders arrive at the market every week, their average sojourn is one
week, and there is a satisfactory trade between two random agents in the market with
probability 0.05; that is, d = 5. Then our welfare analysis implies that if the cost
associated with waiting for one week is less than 10% of the surplus from a typical
trade, then the Patient(α) algorithm, for a tuned value of α, is socially preferred to
CHAPTER 2. DYNAMIC MATCHING MARKETS 27
the Greedy algorithm.
When agents discount the future, how should the planner trade off the frequency
of transactions and the thickness of the market? To answer this, we characterize
the optimal trade frequency under the Patient(α) algorithm. Recall that under this
algorithm each agent’s exponential clock ticks at rate 1α
, so we search their neighbors
for a potential match at rate α := 1+ 1α
. The optimal α, the trade frequency, is stated
in the following theorem.
Theorem 2.2.10. Given d, δ, as m,T → ∞, there exists d ∈ [d/2, d] as a function
of m, d27 such that the Patient( 1max{α,1}−1
) algorithm where α is the solution of
δ −(δ + d+
dδ
α
)e−
dα = 0, (2.2.1)
attains the largest welfare among all Patient(α) algorithms. In particular, if δ < d/4,
then d/ log(d/δ) ≤ α ≤ d/ log(2d/δ).
In addition, if δ < (d− 1)/2, then α∗ is a non-decreasing function of δ and d.
Figure 2.2 illustrates max{α, 1} as a function of δ. As one would expect, the
optimal trade frequency is increasing in δ. Moreover, Theorem 2.2.10 indicates that
the optimal trade frequency is increasing in d. In Subsection 2.2.1, we showed that
L(Patient) is exponentially smaller in d than L(Greedy). This may suggest that wait-
ing is mostly valuable in dense graphs. By contrast, Theorem 2.2.10 shows that one
should wait longer as the graph becomes sparser. Intuitively, an algorithm performs
well if, whenever it searches neighbors of a critical agent for a potential match, it can
find a match with very high probability. This probability is a function of both the
pool size and d. When d is smaller, the pool size should be larger (i.e. the trade
frequency should be lower) so that the probability of finding a match remains high.
For larger values of d, on the other hand, a smaller pool size suffices.
Finally, we note that Patient (i.e. α = ∞) is the optimal policy for a range of
parameter values. To see why, suppose agents discount their welfare, but never perish.
The planner would still wish to match them at some positive rate. For a range of
27More precisely, d := xd/m where x is the solution of equation (A.3.7).
CHAPTER 2. DYNAMIC MATCHING MARKETS 28
0 0.05 0.1 0.15 0.2 0.25
0.8
1
1.2
1.4
1.6
1.8
δ
α∗
d=15
d=12.5
d=10
d=7.5
d=5
Figure 2.2: Optimal trade frequency (max{α, 1}) as a function of the discount rate fordifferent values of d. Our analysis shows that the optimal trade frequency is smallerin sparser graphs.
parameters, this positive rate is less than 1. In a world where agents perish, α is
bounded below by 1, and we have a corner solution for such parameter values.
Proof Overview. [Theorem 2.2.9 and Theorem 2.2.10] We first show that for large val-
ues of m and T , (i) W(Greedy) ≤ 1− 12d+1
, and (ii) W(Patient(α)) ' 22−e−d/2α+δ/α
(1−e−d/2α), where α = 1 + 1/α. The proof of (i) is very simple: 1/(2d + 1) fraction of
agents perish under the Greedy algorithm. Therefore, even if all of the matched
agents receive a utility of 1, the social welfare is no more than 1− 1/(2d+ 1).
The proof of (ii) is more involved. The idea is to define a random variable Xt
representing the potential utility of the agents in the pool at time t, i.e., if all agents
who are in the pool at time t get matched immediately, then they receive a total
utility of Xt. We show that Xt can be estimated with a small error by studying the
evolution of the system through a differential equation. Given Xt and the pool size at
time t, Zt, the expected utility of an agent that is matched at time t is exactly Xt/Zt.
Using our concentration results on Zt, we can then compute the expected utility of
CHAPTER 2. DYNAMIC MATCHING MARKETS 29
the agents that are matched in any interval [t, t + dt]. Integrating over all t ∈ [0, T ]
proves the claim. (See Section 2.6 for an exact analysis of welfare under the Patient
algorithm)
To prove Theorem 2.2.10, we characterize the unique global maximum of W(Patient(α)).
The key point is that the optimum value of α (= 1 + 1/α) is less than 1 for a range
of parameters. However, since α ≥ 0, we must have that α ∈ [1,∞). Therefore,
whenever the solution of equation (2.2.1) is less than 1, the optimal α is 1 and we
have a corner solution; i.e. setting α =∞ (running the Patient algorithm) is optimal.
2.2.3 Information and Incentive-Compatibility
Up to this point, we have assumed that the planner knows the set of critical agents;
i.e. he has accurate short-horizon information about agent departures. We now relax
this assumption in both directions.
Suppose that the planner does not know the set of critical agents. That is, the
planner’s policy may depend on the graph Gt, but not the set of critical agents Act .
Recall that OPT is the optimum algorithm subject to these constraints.
Theorem 2.2.11. For d ≥ 2, as T,m→∞,
1
2d+ 1≤ L(OPT) ≤ L(Greedy) ≤ log(2)
d,
Theorem 2.2.11 shows that the loss of OPT and Greedy are relatively close. This
indicates that waiting and criticality information are complements, in that waiting
to thicken the market is substantially valuable only when the planner can identify
urgent cases. Observe that OPT could in principle wait to thicken the market, but
the gains from doing so (compared to running the Greedy algorithm) are not large.
Under these new information assumptions, we once more find that local algorithms
can perform close to computationally intensive global optima.
CHAPTER 2. DYNAMIC MATCHING MARKETS 30
Moreover, Theorem 2.2.4 and Theorem 2.2.11 together show that criticality infor-
mation is valuable, since the loss of Patient, which naıvely exploits criticality infor-
mation, is exponentially smaller than the loss of OPT, the optimal algorithm without
this information.
What if the planner knows more than just the set of critical agents? For instance,
the planner may have long-horizon forecasts of agent departure times, or the planner
may know that certain agents are more likely to have acceptable transactions in
future than other agents28. However, Theorem 2.2.6 shows that no expansion of
the planner’s information set yields a better-than-exponential loss. This is because
L(OMN) is the loss under a maximal expansion of the planner’s information; the case
where the planner has perfect foreknowledge of the future.
Taken together, these results suggest that criticality information is particularly
valuable. This information is necessary to achieve exponentially small loss, and no
expansion of information enables an algorithm to achieve better-than-exponential
loss.
However, in many settings, it is plausible that agents have privileged insight into
their own departure timings. In such cases, agents may have incentives to misreport
whether they are critical, in order to increase their chance of getting matched or to
decrease their waiting time. We now exhibit a truthful mechanism without transfers
that elicits such information from agents, and implements the Patient(α) algorithm.
We assume that agents are fully rational and know the underlying parameters,
but they do not observe the actual realization of the stochastic process. That is,
agents observe whether they are critical, but do not observe Gt, while the planner
observes Gt but does not observe which agents are critical. Consequently, agents’
strategies are independent of the realized sample path. Our results are sensitive to
this assumption29; for instance, if the agent knew that she had a neighbor, or knew
28In our model, the number of acceptable transactions that a given agent will have with the nextN agents to arrive is Bernoulli distributed. If the planner knows beforehand whether a given agent’srealization is above or below the 50th percentile of this distribution, it is as though agents havedifferent ‘types’.
29This assumption is plausible in many settings; generally, centralized brokers know more aboutthe current state of the market than individual traders. Indeed, frequently agents approach central-ized brokers because they do not know who is available to trade with them.
CHAPTER 2. DYNAMIC MATCHING MARKETS 31
that the pool at that moment was very large, she would have an incentive under our
mechanism to falsely report that she was critical.
The truthful mechanism, Patient-Mechanism(α), is described below.
Definition 2.2.12 (Patient-Mechanism(α)). Assign independent exponential clocks
with rate 1/α to each agent a, where α ∈ [0,∞). Ask agents to report when they
get critical. If agent’s exponential clock ticks or if she reports becoming critical, the
market-maker attempts to match her to a random neighbor. If the agent has no
neighbors, the market-maker treats her as if she has perished, i.e., she will never be
matched again.
Each agent a selects a mixed strategy by choosing a function ca(·); at the interval
[t, t + dt] after her arrival, if she is not yet critical, she reports being critical with
rate ca(t)dt, and when she truly becomes critical she reports that immediately. Our
main result in this section asserts that if agents are not too impatient, then the
Patient-Mechanism(α) is incentive-compatible in the sense that the truthful strategy
profile is a strong ε-Nash equilibrium.30
Theorem 2.2.13. Suppose that the market is in the stationary distribution and31
d = polylog(m). Let α = 1/α + 1 and β = α(1 − d/m)m/α. Then, for 0 ≤ δ ≤ β,
ca(t) = 0 for all a, t (i.e., truthful strategy profile) is a strong ε-Nash equilibrium for
Patient-Mechanism(α), where ε→ 0 as m→∞.
If d ≥ 2 and 0 ≤ δ ≤ e−d/2, the truthful strategy profile is a strong ε-Nash
equilibrium for Patient-Mechanism(∞), where ε→ 0 as m→∞.
Proof Overview. There is a hidden obstacle in proving that truthful reporting is
incentive-compatible: Even if one assumes that the market is in a stationary distri-
bution at the point an agent enters, the agent’s beliefs about pool size may change
as time passes. In particular, an agent makes inferences about the current distribu-
tion of pool size, conditional on not having been matched yet, and this conditional
30Any strong ε-Nash equilibrium is an ε-Nash equilibrium. For a definition of strong ε-Nashequilibrium, see Definition 2.7.1.
31polylog(m) denotes any polynomial function of log(m). In particular, d = polylog(m) if d is aconstant independent of m.
CHAPTER 2. DYNAMIC MATCHING MARKETS 32
distribution is different from the stationary distribution. This makes it difficult to
compute the payoffs from deviations from truthful reporting. We tackle this problem
by using the concentration bounds from Proposition 2.5.9, and focusing on strong
ε-Nash equilibrium, which allows small deviations from full optimality.
The intuition behind this proof is that an agent can be matched in one of two
ways under Patient-Mechanism(∞): Either she becomes critical, and has a neighbor,
or one of her neighbors becomes critical, and is matched to her. By symmetry, the
chance of either happening is the same, because with probability 1, every matched
pair consists of one critical agent and one non-critical agent. When an agent declares
that she is critical, she is taking her chance that she has a neighbor in the pool right
now. By contrast, if she waits, there is some probability that another agent will
become critical and be matched to her. Consequently, for small δ, agents will opt to
wait.
2.2.4 Technical Contributions
As alluded to above, most of our results follow from concentration results on the
distribution of the pool size for each of the online algorithms that are stated in
Proposition 2.5.5 and Proposition 2.5.9. In this last part we describe ideas behind
these crucial results.
For analyzing many classes of stochastic processes one needs to prove concentra-
tion bounds on functions defined on the underlying process by means of central limit
theorems, Chernoff bounds or Azuma-Hoeffding bounds. In our case many of these
tools fail. This is because we are interested in proving that for any large time t, a
given function is concentrated in an interval whose size depend only on d,m and not
t. Since t can be significantly larger than d,m a direct proof fails.
In contrast we observe that Zt is a Markov Chain for a large class of online
algorithms. Building on this observation, first we show that the underlying Markov
Chain has a unique stationary distribution and it mixes rapidly. Then we use the
stationary distribution of the Markov Chain to prove our concentration bounds.
However, that is not the end of the story. We do not have a closed form expression
CHAPTER 2. DYNAMIC MATCHING MARKETS 33
for the stationary distribution of the chain, because we are dealing with an infinite
state space continuous time Markov Chain where the transition rates are complex
functions of the states. Instead, we use the following trick. Suppose we want to prove
that Zt is contained in an interval [k∗ − f(m, d), k∗ + f(m, d)] for some k∗ ∈ N with
high probability, where f(m, d) is a function of m, d that does not depend on t. We
consider a sequence of pairs of states P1 := (k∗ − 1, k∗ + 1), P2 := (k∗ − 2, k∗ + 2),
etc. We show that if the Markov Chain is at any of the states of Pi, it is more likely
(by an additive function of m, d) that it jumps to a state of Pi−1 as opposed to Pi+1.
Using balance equations and simple algebraic manipulations, this implies that the
probability of states in Pi geometrically decrease as i increases. In other words Zt is
concentrated in a small interval around k∗. We believe that this technique can be
used in studying other complex stochastic processes.
2.3 Performance of the Optimum and Periodic Al-
gorithms
In this section we lower-bound the loss of the optimum solutions in terms of d. In
particular, we prove the following theorems.
Theorem 2.3.1. If m > 10d, then for any T > 0
L(OPT) ≥ 1
2d+ 1 + d2/m.
Theorem 2.3.2. If m > 10d, then for any T > 0,
L(OMN) ≥ e−d−d2/m
d+ 1 + d2/m
Before proving the above theorems, it is useful to study the evolution of the system
in the case of the inactive algorithm, i.e., where the online algorithm does nothing
and no agents ever get matched. We later use this analysis in this section, as well as
Section 2.4 and Section 2.5.
CHAPTER 2. DYNAMIC MATCHING MARKETS 34
We adopt the notation At and Zt to denote the agents in the pool and the pool
size in this case. Observe that by definition for any matching algorithm and any
realization of the process,
Zt ≤ Zt. (2.3.1)
Using the above equation, in the following fact we show that for any matching algo-
rithm E [Zt] ≤ m.
Proposition 2.3.3. For any t0 ≥ 0,
P[Zt0 = `
]≤ m`
`!.
Therefore, Zt is distributed as a Poisson random variable of rate m(1− e−t0), so
E[Zt0
]= (1− e−t0)m.
Proof. Let K be a random variable indicating the number agents who enter the pool
in the interval [0, t0]. By Bayes rule,
P[Zt0 = `
]=∞∑k=0
P[Zt0 = `,K = k
]=∞∑k=0
P[Zt0 = `|K = k
]· (mt0)ke−mt0
k!,
where the last equation follows by the fact that arrival rate of the agents is a Poisson
random variable of rate m.
Now, conditioned on the event that an agent a arrives in the interval [0, t0], the
probability that she is in the pool at time t0 is at least,
P [Xai = 1] =
∫ t0
t=0
1
t0P [s(ai) ≥ t0 − t] dt =
1
t0
∫ t0
t=0
et−t0dt =1− e−t0
t0.
Therefore, conditioned on K = k, the distribution of the number of agents at time t0
is a Binomial random variable B(k, p), where p := (1− e−t0)/t0. Let µ = m(1− e−t0),
CHAPTER 2. DYNAMIC MATCHING MARKETS 35
we have
P[Zt0 = `
]=
∞∑k=`
(k
`
)· p` · (1− p)k−` (mt0)ke−mt0
k!
=∞∑k=`
mke−mt0
`!(k − `)!(1− e−t0)`(t0 − 1 + e−t0)k−`
=m`e−mt0µ`
`!
∞∑k=`
(mt0 − µ)k−`
(k − `)!=µ`e−µ
`!.
2.3.1 Loss of the Optimum Online Algorithm
In this section, we prove Theorem 2.3.1. Let ζ be the expected pool size of the OPT,
ζ := Et∼unif[0,T ] [Zt]
Since OPT does not know Act , each critical agent perishes with probability 1. There-
fore,
L(OPT) =1
m · TE[∫ T
t=0
Ztdt
]=
ζT
mT= ζ/m. (2.3.2)
To finish the proof we need to lower bound ζ by m/(2d+ 1 + d2/m). We provide an
indirect proof by showing a lower-bound on L(OPT) which in turn lower-bounds ζ.
Our idea is to lower-bound the probability that an agent does not have any ac-
ceptable transactions throughout her sojourn, and this directly gives a lower-bound
on L(OPT) as those agents cannot be matched under any algorithm. Fix an agent
a ∈ A. Say a enters the market at a time t0 ∼ unif[0, T ], and s(a) = t, we can write
P [N(a) = ∅] =
∫ ∞t=0
P [s(a) = t] · E[(1− d/m)|At0 |
]· E[(1− d/m)|A
nt0,t+t0
|]dt(2.3.3)
To see the above, note that a does not have any acceptable transactions, if she doesn’t
have any neighbors upon arrival, and none of the new agents that arrive during her
CHAPTER 2. DYNAMIC MATCHING MARKETS 36
sojourn are not connected to her. Using the Jensen’s inequality, we have
P [N(a) = ∅] ≥∫ ∞t=0
e−t · (1− d/m)E[Zt0 ] · (1− d/m)E[|Ant0,t+t0 |]dt
=
∫ ∞t=0
e−t · (1− d/m)ζ · (1− d/m)mtdt
The last equality follows by the fact that E[|Ant0,t+t0|
]= mt. Since d/m < 1/10,
1− d/m ≥ e−d/m−d2/m2
,
L(OPT) ≥ P [N(a) = ∅] ≥ e−ζ(d/m+d2/m2)
∫ ∞t=0
e−t(1+d+d2/m)dt ≥ 1− ζ(1 + d/m)d/m
1 + d+ d2/m(2.3.4)
Putting (2.3.2) and (2.3.4) together, for β := ζd/m we get
L(OPT) ≥ max{1− β(1 + d/m)
1 + d+ d2/m,β
d} ≥ 1
2d+ 1 + d2/m
where the last inequality follows by letting β = d2d+1+d2/m
be the minimizer of the
middle expression.
2.3.2 Loss of the Omniscient Algorithm
In this section, we prove Theorem 2.3.2. This demonstrates that, in the case that
the planner observes the critical agents, no policy can yield a faster-than-exponential
decrease in losses, as a function of the average degree of each agent.32
The proof is very similar to Theorem 2.3.1. Let ζ be the expected pool size of the
OMN,
ζ := Et∼unif[0,T ] [Zt] .
By (2.3.1) and Proposition 2.3.3,
ζ ≤ Et∼unif[0,T ]
[Zt
]≤ m.
32This, in fact, proves a much more powerful claim: It shows that even if the planner knows whenagents are going to be critical upon their arrival, she still cannot to faster-than-exponential decreasein losses.
CHAPTER 2. DYNAMIC MATCHING MARKETS 37
Note that (2.3.2) does not hold in this case because the omniscient algorithm knows
the set of critical agents at time t.
Now, fix an agent a ∈ A, and let us lower-bound the probability that N(a) = ∅.Say a enters the market at time t0 ∼ unif[0, T ] and s(a) = t, then
P [N(a) = ∅] =
∫ ∞t=0
P [s(a) = t] · E[(1− d/m)Zt0
]· E[(1− d/m)|A
nt0,t+t0
|]dt
≥∫ ∞t=0
e−t(1− d/m)ζ+mtdt ≥ e−ζ(1+d/m)d/m
1 + d+ d2/m≥ e−d−d
2/m
1 + d+ d2/m.
where the first inequality uses the Jensen’s inequality and the second inequality uses
the fact that when d/m < 1/10, 1− d/m ≥ e−d/m−d2/m2
.
2.4 Modeling an Online Algorithm as a Markov
Chain
2.4.1 Background
As an important preliminary, we establish that under both of the Patient and Greedy
algorithms the random processes Zt are Markovian, have unique stationary distribu-
tions, and mix rapidly to the stationary distribution. Before getting into the details
we provide a brief overview on continuous time Markov Chains. We refer interested
readers to [64, 56] for detailed discussions.
Let Zt be a continuous time Markov Chain on the non-negative integers (N) that
starts from state 0. For any two states i, j ∈ N , we assume that the rate of going
from i to j is ri→j ≥ 0. The rate matrix Q ∈ N× N is defined as follows,
Q(i, j) :=
ri→j if i 6= j,∑k 6=i−ri→k otherwise.
Note that, by definition, the sum of the entries in each row of Q is zero. It turns out
CHAPTER 2. DYNAMIC MATCHING MARKETS 38
that (see e.g., [64, Theorem 2.1.1]) the transition probability in t units of time is,
etQ =∞∑i=0
tiQi
i!.
Let Pt := etQ be the transition probability matrix of the Markov Chain in t time
units. It follows that,
d
dtPt = PtQ. (2.4.1)
In particular, in any infinitesimal time step dt, the chain moves based on Q · dt.A Markov Chain is irreducible if for any pair of states i, j ∈ N , j is reachable
from i with a non-zero probability. Fix a state i ≥ 0, and suppose that Zt0 = i, and
let T1 be the first jump out of i (note that T1 is distributed as an exponential random
variable). State i is positive recurrent iff
E [inf{t ≥ T1 : Zt = i}|Zt0 = i] <∞ (2.4.2)
The ergodic theorem [64, Theorem 3.8.1] entails that a continuous time Markov
Chain has a unique stationary distribution if and only if it has a positive recurrent
state.
