Communication and Prices in Economic Mechanisms

Ilya Segal

Four lectures

1. Communication costs in economics

2. Communication and prices in social choice

3. Market design applications

4. Communication cost of incentives

Communication costs in economics

“Market Design” Problems• Find allocations satisfying social goals when agents

have private knowledge (e.g. of own preferences)– Combinatorial auctions: FCC spectrum, landing slots, …– Matching: candidates to jobs, internships, …– Kidney exchange

• Incentives?– can be verified with a Direct Revelation Mechanism

(e.g., Vickrey-Clarke-Groves)

• But full revelation is costly and is avoided in most real-life mechanisms

• Instead: sequential mechanisms – e.g., iterative auctions, interviews,… - to grope for equilibrium

Hayek’s (1945) Critique of Lange-Lerner socialism

• Knowledge of “particular circumstances of time and place” too enormous to transmit to a central planner

“Ultimate decisions must be left to the people familiar with these circumstances”

• Information needed to coordinate individual actions can be summarized in prices

• “Nobody has yet succeeded in designing an alternative system”

Costs of Revelation

1. Communication costs: – E.g., combinatorial auctions: 2L bundle values – Measured in bits (communication complexity)

or real numbers (message space dimension)2. Cost of agents evaluating own preferences

– E.g., job interviews, flyouts, …3. More deviations when commitment is

imperfect (short-term contracts, cheap talk)– E.g., revealed info may be exploited by

auctioneer or competitors in the future

Communication• N = set of agents, X = set of outcomes • Agent i privately observes his “type” Ri 2 Ri

• State: R = (R1, …, RN) 2 R1£…£ RN = R• Want to implement choice rule F: R X• F(R) = “optimal” outcomes in state R• Sequential communication may save over simultaneous• Communication Protocol =

1. Extensive-form message game 2. Outcomes assigned to terminal nodes3. Agents’ strategies (type-contingent)

• Do not require incentive-compatibility (e.g., agents could be computers)

Communication

1

2

2

1

aa

bb

cc

d

d

a a

d

d

• State space partitioned into product sets• In every state R must implement an outcome in F(R)• Characterizing such protocols is hard

Agent 2’s type

Agent 1’s type

• Announces message m 2 M

• Each agent accepts or rejects m based on his own type

• Acceptance by all agents verifies x 2 F(R)

• Verification in each state R

m

x

Agent 2’s type

Agent 1’s type

Verification (Nondeterminism)• Omniscient oracle knows state R but must

prove to an outsider that x 2 F(R)

Verification: Formalism

• Protocol Г = M, μ, h • M is a message space • μ: R M is a message correspondence s.t.

“Privacy Preservation:”

μ(R) = \i μi(Ri) R • h: M X is outcome function • Г verifies F if h(μ(R)) ½ F(R) R• Г fully verifies F if h(μ(R)) = F(R) R

Why Verification?

• Any deterministic protocol can be verified by oracle sending messages in agents’ stead (i.e., M = terminal nodes of the protocol)

verification costs · communication costs

• Economic example – Walrasian equilibrium: message = (allocation, prices)

• Steady state of a communication process (e.g., tatonnement, auctions, deferred acceptance algorithms)?

Measuring communication costs

• Discrete: number of bits (communication complexity)

– E.g., oracle needs log2M bits to encode message

• Continuous: number of real numbers– How many reals are needed to encode a message from

M – dimension of M?• Agents’ costs of evaluating own “types”

– E.g., number of job interviews, dates, sorting…• “Privacy” – reducing revelation of agents’ types

E.g., revealed info may be exploited in the future

• Problem - Peano space-filling curve: [0,1] [0,1]n

• Inverse Peano function describes [0,1]n with one number

• Note: Peano function is continuous (its inverse is not)

Measuring Dimension: Problem• Intuition: dimension of M = “number of real

numbers” needed to describe points in M

Questions:

1. How to define the dimension of a “space”?

2. Which transformations preserve this dimension?

Two classes of approaches:

• Topological dimensions

• Metric dimensions

Topological dimension - “inductive”

• “We say that space is 3-dimensional because the walls of a prison are 2-dimensional”

• M is topological space (defined “open sets”)• ind()= - 1• “ind M k” if every point in M lies in an

open set whose boundary has “ind k – 1”• ind M = min. integer k s.t.“ind M k”

Preservation of topological dimension

Proposition: If f: X Y is a continuous 1-to-1 function then ind Y ind X

• Proof idea: f(X) has fewer open sets than X – “harder to imprison” points

• Could have ind Y > ind X if f -1 is not continuous (e.g., Peano function)