Let π : N → R+ be the stationary distribution of a Markov chain. It follows by
the definition that for any t ≥ 0, Pt = πPt. The balance equations of a Markov chain
say that for any S ⊆ N ,
∑i∈S,j /∈S
π(i)ri→j =∑
i∈S,j /∈S
π(j)rj→i. (2.4.3)
Let zt(.) be the distribution of Zt at time t ≥ 0, i.e., zt(i) := P [Zt = i] for any
integer i ≥ 0. For any ε > 0, we define the mixing time (in total variation distance)
of this Markov Chain as follows,
τmix(ε) = inf{t : ‖zt − π‖TV :=
∞∑k=0
|π(k)− zt(k)| ≤ ε}. (2.4.4)
CHAPTER 2. DYNAMIC MATCHING MARKETS 39
2.4.2 Markov Chain Characterization
First, we argue that Zt is a Markov process under the Patient and Greedy algorithms.
This follows from the following simple observation.
Proposition 2.4.1. Under either of Greedy or Patient algorithms, for any t ≥ 0,
conditioned on Zt, the distribution of Gt is uniquely defined. So, given Zt, Gt is
conditionally independent of Zt′ for t′ < t.
Proof. Under the Greedy algorithm, at any time t ≥ 0, |Et| = 0. Therefore, condi-
tioned on Zt, Gt is an empty graph with |Zt| vertices.
For the Patient algorithm, note that the algorithm never looks at the edges be-
tween non-critical agents, so the algorithm is oblivious to these edges. It follows that
under the Patient algorithm, for any t ≥ 0, conditioned on Zt, Gt is an Erdos-Renyi
random graph with |Zt| vertices and parameter d/m.
The following is the main theorem of this section.
Theorem 2.4.2. For the Patient and Greedy algorithms and any 0 ≤ t0 < t1,
P [Zt1|Zt for 0 ≤ t < t1] = P [Zt1|Zt for t0 ≤ t < t1] .
The corresponding Markov Chains have unique stationary distributions and mix in
time O(log(m) log(1/ε)) in total variation distance:
τmix(ε) ≤ O(log(m) log(1/ε)).
The proof of the theorem can be found in the Appendix A.2.
This theorem is crucial in justifying our focus on long-run results in Section 2.2,
since these Markov chains converge very rapidly (in O(log(m)) time) to their station-
ary distributions.
CHAPTER 2. DYNAMIC MATCHING MARKETS 40
2.5 Performance Analysis
In this section we upper bound L(Greedy) and L(Patient) as a function of d, and we
upper bound L(Patient(α)) as a function of d and α.
We prove the following three theorems.33
Theorem 2.5.1. For any ε ≥ 0 and T > 0,
L(Greedy) ≤ log(2)
d+τmix(ε)
T+ 6ε+O
( log(m/d)√dm
), (2.5.1)
where τmix(ε) ≤ 2 log(m/d) log(2/ε).
Theorem 2.5.2. For any ε > 0 and T > 0,
L(Patient) ≤ maxz∈[1/2,1]
(z + O(1/
√m))e−zd +
τmix(ε)
T+εm
d2+ 2/m, (2.5.2)
where τmix(ε) ≤ 8 log(m) log(4/ε).
Theorem 2.5.3. Let α := 1/α + 1. For any ε > 0 and T > 0,
L(Patient(α)) ≤ maxz∈[1/2,1]
(z + O(
√α/m)
)e−zd/α +
τmix(ε)
αT+εmα
d2+ 2α/m,
where τmix(ε) ≤ 8 log(m/α) log(4/ε).
We will prove Theorem 2.5.1 in Subsection 2.5.1, Theorem 2.5.2 in Subsection 2.5.2
and Theorem 2.5.3 in Subsection 2.5.3.
2.5.1 Loss of the Greedy Algorithm
In this part we upper bound L(Greedy). We crucially exploit the fact that Zt is a
Markov Chain and has a unique stationary distribution, π : N → R+. Our proof
33We use the operators O and O in the standard way. That is, f(m) = O(g(m)) iff there existsa positive real number N and a real number m0 such that |f(m)| ≤ N |g(m)| for all m ≥ m0. O issimilar but ignores logarithmic factors, i.e. f(m) = O(g(m)) iff f(m) = O(g(m) logk g(m)) for somek.
CHAPTER 2. DYNAMIC MATCHING MARKETS 41
proceeds in three steps: First, we show that L(Greedy) is bounded by a function of
the expected pool size. Second, we show that the stationary distribution is highly
concentrated around some point k∗, which we characterize. Third, we show that k∗
is close to the expected pool size.
Let ζ := EZ∼µ [Z] be the expected size of the pool under the stationary distribution
of the Markov Chain on Zt. First, observe that if the Markov Chain on Zt is mixed,
then the agents perish at the rate of ζ, as the pool is almost always an empty graph
under the Greedy algorithm. Roughly speaking, if we run the Greedy algorithm for
a sufficiently long time then Markov Chain on size of the pool mixes and we get
L(Greedy) ' ζm
. This observation is made rigorous in the following lemma. Note
that as T and m grow, the first three terms become negligible.
Lemma 2.5.4. For any ε > 0, and T > 0,
L(Greedy) ≤ τmix(ε)
T+ 6ε+
1
m2−6m +
EZ∼π [Z]
m.
The theorem is proved in the Appendix A.3.1.
The proof of the above lemma involves lots of algebra, but the intuition is as
follows: The EZ∼π [Z]m
term is the loss under the stationary distribution. This is equal to
L(Greedy) with two approximations: First, it takes some time for the chain to transit
to the stationary distribution. Second, even when the chain mixes, the distribution
of the chain is not exactly equal to the stationary distribution. The τmix(ε)T
term
provides an upper bound for the loss associated with the first approximation, and the
term (6ε + 1m
2−6m) provides an upper bound for the loss associated with the second
approximation.
Given Lemma 2.5.4 , in the rest of the proof we just need to get an upper bound for
EZ∼π [Z]. Unfortunately, we do not have any closed form expression of the stationary
distribution, π(·). Instead, we use the balance equations of the Markov Chain defined
on Zt to characterize π(·) and upper bound EZ∼π [Z].
Let us rigorously define the transition probability operator of the Markov Chain
on Zt. For any pool size k, the Markov Chain transits only to the states k + 1 or
CHAPTER 2. DYNAMIC MATCHING MARKETS 42
kk − 1 k + 1
Figure 2.3: An illustration of the transition paths of the Zt Markov Chain under theGreedy algorithm
k − 1. It transits to state k + 1 if a new agent arrives and the market-maker cannot
match her (i.e., the new agent does not have any edge to the agents currently in the
pool) and the Markov Chain transits to the state k − 1 if a new agent arrives and
is matched or an agent currently in the pool gets critical. Thus, the transition rates
rk→k+1 and rk→k−1 are defined as follows,
rk→k+1 := m(
1− d
m
)k(2.5.3)
rk→k−1 := k +m(
1−(
1− d
m
)k). (2.5.4)
In the above equations we used the fact that agents arrive at rate m, they perish at
rate 1 and the probability of an acceptable transaction between two agents is d/m.
Let us write down the balance equation for the above Markov Chain (see equation
(2.4.3) for the full generality). Consider the cut separating the states 0, 1, 2, . . . , k−1
from the rest (see Figure 2.3 for an illustration). It follows that,
π(k − 1)rk−1→k = π(k)rk→k−1. (2.5.5)
Now, we are ready to characterize the stationary distribution π(·). In the following
proposition we show that there is a number z∗ ≤ log(2)m/d such that under the
stationary distribution, the size of the pool is highly concentrated in an interval of
length O(√m/d) around z∗.34
Proposition 2.5.5. There exists m/(2d + 1) ≤ k∗ < log(2)m/d such that for any
34In this paper, log x refers to the natural log of x.
CHAPTER 2. DYNAMIC MATCHING MARKETS 43
σ > 1,
Pπ[k∗ − σ
√2m/d ≤ Z ≤ k∗ + σ
√2m/d
]≥ 1−O(
√m/d)e−σ
2
.
Proof. Let us define f : R→ R as an interpolation of the difference of transition rates
over the reals,
f(x) := m(1− d/m)x − (x+m(1− (1− d/m)x)).
In particular, observe that f(k) = rk→k+1−rk→k−1. The above function is a decreasing
convex function over non-negative reals. We define k∗ as the unique root of this
function. Let k∗min := m/(2d+ 1) and k∗max := log(2)m/d. We show that f(k∗min) ≥ 0
and f(k∗max) ≤ 0. This shows that k∗min ≤ k∗ < k∗max.
f(k∗min) ≥ −k∗min −m+ 2m(1− d/m)k∗min ≥ 2m
(1− k∗mind
m
)− k∗min −m = 0,
f(k∗max) ≤ −k∗max −m+ 2m(1− d/m)k∗max ≤ −k∗max −m+ 2me−(k∗max)d/m = −k∗max ≤ 0.
In the first inequality we used equation (A.1.4) from Section A.1.
It remains to show that π is highly concentrated around k∗. We prove this in
several steps.
Lemma 2.5.6. For any integer k ≥ k∗
π(k + 1)
π(k)≤ e−(k−k∗)d/m.
And, for any k ≤ k∗, π(k − 1)/π(k) ≤ e−(k∗−k+1)d/m.
Proof. For k ≥ k∗, by (2.5.3), (2.5.4), (2.5.5),
π(k)
π(k + 1)=
(k + 1) +m(1− (1− d/m)k+1)
m(1− d/m)k=k − k∗ + 1−m(1− d/m)k+1 + 2m(1− d/m)k
∗
m(1− d/m)k
CHAPTER 2. DYNAMIC MATCHING MARKETS 44
where we used the definition of k∗. Therefore,
π(k)
π(k + 1)≥ −(1− d/m) +
2
(1− d/m)k−k∗≥ 1
(1− d/m)k−k∗≥ e−(k∗−k)d/m
where the last inequality uses 1 − x ≤ e−x. Multiplying across the inequality yields
the claim. Similarly, we can prove the second conclusion. For k ≤ k∗,
π(k − 1)
π(k)=
k − k∗ −m(1− d/m)k + 2m(1− d/m)k∗
m(1− d/m)k−1
≤ −(1− d/m) + 2(1− d/m)k∗−k+1 ≤ (1− d/m)k
∗−k+1 ≤ e−(k∗−k+1)d/m,
where the second to last inequality uses k ≤ k∗.
By repeated application of the above lemma, for any integer k ≥ k∗, we get35
π(k) ≤ π(k)
π(dk∗e)≤ E
(− d
m
k−1∑i=dk∗e
(i− k∗))≤ E(−d(k − k∗ − 1)2/2m). (2.5.6)
We are almost done. For any σ > 0,
∞∑k=k∗+1+σ
√2m/d
π(k) ≤∞∑
k=k∗+1+σ√
2m/d
e−d(k−k∗−1)2/2m =∞∑k=0
e−d(k+σ√
2m/d)2/2m
≤ e−σ2
min{1/2, σ√d/2m}
The last inequality uses equation (A.1.1) from Appendix A.1. We can similarly upper
bound∑k∗−σ
√2m/d
k=0 π(k).
Proposition 2.5.5 shows that the probability that the size of the pool falls outside
an interval of length O(√m/d) around k∗ drops exponentially fast as the market size
grows. We also remark that the upper bound on k∗ becomes tight as d goes to infinity.
35dk∗e indicates the smallest integer larger than k∗.
CHAPTER 2. DYNAMIC MATCHING MARKETS 45
The following lemma exploits Proposition 2.5.5 to show that the expected value
of the pool size under the stationary distribution is close to k∗.
Lemma 2.5.7. For k∗ as in Proposition 2.5.5 ,
EZ∼π [Z] ≤ k∗ +O(√m/d log(m/d)).
This lemma is proved in the Appendix A.3.2.
Now, Theorem 2.5.1. follows immediately by Lemma 2.5.4 and Lemma 2.5.7
because we have
EZ∼π [Z]
m≤ 1
m(k∗ +O(
√m logm)) ≤ log(2)
d+ o(1)
2.5.2 Loss of the Patient Algorithm
Throughout this section we use Zt to denote the size of the pool under Patient. Let
π : N → R+ be the unique stationary distribution of the Markov Chain on Zt, and
let ζ := EZ∼π [Z] be the expected size of the pool under that distribution.
Once more our proof strategy proceeds in three steps. First, we show that
L(Patient) is bounded by a function of EZ∼π[Z(1− d/m)Z−1
]. Second, we show that
the stationary distribution of Zt is highly concentrated around some point k∗. Third,
we use this concentration result to produce an upper bound for EZ∼π[Z(1− d/m)Z−1
].
By Proposition 2.4.1, at any point in time Gt is an Erdos-Reyni random graph.
Thus, once an agent becomes critical, he has no acceptable transactions with proba-
bility (1− d/m)Zt−1. Since each agent becomes critical with rate 1, if we run Patient
for a sufficiently long time, then L(Patient) ≈ ζm
(1− d/m)ζ−1. The following lemma
makes the above discussion rigorous.
Lemma 2.5.8. For any ε > 0 and T > 0,
L(Patient) ≤ 1
mEZ∼π
[Z(1− d/m)Z−1
]+τmix(ε)
T+εm
d2.
Proof. See Appendix A.3.3.
CHAPTER 2. DYNAMIC MATCHING MARKETS 46
k + 1k k + 2
Figure 2.4: An illustration of the transition paths of the Zt Markov Chain under thePatient Algorithm
So in the rest of the proof we just need to lower bound EZ∼π[Z(1− d/m)Z−1
].
As in the Greedy case, we do not have a closed form expression for the stationary
distribution, π(·). Instead, we use the balance equations of the Markov Chain on Zt
to show that π is highly concentrated around a number k∗ where k∗ ∈ [m/2,m].
Let us start by defining the transition probability operator of the Markov Chain
on Zt. For any pool size k, the Markov Chain transits only to states k + 1, k − 1, or
k − 2. The Markov Chain transits to state k + 1 if a new agent arrives, to the state
k − 1 if an agent gets critical and the the planner cannot match him, and it transits
to state k − 2 if an agent gets critical and the planner matches him.
Remember that agents arrive with the rate m, they become critical with the rate
of 1 and the probability of an acceptable transaction between two agents is d/m.
Thus, the transition rates rk→k+1, rk→k−1, and rk→k−2 are defined as follows,
rk→k+1 := m (2.5.7)
rk→k−1 := k(
1− d
m
)k−1
(2.5.8)
rk→k−2 := k(
1−(
1− d
m
)k−1). (2.5.9)
Let us write down the balance equation for the above Markov Chain (see equation
(2.4.3) for the full generality). Consider the cut separating the states 0, 1, 2, . . . , k
from the rest (see Figure 2.4 for an illustration). It follows that
π(k)rk→k+1 = π(k + 1)rk+1→k + π(k + 1)rk+1→k−1 + π(k + 2)rk+2→k (2.5.10)
CHAPTER 2. DYNAMIC MATCHING MARKETS 47
Now we can characterize π(·). We show that under the stationary distribution,
the size of the pool is highly concentrated around a number k∗ where k∗ ∈ [m/2,m].
Remember that under the Greedy algorithm, the concentration was around k∗ ∈[ m2d+1
, log(2)md
], whereas here it is at least m/2.
Proposition 2.5.9 (Patient Concentration). There exists a number m/2− 2 ≤ k∗ ≤m− 1 such that for any σ ≥ 1,
Pπ[k∗ − σ
√4m ≤ Z
]≥ 1− 2
√me−σ
2
, P[Z ≤ k∗ + σ
√4m]≥ 1− 8
√me− σ2√m
2σ+√m .
Proof Overview. The proof idea is similar to Proposition 2.5.5. First, let us rewrite
(2.5.10) by replacing transition probabilities from (2.5.7), (2.5.8), and (2.5.9):
mπ(k) = (k + 1)π(k + 1) + (k + 2)(
1−(
1− d
m
)k+1)π(k + 2) (2.5.11)
Let us define a continous f : R→ R as follows,
f(x) := m− (x+ 1)− (x+ 2)(1− (1− d/m)x+1). (2.5.12)
It follows that
f(m− 1) ≤ 0, f(m/2− 2) > 0,
so f(.) has a root k∗ such that m/2− 2 < k∗ < m. In the rest of the proof we show
that the states that are far from k∗ have very small probability in the stationary
distribution, which completes the proof of Proposition 2.5.9. This part of the proof
involves lost of algebra and is essentially very similar to the proof of the Proposi-
tion 2.5.5. We refer the interested reader to the Subsection A.3.4 for the complete
proof of this last step.
Since the stationary distribution of Zt is highly concentrated around k∗ ∈ [m/2−2,m− 1] by the above proposition, we get an upper-bound for EZ∼π
[Z(1− d/m)Z
],
which is proved in the Appendix A.3.5.
CHAPTER 2. DYNAMIC MATCHING MARKETS 48
Lemma 2.5.10. For any d ≥ 0 and sufficiently large m,
EZ∼π[Z(1− d/m)Z
]≤ max
z∈[m/2,m](z + O(
√m))(1− d/m)z + 2.
Now Theorem 2.5.2 follows immediately by combining Lemma 2.5.8 and Lemma 2.5.10.
2.5.3 Loss of the Patient(α) Algorithm
Our idea is to slow down the process and use Theorem 2.5.2 to analyze the Patient(α)
algorithm. More precisely, instead of analyzing Patient(α) algorithm on a (m, d, 1)
market we analyze the Patient algorithm on a (m/α, d/α, 1) market. First we need
to prove a lemma on the equivalence of markets with different criticality rates.
Definition 2.5.11 (Market Equivalence). An α-scaling of a dynamic matching mar-
ket (m, d, λ) is defined as follows. Given any realization of this market, i.e., given
(Act , Ant , E) for any 0 ≤ t ≤ ∞, we construct another realization (Act
′, Ant′, E ′) with
(Act′, Ant
′) = (Acα·t, Anα·t) and the same set of acceptable transactions. We say two dy-
namic matching markets (m, d, λ) and (m′, d′, λ′) are equivalent if one is an α-scaling
of the other.
It turns out that for any α ≥ 0, and any time t, any of the Greedy, Patient or
Patient(α) algorithms (and in general any time-scale independent online algorithm)
the set of matched agents at time t of a realization of a (m, d, λ) matching market is
the same as the set of matched agents at time αt of an α-scaling of that realization.
The following proposition makes this rigorous.
Proposition 2.5.12. For any m, d, λ the (m/λ, d/λ, 1) matching market is equivalent
to the (m, d, λ) matching market.36
Now, Theorem 2.5.3 follows simply by combining the above proposition and The-
orem 2.5.2. First, by the additivity of the Poisson process, the loss of the Patient(α)
algorithm in a (m, d, 1) matching market is equal to the loss of the Patient algorithm
36The proof is by inspection.
CHAPTER 2. DYNAMIC MATCHING MARKETS 49
in a (m, d, α) matching market, where α = 1/α + 1. Second, by the above proposi-
tion, the loss of the Patient algorithm on a (m, d, α) matching market at time T is
the same as the loss of this algorithm on a (m/α, d/α, 1) market at time αT . The
latter is upper-bounded in Theorem 2.5.2.
2.6 Welfare and Optimal Waiting Time under Dis-
counting
Theorem 2.6.1. There is a number m/2 − 2 ≤ k∗ ≤ m (as defined in Proposi-
tion 2.5.9) such that for any T ≥ 0, δ ≥ 0 and ε < 1/2m2,
W(Patient) ≥ T − T0
T
( 1− qk∗+O(√m)
1 + δ/2− 12qk∗−O(
√m)− O(m−3/2)
)W(Patient) ≤ 2T0
T+T − T0
T
( 1− qk∗−O(√m)
1 + δ/2− 12qk∗+O(
√m)
+ O(m−3/2))
where T0 = 16 log(m) log(4/ε). As a corollary, for any α ≥ 0, and α = 1/α + 1,
W(Patient(α)) ≥ T − T0
T
( 1− qk∗/α+O(√m)
1 + δ/2α− 12qk∗/α−O(
√m)− O(m−3/2)
)W(Patient(α)) ≤ 2T0
T+T − T0
T
( 1− qk∗/α−O(√m)
1 + δ/2α− 12qk∗/α+O(
√m)
+ O(m−3/2))
Theorem 2.6.2. If m > 10d, for any T ≥ 0,
W(Greedy) ≤ 1− 1
2d+ 1 + d2/m.
Say an agent a is arrived at time ta(a). We let Xt be the sum of the potential
utility of the agents in At:
Xt =∑a∈At
e−δ(t−ta(a)),
i.e., if we match all of the agents currently in the pool immediately, the total utility
CHAPTER 2. DYNAMIC MATCHING MARKETS 50
that they receive is exactly Xt.
For t0, ε > 0, let Wt0,t0+ε be the expected total utility of the agents who are
matched in the interval [t0, t0 + ε]. By definition the social welfare of an online
algorithm, we have:
W(Patient) = E[
1
T
∫ T
t=0
Wt,t+dtdt
]=
1
T
∫ T
t=0
E [Wt,t+dt] dt
2.6.1 Welfare of the Patient Algorithm
In this section, we prove Theorem 2.6.1. All agents are equally likely to become
critical at each moment. From the perspective of the planner, all agents are equally
likely to be the neighbor of a critical agent. Hence, the expected utility of each of the
agents who are matched at time t under the Patient algorithm is Xt/Zt. Thus,
W(Patient) =1
mT
∫ T
t=0
E[2Xt
ZtZt(1− (1− d/m)Zt)dt
]=
2
mT
∫ T
t=0
E[Xt(1− (1− d/m)Zt)
]dt
(2.6.1)
First, we prove the following lemma.
Lemma 2.6.3. For any ε < 1/2m2 and t ≥ τmix(ε),
E [Xt](
1−qk∗+O(√m))−O(m−1/2) ≤ E
[Xt(1− qZt)
]≤ E [Xt]
(1−qk∗−O(
√m))
+O(m−1/2).
The proof of above lemma basically follows from the concentration inequalities
proved in Proposition 2.5.9. See Section A.4 for the details.
It remains to estimate E [Xt]. This is done in the following lemma.
Lemma 2.6.4. For any ε < 1/2m2, t1 ≥ 16 log(m) log(4/ε) ≥ 2τmix(ε),
m−O(m−1/2)
2 + δ − qk∗−O(√m)≤ E [Xt1 ] ≤ m+O(m−1/2)
2 + δ − qk∗+O(√m)
CHAPTER 2. DYNAMIC MATCHING MARKETS 51
Proof. Let η > 0 be very close to zero (eventually we let η → 0). Since we have a
(m, d, 1) matching market, using equation (2.4.1) for any t ≥ 0 we have,
E [Xt+η|Xt, Zt] = Xt(e−ηδ) +mη − ηZt
(Xt
ZtqZt)− 2ηZt
(Xt
Zt(1− qZt)
)±O(η2)
The first term in the RHS follows from the exponential discount in the utility of the
agents in the pool. The second term in the RHS stands for the new arrivals. The
third term stands for the perished agents and the last term stands for the the matched
agents. We use the notation A = B ± C to denote B − C ≤ A ≤ B + C.
We use e−x = 1− x+O(x2) and rearrange the equation to get,
E [Xt+η|Xt, Zt] = mη +Xt − η(1 + δ)Xt − ηXt(1− qZt)±O(η2).
Using Lemma 2.6.3 for any t ≥ τmix(ε) we can estimate E[Xt(1− qZt)
]. Taking
expectation from both sides of the above inequality we get,
E [Xt+ε]− E [Xt]
η= m− E [Xt] (2 + δ − qk∗±O(
√m))±O(m−1/2)−O(η)
Letting η → 0, and solving the above differential equation from τmix(ε) to t1 we get
E [Xt1 ] =m±O(m−1/2)
2 + δ − qk∗±O(√m)
+ C1E(− (δ + 2− qk∗±O(
√m))(t1 − τmix(ε))
).
Now, for t1 = τmix(ε) we use the initial condition E[Xτmix(ε)
]≤ E
[Zτmix(ε)
]≤ m,
and we can let C1 ≤ m. Finally, since t1 ≥ 2τmix(ε) and t1/2 ≥ 2 log(m) we can
upper-bound the latter term with O(m−1/2).
Let T0 = 16 log(m) log(4/ε). Since (for any matching algorithm) the sum of the
CHAPTER 2. DYNAMIC MATCHING MARKETS 52
utilities of the agents that leave the market before time T0 is at most mT0 in expec-
tation, by the above two lemmas, we can write
W(Patient) =2
mT
(mT0 +
∫ T
T0
E[Xt(1− qZt)
]dt)
≤ 2T0
T+
2
mT
∫ T
T0
(m(1− qk∗−O(√m))
δ + 2− qk∗+O(√m)
+ O(m−1/2))dt
≤ 2T0
T+T − T0
T
( 1− qk∗−O(√m)
1 + δ/2− 12qk∗+O(
√m)
+ O(m−3/2))
Similarly, since the sum of the utilities of the agents that leave the market by time
T0 is non-negative, we can show that
W(Patient) ≥ T − T0
T
( 1− qk∗+O(√m)
1 + δ/2− 12qk∗−O(
√m)− O(m−3/2)
)2.6.2 Welfare of the Greedy Algorithm
Here, we upper-bound the welfare of the optimum online algorithm, OPT, and that
immediately upper-bounds the welfare of the Greedy algorithm. Recall that by The-
orem 2.3.1, for any T >, 1/(2d + 1 + d2/m) fraction of the agents perish in OPT.
On the other hand, by definition of utility, we receive a utility at most 1 from any
matched agent. Therefore, even if all of the matched agents receive a utility of 1, (for
any δ ≥ 0)
W(Greedy) ≤W(OPT) ≤ 1− 1
2d+ 1 + d2/m.
2.7 Incentive-Compatible Mechanisms
In this section we design a dynamic mechanism to elicit the departure times of agents.
As alluded to in Subsection 2.2.3, we assume that agents only have statistical knowl-
edge about the rest of the market: That is, each agent knows the market parameters
(m, d, 1), her own status (present, critical, perished), and the details of the dynamic
mechanism that the market-maker is executing. Agents do not observe the graph Gt.
CHAPTER 2. DYNAMIC MATCHING MARKETS 53
Each agent a chooses a mixed strategy, that is she reports getting critical at an
infinitesimal time [t, t+dt] with rate ca(t)dt. In other words, each agent a has a clock
that ticks with rate ca(t) at time t and she reports criticality when the clock ticks. We
assume each agent’s strategy function, ca(·) is well-behaved, i.e., it is non-negative,
continuously differentiable and continuously integrable. Note that since the agent
can only observe the parameters of the market ca(·) can depend on any parameter
in our model but this function is constant in different sample paths of the stochastic
process.
A strategy profile C is a vector of well-behaved functions for each agent in the
market, that is, C = [ca]a∈A. For an agent a and a strategy profile C, let E [uC(a)]
be the expected utility of a under the strategy profile C. Note that for any C, a,
0 ≤ E [uC(a)] ≤ 1. Given a strategy profile C = [ca]a∈A, let C − ca + ca denote a
strategy profile same as C but for agent a who is playing ca rather than ca. The
following definition introduces our solution concept.
Definition 2.7.1. A strategy profile C is a strong ε-Nash equilibrium if for any agent
a and any well-behaved function ca(.),
1− E [uC(a)] ≤ (1 + ε)(1− E [uC−ca+ca ]).
Note that the solution concept we are introducing here is different from the
usual definition of an ε-Nash equilibrium, where the condition is either E [uC(a)] ≥E [uC−ca+ca ] − ε, or E [uC(a)] ≥ (1 − ε)E [uC−ca+ca ]. The reason that we are using
1 − E [uC(a)] as a measure of distance is because we know that under Patient(α) al-
gorithm, E [uC(a)] is very close to 1, so 1−E [uC(a)] is a lower-order term. Thus, this
definition restricts us to a stronger equilibrium concept, which requires us to show
that in equilibrium agents can neither increase their utilities, nor the lower-order
terms associated with their utilities by a factor of more than ε.
Throughout this section let k∗ ∈ [m/2−2,m−1] be the root of (A.3.7) as defined
in Proposition 2.5.9, and let β := (1− d/m)k∗. In this section we show that if δ (the
discount rate) is no more than β, then the strategy vector ca(t) = 0 for all agents
a and t is an ε-mixed strategy Nash equilibrium for ε very close to zero. In other
CHAPTER 2. DYNAMIC MATCHING MARKETS 54
words, if all other agents are truthful, an agent’s utility from being truthful is almost
as large as any other strategy.
Theorem 2.7.2. If the market is at stationary and δ ≤ β, then ca(t) = 0 for all a, t
is a strong O(d4 log3(m)/√m)-Nash equilibrium for Patient-Mechanism(∞).
By our market equivalence result (Proposition 2.5.12), Theorem 2.7.2 leads to the
following corollary.
Corollary 2.7.3. Let α = 1/α+1 and β(α) = α(1−d/m)m/α. If the market is at sta-
tionary and δ ≤ β(α), then ca(t) = 0 for all a, t is a strong O((d/α)4 log3(m/α)/√m/α)-
Nash equilibrium for Patient-Mechanism(α).
The proof of the above theorem is involved but the basic idea is very easy. If
an agent reports getting critical at the time of arrival she will receive a utility of
1 − β. On the other hand, if she is truthful (assuming δ = 0) she will receive about
1 − β/2. In the course of the proof we show that by choosing any strategy vector
c(·) the expected utility of an agent interpolates between these two numbers, so it is
maximized when she is truthful.
The precise proof of the theorem is based on Lemma 2.7.4. In this lemma, we
upper-bound the the utility of an agent for any arbitrary strategy, given that all other
agents are truthful.
Lemma 2.7.4. Let Z0 be in the stationary distribution. Suppose a enters the market
at time 0. If δ < β, and 10d4 log3(m) ≤√m, then for any well-behaved function c(.),
E [uc(a)] ≤ 2(1− β)
2− β + δ+O
(d4 log3(m)/
√m)β,
Proof. We present the sketch of the proof here. The full proof can be found in
Section A.5.
For an agent a who arrives the market at time t0, let P [a ∈ At+t0 ] be the proba-
bility that agent a is in the pool at time t+ t0. Observe that an agent gets matched
in one of the following two ways: First, a becomes critical in the interval [t, t+ ε] with
probability ε ·P [a ∈ At] (1+ c(t)) and if she is critical she is matched with probability
CHAPTER 2. DYNAMIC MATCHING MARKETS 55
E[(1− (1− d/m)Zt−1|a ∈ At
]. Second, a may also get matched (without being crit-
ical) in the interval [t, t + ε]. Observe that if an agent b ∈ At where b 6= a becomes
critical she will be matched with a with probability (1 − (1 − d/m)Zt−1)/(Zt − 1).
Therefore, the probability that a is matched at [t, t+ ε] without being critical is
P [a ∈ At] · E[ε · (Zt − 1)
1− (1− d/m)Zt−1
Zt − 1|a ∈ At
]= ε · P [a ∈ At]E
[1− (1− d/m)Zt−1|a ∈ At
],
and the probability of getting matched at [t, t+ ε] is:
ε(2 + c(t))E[1− (1− d/m)Zt−1|a ∈ At
]P [a ∈ At] .
Based on this expression, for any strategy of agent a we have,
E [uc(a)] ≤ β
m+
∫ t∗
t=0
(2 + c(t))E[1− (1− d/m)Zt−1|a ∈ At
]P [a ∈ At] e−δtdt
where t∗ is the moment where the expected utility that a receives in the interval
[t∗,∞) is negligible, i.e., in the best case it is at most β/m.
In order to bound the expected utility, we need to bound P [a ∈ At+t0 ]. We do this
by writing down the dynamical equation of P [a ∈ At+t0 ] evolution, and solving the as-
sociated differential equation. In addition, we need to study E[1− (1− d/m)Zt−1|a ∈ At
]to bound the utility expression. This is not easy in general; although the distribu-
tion of Zt remains stationary, the distribution of Zt conditioned on a ∈ At can be a
very different distribution. Therefore, we prove simple upper and lower bounds on
E[1− (1− d/m)Zt−1|a ∈ At
]using the concentration properties of Zt. The details
of all these calculations are presented in Section A.5, in which we finally obtain the
following closed form upper-bound on the expected utility of a:
E [uc(a)] ≤ 2dσ5
√mβ +
∫ ∞t=0
(1− β)(2 + c(t))E(−∫ t
τ=0
(2 + c(τ)− β)dτ)e−δtdt.(2.7.1)
CHAPTER 2. DYNAMIC MATCHING MARKETS 56
Finally, we show that the right hand side is maximized by letting c(t) = 0 for all
t. Let Uc(a) be the right hand side of the above equation. Let c be a function that
maximizes Uc(a) which is not equal to zero. Suppose c(t) 6= 0 for some t ≥ 0. We
define a function c : R+ → R+ and we show that if δ < β, then Uc(a) > Uc(a). Let c
be the following function,
c(τ) =
c(τ) if τ < t,
0 if t ≤ τ ≤ t+ ε,
c(τ) + c(τ − ε) if t+ ε ≤ τ ≤ t+ 2ε,
c(τ) otherwise.
In words, we push the mass of c(.) in the interval [t, t + ε] to the right. We remark
that the above function c(.) is not necessarily continuous so we need to smooth it
out. The latter can be done without introducing any errors and we do not describe
the details here. In Section A.5, we show that Uc(a)− Uc(a) is non-negative as long
as δ ≤ β, which means that the maximizer of Uc(a) is the all zero function. Plugging
in c(t) = 0 into (2.7.1) completes the proof of Lemma 2.7.4.
The proof of Theorem 2.7.2 follows simply from the above analysis.
Proof of Theorem 2.7.2. All we need to do is to lower-bound the expected utility of
an agent a if she is truthful. We omit the details as they are essentially similar. So,
if all agents are truthful,
E [u(a)] ≥ 2(1− β)
2− β + δ−O
(d4 log3(m)√m
)β.
This shows that the strategy vector corresponding to truthful agents is a strong
O(d4 log3(m)/√m)-Nash equilibrium.
CHAPTER 2. DYNAMIC MATCHING MARKETS 57
2.8 Concluding Discussions
In this paper, we developed a model of dynamic matching markets to investigate the
features of good policy. This paper innovates by accounting for stochastic departures
and analyzing the problem under a variety of information conditions. Rather than
modeling market thickness via a fixed match-function, it explicitly accounts for the
network structure that affects the planner’s options. This allows market thickness to
emerge as an endogenous phenomenon, responsive to the underlying constraints.
In this part of the paper, to connect our findings to real-world markets, we first
review the key insights of our analysis and then discuss how relaxing our modeling
assumptions would have implications for our results. At the end, we recommend some
promising extensions.
2.8.1 Insights of the Paper
There are many real-world market design problems where the timing of transactions
must be decided by a policymaker. These include paired kidney exchanges, dating
agencies, and online labor markets such as oDesk. In such markets, policymakers
face a trade-off between the speed of transactions and the thickness of the market.
It is natural to ask, “Does it matter when transactions occur? How much does it
matter?” The first insight of this paper is that waiting to thicken the market can
yield substantial welfare gains. In addition, we find that naıve local algorithms that
choose the right time to match can come close to optimal benchmarks that exploit
the whole graph structure. This shows that the right timing decision is a first order
concern in dynamic matching markets.
A key finding of our analysis is that information and waiting time are comple-
ments: If the planner lacks information about agents’ departure times, losses are
relatively large compared to simple algorithms that have access to that information.
This shows that having access short-horizon information about departure times is
especially valuable to the planner. When the urgency of individual cases is private
information, we exhibit a mechanism without transfers that elicits such information
from sufficiently patient agents.
CHAPTER 2. DYNAMIC MATCHING MARKETS 58
These results suggest that the dimension of time is a first-order concern in many
matching markets, with welfare implications that static models do not capture. They
also suggest that policymakers would reap large gains from acquiring timing infor-
mation about agent departures, such as by paying for predictive diagnostic testing or
monitoring agents’ outside options.
2.8.2 Discussion of Assumptions
In order to make this setting analytically tractable, we have made several important
simplifying assumptions. Here we discuss how relaxing those assumptions would have
implications for our results.
First, we have assumed that agents are ex ante homogeneous: They have in ex-
pectation the same average degree.37 What would happen if the planner knew that
certain agents currently in the pool were more likely to have edges with future agents?
Clearly, algorithms that treat heterogeneous agents identically could be inefficient.
However, it is an open question whether there are local algorithms, sensitive to indi-
vidual heterogeneity, that are close to optimal.
Second, we have assumed that agents’ preferences are binary: All acceptable
matches are equally good. In many settings, acceptable matches may vary in quality.
We conjecture that this would reinforce our existing results, since waiting to thicken
the market could allow planners to make better matches, in addition to increasing
the size of the matching.38
Third, we have assumed that agents have the memoryless property; that is, they
become critical at some constant Poisson rate. One might ask what would be different
if the planner knew ahead of time which agents would be long - or short - lived. Our
37Note, however, that agents are ex post heterogeneous as they have different positions in thetrade network.
38To take one natural extension, suppose that the value of an acceptable match is v for both agentsinvolved, where v is a random variable drawn iid across pairs of agents from some distribution F (·).Suppose that the Greedy and Patient algorithms are modified to select the highest match-valueamong the acceptable matches. Then the value to a matched agent under Greedy is (roughly)the highest among N draws from F (·), where N is distributed Binomial(k∗Greedy,
dm ). By contrast,
the value to a matched agent under Patient is (roughly) the highest among N draws from F (·),where N is distributed Binomial(k∗Patient,
dm ). By our previous arguments, k∗Patient > k∗Greedy, so this
strengthens our result.
CHAPTER 2. DYNAMIC MATCHING MARKETS 59
performance bounds on the Omniscient algorithm provide a partial answer to this
question: Such information may be gainful, but a large proportion of the gains can
be realized via the Patient algorithm, which uses only short-horizon information about
agent’s departure times.
In addition, our assumption on agents’ departure processes can be enriched by
assuming agents have a range of sequential states, while an independent process spec-
ifies transition rates from one state to the next, and agents who are at the “final”
state have some exogenous criticality rate. The full analysis of optimal timing under
discounting in such environment is a subject of further research. Nevertheless, our
results suggest that for small waiting costs, if the planner observes critical agents, the
Patient algorithm is close-to-optimal, and if the planner cannot observe the critical
agents, waiting until agents transit to the final state (as there is no risk of agents
perishing before that time) and then greedily matching those agents who have a risk
of perishing is close-to-optimal.
Finally, our theoretical bounds on error terms, O(√m), are small compared to the
market size only if the market is relatively large. What would happen if the market
is small, e.g. if m < 100? To check the robustness of our results to the large market
assumptions, we simulated our model for small markets. Our simulation results (see
Section A.6) show that our results continue to hold for very small markets as well.
2.8.3 Further Extensions
We suggest some promising extensions. First, one could generalize the model to allow
multiple types of agents, with the probability of an acceptable transaction differing
across type-pairs. This could capture settings where certain agents are known ex ante
to be less likely to have acceptable trades than other agents, as is the case for patients
with high Panel Reactive Antibody (PRA) scores in paired kidney exchanges. The
multiple-types framework also contains bipartite markets as a special case.
Second, one could adopt more gradual assumptions about agent departure pro-
cesses; agents could have a range of individual states with state-dependent perishing
rates, and an independent process specifying transition rates between states. Our
CHAPTER 2. DYNAMIC MATCHING MARKETS 60
model, in which agents transition to a critical state at rate λ and then perish immi-
nently, is a limit case of the multiple-states framework.
Third, it would be interesting to enrich the space of preferences in the model,
such as by allowing matches to yield a range of payoffs. Further insights may come
by making explicit the role of price in dynamic matching markets. It is not obvious
how to do so, but similar extensions have been made for static models [49, 42], and
the correct formulation may seem obvious in retrospect.
Much remains to be done in the theory of dynamic matching. As market design
expands its reach, re-engineering markets from the ground up, economists will in-
creasingly have to answer questions about the timing and frequency of transactions.
Many dynamic matching markets have important features (outlined above) that we
have not modeled explicitly. We offer this paper as a step towards systematically
understanding matching problems that take place across time.
Chapter 3
Random Allocation Mechanisms
3.1 Introduction
When cash transfers are limited and goods are indivisible, it can sometimes be im-
possible to allocate goods in an envy-free (“fair”) way. This challenge is faced, for
example, when assigning students to courses, cadets to military bases, or setting a
competitive sports schedule. Early economic studies of this problem by Hylland and
Zeckhauser (1979, HZ) [43] and Bogomolnaia and Moulin (2001, BM) [19] assume
that each agent must receive just a single good and show that it is then possible
to allocate the probabilities of receiving each good in an efficient, envy-free manner.
In a recent paper, Budish, Che, Kojima, and Milgrom (2013, BCKM) [24] propose
expanding this approach to a wider set of multi-item allocation problems in which
the constraints may be more complex than merely a set of one-item-per-person con-
straints. For example, in course allocation, a student may wish to have at least one
class in science and one in humanities in a particular term. They show that for any
expected allocation that satisfies all the constraints, if the constraints have a par-
ticular “bihierarchy” structure, then the expected allocation can always be achieved
by randomizing among pure allocations in which each fractional expected allocation
is rounded up or down to an adjacent integer and all the constraints are simultane-
ously satisfied. When the conditions are satisfied, this sometimes makes it possible to
use mechanisms that select efficient, envy-free expected allocations and to implement
61
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 62
those through randomization.
However, BCKM also found that the bihierarchy condition can be a necessary con-
dition, and that can rule out some potential applications. For instance, the condition
is violated in course allocation if there are both curricular limitations (“at most one
math course”) and scheduling limitations (“only one course beginning at 10am”), and
in school choice if a school with limited capacity has both gender and racial diversity
constraints.