Metric dimension (ball covering)• M is a metric space (has distances)• NM() = min. number of -balls needed to cover M• E.g. N [0,1]n () ~ A (1/)n

• In general, dim M = limsup+0 ln NM()/ln (1/)– To cover unbounded S could allow to vary ball size

(“Hausdorff dimension”)

• dim M ind M: – M = irrational numbers in [0,1] dim M = 1, ind M = 0

• dim M may be “fractal” – e.g., dim (China’s coastline) = 1.16

Interpretation of metric dimension: Approximation with bits

• NM () = min. number of messages needed to approximate points from M within (“discretize”)

• log2NM () = min. number of bits needed to encode such messages

• Thus can approximate points from M within using ~ dim M log2(1/) bits as +0

• Thus dim M relates to communication complexity of approximating M with bits

• But: dim M may not represent the hardness of approximation for a fixed > 0, or for +0 slowly as other parameters grow (will have example)

Preservation of metric dimension

• Proposition: If f: X M is a 1-to-1 function and f -1 is Lipschitz

(i.e., A >0: dist(x, x) Adist(f (x), f (x)) x,x) then dim M dim X• Proof: f “does not shrink” distances except by a

factor• Continuity of f -1 not enough (e.g. let f be inverse

Peano)

“Fooling set” dimension bounds • Rf R is a fooling set if R,R Rf,

RR (R) (R) = • Proposition: Suppose Rf is a fooling

set. • (a) If has a (local) continuous

selection, ind M ind Rf.• (b) If has a (local) selection whose

inverse is Lipschitz continuous, then dim M dim Rf.

So, which dimension is better?• Use topological dimension, restrict to have a

continuous selection?– Mount-Reiter, Walker, etc. – Rules out discrete messages (e.g., indivisible

allocations)

• Use metric dimension, restrict to have a selection whose inverse is Lipschitz continuous?– Hurwicz 1977, Nisan-Segal– Interpretation: small errors in message transmission

should not yield large mistakes– Need not rule out any mechanisms - can define a

“right” metric on any M to have Lipschitz continuity– Allows “fast” discrete approximations

Lecture 2: Communication and Prices in Social Choice

Hayek’s (1945) Critique of socialism

• Knowledge of “particular circumstances of time and place” too enormous to transmit to a central planner

“Ultimate decisions must be left to the people familiar with these circumstances”

• Information needed to coordinate individual actions can be summarized in prices

• “Nobody has yet succeeded in designing an alternative system”

Is it necessary to use prices?• Fundamental Welfare Thms: Supporting prices (1)

are sufficient for efficiency and (2) can be constructed with full information about the economy– But then can compute efficient allocation directly

• Hurwicz, Mount-Reiter: In a convex economy with distributed preference information, Walrasian equilibrium is “dimensionally minimal” among “regular” mechanisms verifying efficiency

• Did not rule out mechanisms that don’t use prices• Inapplicable to market design:

– Nonconvexities, discrete decisions– Other goals: approximation, group stability, fairness, …– Other communication costs: bits, evaluations, …

• Message m is less informative than message m'if m' accepted m accepted.

• m is a minimally informative message verifying outcome x if any less informative message verifying x is equivalent to m.

• Such messages minimize communication costs:– Size of M - number of bits or reals to encode a message

– Preference evaluation costs, privacy loss, etc.

Informativeness: Partial Order

m

m'

x

Allocate an Object between 2 Agents

• Messages verifying “2”

• Minimally informative messages verifying “2”

1

2

• Equivalent to announcing supporting equilibrium price p

• Each p must be used (in a diagonal state) need infinitely many messages (continuum)

mp

pAgent 2’s value

Agent 1’s value

General Results• Characterize social choice problems (social goals

and preference domains) for which it is necessary to find supporting prices

• Algorithm deriving the form of prices (budget sets) that verify solutions to a given problem with minimal information

• Price space yields communication cost• Selected applications:

– Pareto efficiency in convex economies

– Exact or approximate surplus-maximization

– Stable many-to-one matchings

• Extra communication cost of incentives

• N = set of agents, X = set of outcomes

• Agent i’s “type” is a preference relation Ri over X

• State: R = (R1, …, RN) 2 R1£…£ RN = R

• Choice rule F: R X

• F(R) = “optimal” outcomes in state R

• Protocol verifies F if R 2 R x 2 F(R) m 2 M that is acceptable in state R and verifies x

Social Choice Problem

fully

• Oracle’s message:– Proposed outcome x X

– Budget set Bi X for each agent i

• Agent i accepts iff x is his optimal choice from Bi

i.e., Bi L(x, Ri) {y 2 X: x Ri y}

• Acceptance by all agents (equilibrium) must verify x

L(x, R1)

L(x, R2)

x

Verification by budget equilibria

B1

B2

• Full verification: R 2 R x 2 F(R) budget equilibrium (B,x) verifying x in state R

• E.g., Fundamental Welfare Theorems: Any Pareto efficient allocation in a convex economy can be verified with a Walrasian equilibrium– Extend to other social choice problems?