The goal of this paper is to expand the BCKM’s approach to a much more general
class of allocation problems by reconceptualizing the role of constraints. Our anal-
ysis shows that many more constraints can be managed if some of them are “soft”,
in the sense that they can bear small errors with relatively small costs. More pre-
cisely, the innovation of this paper is to partition the full set of constraints into a
set of hard constraints that must always be satisfied exactly, and a set of soft con-
straints that should be satisfied with high probability, at least approximately. The
main theorem of the paper identifies a rich constraint structure that is approximately
implementable, meaning that if an expected allocation satisfies all the constraints,
then it can be implemented by randomizing among pure allocations that satisfy all
the hard constraints and satisfy the soft constraints with only very small errors.
The importance of this result arises from the way it expands potential applications.
In the school choice example, the requirement that each student must be assigned
to exactly one school is (in our conception) a hard constraint that must be satisfied,
but the requirement that 50% of students in a school live in the walk zone may be
a soft constraint – if necessary, 48% will do. Allowing this flexibility is particularly
important when the constraints are inconsistent, and in other cases it provides greater
scope for accommodating individual student preferences. In particular, we employ
this result to fix the ex post unfairness of the random serial dictatorship mechanism,
while maintaining its strategy-proofness.
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 63
3.1.1 Model and Contributions
In this paper, we analyze a general model of matching with indivisible objects. Sec-
tion 3.2 introduces the building blocks of our model. In Subsection 3.2.1, we propose
a new notion of approximate implementation. The key conceptual feature of our
model is that we partition the set of constraints into a set of hard constraints that
are inflexible and a set of soft constraints that are flexible, and we call it a hard-soft
partitioned constraint set. We say that a hard-soft partitioned constraint set is ap-
proximately implementable if for any feasible fractional assignment that satisfies both
hard and soft constraints, there exists a lottery (probability distribution) over pure
assignments such that the following three properties hold: (i) the expected value of
the lottery is equal to the fractional assignment, (ii) the outcome of the lottery sat-
isfies hard constraints, and (iii) the outcome of the lottery satisfies soft constraints
with very small errors1. We then ask: What kind of hard-soft partitioned constraint
structures are approximately implementable?
The main theoretical contribution of the paper is stated in Theorem 3.3.1. This
theorem has two key elements. First, it identifies a rich structure for soft constraints
under which the whole constraint structure is approximately implementable, given
that the structure of hard constraints is the same maximal structure introduced in
BCKM – the “bihierarchical” structure. The structure that we identify for soft con-
straints has several applications in real-life allocation problems.
The second key element of the main theorem arises from its constructive proof. We
prove Theorem 3.3.1 in the Section B.1 by constructing a novel matching algorithm
which approximately implements any given feasible fractional assignment. We invent
a matrix operation that takes a fractional assignment as its input and (randomly)
generates another assignment with fewer fractional elements as its output. By itera-
tive applications of this operation, an integral assignment is generated.2 Throughout
1This requirement will be rigorously defined in Subsection 3.2.1. To see a numerical example,consider a school with 2000 seats. Our framework can guarantee that the probability of assigningmore than 2200 students to the school is at most 0.0013. In particular, the probability of an errorin the satisfaction of soft constraints goes to 0 (with an exponential rate) as the ‘goal’ gets larger.
2It is worth mentioning that the randomized mechanism stops in polynomial time, which is animportant requirement for a practical matching algorithms in relatively large markets.
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 64
this stochastic process, the expected value of the assignment does not change and
hard constraints are satisfied at all iterations. We then apply probabilistic concentra-
tion bounds to our randomized mechanism in order to prove that soft constraints are
satisfied with very small errors. It is worth mentioning that the previous literature on
the economic theory of implementation relies on the Birkhoff-von Neumann theorem
[17, 77] (in HZ and BM) or a generalization thereof (in BCKM). In contrast, our
random mechanism, as mentioned above, exploits a quite different set of probabilistic
tools.
We then discuss two main corollaries of the main theorem with many implications
in real-world marketplaces. Our first corollary is stated in Subsection 3.3.2. The
corollary asserts that if the structure of hard constraints is hierarchical (as opposed to
bihierarchical), then for any general structure of soft constraints, the whole constraint
structure is approximately implementable. Several applications of this corollary are
discussed in Subsection 3.4.1. For instance, in the school choice setting or the course
allocation problem, market-makers can respect a hierarchy of student-side constraints
exactly, while approximately satisfying any other constraints.
The second important corollary of Theorem 3.3.1 is presented in Subsection 3.3.3.
This corollary is about local constraint structures. A constraint is local if it relates
to a single agent or a single object. For instance, “schools s1 and s2 cannot admit
more than q students in total” is not local because it involves multiple schools and
multiple students. Our second corollary states that given a set of hard constraints
only of capacity constraints (e.g. if the number of available seats in schools cannot be
violated and all students should be assigned to exactly one school), any set of local
soft constraints can be approximately satisfied. We discuss several applications for
which such a constraint structure is natural. In the school choice setting, for example,
this corollary can guarantee that each student is assigned to exactly one school and
schools’ capacity constraints are not violated, while racial, diversity, and walk zone
constraints are satisfied, at least approximately.
In Subsection 3.3.4, we state a fully generalized version of our main theorem and
show that given a bihierarchy of hard constraints, any set of soft constraints can
be approximately satisfied, but with a weaker notion of approximate satisfaction.
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 65
This finding shows that if the set of hard constraints has its maximal form (i.e. it is
bihierarchical) then one should either restrict the structure of soft constraints to what
we identified in the main theorem, or use a weaker notion of approximate satisfaction
for soft constraints with a more complicated structure. It is, however, worthwhile
to emphasize that even under this weaker notion, our quantitative bounds guarantee
that the probability of violating a soft constraint goes to 0 (with an exponential rate)
as the right-hand side of the constraint grows.
The tools that we exploit in constructing our randomized mechanism provide us
with a luxury that could not be afforded by previous tools, because our matching
algorithm guarantees that for any arbitrary set of “weights” for the elements of soft
constraints, they will be satisfied with only very small errors. This has multiple
interesting applications. First, the literature often considers constraints such as “the
sum of all African-American students assigned to school 1 should be at least q”, in
which all African-American students have equal weights. In our model, however, a
soft constraint can require “the sum of male African-Americans plus k times the sum
of female African-Americans should be at least q”. Second, in implementing walk-
zone requirements, one can directly incorporate each student’s distance from different
schools into the constraints; see Subsection 3.4.2.
The second application of our “generalized weights” setting is stated in Subsec-
tion 3.4.3, in which we discuss ex post properties of our randomized mechanism.
Market-makers are concerned with ex post properties of allocation mechanisms be-
cause ex ante fairness does not guarantee ex post fairness. Our main result in this
section guarantees that under our proposed randomized mechanism, ex post utilities
of the agents (or objects) are approximately equal to their ex ante utilities, and ex
post social welfare is approximately equal to the ex ante social welfare.
In Section 3.5, we employ our utility and welfare guarantees to improve two clas-
sical allocation mechanisms, namely the random serial dictatorship (RSD)3 and the
pseudo-market mechanisms. It is well-known that RSD with multi-unit demand is ex
ante fair and strategy-proof, but ex post (very) unfair. We fix the ex post unfairness
3RSD works as follows: draw a fair random ordering of the agents and then let agents select theirmost favorite bundle of objects (among those remaining) according to the realized random orderingwithout violating the constraints.
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 66
of RSD by constructing the expected allocation associated with the RSD, and then
approximately implementing it by employing our main theorem. The mechanism
remains strategy-proof by a revelation principle argument, and our utility bounds
guarantee that the ex post allocation is “approximately” fair. Next, in ??, we employ
the competitive equilibrium from equal incomes (CEEI) mechanism introduced in
BCKM to construct a fractional assignment, and then employ our techniques to (ap-
proximately) satisfy many more constraints. Our implementation technique, more-
over, guarantees that ex post utilities are approximately equal to ex ante utilities.
Since the competitive equilibrium allocation is envy-free, it then follows that the ex
post allocation is “approximately envy-free”.4
3.1.2 Related Work
Randomization is commonplace in everyday life and has been studied in various set-
tings such as school choice, course allocation, and house allocation [1, 2, 21]. Perhaps
the most practically popular random mechanism is to draw a fair random ordering
of agents and then let the agents select their most favorite object (among those re-
maining) according to the realized random ordering without violating the constraints.
This mechanism, which is known as Random Serial Dictatorship (RSD) is a desirable
mechanism as it is strategy-proof and ex post Pareto efficient [2, 15, 29]. Nevertheless,
RSD is ex ante inefficient, ex post (highly) unfair, and cannot handle lower quotas
[19, 50, 40]. Manea (2009) [58], Kojima tand Manea (2010) [52], and Che and Kojima
(2010) [27] compare PS and RSD and analyze their connections in large markets.
The idea to construct a fractional assignment and then implementing it by a lot-
tery over pure assignments was first introduced in Hylland and Zeckhauser (1979)
[43] for cardinal utilities. Bogomolnaia and Moulin (2001) [19] constructed a mecha-
nism, the Probabilistic Serial Mechanism (PS), for ordinal utilities based on the same
4This paper has benefited from helpful comments of many people. The invaluable constantguidance of Paul Milgrom has been essential. We thank Alex Wolitzky for his constructive commentson the earlier drafts of the paper. We thank Al Roth for his valuable suggestions. We are also gratefulto Gabriel Carroll, Kareem Elnahal, Fuhito Kojima, Matthew Jackson, Bobby Pakzad-Hurson, andIlya Segal for their great suggestions, and several seminar participants for their comments andfeedback. All errors are ours.
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 67
trick. Both papers model one-to-one matching markets with no other constraints.
Budish, Che, Kojima, and Milgrom [2013] [24] build on those two papers by consid-
ering a richer constraint structure.5 Our paper extends this literature by designing a
randomized mechanism which can accommodate a much richer class of constraints.
The approximate satisfaction of constraints has been studied in a few recent pa-
pers. Budish (2011) studies the problem of combinatorial assignment by introducing
a notion of approximate competitive equilibrium from equal incomes, which treats
course capacities as flexible constraints [21]. Ehlers et al (2011) introduce a deferred
acceptance algorithm with soft bounds in which they adjust group-specific lower and
upper bounds to achieve a fair and non-wasteful mechanism [32]. There are some
key points that separate our paper from these works. First, we propose a frame-
work which can handle “overlapping” constraints. For instance, in the school choice
setting, we can accommodate racial, gender, and walk-zone priority constraints si-
multaneously. Second, we provide a rich language for the market-maker to declare
a partitioned constraint set, which contains both flexible and inflexible constraints.
Third, our mechanism runs in polynomial time, whereas the approach introduced in
[21] is computationally hard6.
The problem of reduced-form implementation in the auction literature is also
related to our work [60, 20, 26]. In this problem, an interim allocation, which describes
the marginal probabilities of each bidder obtaining the good as a function of his type,
is constructed. Then, same as our problem, the question that is asked here is: which
interim allocations can be implemented by a lottery over feasible pure allocations?
The approximate satisfaction of constraints, however, is not studied in that literature.
The trick to approximately implement a fractional allocation has been employed
in Nguyen, Peivandi, and Vohra (2014) [63] as well, in which they model a one-
sided matching market with complementarities. Their method is both conceptually
and technically different from ours. From a conceptual perspective, their goal is to
5In a recent work, Pycia and Unver [68] study a more general structure (the Totally Unimodularor TU structure) and show that they can accommodate constraints such as strategy-proofness andenvy-freeness as linear constraints as long as they fit into the TU structure. Our approach isconceptually different from theirs since we consider flexible constraints (i.e. goals) which may notfit into the TU structure.
6It cannot be solved in polynomial time [69].
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 68
handle complementarities in a framework with only capacity constraints, whereas
our paper is concerned with implementing generalized constraint structures. From
a technical perspective, they employ a different iterative rounding technique to first
prove the existence of the lottery, and then to construct the lottery, their paper uses
the “ellipsoid method” which implements an assignment with an expected value that
is arbitrarily close (but not exactly equal) to the original fractional assignment–unlike
our method, which employs a different technique, and implements the assignment
exactly. Their approximation bounds are additive, rather than probabilistic. The
reason that they are able to provide a small additive upper bound for the violation of
capacities in [63] is that they do not have any constraints except capacity constraints
and their framework does not include any intersecting constraints.
In addition, various rounding techniques has been developed in the computer
science literature, for instance, see [28, 74]. The techniques used in [28, 74] inspired
our design. Their rounding techniques, however, are specifically designed for the job
scheduling problem. As a result, their randomized algorithms accommodate neither
non-local soft constraints, nor (bi)hierarchical hard constraints.
3.2 Setup
Consider an environment in which a finite set of objects O has to be allocated to
a finite set of agents N . We denote the set of agent-object pairs, N × O, by E,
where each (n, o) ∈ E is an edge. Sometimes we use ‘e’ to denote edges. A pure
assignment is defined by a non-negative matrix X = [Xno] where each Xno ∈ {0, 1}denotes the amount of object o which is assigned to agent n for all (n, o) ∈ E. We
require the matrix to be binary valued to capture the indivisibility of the objects.
A block B ⊆ E is a subset of edges. A constraint S is a triple (B, qB, qB),
which is a block B associated with a vector of integer quotas (qB, qB) as the floor
and ceiling quotas on B. A structure is a subset E ⊆ 2E; i.e. a collection of blocks.
A constraint set is a set of constraints. Let q = [(qB, qB)B∈E ].
We say that X is feasible with respect to (E ,q) (or simply, with respect to Ewhen q is clearly known from the context) if
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 69
Figure 3.1: Model framework
qB≤∑e∈B
Xe ≤ qB ∀B ∈ E . (3.2.1)
We call a block B ∈ E agent k’s capacity block when B = {Xkj|j ∈ O}.Similarly, we call a block B ∈ E an object m’s capacity block when B = {Xim|i ∈N} (See Figure 3.2). A capacity constraints is a constraint (B, q
B, qB), where
B ∈ E is a capacity block. We sometimes refer to capacity constraints of agents and
objects as row constraints and column constraints, respectively.
Figure 3.2: Capacity blocks
A fractional assignment is defined by a matrix x = [xno], where each xno ∈ [0, 1]
is the quantity of object o assigned to agent n. To distinguish between pure and
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 70
fractional assignments, we usually use X to denote a pure assignment and x for a
fractional assignment. We use the term expected assignment to address fractional
assignments occasionally.
Given a structure E and associated quotas q, a fractional assignment matrix x is
implementable under quotas q if there exist positive numbers λ1, . . . , λK , which
sum up to one, and pure assignments X1, . . . , XK , which are feasible under q, such
that
x =K∑i=1
λiXi.
We also say that a structure E is universally implementable if, for any quotas
q = (qB, qB)B∈E , every fractional assignment matrix satisfying q is implementable
under q.
The main existing theoretical result on the implementability of a structure is
introduced in the BCKM’s paper [24], where they identify bihierarchy as a sufficient
condition for universally implementability of a structure. More precisely, a structure
H is a hierarchy if for every pair of blocks B and B′ in H, we have that B′ ⊂ B or
B ⊂ B′ or B ∩B′ = ∅. A simple hierarchy is depicted in Figure 3.3. A structure H is
a bihierarchy if there exists two hierarchies H1 and H2 such that H1 ∩H2 = ∅ and
H = H1 ∪ H2. The following theorem identifies a sufficient, and almost necessary,
condition under which a structure is universally implementable.
Figure 3.3: A hierarchy
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 71
Theorem 3.2.1. [BCKM, 2013] If a structure H is a bihierarchy, then it is univer-
sally implementable. In addition, if H contains all agents and objects capacity blocks,
then it is universally implementable if and only if it is a bihierarchy.
3.2.1 Approximate Implementation
In many assignment problems, the involved constraints have multiple intersections
and the bihierarchy assumption fails. The following two examples clarify the bihier-
archy limitations.
Example 3.2.2. Consider a course allocation setting, where students are required to
take at least one of the two courses {f1, f2} and at most one of the finance course
f1 and the microeconomics course m. It is easy to verify that together with courses’
capacities, these constraint blocks do not form a bihierarchy.
Example 3.2.3. In the Boston School Program in 2012, fifty percent of each school
seats were set aside for walk-zone priority students. Consider a school which also has
a group-specific quota on female students. Together with the requirement that each
student should be assigned to one school, these blocks do not form a bihierarchy. (See
Figure 3.4 for an illustration)
Figure 3.4: Overlapping constraints may not fit into the bihierarchical structure. Inthe school choice setting, for instance, walk-zone priorities and gender (or racial)diversity requirements are inconsistent with the bihierarchy.
We overcome the limitations of the bihierarchical structure by reconceptualizing
the role of constraints. We show that by treating some constraints as goals rather
than inflexible constraints, we can accommodate many more constraints. In the school
choice setting, for example, a slight violation in racial or gender diversity goals is not
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 72
infinitely costly. When such goals are modeled as inflexible constraints, it is implicitly
assumed that even a slight violation is infinitely costly, which is not often the case.
More precisely, we accommodate both flexible and inflexible constraints into the allo-
cation problem by proposing the following framework: We ask the market-maker to
partition the full set of constraints into a set of hard constraints that must be satisfied
exactly and a set of soft constraints that must be satisfied, at least approximately.
Accordingly, the structure will be partitioned into two sets: a set of hard blocks,
H, which are blocks of inflexible constraints, and a set of soft blocks, S, which are
blocks of flexible constraint. We refer to E = H ∪ S as a hard-soft partitioned
structure, or simply a partitioned structure.
Another novel generalization that we consider in this paper is that in our model,
elements of soft constraints can have a more general form compared to hard con-
straints. More precisely, for a soft block B′, we say X is feasible with respect to B′
if:
qB′≤∑e∈B′
weXe ≤ qB′
where we can take any arbitrary non-negative value and qB′
and qB′ can be any
non-negative real number. Recall that, similar to BCKM, for a hard block B, we
require we = 1 for all e ∈ B and qB
and qB are restricted to be non-negative integers.
This generalization expands the scope of practical applications of the model; this will
be discussed in Subsection 3.4.1.
Our goal in this paper is to identify structural conditions imposed on H and Sunder which E = H ∪ S is “approximately implementable”. In the following, we
rigorously define the notion of approximate implementation.
Definition 3.2.4. Given a hard-soft partitioned structure E = H ∪ S, we say E is
Approximately Implementable if for any vector of quotas q and any expected
assignment x which is feasible with respect to (E ,q), there exists a lottery (probability
distribution) over pure assignments X1, . . . , XK such that, if we denote the outcome
of the lottery by the random variable X, the following properties hold:
P1. Assignment Preservation: E[X] = x.
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 73
P2. Exact Satisfaction of Hard Constraints: All constraints in H are satisfied.
P3. Approximate Satisfaction of Soft Constraints: For any soft block B ∈ Swith
∑e∈B xe = µ and for any ε > 0, we have
Pr(dev+ ≥ εµ) ≤ e−µε2
3 (3.2.2)
Pr(dev− ≥ εµ) ≤ e−µε2
2 (3.2.3)
where dev+ and dev− are defined as follows:
dev+ = max(0,∑e∈B
Xe − µ)
dev− = max(0, µ−
∑e∈B
Xe
)Property 1 simply states that there exists a lottery which implements x. Property
2 states that hard constraints are satisfied with no error. Property 3 is the key part
of the definition which quantifies our notion of approximation.
Property 3 is the core conceptual part of this definition. Property 3 is appealing as
it guarantees that the probability of violating constraints is exponentially decreasing
in µ and goes to 0 very rapidly. Hence, for large goals (i.e. constraints with large right-
hand or large left-hand sides), the probability of violating them by a factor greater
than ε is very small. Property 3 also guarantees that the probability of observing
very bad events (violating constraints by a large multiplicative factor ε) decreases
exponentially as ε grows. For example, in a school with 2000 seats, the probability of
admitting more than 2100 students is bounded above by 0.19, and the probability of
admitting more than 2200 students is no more than 0.0013.
3.3 The Main Theorem
We present our main theoretical result in this section. Given a partitioned structure
E = H∪S, our goal in this section is to identify structures for H and S under which
E is approximately implementable in the sense of Definition 3.2.4.
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 74
First of all, note that Theorem 3.2.1 shows that even if S = ∅ (i.e., there are no
soft constraints), in order for E to be implementable, bihierarchy is a sufficient and
(almost) necessary condition7 for H. In other words, the bihierarchy is the weakest
condition we can impose on hard constraints. We maintain this maximal structure
and let hard blocks form a bihierarchy; i.e., we assume H = H1 ∪ H2, where H1
and H2 are two hierarchies. Then, given a bihierarchical hard structure, we aim to
identify a structural condition, if any, for soft blocks S under which E = H ∪ S is
approximately implementable.
In the corollaries of our main theorem, we show that by imposing further restric-
tions on the structure of hard blocks, one can approximately implement a fully general
set of soft constraints.
Figure 3.5: The solid blocks form a hierarchy H1. The dashed blocks are all in thedeepest level of H1. A block that, for example, contains X32 and X33 is not in thedeepest level of H1.