L(x, R1)

L(x, R2)

x

Verification by budget equilibria

R 

B1

B2

B2

x

B1 L(x, R1)

L(x, R2)

R'  R 

• Larger budget sets more informative equilibrium

Definition: F is monotonic if RR xF(R) R'R, L(x, Ri) L(x, Ri') i xF(R').Theorem: F is fully verified with a budget

equilibrium protocol F is monotonic. Williams (1986), Greenberg (1990), Miyagawa (2002)

Verification by budget equilibria

• Knowing R, could compute an optimal outcome directly, without using supporting budget sets

• Can we construct budget equilibrium not just with full info, but from any verifying communication?

Necessity of budget equilibria?

L(x, R1)

L(x, R2)

x

B1

B2

R 

m

Definition: F has the Budget Revelation Property if for any message verifying x there exists a less informative budget equilibrium (B,x) verifying x.

Theorem: F satisfies BRP F is Intersection-Monotonic (stronger than monotonicity).

L(x, R1)

L(x, R2)

x

Necessity of budget equilibria?

m

B1

B2

ÅR2m L( x, R2) =

ÅR2m L( x, R1) =

Some choice rules satisfying BRP

• Pareto efficiency• Approximate Pareto efficiency• The core• Stable matching• Envy-free• More generally: Any “CU” rule described

by coalitions’ blocking sets (x,S) X –outcomes coalition S N can use

to block candidate outcome x 2 X– x F(R) no coalition S N has a strict

Pareto improvement over x within (x,S)

Venn Diagram

M = verified with budget protocol

IM = BRP CUPareto 

Core 

Approx. Pareto No-Envy

 

Social Choice Rules: Boolean Representation

• “x F(R)” can be written as a boolean formula with R atoms {z Ri y}y,z X, i N describing R– Using “conjunctions” ( = AND), “disjunctions” ( =

OR), “negations” ( = NOT)• F is monotonic need only use atoms {x

Ri y}y X, i N , no negations• can be written in “Monotone Disjunctive

Normal Form” = ( (i,y) (x Ri y)) • RF-1(x), MDNF can be verified with one

of its clauses - a “budget equilibrium” (cf. Theorem 1)

Restrictions imposed by IM

• Equivalently, for monotonic rules, “x F(R)” can also be written in “Monotone Conjunctive Normal Form” = ((i,y) (x Ri y))

• F is Intersection Monotonic can be written as MCNF whose disjunctive clauses don’t contain (x Ri y) (x Ri z) with y z

• Intuition: if “x F(R)” is preserved by taking y or z individually out of L(x,Ri), should be preserved by taking out y and z together

Restrictions imposed by CU

• F if CU “x F(R)” can be written as MCNF =

((i,y) (x Ri y)) s.t. each clause has a single y – i.e., takes the form iS (x Ri y)– Interpretation: coalition S does not want to block x with y – i.e., does not contain (x Ri y)(x Rj z) with y z

• Characterizes monotonic choice rule that are “binary,” which means– calculate social relation Σ between x,y from agents’

preferences x Ri y (e.g., Σ = “not blocked by”) – IIA– Choose maximal elements in Σ from X– Note: if we required Σ to be rational we would hit Arrow’s

Impossibility Theorem

Lecture 3: Market Design Applications

Definition: F has the Budget Revelation Property if for any message verifying x there exists a less informative budget equilibrium (B,x) verifying x.

Theorem: F satisfies BRP F is Intersection-Monotonic.• Stronger than monotonicity, but satisfied by many choice rules –

e.g. Pareto, approx. Pareto, core, stability, envy-free – all CU rules.

Necessity of budget equilibria:

L(x, R1)

L(x, R2)

x

B1

B2

m

Intuition

• Social goals “congruent” with private preferences minimize communication by asking agents to act on their knowledge selfishly within budget sets (cf. Hayek)

• Design budget sets to coordinate choices and attain social goals with minimal communication

• Budget Revelation Property use budget equilibria• Minimize informativeness = shrink budget sets

“critical” states R:

Bi = L(x, Ri) = \ R'i Ri: xF(R'i, R-i) L(x, R'i)i

• In such R, (B1,…, BN, x) is a unique budget equilibrium verifying x (up to equivalence)

Minimally Informative Verifying Messages

R  m

xB2

L(x, R1)

L(x, R2)

B1

R

Market Design: Roadmap• Use budget-shrinking algorithm to construct min.

informative budget equilibria (B,x) verifying x• If Bi = L(x, Ri) i for some R 2 R (“critical state”)

minimal message space for full verification of F• To bound below simple verification cost - use tricks,

e.g., – Restrict to states R in which F(R) is a singleton– Construct a fooling set whose states can’t share a verifying

budget equilibrium• Verification cost very high problem is hopeless• Verification cost low Can it be achieved in a

deterministic, incentive-compatible protocol?