3.3.1 The Structure of Soft Blocks
Now we show that if H forms a bihierarchy, there exists a rich structure for the soft
blocks S under which E = H∪S is approximately implementable. To do so, we need
to define a few helpful concepts. For a block B ∈ S, we say that B is in the deepest
level of H1 if for any block C ∈ H1, either B ⊆ C or B ∩C = ∅. (See Figure 3.5 for
7We use the term “almost” because it is not a necessary condition in general, but it is necessaryin two-sided matching markets with finite capacities.
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 75
an illustration) We also say that B ∈ S is in the deepest level of a bihierarchy
H = H1∪H2 if it is in the deepest level of either of H1 or H2. The following theorem
states our main result.
Theorem 3.3.1. [The Main Theorem] Let E = H ∪ S be a hard-soft partitioned
structure such that H is a bihierarchy and any block in S is in the deepest level of H.
Then, E is approximately implementable.
Proof Overview. We present only an overview of the proof here. The full proof
can be found in the Section B.1. The proof is constructive; that is, we propose a ran-
domized mechanism that, given a partitioned structure with properties as described
in Theorem 3.3.1, approximately implements a given feasible fractional assignment.
To do so, let us define a constraint to be tight if it is binding, and to be floating
otherwise. This definition applies to the implicit constraints 0 ≤ xe ≤ 1 for all e ∈ E.
The core of our randomized mechanism is a probabilistic operation that we de-
sign, called Operation X . We iteratively apply Operation X to the initial fractional
assignment until a pure assignment is generated. At each iteration t, which is sym-
bolically depicted in Figure 3.3.1, the fractional assignment xt is converted to xt+1 in
a way such that: (1) the number of floating constraints decreases, (2) E(xt+1|xt) = xt,
and (3) xt+1 is feasible with respect to H. The first property guarantees that after a
finite (and small) number of iterations8, the obtained assignment is pure. The second
property ensures that the resulting pure assignment is equal to the original fractional
assignment in expectation. The third property guarantees that all hard constraints
are satisfied throughout the whole process of the mechanism.
Figure 3.6: A symbolic representation of our iterative mechanism. At each iterationt, the fractional assignment xt is converted to xt+1, and this continues until a pureassignment is generated.
8Our randomized mechanism stops after at most |H|+ |E| iterations.
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 76
In the last step, we prove that after iterative applications of Operation X , soft
constraints are approximately satisfied. Roughly speaking, we design Operation Xin such a way that it never increases (or decreases) two (or more) elements of a soft
constraint at the same iteration. Consequently, elements of each soft block become
“negatively correlated”. Negative correlation then allows us to employ probabilistic
concentration bounds to prove that soft constraints are approximately satisfied.
In Section B.1, we design Operation X and prove its desired properties.
3.3.2 Corollary 1: Fully General Soft Structure
The first corollary of the main theorem asserts that if H forms a single hierarchy
(rather than two hierarchies), then for any arbitrary set of soft constraints, E [=]H∪Sis approximately implementable.
Corollary 3.3.2 (of Theorem 3.3.1). Let E = H ∪ S be a hard-soft partitioned con-
straint set, where H is a single hierarchy; i.e., H1 = ∅ or H2 = ∅. Then, for all
S ⊆ 2E, E is approximately implementable.
Proof. By assumption, at least one of the H1 or H2 is empty. Without loss of gen-
erality, suppose H1 = ∅. We add a “dummy” constraint to H1, which contains all
the elements, i.e. the constraint 0 ≤∑
e∈E xe < ∞. Obviously, any soft constraint
block is in the deepest level of H1. Hence, by Theorem 3.3.1, E is approximately
implementable.
This corollary illustrates the trade-off between the richness of hard and soft struc-
tures. On the one hand, if H forms a bihierarchy (which is almost the richest possible
structure for H), then by Theorem 3.3.1, S needs to be in the deepest level of H. On
the other hand, if H forms a hierarchy (which is not as rich as bihierarchy), then Swould have its richest possible structure: a fully general structure.
3.3.3 Corollary 2: Local Structure
Our second corollary considers a specific structure for H, involving only capacity
blocks. Recall that an agent or object’s capacity (or an agent’s row and an object’s
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 77
Figure 3.7: A local structure can contain any kind of blocks, as long as the blocksare subsets of columns or rows; i.e. they involve a single object and possibly multipleagents or a single agent and possibly multiple objects.
column) block is the block that involves all the elements in the row (column) cor-
responding to that agent (object) in the assignment matrix (See Figure 3.2). We
say that a structure is local if each block involves one agent with possibly multiple
objects or one object with possibly multiple agents, but not multiple agents and mul-
tiple objects at the same time. In this case, if S is a local structure, then E = H∪ Sis approximately implementable.
To fix ideas, let E be a structure such that E = H ∪ S, where H = H1 ∪ H2, H1
contains all row blocks, and H2 contains all column blocks. Also, let S contain only
sub-row or sub-column blocks, which we formally define below.9
Definition 3.3.3. The structure corresponding to an agent n ∈ N , denoted by E(n),
is the set of all blocks B ∈ E such that B can be represented as
qB≤∑j∈Z
xnj ≤ qB
for some subset Z ⊆ O. By adapting this definition in a natural way, we also define
the structure corresponding to an object o ∈ O, and denote it by E(o).
9This model of ‘local’ structures, which is a special case of our model, has been studied in [74]as well.
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 78
Definition 3.3.4. A structure E is local if
E =⋃
v∈N∪O
E(v).
In words, a structure is local if all blocks are sub-row or sub-column blocks. Note
that for any v ∈ N ∪ O, there are no restrictions on the structure of the blocks in
E(v), e.g. they may have intersections. An example of a local structure is depicted
in Figure 3.7.
Corollary 3.3.5 (of Theorem 3.3.1). Let E = H ∪ S be a structure such that His the set of all capacity blocks and S is a local structure. Then E is approximately
implementable.
Proof. Since H is the set of all capacity blocks and S is local, any block in S is in
the deepest level of H1 or H2. The corollary follows from Theorem 3.3.1.
3.3.4 Generalized Structures
In this section, we generalize our result further and show that even if a constraint is not
in the deepest level the bihierarchy of hard constraints, it can still be approximately
satisfied with a slightly weaker notion of approximate satisfaction.
First, we need to define a new concept which, intuitively, describes the level of
complexity of the structure of a soft constraint. Consider a bihierarchy H = H1∪H2.
For a block B ∈ S, we say that B is in depth k of hierarchy H1 if B can be
partitioned into k subsets B1, B2, · · · , Bk such that all are in the deepest level of
H1, and moreover, no partitioning of B into k − 1 subsets satisfies this property
(See Figure 3.8 for an illustration). We also say that B ∈ S is in the depth k of
bihierarchy H = H1 ∪H2 if it is in depth k of either of H1 or H2.
The following theorem states that any soft constraint is approximately imple-
mentable, and the probabilistic guarantee that we provide for the error size depends
on the depth of the block corresponding to the constraint.
Theorem 3.3.6. Let E = H ∪ S be a hard-soft partitioned structure such that H is
a bihierarchy. Then, for any vector of quotas q and any expected assignment x which
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 79
Figure 3.8: The block B = {X13, X14, X23, X24} is in depth 2 of the depicted hierar-chy, since it can be partitioned to two subsets, B1 = {X13, X23} and B2 = {X14, X24}both of which are in the deepest level of the hierarchy.
is feasible with respect to (E ,q), there exists a lottery (probability distribution) over
pure assignments X1, . . . , XK such that, if we denote the outcome of the lottery by
the random variable X, the following properties are satisfied:
P1. E[X] = x.
P2. All constraints in H are satisfied.
P3. For any ε > 0 and for any soft block B ∈ S which is in depth k of H, if∑e∈B xe = µ, then we have
Pr(dev+ ≥ εµ) ≤ k · e−µε2
3k (3.3.1)
Pr(dev− ≥ εµ) ≤ k · e−µε2
2k (3.3.2)
where dev+ and dev− are defined as follows:
dev+ = max(0,∑e∈B
Xe − µ)
dev− = max(0, µ−
∑e∈B
Xe
)We prove this theorem in ??.
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 80
This theorem clarifies the natural trade-off between the complexity of the structure
of hard constraints and the the probabilistic guarantees that we provide for soft
constraints: The richer the former, the weaker the latter becomes. On the one hand,
we have seen in Corollary 3.3.2 that if the structure of hard blocks is hierarchical,
then any set of soft constraints is approximately implementable. Theorem 3.3.6, on
the other hand, shows that if the market-maker insists on having a bihierarchical hard
structure then he should either restrict the structure of soft blocks to the deepest level
of the bihierarchy, or he should weaken the probabilistic guarantees for satisfying the
‘goals’ that are not in the deepest level of the bihierarchy.
3.4 Applications
In this section, we discuss several applications of our results. We first start by dis-
cussing applications of our results in the school choice environment. We then intro-
duce a novel way to handle walk-zone priority quotas based on students’ distance to
a different schools. Next, we show how our framework can provide appealing ex post
guarantees for the efficiency and fairness of the final allocation. Finally, we modify
the popular Random Serial Dictatorship mechanism in multi-unit demand settings to
make in ex post approximately fair.
3.4.1 Diversity Requirements in School Choice
Consider a school choice setting, where n students are to be assigned to k schools.
Several types of constraints naturally arise in this market. A few examples are:
• Capacity constraints of schools and students. These constraints require that
each student must be assigned to exactly one school, and that each school has
limited number of available seats.
• Walk-zone priorities. Families get walk-zone priority to any school within their
walk-zone. For instance, schools in the Boston School Program are required to
assign fifty percent of their seats to students within the walk-zone. The other
half is open to everyone, including those in the walk-zone.
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 81
• Affirmative action policies. Affirmative action is defined as “positive steps taken
to increase the representation of women and minorities in areas of employment,
education, and culture from which they have been historically excluded.”[3] One
goal of such policies is to increase diversity, and balancing out the social effects
that weaken specific groups.10 Affirmative action policies are usually imple-
mented as minimum quotas on students within a minority group.11
• Grade-based quotas. Schools may have grade-based diversity policies. For in-
stance, New York City’s Educational Option program has quotas based on test
scores [1].
The bihierarchy assumption often fails when multiple constraints such as these
exist. For example, if a school has minimum quotas on both female students and
walk-zone priority students, as in Example 3.2.3, the blocks associated with these two
constraints overlap and the bihierarchy assumption fails. In contrast, our framework
can accommodate the above-mentioned constraints even if their blocks overlap.
The importance of our result is amplified by noting that school-side constraints
are somewhat flexible, since a school may be willing to go a bit over capacity in order
to satisfy gender or racial diversity requirements. In the following, we show one way
to apply both Corollary 3.3.2 and Corollary 3.3.5 into the school choice setting.
Corollary 3.3.2 in school choice setting: Let H be a single hierarchy, which
includes all the blocks of student-side inflexible constraints. A natural student-side
hard constraint is that each student should be assigned to exactly one school. Hence,
one can define H to be the set of all student-side capacity blocks, where qB
= qB =
1 for all B ∈ H. Now, by Corollary 3.3.2, any general set of constraints can be
approximately satisfied.
It is worth emphasizing that Corollary 3.3.2 allows for multiple schools to be
involved in the same constraint (see Figure 3.9 for an illustration), which is important
in some applications. For instance, in New York City public schools, a considerable
10 Another argument in favor of affirmative action policies is that they increase structural inte-gration, which “serves the ideal of equal opportunity.”[45]
11See [39, 4, 53] for detailed theoretical analysis of affirmative action policies.
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 82
Figure 3.9: In the school choice problem, when hard constraints require each studentto be assigned to exactly one school, the set of hard blocks forms a single hierarchy.Consequently, by Corollary 3.3.2, any set of soft constraints can be approximatelyimplemented.
fraction of these schools are co-located12. Consequently, they have joint quotas such
as an upper quota on the number of students who can be inside a school’s building at
any point in time. By Corollary Corollary 3.3.2, these “joint” blocks can be included
in S and our mechanism can satisfy them with very small errors.
Corollary 3.3.5 in school choice setting: If a violation in both row and column
constraints is very costly, then Corollary 3.3.5 can be more useful than Corollary 3.3.2.
In this case, we define H to be the set of all row and column blocks. Obviously, H is
a bihierarchy. Now we can apply Corollary 3.3.5 to guarantee that for any local S,
E = H ∪ S is approximately implementable.
3.4.2 Distance-based Walk-zone Priorities
Policy-makers in the school choice systems are often interested in prioritizing stu-
dents based on their distance. A typical way to implement walk-zone priorities is
partitioning the city into artificial zones and imposing lower quotas on the number of
students who live in the same zone as schools. This method treats students who live
12See http://www.nyccharterschools.org/sites/default/files/resources/Facts_
Colocation.pdf
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 83
just inside and outside of each zone’s border very differently, as it assigns a weight 1
to students who live inside and a weight 0 to students who live outside.
A more natural way to include distance-based priorities in the school choice prob-
lem is to assign weights to each student-school pair (s, h) based on student s’s distance
to school h. More formally, one can impose qB≤∑
(s,h)∈B d(s,h)x(s,h) ≤ qB as a soft
constraint, where d(s,h) is either the distance of student s from school h, or any other
“penalty function”.13 It is very straightforward to see that this can be accommodated
into our framework, as soft constraints can have real-valued coefficients.
3.4.3 Ex post Guarantees
In practice, the indivisibility of the objects and (possibly) the lack of monetary trans-
fers make the allocation of resources likely to be asymmetric and unfair. One of the
main motivations for randomization is to restore fairness by constructing an ex ante
fair allocation. Nevertheless, given a fair fractional allocation, there could be very
large discrepancies in realized utilities.
Our next application concerns ex post properties of our proposed randomized
mechanism. We show that the mechanism approximately maintains the fairness and
efficiency properties of the original fractional assignment. Then in the next section,
we employ those guarantees to refine the classical random serial dictatorship (RSD)
mechanism. In particular, we show that by adding utility goals (as soft constraints)
to the RSD mechanism, we can fix its ex post unfairness, while it remains strategy-
proof. We also employ the same guarantees to refine the “pseudo-market mechanism”
(introduced in HZ and BCKM) in ??. Our expansion helps us to manage a richer set
of constraints, and to provide ex post guarantees for the utilities.
The following definition captures our main notion of our approximate guarantees
for utilities and welfare.
Definition 3.4.1. A random variable x is approximately lower-bounded by a con-
stant µ (denoted by µ . x) if the following two conditions hold:
13d(s,h) can incorporate considerations such as how accessible school h is for student s by publictransportation, as well.
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 84
1. E(x) = µ
2. Pr(x ≤ µ(1− ε)
)≤ e−µε
2/2
In words, a random variable x is approximately lower-bounded by a constant µ if
x is equal to µ in expectation and the probability of x being less than µ(1− ε) is very
small, for any ε > 0. It is clear that if µ = 0, then any random variable for which
E(x) = 0 is approximately lower-bounded by 0. As will be clear soon, this definition
is particularly interesting for larger values of µ.
Our general framework provides a simple way to get lower bounds for ex post
utilities. We show that if the ex ante assignment is “fair” (e.g., if it respects the
equal treatment of equals or if it is envy-free), then the ex post allocation of our
mechanism remains fair, at least approximately.
To fix ideas, consider an environment where the set of hard blocks H forms a
single hierarchy14. We impose no restriction on the soft structure S. We assume
that utilities are Von Neumann-Morgenstern utilities and are additive; that is, there
exist values (uik)k∈O such that an agent i’s utility from an allocation x, with i’th
row xi = (xi1, xi2, · · · , xi|O|), is vix =∑|O|
k=1 xikuik, where, without loss of generality,
uik ∈ [0, 1] for all i, k. Also, let W (x) =∑|N |
i=1 vix be the social welfare associated
with allocation x.In the following theorem, we guarantee that the ex post utility of
any agent i and the ex post social welfare are approximately lower-bounded by vix
and W (x), respectively.
Theorem 3.4.2 (Utility and Welfare Bounds). Any feasible fractional assignment x
is approximately implementable in such a way that for each i, if X is the outcome of
the lottery, then vix . viX and W (x) . W (X).
Proof. The idea of the proof is to add the following artificial constraints for the social
welfare and for the utility of agents to the soft constraint set:
|O|∑k=1
Xikuik ≥ vix ∀i ∈ N,
14This assumption is for expositional clarity. In fact, it is enough if every all-row blocks are in thedeepest level of H.
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 85
|N |∑i=1
|O|∑k=1
xikuik ≥ W (x).
Since hard blocks form a single hierarchy, the blocks associated with these new
constraints are in the deepest level of the empty hierarchy of the hard structure. The
proof follows immediately from Theorem 3.3.1.
Remark 3.4.3. Theorem 3.4.2 provides lower bounds that are interesting when vix
is relatively large, which is the case when each agent is (in expectation) allocated
to several objects (since uik’s are normalized to be in [0, 1]). Therefore, in settings
such as school choice, our bounds are not practically interesting for providing fairness
among students. In fact, it is clear that because each student is assigned to a single
school, guaranteeing an envy-free ex post allocation is nearly impossible. Nevertheless,
our bounds give strong ex post guarantees for schools, or in general, for when agents
(objects) are assigned to a large number of objects (agents). Note that we can define
the “utility of the schools” similar to that of students; that is, vxj =∑|N |
k=1 xkjukj is
the utility of object j from assignment x, where (ukj)k∈N is the value of agent k for
object j. In addition, since W (x) is the sum of all utilities of the agents and thus
often has a large value relative to individual agents’ utilities, the bound provided in
Theorem 3.4.2 for W (X) is a strong probabilistic bound.
3.5 Fixing Random Serial Dictatorship
Random serial dictatorship is one of the most practically popular mechanisms for
the allocation of indivisible objects. RSD works as follows: The planner draws an
ordering of agents uniformly at random and then lets the agents select their favorite
bundle of objects (among those remaining without violating the constraints) one by
one according to the realized random ordering. In Subsection 3.1.2, we discussed
that although this mechanism is strategy-proof and ex ante fair15, it is ex ante in-
efficient and ex post (very) unfair. Che and Kojima (2010) [27] shows that the ex
ante inefficiency disappears in large markets. Ex post unfairness, however, remains
15The RSD mechanism satisfies the ‘equal treatment of equals’ and the ‘SD envy-freeness’ criteria.
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 86
a serious issue for the RSD mechanism since agents with best priorities can choose
most favorite items. The following example clarifies this problem.
Example 3.5.1. Consider a course allocation setting, where there are two students,
s1 and s2 each planning to take two coerces. Suppose there are four different courses,
c1, c2, c3 and c4, each with capacity of 1. Let us assume that both students prefer c1
and c2 the most.
Now if we run the RSD mechanism and choose one of the two random orderings
with equal probability. This mechanism is obviously ex ante fair, in the sense that it
is treating students in a symmetric fashion. Yet, the student with the best priority
will take c1 and c2 and the other student has no choice but to take c3 and c4, which
is ex post very unfair.
Consider the same model as in Subsection 3.4.3 in which agents have additive
utilities over all subsets of objects and suppose all constraints’ lower quotas are set to
zero. For any k ∈ {1, 2, · · · , N !}, let πk be a priority ordering of agents. We introduce
a new mechanism, the Approximate Random Serial Dictatorship (ARSD) mechanism,
and prove that this mechanism is strategy-proof, ex ante fair, and ex post approxi-
mately fair. The idea is simple: the RSD mechanism induces an ex ante assignment,
which is potentially fractional. We ask for agents’ preferences, construct the expected
assignment induced by the RSD, and then employ our randomized mechanism based
on the Operation X to implement it. We formally define ARSD below.
The Approximate Random Serial Dictatorship Mechanism (ARSD)
1. Agents report their ordinal preferences over individual objects.
2. Construct the expected random serial dictatorship assignment xrsd in the fol-
lowing way: run the serial dictatorship mechanism, with prioritizing agents
according to πk, and without violating any of the (hard and soft) constraints.
Denote the resulting pure assignment by Xk. Let xrsd = 1N !
∑N !1 Xk.
3. The mechanism approximately implements xrsd.
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 87
The following theorem shows that ARSD is strategy-proof, and in contrast to
the standard RSD, the realized utilities of the agents is approximately equal to their
expected utilities.
Theorem 3.5.2. The ARSD mechanism is strategy-proof and respects equal treatment
of equals16. Moreover, ex post utilities of the agents are approximately lower-bounded
by their ex ante utilities.
Proof. Since xrsd is the fractional assignment induced by the RSD and this mechanism
is strategy-proof, a straightforward argument based on the revelation principle clari-
fies that ARSD is also strategy-proof since it implements xrsd. Note that, in expecta-
tion, we are exactly implementing xrsd and there are no approximations in this step.
The second part of the theorem, that ex post utilities of the agents are approximately
lower-bounded by their ex post utilities, follows immediately from Theorem 3.4.2.
3.6 Conclusion
We study the mechanism design problem of allocating indivisible objects to agents
in a setting where cash transfers are precluded and the final allocation needs to
satisfy some constraints. One efficient and ex ante fair solution to this problem is the
“expected assignment” method, in which the mechanism first finds a feasible fractional
assignment, and then implements that fractional assignment by running a lottery over
feasible pure assignment. Previous literature have characterized a maximal ‘constraint
structure’ that can be accommodated into the expected assignment method. Such
structure rules out many real-world applications. We show that by reconceptualizing
the role of constraints and treating some of them as goals rather than hard constraints,
one can accommodate many more constraints.
The main theorem of the paper identifies a rich constraint structure that is ap-
proximately implementable, meaning that any expected assignment that satisfies both
hard constraints and soft constraints (i.e. goals) can be implemented by a lottery over
16An allocation mechanism respects equal treatment of equals if agents with the same utilities overbundles of objects have the same allocations.
CHAPTER 3. RANDOM ALLOCATION MECHANISMS 88
nearby pure assignments in a way such that hard constraints can be exactly satisfied
and goals can be satisfied with only very small errors. As a corollary of the main the-
orem, we show that if the structure of hard constraints is hierarchical, then any set of
goals can be approximately satisfied. This allows us to significantly expand potential
applications of the expected assignment method. For instance, in the school choice
setting, we can accommodate racial, gender, and walk-zone priority constraints at the
same time.
The key technical novelty of this study is the randomized mechanism that we
design in order to implement a fractional assignment. We quantify the violations
in soft constraints by applying probabilistic concentration bounds. This framework
helps us to preserve some desirable properties of the expected allocation in the ex
post allocation. For instance, an envy-free or efficient expected allocation remains
approximately envy-free and efficient ex post. By applying the same technique, we
introduce a new way to implement walk-zone requirements in which, rather than
setting a quota on students from a specific ‘walk-zone’, we define a penalty function
based on each students’ distance to each school. In this way, students who live just
inside and outside of a specific walk-zone are not treated differently.
We exploit the same technique to modify the random serial dictatorship mech-
anism by making it ex post (approximately) fair. This is done by constructing the
ex ante assignment associated with RSD, and implementing it by our randomized
mechanism.
We are hopeful that the proposed framework for partitioning constraints and
the randomized mechanism we designed will pave the way for designing improved
allocation mechanisms in practice.
Appendix A
Missing Proofs From Chapter 2
A.1 Auxiliary Inequalities
In this section we prove several inequalities that are used throughout the paper. For
any a, b ≥ 0,
∞∑i=a
e−bi2
=∞∑i=0
e−b(i+a)2 ≤∞∑i=0
e−ba2−2iab = e−ba
2∞∑i=0
(e−2ab)i
=e−ba
2
1− e−2ab≤ e−ba
2
min{ab, 1/2}.(A.1.1)
The last inequality can be proved as follows: If 2ab ≤ 1, then e−2ab ≤ ab, otherwise
e−2ab ≤ 1/2.
For any a, b ≥ 0,
∞∑i=a
(i− 1)e−bi2 ≤
∫ ∞a−1
xe−bx2
dx =−1
2be−bx
2 |∞a−1=e−b(a−1)2
2b. (A.1.2)
For any a ≥ 0 and 0 ≤ b ≤ 1,
∞∑i=a
ie−bi = e−ba∞∑i=0
(i+ a)e−bi = e−ba( a
1− e−b+
1
(1− e−b)
)≤ e−ba(2ba+ 4)
b2.(A.1.3)
89
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 90
The Bernoulli inequality states that for any x ≤ 1, and any n ≥ 1,
(1− x)n ≥ 1− xn. (A.1.4)
Here, we prove for integer n. The above equation can be proved by a simple induction
on n. It trivially holds for n = 0. Assuming it holds for n we can write,
(1− x)n+1 = (1− x)(1− x)n ≥ (1− x)(1− xn) = 1− x(n+ 1) + x2n ≥ 1− x(n+ 1).
A.2 Proof of Theorem 2.4.2
A.2.1 Stationary Distributions: Existence and Uniqueness
In this part we show that the Markov Chain on Zt has a unique stationary distribution
under each of the Greedy and Patient algorithms. By Proposition 2.4.1, Zt is a Markov
chain on the non-negative integers (N ) that starts from state zero.
First, we show that the Markov Chain is irreducible. First note that every state
i > 0 is reachable from state 0 with a non-zero probability. It is sufficient that i
agents arrive at the market with no acceptable bilateral transactions. On the other
hand, state 0 is reachable from any i > 0 with a non-zero probability. It is sufficient
that all of the i agents in the pool become critical and no new agents arrive at the
market. So Zt is an irreducible Markov Chain.
Therefore, by the ergodic theorem it has a unique stationary distribution if and
only if it has a positive recurrent state [64, Theorem 3.8.1]. In the rest of the proof
we show that state 0 is positive recurrent. By (2.3.1) Zt = 0 if Zt = 0. So, it is
sufficient to show
E[inf{t ≥ T1 : Zt = 0}|Zt0 = 0
]<∞. (A.2.1)
It follows that Zt is just a continuous time birth-death process onN with the following
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 91
transition rates,
rk→k+1 = m and rk→k−1 := k (A.2.2)
It is well known (see e.g. [38, p. 249-250]) that Zt has a stationary distribution if and
only if∞∑k=1
r0→1r1→2 . . . rk−1→k
r1→0 . . . rk→k−1
<∞.
Using (A.2.2) we have
∞∑k=1
r0→1r1→2 . . . rk−1→k
r1→0 . . . rk→k−1
=∞∑k=1
mk
k!= em − 1 <∞
Therefore, Zt has a stationary distribution. The ergodic theorem [64, Theorem 3.8.1]
entails that every state in the support of the stationary distribution is positive recur-
rent. Thus, state 0 is positive recurrent under Zt. This proves (A.2.1), so Zt is an
ergodic Markov Chain.
A.2.2 Upper bounding the Mixing Times
In this part we complete the proof of Theorem 2.4.2 and provide an upper bound the
mixing of Markov Chain Zt for the Greedy and Patient algorithms. Let π(.) be the
stationary distribution of the Markov Chain.
A.2.3 Mixing time of the Greedy Algorithm
We use the coupling technique (see [56, Chapter 5]) to get an upper bound for the
mixing time of the Greedy algorithm. Suppose we have two Markov Chains Yt, Zt
(with different starting distributions) each running the Greedy algorithm. We define
a joint Markov Chain (Yt, Zt)∞t=0 with the property that projecting on either of Yt and
Zt we see the stochastic process of Greedy algorithm, and that they stay together at
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 92
all times after their first simultaneous visit to a single state, i.e.,
if Yt0 = Zt0 , then Yt = Zt for t ≥ t0.
Next we define the joint chain. We define this chain such that for any t ≥ t0,
|Yt − Zt| ≤ |Yt0 − Zt0|. Assume that Yt0 = y, Zt0 = z at some time t0 ≥ 0, for
y, z ∈ N . Without loss of generality assume y < z (note that if y = z there is
nothing to define). Consider any arbitrary labeling of the agents in the first pool
with a1, . . . , ay, and in the second pool with b1, . . . , bz. Define z + 1 independent
exponential clocks such that the first z clocks have rate 1, and the last one has rate
m. If the i-th clock ticks for 1 ≤ i ≤ y, then both of ai and bi become critical (recall
that in the Greedy algorithm the critical agent leaves the market right away). If
y < i ≤ z, then bi becomes critical, and if i = z+1 new agents ay+1, bz+1 arrive to the
markets. In the latter case we need to draw edges between the new agents and those
currently in the pool. We use z independent coins each with parameter d/m. We use
the first y coins to decide simultaneously on the potential transactions (ai, ay+1) and
(bi, bz+1) for 1 ≤ i ≤ y, and the last z− y coins for the rest. This implies that for any
1 ≤ i ≤ y, (ai, ay+1) is an acceptable transaction iff (bi, bz+1) is acceptable. Observe
that if ay+1 has at least one acceptable transaction then so has bz+1 but the converse
does not necessarily hold.
It follows from the above construction that |Yt − Zt| is a non-increasing function
of t. Furthermore, this value decreases when either of the agents by+1, . . . , bz become
critical (we note that this value may also decrease when a new agent arrives but we
do not exploit this situation here). Now suppose |Y0 − Z0| = k. It follows that the
two chains arrive to the same state when all of the k agents that are not in common
become critical. This has the same distribution as the maximum of k independent
exponential random variables with rate 1. Let Ek be a random variable that is the
maximum of k independent exponentials of rate 1. For any t ≥ 0,
P [Zt 6= Yt] ≤ P[E|Y0−Z0| ≥ t
]= 1− (1− e−t)|Y0−Z0|.
Now, we are ready to bound the mixing time of the Greedy algorithm. Let zt(.)
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 93
be the distribution of the pool size at time t when there is no agent in the pool at
time 0 and let π(.) be the stationary distribution. Fix 0 < ε < 1/4, and let β ≥ 0
be a parameter that we fix later. Let (Yt, Zt) be the joint Markov chain that we
constructed above where Yt is started at the stationary distribution and Zt is started
at state zero. Then,
‖zt − π‖TV ≤ P [Yt 6= Zt] =∞∑i=0
π(i)P [Yt 6= Zt|Y0 = i]
≤∞∑i=0
π(i)P [Ei ≥ t]
≤βm/d∑i=0
(1− (1− e−t)βm/d) +∞∑
i=βm/d
π(i)
≤ β2m2
d2e−t + 2e−m(β−1)2/2d
where the last inequality follows by equation (A.1.4) and Proposition 2.5.5. Letting
β = 1 +√
2 log(2/ε) and t = 2 log(βm/d) · log(2/ε) we get ‖zt − π‖TV ≤ ε, which
proves the theorem.
A.2.4 Mixing time of the Patient Algorithm
It remains to bound the mixing time of the Patient algorithm. The construction of
the joint Markov Chain is very similar to the above construction except some caveats.
Again, suppose Yt0 = y and Zt0 = z for y, z ∈ N and t0 ≥ 0 and that y < z. Let
a1, . . . , ay and b1, . . . , bz be a labeling of the agents. We consider two cases.
Case 1) z > y+ 1. In this case the construction is essentially the same as the Greedy
algorithm. The only difference is that we toss random coins to decide on
acceptable bilateral transactions at the time that an agent becomes critical
(and not at the time of arrival). It follows that when new agents arrive the
size of each of the pools increase by 1 (so the difference remains unchanged).
If any of the agents by+1, . . . , bz become critical then the size of second pool
decrease by 1 or 2 and so is the difference of the pool sizes.
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 94
Case 2) z = y+1. In this case we define a slightly different coupling. This is because,
for some parameters and starting values, the Markov chains may not visit the
same state for a long time for the coupling defined in Case 1 . If z � m/d,
then with a high probability any critical agent gets matched. Therefore, the
magnitude of |Zt − Yt| does not quickly decrease (for a concrete example,
consider the case where d = m, y = m/2 and z = m/2 + 1). Therefore, in
this case we change the coupling. We use z+2 independent clocks where the
first z are the same as before, i.e., they have rate 1 and when the i-th clock
ticks bi (and ai if i ≤ y) become critical. The last two clocks have rate m,
when the z + 1-st clock ticks a new agent arrives to the first pool and when
z + 2-nd one ticks a new agent arrives to the second pool.
Let |Y0 − Z0| = k. By the above construction |Yt − Zt| is a decreasing function of
t unless |Yt − Zt| = 1. In the latter case this difference goes to zero if a new agent
arrives to the smaller pool and it increases if a new agent arrives to the bigger pool.
Let τ be the first time t where |Yt − Zt| = 1. Similar to the Greedy algorithm, the
event |Yt − Zt| = 1 occurs if the second to maximum of k independent exponential
random variables with rate 1 is at most t. Therefore,
P [τ ≤ t] ≤ P [Ek ≤ t] ≤ (1− e−t)k
Now, suppose t ≥ τ ; we need to bound the time it takes to make the difference
zero. First, note that after time τ the difference is never more than 2. Let Xt be
the (continuous time) Markov Chain illustrated in Figure A.1 and suppose X0 = 1.
Using m ≥ 1, it is easy to see that if Xt = 0 for some t ≥ 0, then |Yt+τ − Zt+τ | = 0
(but the converse is not necessarily true). It is a simple exercise that for t ≥ 8,
P [Xt 6= 0] =∞∑k=0
e−ttk
k!2−k/2 ≤
t/4∑k=0
e−ttk
k!+ 2−t/8 ≤ 2−t/4 + 2−t/8. (A.2.3)
Now, we are ready to upper-bound the mixing time of the Patient algorithm. Let
zt(.) be the distribution of the pool size at time t where there is no agent at time 0,
and let π(.) be the stationary distribution. Fix ε > 0, and let β ≥ 2 be a parameter
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 95
0 1 2
1
1
1
Figure A.1: A three state Markov Chain used for analyzing the mixing time of thePatient algorithm.
that we fix later. Let (Yt, Zt) be the joint chain that we constructed above where Yt
is started at the stationary distribution and Zt is started at state zero.
‖zt − π‖TV ≤ P [Zt 6= Yt] ≤ P [τ ≤ t/2] + P [Xt ≤ t/2]
≤∞∑i=0
π(i)P [τ ≤ t/2|Y0 = i] + 2−t/8+1
≤ 2−t/8+1 +∞∑i=0
π(i)(1− (1− e−t/2)i)
≤ 2−t/8+1 +
βm∑i=0
(it/2) +∞∑
i=βm
π(i)
≤ 2−t/8+1 +β2m2t
2+ 6e−(β−1)m/3.
where in the second to last equation we used equation (A.1.4) and in the last equation
we used Proposition 2.5.9. Letting β = 10 and t = 8 log(m) log(4/ε) implies that
‖zt − π‖TV ≤ ε which proves Theorem 2.4.2.
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 96
A.3 Proofs from Section 2.5
A.3.1 Proof of Lemma 2.5.4
Proof. By Proposition 2.3.3, E [Zt] ≤ m for all t, so
L(Greedy) =1
m · TE[∫ T
t=0
Ztdt
]=
1
mT
∫ T
t=0
E [Zt] dt
≤ 1
mTm · τmix(ε) +
1
mT
∫ T
t=τmix(ε)
E [Zt] dt (A.3.1)
where the second equality uses the linearity of expectation. Let Zt be the number of
agents in the pool at time t when we do not match any pair of agents. By (2.3.1),
P [Zt ≥ i] ≤ P[Zt ≥ i
].
Therefore, for t ≥ τmix(ε),
E [Zt] =∞∑i=1
P [Zt ≥ i] ≤6m∑i=0
P [Zt ≥ i] +∞∑
i=6m+1
P[Zt ≥ i
]≤
6m∑i=0
(PZ∼π [Z ≥ i] + ε) +∞∑
i=6m+1
∞∑`=i
m`
`!
≤ EZ∼π [Z] + ε6m+∞∑
i=6m+1
2mi
i!
≤ EZ∼π [Z] + ε6m+4m6m
(6m)!(A.3.2)
≤ EZ∼π [Z] + ε6m+ 2−6m. (A.3.3)
where the second inequality uses P[Zt = `
]≤ m`/`! of Proposition 2.3.3 and the
last inequality follows by the Stirling’s approximation1 of (6m)!. Putting (A.3.1) and
1Stirling’s approximation states that
n! ≥√
2πn(ne
)n.
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 97
(A.3.3) proves the lemma.
A.3.2 Proof of Lemma 2.5.7
Proof. Let Let ∆ ≥ 0 be a parameter that we fix later. We have,
EZ∼π [Z] ≤ k∗ + ∆ +∞∑
i=k∗+∆+1
iπ(i). (A.3.4)
By equation (2.5.6),
∞∑i=k∗+∆+1
iπ(i) =∞∑
i=∆+1
e−d(i−1)2/2m(i+ k∗)
=∞∑i=∆
e−di2/2m(i− 1) +
∞∑i=∆
e−di2/2m(k∗ + 2)
≤ e−d(∆−1)2/2m
d/m+ (k∗ + 2)
e−d∆2/2m
min{1/2, d∆/2m}, (A.3.5)
where in the last step we used equations (A.1.1) and (A.1.2). Letting ∆ := 1 +
2√m/d log(m/d) in the above equation, the right hand side is at most 1. The lemma
follows from (A.3.4) and the above equation.
A.3.3 Proof of Lemma 2.5.8
Proof. By linearity of expectation,
L(Patient) =1
m · TE[∫ T
t=0
Zt(1− d/m)Zt−1dt
]=
1
m · T
∫ T
t=0
E[Zt(1− d/m)Zt−1
]dt.
Since for any t ≥ 0, E[Zt(1− d/m)Zt−1
]≤ E [Zt] ≤ E
[Zt
]≤ m, we can write
L(Patient) ≤ τmix(ε)
T+
1
m · T
∫ T
t=τmix(ε)
∞∑i=0
(π(i) + ε)i(1− d/m)i−1dt
≤ τmix(ε)
T+
EZ∼π[Z(1− d/m)Z−1
]m
+εm
d2
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 98
where the last inequality uses the identity∑∞
i=0 i(1− d/m)i−1 = m2/d2.
A.3.4 Proof of Proposition 2.5.9
Let us first rewrite what we derived in the proof overview of this proposition in the
main text. The balance equations of the Markov chain associated with the Patient
algorithm can be written as follows by replacing transition probabilities from (2.5.7),
(2.5.8), and (2.5.9) in (2.5.10):
mπ(k) = (k + 1)π(k + 1) + (k + 2)(
1−(
1− d
m
)k+1)π(k + 2) (A.3.6)
Now define a continous f : R→ R as follows,
f(x) := m− (x+ 1)− (x+ 2)(1− (1− d/m)x+1). (A.3.7)
It follows that
f(m− 1) ≤ 0, f(m/2− 2) > 0,
which means that f(.) has a root k∗ such that m/2− 2 < k∗ < m. In the rest of the
proof we show that the states that are far from k∗ have very small probability in the
stationary distribution
In order to complete the proof of Proposition 2.5.9, we first prove the following
useful lemma.
Lemma A.3.1. For any integer k ≤ k∗,
π(k)
max{π(k + 1), π(k + 2)}≤ e−(k∗−k)/m.
Similarly, for any integer k ≥ k∗, min{π(k+1),π(k+2)}π(k)
≤ e−(k−k∗)/(m+k−k∗).
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 99
Proof. For k ≤ k∗, by equation (A.3.6),
π(k)
max{π(k + 1), π(k + 2)}≤ (k + 1) + (k + 2)(1− (1− d/m)k+1)
m
≤ (k − k∗) + (k∗ + 1) + (k∗ + 2)(1− (1− d/m)k∗+1)
m
= 1− k∗ − km
≤ e−(k∗−k)/m,
where the last equality follows by the definition of k∗ and the last inequality uses
1− x ≤ e−x. The second conclusion can be proved similarly. For k ≥ k∗,
min{π(k + 1), π(k + 2)}π(k)
≤ m
(k + 1) + (k + 2)(1− (1− d/m)k+1)
≤ m
(k − k∗) + (k∗ + 1) + (k∗ + 2)(1− (1− d/m)k∗+1)
=m
m+ k − k∗= 1− k − k∗
m+ k − k∗≤ e−(k−k∗)/(m+k−k∗).
where the equality follows by the definition of k∗.
Now, we use the above claim to upper-bound π(k) for values k that are far from
k∗. First, fix k ≤ k∗. Let n0, n1, . . . be sequence of integers defined as follows: n0 = k,
and ni+1 := arg max{π(ni + 1), π(ni + 2)} for i ≥ 1. It follows that,
π(k) ≤∏
i:ni≤k∗
π(ni)
π(ni+1)≤ E
(−∑
i:ni≤k∗
k∗ − nim
)(A.3.8)
≤ E(−
(k∗−k)/2∑i=0
2i
m
)≤ e−(k∗−k)2/4m, (A.3.9)
where the second to last inequality uses |ni − ni−1| ≤ 2.
Now, fix k ≥ k∗ + 2. In this case we construct the following sequence of integers,
n0 = bk∗ + 2c, and ni+1 := arg min{π(ni + 1), π(ni + 2)} for i ≥ 1. Let nj be the
largest number in the sequence that is at most k (observe that nj = k− 1 or nj = k).
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 100
We upper-bound π(k) by upper-bounding π(nj),
π(k) ≤ m · π(nj)
k≤ 2
j−1∏i=0
π(ni)
π(ni+1)≤ 2E
(−
j−1∑i=0
ni − k∗
m+ ni − k∗)
≤ 2E(−
(j−1)/2∑i=0
2i
m+ k − k∗)≤ 2E
(−(k − k∗ − 1)2
4(m+ k − k∗)
).
(A.3.10)
To see the first inequality note that if nj = k, then there is nothing to show; otherwise
we have nj = k−1. In this case by equation (A.3.6), mπ(k−1) ≥ kπ(k). The second
to last inequality uses the fact that |ni − ni+1| ≤ 2.
We are almost done. The proposition follows from (??) and (A.3.9). First, for
σ ≥ 1, let ∆ = σ√
4m, then by equation (A.1.1)
k∗−∆∑i=0
π(i) ≤∞∑i=∆
e−i2/4m ≤ e−∆2/4m
min{1/2,∆/4m}≤ 2√me−σ
2
.
Similarly,
∞∑i=k∗+∆
π(i) ≤ 2∞∑
i=∆+1
e−(i−1)2/4(i+m) ≤ 2∞∑i=∆
e−i/(4+√
4m/σ)
≤ 2e−∆/(4+
√4m/σ)
1− e−1/(4+√
4m)≤ 8√me
−σ2√m2σ+√m
This completes the proof of Proposition 2.5.9.
A.3.5 Proof of Lemma 2.5.10
Proof. Let ∆ := 3√m log(m), and let β := maxz∈[m/2−∆,m+∆] z(1− d/m)z.
EZ∼π[Z(1− d/m)Z
]≤ β +
m/2−∆−1∑i=0
m
2π(i)(1− d/m)i (A.3.11)
+∞∑
i=m+∆
iπ(i)(1− d/m)m (A.3.12)
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 101
We upper bound each of the terms in the right hand side separately. We start with
upper bounding β. Let ∆′ := 4(log(2m) + 1)∆.
β ≤ maxz∈[m/2,m]
z(1− d/m)z +m/2(1− d/m)m/2((1− d/m)−∆ − 1) + (1− d/m)m∆
≤ maxz∈[m/2,m]
(z + ∆′ + ∆)(1− d/m)z + 1. (A.3.13)
To see the last inequality we consider two cases. If (1 − d/m)−∆ ≤ 1 + ∆′/m then
the inequality obviously holds. Otherwise, (assuming ∆′ ≤ m),
(1− d/m)∆ ≤ 1
1 + ∆′/m≤ 1−∆′/2m,
By the definition of β,
β ≤ (m+ ∆)(1− d/m)m/2−∆ ≤ 2m(1−∆′/2m)m/2∆−1 ≤ 2me∆′/4∆−1 ≤ 1.
It remains to upper bound the second and the third term in (A.3.12). We start
with the second term. By Proposition 2.5.9,
m/2−∆−1∑i=0
π(i) ≤ 1
m3/2. (A.3.14)
where we used equation (A.1.1). On the other hand, by equation (??)
∞∑i=m+∆
iπ(i) ≤ e−∆/(2+√m)(
m
1− e−1/(2+√m)
+2∆ + 4
1/(2 +√m)2
) ≤ 1√m. (A.3.15)
where we used equation (A.1.3).
The lemma follows from (A.3.12), (A.3.13), (A.3.14) and (A.3.15).
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 102
A.4 Proofs from Section 2.6
A.4.1 Proof of Lemma 2.6.3
Let ∆ := 3√m log(m). Let E be the event that Zt ∈ [k∗−∆, k∗+ ∆]. First, we show
the following inequality and then we upper-bound E[Xt|E
]P[E].
E [Xt] (1− qk∗+∆)− E[Xt|E
]P[E]≤ E
[Xt(1− qZt)
](A.4.1)
≤ E [Xt] (1− qk∗−∆) + E[Xt|E
]P[E]
(A.4.2)
We prove the right inequality and the left can be proved similarly.
By definition of expectation,
E[Xt(1− qZt)
]= E
[Xt(1− qZt)|E
]· P [E ] + E
[Xt(1− qZt)|E
]· P[E]
≤ E [Xt|E ] (1− qk∗−∆) + E[Xt(1− qZt)|E
]· P[E]
Now, for any random variable X and any event E we have E [X|E ] · P [E ] = E [X] −E[X|E
]· P[E]. Therefore,
E[Xt(1− qZt)
]≤ (1− qk∗−∆)(E [Xt]− E
[Xt|E
]· P[E]) + E
[Xt(1− qZt)|E
]· P[E]
≤ E [Xt] (1− qk∗−∆) + E[Xt|E
]P[E]
where we simply used the non-negativity of Xt and that (1− qk∗−∆) ≤ 1. This proves
the right inequality of (??). The left inequality can be proved similarly.
It remains to upper-bound E[Xt|E
]P[E]. Let π(.) be the stationary distribution
of the Markov Chain Zt. Since by definition of Xt, Xt ≤ Zt with probability 1,
E[Xt|E
]P[E]≤ E
[Zt|E
]P[E]
≤k∗−∆∑i=0
i(π(i) + ε) +6m∑
i=k∗+∆
i(π(i) + ε) +∞∑
i=6m+1
i · P[Zt = i
]where the last term uses the fact that Zt is at most the size of the pool of the inactive
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 103
policy at time t, i.e., P [Zt = i] ≤ P[Zt = i
]for all i > 0. We bound the first term of
RHS using Proposition 2.5.9, the second term using (A.3.15) and the last term using
Proposition 2.3.3.
E[Xt|E
]P[E]≤ 4√
m+ 6mε+
∞∑i=6m
mi
i!≤ 4√
m+
3
m+ 2−6m.
A.5 Proofs from Section 2.7
A.5.1 Proof of Lemma 2.7.4
In this section, we present the full proof of Lemma 2.7.4. We prove the lemma by
writing a closed form expression for the utility of a and then upper-bounding that
expression.
In the following claim we study the probability a is matched in the interval [t, t+ε]
and the probability that it leaves the market in that interval.
Claim A.5.1. For any time t ≥ 0, and ε > 0,
P [a ∈Mt,t+ε] = ε · P [a ∈ At] (2 + c(t))E[1− (1− d/m)Zt|a ∈ At
]±O(ε2)
(A.5.1)
P [a /∈ At+ε, a ∈ At] = P [a ∈ At] (1− ε(1 + c(t) + E[1− (1− d/m)Zt−1|a ∈ At
])±O(ε2))
(A.5.2)
Proof. The claim follows from two simple observations. First, a becomes critical
in the interval [t, t + ε] with probability ε · P [a ∈ At] (1 + c(t)) and if he is critical
he is matched with probability E[(1− (1− d/m)Zt−1|a ∈ At
]. Second, a may also
get matched (without getting critical) in the interval [t, t + ε]. Observe that if an
agent b ∈ At where b 6= a gets critical she will be matched with a with probability
(1− (1−d/m)Zt−1)/(Zt−1),. Therefore, the probability that a is matched at [t, t+ ε]
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 104
without getting critical is
P [a ∈ At] · E[ε · (Zt − 1)
1− (1− d/m)Zt−1
Zt − 1|a ∈ At
]= ε · P [a ∈ At]E
[1− (1− d/m)Zt−1|a ∈ At
]The claim follows from simple algebraic manipulations.
We need to study the conditional expectation E[1− (1− d/m)Zt−1|a ∈ At
]to use
the above claim. This is not easy in general; although the distribution of Zt remains
stationary, the distribution of Zt conditioned on a ∈ At can be a very different distri-
bution. So, here we prove simple upper and lower bounds on E[1− (1− d/m)Zt−1|a ∈ At
]using the concentration properties of Zt. By the assumption of the lemma Zt is at
stationary at any time t ≥ 0. Let k∗ be the number defined in Proposition 2.5.9, and
β = (1− d/m)k∗. Also, let σ :=
√6 log(8m/β). By Proposition 2.5.9, for any t ≥ 0,
E[1− (1− d/m)Zt−1|a ∈ At
]≤ E
[1− (1− d/m)Zt−1|Zt < k∗ + σ
√4m, a ∈ At
]+ P
[Zt ≥ k∗ + σ
√4m|a ∈ At
]≤ 1− (1− d/m)k
∗+σ√
4m +P[Zt ≥ k∗ + σ
√4m]
P [a ∈ At]
≤ 1− β + β(1− (1− d/m)σ√
4m) +8√me−σ
2/3
P [a ∈ At]
≤ 1− β +2σdβ√m
+β
m2 · P [a ∈ At](A.5.3)
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 105
In the last inequality we used (A.1.4) and the definition of σ. Similarly,
E[1− (1− d/m)Zt−1|a ∈ At
]≥ E
[1− (1− d/m)Zt−1|Zt ≥ k∗ − σ
√4m, a ∈ At
]· P[Zt ≥ k∗ − σ
√4m|a ∈ At
]≥ (1− (1− d/m)k
∗−σ√
4m)P [a ∈ At]− P
[Zt < k∗ − σ
√4m]
P [a ∈ At]
≥ 1− β − β((1− d/m)−σ√
4m − 1)− 2√me−σ
2
P [a ∈ At]
≥ 1− β − 4dσβ√m− β3
m3 · P [a ∈ At](A.5.4)
where in the last inequality we used (A.1.4), the assumption that 2dσ ≤√m and the
definition of σ.
Next, we write a closed form upper-bound for P [a ∈ At]. Choose t∗ such that∫ t∗t=0
(2 + c(t))dt = 2 log(m/β). Observe that t∗ ≤ log(m/β) ≤ σ2/6. Since a leaves
the market with rate at least 1 + c(t) and at most 2 + c(t), we can write
β2
m2= E
(−∫ t∗
t=0
(2 + c(t))dt)≤ P [a ∈ At∗ ] ≤ E
(−∫ t∗
t=0
(1 + c(t))dt)≤ β
m(A.5.5)
Intuitively, t∗ is a moment where the expected utility of that a receives in the interval
[t∗,∞) is negligible, i.e., in the best case it is at most β/m.
By Claim A.5.1 and (??), for any t ≤ t∗,
P [a ∈ At+ε]− P [a ∈ At]ε
≤ −P [a ∈ At](
2 + c(t)− β − 4dσβ√m− β3
m3 · P [a ∈ At]±O(ε)
)≤ −P [a ∈ At]
(2 + c(t)− β − 5dσβ√
m±O(ε)
)where in the last inequality we used (A.5.5). Letting ε → 0, for t ≤ t∗, the above
differential equation yields,
P [a ∈ At] ≤ E(−∫ t
τ=0
(2 + c(τ)− β)dτ)
+2dσ3β√
m. (A.5.6)
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 106
where in the last inequality we used t∗ ≤ σ2/6, ex ≤ 1 + 2x for x ≤ 1 and lemma’s
assumption 5dσ2 ≤√m .
Now, we are ready to upper-bound the utility of a. By (A.5.5) the expected utility
that a gains after t∗ is no more than β/m. Therefore,
E [uc(a)] ≤ β
m+
∫ t∗
t=0
(2 + c(t))E[1− (1− d/m)Zt−1|a ∈ At
]P [a ∈ At] e−δtdt
≤ β
m+
∫ t∗
t=0
(2 + c(t))((1− β)P [a ∈ At] + β/√m)e−δtdt
≤ β
m+
∫ t∗
t=0
(2 + c(t))(
(1− β)E(−∫ t
τ=0
(2 + c(τ)− β)dτ)
+3dσ3
√mβ)e−δtdt
≤ 2dσ5
√mβ +
∫ ∞t=0
(1− β)(2 + c(t))E(−∫ t
τ=0
(2 + c(τ)− β)dτ)e−δtdt.
In the first inequality we used equation (??), in second inequality we used equation
(A.5.6), and in the last inequality we use the definition of t∗. We have finally obtained
a closed form upper-bound on the expected utility of a.
Let Uc(a) be the right hand side of the above equation. Next, we show that Uc(a)
is maximized by letting c(t) = 0 for all t. This will complete the proof of Lemma 2.7.4.
Let c be a function that maximizes Uc(a) which is not equal to zero. Suppose c(t) 6= 0
for some t ≥ 0. We define a function c : R+ → R+ and we show that if δ < β, then
Uc(a) > Uc(a). Let c be the following function,
c(τ) =
c(τ) if τ < t,
0 if t ≤ τ ≤ t+ ε,
c(τ) + c(τ − ε) if t+ ε ≤ τ ≤ t+ 2ε,
c(τ) otherwise.
In words, we push the mass of c(.) in the interval [t, t + ε] to the right. We remark
that the above function c(.) is not necessarily continuous so we need to smooth it
out. The latter can be done without introducing any errors and we do not describe
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 107
the details here. Let S :=∫ tτ=0
(1 + c(t) + β)dτ . Assuming c′(t)� 1/ε, we have
Uc(a)− Uc(a) ≥ −ε · c(t)(1− β)e−Se−δt + ε · c(t)(1− β)e−S−ε(2−β)e−δ(t+ε)
+ε(1− β)(2 + c(t+ ε))(e−S−ε(2−β)e−δ(t+ε) − e−S−ε(2+c(t)−β)e−δ(t+ε))
= −ε2 · c(t)(1− β)e−S−δt(2− β + δ) + ε2(1− β)(2 + c(t+ ε))e−S−δtc(t)
≥ ε2 · (1− β)e−S−δtc(t)(β − δ).
Since δ < β by the lemma’s assumption, the maximizer of Uc(a) is the all zero
function. Therefore, for any well-behaved function c(.),
E [uc(a)] ≤ 2dσ5
√mβ +
∫ ∞t=0
2(1− β)E(−∫ t
τ=0
(2− β)dτ)e−δtdt
≤ O(d4 log3(m)√
m)β +
2(1− β)
2− β + δ.
In the last inequality we used that σ = O(√
log(m/β)) and β ≤ e−d. This completes
the proof of Lemma 2.7.4.
A.6 Small Market Simulations
In Proposition 2.5.5 and Proposition 2.5.9, we prove that the Markov chains of the
Greedy and Patient algorithms are highly concentrated in intervals of size O(√m/d)
and O(√m), respectively. These intervals are plausible concentration bounds when
m is relatively large. In fact, most of our theoretical results are interesting when
markets are relatively large. Therefore, it is natural to ask: What if m is relatively
small? And what if the d is not small relative to m?
Figure A.2 depicts the simulation results of our model for small m and small T . We
simulated the market for m = 20 and T = 100 periods, repeated this process for 500
iterations, and computed the average loss for the Greedy, Patient, and the Omniscient
algorithms. As it is clear from the simulation results, the loss of the Patient algorithm
is lower than the Greedy for any d, and in particular, when d increases, the Patient
algorithm’s performance gets closer and closer to the Omniscient algorithm, whereas
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 108
0 2 4 6 8 100
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
d
L(Greedy)
L(Patient)
L(OMN)
Figure A.2: Simulated Losses for m = 20. For very small market sizes and even forrelatively large values of d, the Patient algorithm outperforms the Greedy Algorithm.
APPENDIX A. MISSING PROOFS FROM CHAPTER 2 109
the Greedy algorithm’s loss remains far above both of them.
Appendix B
Missing Proofs From Chapter 3
B.1 Implementation: A Random Mechanism
In this section, we present the complete proof of Theorem 3.3.1. As discussed in the
proof overview of the theorem, the proof is constructive. We will propose an imple-
mentation mechanism (or, equivalently, a lottery) which approximately implements
a partitioned structure that satisfies the properties described in Theorem 3.3.1.
To describe the main idea of our mechanism, we need to introduce the notion
of tight and floating constraints: A constraint is tight if it is binding. This notion
is precisely defined in the following definition. First, for any block B, let x(B) =∑e∈B xe.
Definition B.1.1. A constraint S = (B, qB, qB) is tight if, either x(B) = q
Bor
x(B) = qB; otherwise, S is floating. Similarly, we say that a block B is tight when
the constraint corresponding to it (S in here) is tight.
Note that this definition naturally applies on the (implicit) constraints that for
all e ∈ E, we must have that 0 ≤ xe ≤ 1.
In the core of our randomized mechanism is a stochastic operation that we call
Operation X . We iteratively apply Operation X to the initial fractional assignment.
In each iteration t, which is symbolically depicted in Figure 3.3.1, the fractional as-
signment xt is converted to xt+1 in a way such that: (1) the number of floating
110
APPENDIX B. MISSING PROOFS FROM CHAPTER 3 111
constraints decreases, (2) E(xt+1|xt) = xt, and (3) xt+1 is feasible with respect to H.
The first property guarantees that after a finite (and small) number of iterations1, the
obtained assignment is pure. The second property makes sure that the resulting pure
assignment is equal to the original fractional assignment in expectation. The third
property guarantees that all hard constraints are satisfied throughout the whole pro-
cess of the mechanism. As the last step, we need to show that by iteratively applying
of Operation X , soft constraints are approximately satisfied. This is a more technical
property of Operation X , which we discuss in Subsection B.1.4. Roughly speaking,
we design Operation X in a way such that it never increases (or decreases) two (or
more) elements of a soft constraint at the same iteration. Consequently, elements of
each soft block become “negatively correlated”. It then allows us to employ Chernoff
concentration bounds to prove that soft constraints are approximately satisfied.
In the rest of this section, we design Operation X and prove that it satisfies the
above-mentioned properties.
B.1.1 Definitions
In this section, we introduce the required notions for defining Operation X .
1. For any two links e, e′, a block B is separating e, e′ if B contains exactly one of
them.
2. Given a hierarchy H, a (hard) block B ∈ H is supporting a pair of links (e, e′)
if it is the smallest block (in the number of involved edges) that contains both
e, e′, and moreover, no block in H separates e, e′.
3. We say that a hierarchy H is supporting the pair (e, e′) if there exists a block
in H which supports (e, e′). In particular, if the subset {e, e′} is in the deepest
level of H, then (e, e′) is supported by H.
4. A floating cycle is a sequence e1, . . . , el of distinct edges such that:
• xei is non-integral for all integers i.
1Our randomized mechanism stops after at most |H|+ |E| iterations.
APPENDIX B. MISSING PROOFS FROM CHAPTER 3 112
Figure B.1: A floating cycle of length 6
• (ei, ei+1) is supported by H1 for even integers i.
• (ei, ei+1) is supported by H2 for odd integers i.
where length the of cycle, l, is an even number and i + 1 = 1 for i = l. Figure
B.1 represents a floating cycle of length 6. A floating cycle is said to be minimal
if it does not contain a smaller floating cycle as a subset. We often drop the
minimal phrase and whenever we say a floating cycle, we refer to a minimal
floating cycle, unless otherwise specified.
Next, we define the notion of floating paths; loosely speaking, their structure is
very similar to floating cycles, except in their endpoints. Floating paths start from a
hierarchy and end in the same hierarchy if their length is even, otherwise, they end
in the other hierarchy.
5. A floating path is a sequence e1, e1, . . . , el of distinct edges such that:
• xei is non-integral for all integers i.
• There exists a ∈ {1, 2} such that if we define a = {1, 2}\{a}, then:
APPENDIX B. MISSING PROOFS FROM CHAPTER 3 113
– (ei, ei+1) is supported by Ha for even integers i < l.
– (ei, ei+1) is supported by Ha for odd integers i < l.
• No tight block in Ha contains e1, and no tight block in Hb contains el
where b = a if l is even and b = a if l is odd.
Figure B.2 contains a visual example of a floating path. A floating path is
said to be minimal if it does not contain a smaller floating path as a subset.
Whenever we say a floating path, we refer to a minimal floating path, unless
otherwise specified.
Figure B.2: Example of a floating path: Suppose that in the above fractional assign-ment H1 is the set of row blocks and H2 is the set of column blocks. Also, supposethe lower quotas and upper quotas are set to 0 and 1, respectively. Then, e1, e2, e3 isa (minimal) floating path. However, e1, e4, e3 is not a floating path.
Finally, we introduce the following crucial concept.
Definition B.1.2. Assume we are given a fractional assignment x. For any block B
and any ε > 0, let x↑εB denote a new (fractional) assignment in which the element
of the matrix corresponding to edge e is increased by ε if e ∈ B (i.e. it changes to
xe+ε), and it remains unchanged otherwise. Similarly, let x↓εB denote the fractional
assignment in which the element of the matrix corresponding to edge e is decreased
by ε if e ∈ B (i.e. it changes to xe − ε), and it remains unchanged otherwise.
Example B.1.3. (x↑εB)↓εB′ denotes the fractional assignment in which the value
of any edge e ∈ B − B′ becomes xe + ε, the value of any edge e ∈ B′ − B becomes
xe − ε, and the value of the rest of the edges does not change.
APPENDIX B. MISSING PROOFS FROM CHAPTER 3 114
B.1.2 Operation X
Operation X can be applied on a given floating cycle or a floating path of a frac-
tional assignment x (if none of them exist, then the assignment must be pure by
Lemma B.1.10). We first define this operation for a given floating cycle. Let F =
〈e1, . . . , el〉 be a floating cycle in x. Define
Fo = {ei : i is odd},
Fe = {ei : i is even}.
We call the pair (Fo, Fe) the odd-even decomposition of F . Given two non-negative
reals ε, ε′ (which we describe how to set soon), Operation X generates an assignment
x′ ∈ RN×O in one of the following ways:
• x′ = (x↑εFo)↓εFe with probability ε′
ε+ε′
• x′ = (x↓ε′ Fo)↑ε′ Fe with probability εε+ε′
.
Both ε and ε′ are chosen to be the largest possible numbers such that both of the
assignments (x↑εFo)↓εFe and (x↓ε′ Fo)↑ε′ Fe remain feasible, in the sense that they
satisfy all hard constraints.
The definition of Operation X on a floating path is the same as its definition on a
floating cycle. To summarize, we give a formal definition of the Operation X below.
Definition B.1.4. Consider a fractional assignment x and a floating path or a float-
ing cycle, namely F , given as the inputs to Operation X . Then Operation X gen-
erates a new assignment x′, where x′ = (x ↑ε Fo) ↓ε Fe with probability ε′
ε+ε′and
x′ = (x ↓ε′ Fo) ↑ε′ Fe with probability εε+ε′
, where ε, ε′ are positive numbers chosen to
be the largest possible numbers such that both (x ↑ε Fo) ↓ε Fe and (x ↓ε′ Fo) ↑ε′ Fe are
feasible assignments.
We also denote x′ (which is a random variable) by x l F .
APPENDIX B. MISSING PROOFS FROM CHAPTER 3 115
B.1.3 The Implementation Mechanism
Our implementation mechanism which is based on Operation X is formally defined
below.
The Implementation Mechanism Based on the Operation X :
1. A fractional assignment x is reported to the mechanism.
2. Set i to 1 and let xi = x.
3. Repeat the following as long as xi contains a floating cycle or a floating path:
(a) If xi contains a floating cycle, let F be an arbitrary floating cycle, other-
wise, let F be an arbitrary floating path.
(b) Define xi+1 to be xi l F .
(c) Increase i by one.
4. Report xi as the outcome of the mechanism.
In the rest of this section, we show that the above mechanism approximately
implements x in the sense of Definition 3.2.4.
The first step of the proof is verifying that if the assignment has no floating cycles
or paths, then it is necessarily pure. We prove this claim in Claim B.1.9. The next
step of the proof is to show that Operation X is well-defined in the sense that both ε, ε′
cannot be zero at the same time. We will state and prove this fact in Lemma B.1.10.
Next, we prove the following three important properties of Operation X :
i. The outcome of Operation X satisfies the hard constraints.
ii. Operatoin X satisfies the martingale property, i.e.
E[x l F
∣∣∣ x] = x
iii. The outcome of the Operation X has more tight constraints (compared to x).
APPENDIX B. MISSING PROOFS FROM CHAPTER 3 116
These properties are proved separately in three Lemmas below.
Lemma B.1.5. The outcome of Operation X satisfies the hard constraints.
Proof. By definition, Operation X chooses ε, ε′ such that both of its two possible
outcomes are feasible with respect to H.
Lemma B.1.6. Operatoin X satisfies the martingale property, i.e.
E[x l F
∣∣∣ x] = x
Proof. We prove the lemma by verifying that this property holds for any entry (i, j)
of the assignment matrix, i.e. if (x l F )(i,j) denotes the (i, j)-th element of x l F ,
then we have
E[(x l F )(i,j)
∣∣∣ x] = x(i,j).
In simple words, we prove that operation X does not change the value of entry (i, j)
of the assignment matrix in expectation.
Observe that by the definition of Operation X
E[x l F
∣∣ x] =ε′
ε+ ε′· ((x↑εFo)↓εFe) +
ε
ε+ ε′· ((x↓ε′ Fo)↑ε′ Fe) .
The claim is trivial if (i, j) 6∈ F . So, assume (i, j) ∈ F . Then, we either have
(i, j) ∈ Fo or (i, j) ∈ Fe:
1. If (i, j) ∈ Fo, then Operation X increases x(i,j) by ε with probability ε′
ε+ε′and
decreases it by ε′ with probability εε+ε′
. In this case, the expected amount by
which x(i,j) changes is equal to ε · ε′
ε+ε′− ε′ · ε
ε+ε′= 0.
2. If (i, j) ∈ Fe, then Operation X decreases x(i,j) by ε with probability ε′
ε+ε′, and
increases it by ε′ with probability εε+ε′
. In this case, the expected amount by
which x(i,j) changes is equal to −ε · ε′
ε+ε′+ ε′ · ε
ε+ε′= 0.
This proves the lemma.
APPENDIX B. MISSING PROOFS FROM CHAPTER 3 117
Lemma B.1.7. The outcome of operation X has more tight constraints (compared
to x).
Proof. Suppose F is a floating cycle in x. The proof for the path case is almost
identical. We show that x l F has more tight constraints than x. To do so, we first
show that a tight constraint remains tight after Operations X . Second, we show that
at least one of the floating constraints in x becomes tight in x l F .
To prove the first step, we show that for any tight constraint S, its corresponding
block, B, contains an equal number of elements (edges) from the sets Fo and Fe. This
fact is formally proved below.
Claim B.1.8. Suppose we are given a floating cycle F in the fractional assignment
x, and let (Fo, Fe) be the odd-even decomposition of F . Then, any tight block (in x)
contains an equal number of elements from Fo and Fe.
Proof. Let S = (B, qB, qB) be a tight constraint and w.l.o.g. assume B ∈ H1. Then,
it must be that for any element ei ∈ B ∩ Fe, the element that comes right after ei
in F , i.e. ei+1, belongs to B. This holds because by the definition of floating cycles,
(ei, ei+1) is supported by H1, which means no tight block in H1 separates ei, ei+1.
Consequently, both ei and ei+1 belong to B, or else B itself would separate ei, ei+1.
Therefore, for any element ei ∈ B∩Fe, there exists a distinct element ei+1 ∈ B∩Fowhich corresponds to ei. Similarly, any element in B ∩ Fo corresponds to a distinct
element in B ∩ Fe. This proves the claim.
Now recall that whenever Operation X increases (decreases) the elements in Fo,
it decreases (increases) the elements in Fe. This fact and Claim B.1.8 together imply
that x(B) = (x l F ) (B) (regardless of the choice of ε, ε′). This ensures that any tight
constraint remains tight after operation X .
We now prove the second step, which is to show that at least one of the floating
constraints in x becomes tight in x l F . Observe that any floating constraint S =
(B, qB, qB) provides a positive slack for setting the values of ε, ε′. In simple words,
since S is a floating constraint, we have that qB< x(B) < qB. By this fact, we
can compute the positive upper bounds that S imposes on ε, ε′. Finally, taking the
APPENDIX B. MISSING PROOFS FROM CHAPTER 3 118
minimum of these upper bounds (over all floating constraints S) determines the values
for ε, ε′. We formalize this argument below. Let
s =qB − x(B) ,
s =x(B)− qB,
k =|Fo ∪B| − |Fe ∪B| .
Then, in order to guarantee that x l F satisfies constraint S, the following inequalities
(that can be translated into upper bounds) are imposed on ε, ε′ by Operation X :ε · k ≤ s if k ≥ 0
ε · |k| ≤ s if k < 0(B.1.1)
ε′ · k ≤ s if k ≥ 0
ε′ · |k| ≤ s if k < 0(B.1.2)
Now, let u(S), u′(S) respectively denote the (positive) upper bounds imposed by
Inequalities (B.1.1),(B.1.2) on ε, ε′. By definition of ε, ε′, we have that ε = minS u(S)
and ε′ = minS u′(S) where the minimum is over all the floating constraints S. This
argument implies that:
Claim B.1.9. Operation X chooses ε, ε′ such that ε, ε′ > 0.
Proof. It is enough to show that u(S), u′(S) > 0 for all S. This is implied by noting
that, given a floating constraint S, we have s, s > 0.
The above argument also implies the existence of a floating constraint S1 for
which one of the corresponding inequalities in (B.1.1) is tight. Similarly, there exists
a floating constraint S2 for which one of the corresponding inequalities in (B.1.2) is
tight. These two facts imply that after operation X , either S1 or S2 becomes a tight
constraint.
APPENDIX B. MISSING PROOFS FROM CHAPTER 3 119
To summarize, we first showed that if a constraint is tight, then it remains tight
after operation X . Moreover, we showed that there always exists at least one floating
constraint which becomes tight after operation X . Therefore, the number of tight
constraints decreases, which proves the lemma.
Next, we show that if a fractional assignment contains neither a floating cycle
nor a floating path, then it must be a pure assignment. This guarantees that the
assignment generated by our implementation mechanism is always pure.
Lemma B.1.10. An assignment is pure if and only if it does not contain floating
cycles and floating paths.
Proof. One direction is trivial: if the assignment is pure then it has no floating cycles
or floating paths. We prove the other direction by showing that any assignment x
which is not pure contains a floating path or a floating cycle. Since x is not pure, it
must contain a floating edge e, i.e. an edge e with 0 < xe < 1. We say that a floating
edge e is H1-loose (H2-loose) if no tight block in H1 (H2) contains e. We say that e
is loose if it is either H1-loose or H2-loose.
We need another definition before presenting the proof. Suppose S = (B, qB, qB)
is a tight hard constraint and e is a floating edge in B. Since S is tight, and since
the quotas qB, qB are integral, then B must also contain another floating edge e′. We
denote this edge by p(e, B). If there is more than one such edge, then let p(e, B)
denote one of them arbitrarily.
The proof has two cases, either there is a floating edge which is loose, or there is
no such edge.
Case 1: There exists a loose edge. As the first step of the proof, note that we
are done if there exists a floating edge which is both H1-loose and H2-loose: the edge
would form a floating path of length 1. So, w.l.o.g. suppose there is a floating edge
e which is not H2-loose. In this case, we iteratively construct a floating path that
starts from edge e, i.e. a path F = 〈e1, . . . , el〉 such that e1 = e. At the end, our
iterative construction will either find such a path, or we will find a floating cycle.
APPENDIX B. MISSING PROOFS FROM CHAPTER 3 120
Since e1 is not H2-loose, then there must be a minimal tight block B1 ∈ H2 that
contains e1. Since B1 is tight, and since the quotas are integral, then B1 must also
contain another floating edge p(e1, B1). We extend our (under construction) floating
path by setting e2 = p(e1, B1). Now, if e2 is H1-loose, then 〈e1, e2〉 is a floating
path and the proof is complete. So, suppose e2 is not H1-loose. Consequently, there
must be a minimal tight block B2 ∈ H1 that contains e2. Similar to before, B2 must
contain another floating edge p(e2, B2); we extend F by setting e3 = p(e2, B
2).
By repeating this argument, we can extend F iteratively until the new floating
edge that is added to F , namely ek, either (i) is loose, or (ii) is contained in one of
the previous tight blocks B1, . . . , Bk−1. If case (i) happens, then F is a floating path
and we are done. If case (ii) happens, then we have found a floating cycle: suppose
ek ∈ Bj with j < k. Then, it is straight-forward to verify that 〈ej+1, . . . , ek〉 is a
floating cycle.
Case 2: There is no loose edge. Similar to Case 1, we iteratively construct a
floating cycle F = 〈e1, . . . , el〉. The cycle starts from a floating edge e; initially, we
have e1 = e. Since e1 is not loose, there must be minimal tight blocks B0 ∈ H1
and B1 ∈ H2 such that e1 ∈ B0 and e1 ∈ B1. Then, let e2 = p(e1, B1). Similarly,
since e2 is not loose, there must be a tight block B2 ∈ H1 such that e2 ∈ B2. Let
e3 = p(e2, B2). By applying this argument repeatedly, we can extend F until the
new floating edge that is added to F , namely ek, satisfies ek ∈ Bj for some j with
0 ≤ j < k. Then, it is straight-forward to verify that 〈ej+1, . . . , ek〉 is a floating
cycle.
B.1.4 Approximate Satisfaction of Soft Constraints
Here we prove that soft constraints are approximately satisfied in the sense of Defi-
nition 3.2.4. Loosely speaking, Operation X is designed in a way such that it never
increases (or decreases) two (or more) elements of a soft constraint at the same itera-
tion. Consequently, elements of each soft constraint become “negatively correlated”.
This allows us to employ Chernoff concentration bounds to prove that soft constraints
are approximately satisfied.
APPENDIX B. MISSING PROOFS FROM CHAPTER 3 121
We show the approximate satisfaction of soft constraints by proving two lemmas
below. In the first lemma, we formally (define and) prove that elements of each soft
constraint are “negatively correlated”; the proof uses a negative correlation proof
technique from [74]. Then, in the second lemma, we prove the approximate satisfac-
tion of soft constraints by applying Chernoff concentration bounds. Before stating
the lemmas, we recall the definition of negative correlation.
Definition B.1.11. For an index set B, a set of binary random variables {Xe}e∈Bare negatively correlated if for any subset T ⊆ B we have
Pr
[∏e∈T
Xe = 1
]≤∏e∈T
Pr [Xe = 1] , (B.1.3)
Pr
[∏e∈T
(1−Xe) = 1
]≤∏e∈T
Pr [Xe = 0] . (B.1.4)
Lemma B.1.12. Let {Xe}e∈E denote the set of random variables which represent the
outcome of the implementation mechanism (i.e. the integral assignment); also, let
B be a block corresponding to an arbitrary soft constraint. Then, the set of random
variables {Xe}e∈B are negatively correlated.
Proof. We need to show that (??) and (??) hold for any subset T ⊆ B. We fix an
arbitrary subset T and prove (??) for it; the proof for (??) is identical and follows by
replacing the role of zeros and ones. Since the random variables are binary, we can
prove (??) by showing that
E
[∏e∈T
Xe
]≤∏e∈T
E [Xe] =∏e∈T
xe. (B.1.5)
To prove (??), we introduce a set of random variables {Xe,i} where Xe,i denotes
the value of entry e of the matrix after the i-th execution of operation X . So we
would have Xe,0 = xe for all e. Inductively, we show that for all i:
E
[∏e∈T
Xe,i+1
]≤ E
[∏e∈T
Xe,i
]. (B.1.6)
APPENDIX B. MISSING PROOFS FROM CHAPTER 3 122
The lemma is proved if (??) holds: Assuming that operation X is executed j
times, using (??) we can write
E
[∏e∈T
Xe
]= E
[∏e∈T
Xe,j
]≤ E
[∏e∈T
Xe,0
]=∏e∈T
xe
which shows (??) holds and proves the lemma.
To prove (??), we can alternatively show that
E
[∏e∈T
Xe,i+1
∣∣∣∣∣ {Xe,i}e∈T
]≤∏e∈T
Xe,i. (B.1.7)
We consider three cases to prove (??): since B is in the deepest level of a hierarchy,
then operation X changes either 0, 1, or 2 elements of T . We prove this fact in a
separate claim below.
Claim B.1.13. Suppose T is a block in the deepest level of a hierarchy, then, Oper-
ation X changes either 0, 1, or 2 elements of T .
Proof. W.L.O.G. assume that T is in the deepest level of H1. We prove a stronger
claim. Let T ′ be the largest subset of links that contains T and is in the deepest level
of H1. We prove that Operation X changes at most 2 elements of T ′. To this end,
let F be the floating cycle or path used in Operation X . We need to show that F
contains at most 2 elements of T ′; this proves the claim.
For contradiction, suppose F contains at least 3 elements of T ′. Let the elements
of F be denoted by the sequence e1, . . . , el, and let ei, ej, ek be the first three elements
of T ′ which appear in F , where i < j < k.
First, note that by the definitions of floating cycle and floating path, we must have
that j = i+ 1. We will prove that 〈ej, ej+1 . . . , ek−1, ek〉 makes a floating cycle, which
contradicts with the minimality of F (recall that by definition, operation X always
chooses minimal floating paths and cycles). To this end, first note that (ej, ej+1) is
supported by H2: this holds because ej−1, ej ∈ T ′, which means (ej−1, ej) is supported
by H1. Consequently, (ej, ej+1) must be supported by H2 since F is a floating path
or cycle. Similarly, (ej+1, ej+2) is supported by H1, (ej+2, ej+3) is supported by H2,
APPENDIX B. MISSING PROOFS FROM CHAPTER 3 123
and so on and so forth. Finally, note that (ek, ej) is supported by H1, since ek, ej ∈T ′. This proves that 〈ej, ej+1 . . . , ek−1, ek〉 is a floating cycle, which concludes the
claim.
We continue the proof of lemma by considering each of the three cases separately.
The proof is trivial if Operation X changes 0 elements of T : (??) holds with equality.
So, it remains to consider the two other cases.
First, assume that Operation X changes exactly one element of T , namely e′ ∈ T .
Let T ′ = T\{e′}. Then we have
E
[∏e∈T
Xe,i+1
∣∣∣∣∣ {Xe,i}e∈T
]
=ε′
ε+ ε′· (Xe′,i + ε) ·
∏e∈T ′
Xe,i +ε
ε+ ε′· (Xe′,i − ε′) ·
∏e∈T ′
Xe,i =∏e∈T
Xe,i
which proves (??) with equality in this case. It remains to prove (??) for the case when
Operation X changes exactly 2 elements of T , namely e′, e′′ ∈ T . Let T ′′ = T\{e′, e′′}.Then, w.l.o.g. we can write:
E
[∏e∈T
Xe,i+1
∣∣∣∣∣ {Xe,i}e∈T
]
=ε′
ε+ ε′· (Xe′,i + ε)(Xe′′,i − ε) ·
∏e∈T ′′
Xe,i +ε
ε+ ε′· (Xe′,i − ε′)(Xe′′,i + ε′) ·
∏e∈T ′′
Xe,i
=∏e∈T
Xe,i − εε′ ·∏e∈T ′′
Xe,i
≤∏e∈T
Xe,i
which proves (??) in the third case as well. This finishes the proof of lemma.
Lemma B.1.14. The randomized mechanism based on Operation X satisfies the soft
constraints approximately in the sense of Definition 3.2.4.
Proof. Based on Definition 3.2.4, we need to prove that for any soft constraint defined
APPENDIX B. MISSING PROOFS FROM CHAPTER 3 124
on a block B of the links with∑
e∈B xe = µ, and for any ε > 0, we have
Pr
(∑e∈B
weXe − µ < −εµ
)≤ e−µ
ε2
2 ,
Pr
(∑e∈B
weXe − µ > εµ
)≤ e−µ
ε2
3 .
These probabilistic bounds, as we mentioned before, are known as Chernoff concen-
tration bounds (see Section B.3 for more details). These bounds hold on any set
of binary random variables which are negatively correlated [13]. Lemma B.1.12 just
says that the set of random variables {Xe}e∈B are negatively correlated, which means
Chernoff concentration bounds hold for {Xe}e∈B.
B.2 Average Performance of the Matching Algo-
rithm
In this section, we implement our matching algorithm on an example. The goal of
this example is to show that the average performance of our matching algorithm is
much better than the worst-case bounds that one can theoretically prove. For the
sake of clarity, we use a simple example with multiple intersecting constraints.
Setup of example: Consider a school choice setting, with 10 schools and 10000
students. Suppose each school has a capacity for 1000 students. Also, suppose half
of the students are from the walk-zone of schools 1, 2, 3, 4, and 5 and the other half
are from the walk-zone of schools 6, 7, 8, 9, and 10. Also, half of the students are
categorized as low-socioeconomic status (LSES) students, and half of the students are
male. Suppose all students have the same utility function (or rank-order list) over
schools.
Hard and soft constraints: The only hard constraints imposed on this problem
are “all-row” constraints: All student should be assigned to exactly one school. All
APPENDIX B. MISSING PROOFS FROM CHAPTER 3 125
Figure B.3: The empirical probability of violating a constraint from below by ε%(equivalently, Pr(dev− ≥ εµ)), where µ = 500. The probability is calculated by run-ning the matching algorithm for T = 1000 times and then computing the probabilityof admitting less than 500(1− ε) students.
schools have three diversity goals that we model them as soft constraints: Their goal
is to admit 500 students (i.e. 50% of their capacity) from the students of their own
walk-zone, 500 student from LSES students, and 500 female students.
Fractional assignment: Let x be a fractional assignment, where x(i,j) = 110
for all
pairs (i, j). One can easily show that x satisfies all hard and soft constraints exactly.2
Simulation: We implement this fractional assignment by our matching algorithm
based on Operation X for 1000 times. We then calculate the “empirical probability”
of violating each one of the diversity constraints by a factor of ε = 1%, 2%, · · · , 10%.
Figure B.3 illustrates the empirical probability of admitting less than 500(1 −ε) students of a specific diversity type. As can be seen, the average performance
of our matching algorithm is much better than the worst-case bound that we can
2It is also clear that because of the symmetry of the problem, this assignment is fair and Paretoefficient.
APPENDIX B. MISSING PROOFS FROM CHAPTER 3 126
Figure B.4: The empirical probability of violating a constraint from above by ε%(equivalently, Pr(dev+ ≥ εµ)), where µ = 500. The probability is calculated by run-ning the matching algorithm for T = 1000 times and then computing the probabilityof admitting more than 500(1 + ε) students.
theoretically prove. Figure B.4 illustrates the empirical probability of admitting more
than 500(1 + ε) students of a specific diversity type. Again, the average performance
of our matching algorithm is much better than the theoretical worst-case bound.
B.3 Chernoff Bounds
Let X1, . . . , Xn be a sequence of n independent random binary variables such that
Xi = 1 with probability pi and Xi = 0 with probability 1 − pi. Also, let µ =∑ni=1 E[Xi]. Then for any ε with 0 ≤ ε ≤ 1 we have:
Pr
[n∑i=1
Xi > (1 + ε)µ
]≤ e−ε
2µ/3 (B.3.1)
Pr
[n∑i=1
Xi < (1− ε)µ
]≤ e−ε
2µ/2. (B.3.2)
APPENDIX B. MISSING PROOFS FROM CHAPTER 3 127
Moreover, the above inequalities still hold if the variables X1, . . . , Xn are nega-
tively correlated. (We refer the reader to Definition B.1.11 for the formal definition
of negative correlation)
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