• L goods; Ri over xi 2 R+L

convex, smooth, with a positive utility gradient

Application: Pareto Efficiency in Smooth Convex Exchange Economies

B2

B1

R2

R1

x

Proposition. m is a minimally informative message verifying the Pareto efficiency of an interior allocation in a smooth convex economy m is equivalent to a Walrasian equilibrium.

• Any such equilibrium (B, x) is a unique Walrasian equilibrium supporting x

Number of variables for full verification = (N-1)L quantities + (L – 1) prices

Verification Cost• Fix endowments (N-1)(L-1) quantities + (L – 1) prices =

N(L-1) numbers• Can we verify Pareto efficiency with fewer numbers? • Fooling set of Cobb-Douglas economies (Hurwicz):

– Utilities ui (xi) = Πl xilαil ; with Σl αil = 1 for all i

– Described with N(L -1) parameters

• FOC for (p, x) being a Walrasian eqm:

αil/αik = (pl xil ) /(pk xik) for all l,k,I No two distinct Cobb-Douglas economies share a Walrasian

eqm give a “fooling set” for Pareto efficiency• What about deterministic communication?

– “Tatonnement” converges fast in some but not all economies

Pareto Efficiency with Numeraire

• x = (k, t1,…,tN ); feasibility: Σi ti = 0

• Ri 2 Ri given by quasilinear utility ui(k)+ti

• Efficiency = maxk2K Σi ui(k) (total surplus)

B1

R2

R1

B2

x

t1

t2

k

Proposition. m is a minimally informative message verifying efficiency with numeraire m is equivalent to a price equilibrium with personalized nonlinear prices p 2

NK s.t. i

pi (k) = const on k 2 K. • Each such price must be used in the “critical”

state where Σi ui(k) = const on k 2 K Cost = (N – 1)(K – 1) real numbers = full

revelation of N – 1 agents’ valuations

• Combinatorial auctions: N = 2, K = 2L communication cost ¸ 2L –1 numbers

• N consists of firms (F) and workers (W)

• Matching x = binary relation from F to W in which a worker matches with at most one firm

• Ri 2 Ri depends only on i’s partners

Stable Many-to-one Matching

x

F

W

Stable Matching• Matching is stable if these coalitional

deviations are not strictly Pareto improving:(i) Firm hires some new workers (and fires some)(ii) Worker quits to become unemployed

– This describes an IM choice rule

x

F

W

Minimally Informative Equilibria• A worker’s budget set described by available firms

• A firm’s budget set: available groups of workers

• In a minimally informative equilibrium verifying stability, the groups consist of– The firm’s current workers

– Workers who don’t have the firm in their budget sets

firms’ budget sets, non-combinatorial

x

F

W

Lemma. m is a minimally informative message verifying the stability of matching x m is equivalent to a partitional equilibrium, in which each off-equilibrium match is in either partner’s budget set, but not both.

• Any such equilibrium (B, x) is a unique partitional equilibrium supporting x in state R in which Bi = L(x, Ri) i.

Cost of full verification = 2FW bits• Cf. cost of describing a match x = FW bits• Cf. describing firms’ preferences over 2W groups

=log2(2W)! ~ W 2W bits• Two questions:

1. Is the cost of verification any lower?2. Is the deterministic communication cost higher?

Verification cost is the same:• Take any partitional equilibrium (B, x), and “critical state”

R s.t. Bi = L(x, Ri) i• Further restrict R to ensure that x is unique stable match:

– Each i strictly prefers his current match within Bi – Each i strictly prefers quitting to getting a new partner from Bi

• Can’t have a stable match x ≠ x:– i gets a new partner from Bi i would quit– i gets a new partner j Bi i 2 Bj j would quit– x x x is in all budget sets a coalition would block x by x

(B, x) must be used for verification in state R Communication cost = 2FW bits• Same logic works for 1-to-1 matching (cost ~ FW bits)• This bounds below deterministic communication cost

Deterministic Communication

• This lower bound is almost achieved for substitutable preferences by Gale-Shapley deferred acceptance algorithm:

(· 2FW steps: matches offered, rejected) • Exponentially less than full revelation of firms’

combinatorial preferences• The algorithm also minimizes evaluation costs of the

responding side (whose budget sets are minimized)– With uniformly drawn preferences, on average only 1/3 of

potential partners evaluated

Determinism vs. Nondeterminism• Recall: Deterministic CC in bits ·

nondeterministic CC in bits • Gap is at most exponential – e.g., try each

of the M oracle’s messages for acceptance instead of encoding with log2M bits

• Gap is small in some cases: e.g., matching or auctions with substitute preferences - monotonic “tatonnement” converges quickly

• But there are IM rules with exponential gap

Example of Exponential Gap• 2 agents hire 2 out of 3m workers • Agent 1 is happy if the hired workers share a language• Agent 1 knows privately each worker’s language • Public knowledge: all workers are monolingual, m

languages spoken by a pair of workers, and m languages spoken by a single worker.

• Agent 2 knows privately 2m+1 “capable” workers• Agent 2 is happy 1 if both hired workers are capable• Social goal: make both agents happy. - CU, IM. • Such pair always exists and can be verified by its

announcement - 2 log2 (3m) bits • But deterministic CC is asymptotically proportional to m

– Equivalent to the “Pair-Disjointness.problem”

Average-Case Goals• E.g., approximate expected surplus, given a probability

distribution over states• Example: discrete public good, N agents’ i.i.d. values vi

2{0,1}, Pr{vi = 1} =Cost = c < N. • N large building is efficient with prob. 1• Verifying efficiency requires finding c agents to charge

Lindahl price =1, takes ~ c logN bits – for -approximation guarantee, (k-) log2N bits

“Government solution” approximates expected efficiency without any communication, prices (cf. Groves-Hart)

• Another Example: with many decisions, “authority” may approximate expected efficiency when finding prices would take exponentially longer (cf. Coase, Simon)

• Examples known in which expected-surplus approximation is hard to attain – e.g., combinatorial auctions

Economics without Incentives?• Thought experiment: people are honest.

Would basic economics institutions (markets, firms) remain?

• Minimize communication by asking people to pursue self-interest guided by prices Scope of “market design”

Required price space Lower bounds on communication and evaluation

costs

• Average-case goals may yield non-market institutions (governments, firms)

Lecture 4:Communication Cost of Incentives

So far• Can calculate communication costs of many

economic problems– E.g., use the Budget Revelation Property, construct

min. informative verifying budget equilibria• In some cases, cost is prohibitive, cost of full

revelation of preferences– E.g., combinatorial auctions with general valuations –

need whole combinatorial price space • In other cases, cost is manageable:

– E.g., auctions or matching with “substitute” preferences – enough to use individual prices/budget sets

• Can we then construct an incentive-compatible mechanism?

Incentives and Prices

• “Budget set” in mechanism design: set of agent’s attainable outcomes– E.g. Menus – “Taxation Principle”

• Incentives: agent maximizes utility within his budget set

• Remarkable: even with no incentives (truthful agents), must still describe budget sets, ask agents to maximize own utility within them – For social goals that are “congruent” with preferences

Nash implementation• Agents’ preferences commonly known (but not to designer)• Mechanism g: M1… MN X• (Full) implementation: Set of Nash Equilibria = F(R) R 2 R• m is NE of g (B, x) is a budget equilibrium, where x=g(m),

Bi = {g(mi,m-i): mi 2 Mi} i A Nash protocol is a budget protocol• Theorem 1 Any Nash implementable F is monotonic • Even with symmetric info, selfishness reveal budget sets:• Proposition: Communication cost of Nash implementation

communication cost of full realization– May not hold for extensive-form mechanisms (in which strategies m

are not revealed)

Communication Cost of (1-stage) Nash implementation

• Monotonicity Nash implementability (not every budget equilibrium protocol is a Nash protocol)

• But is “almost” sufficient: • Definition: F has No Veto Power (NVP) if |{i:

xRiy y 2 Y }| N – 1 x 2 F(R).– Trivial in economic applications with N 3

• Maskin (1977): Any monotonic NVP choice rule is Nash implementable

• Proposition: If F is IM and NVP, communication cost of Nash implementation N (communication cost of full realization) + log2N.

Proof: Mechanism• Let = minimal space of budget equilibria fully verifying F• Each agent i announces (Ei, li), where

Ei = (B1i,…, BN

i, xi) 2 , li 2 {1,…, N}

1. E1 =…= EN = (B1,…, BN, x) implement x

2. Ej = (B1,…, BN, x) ji and xi Bi implement x

3. Ej = (B1,…, BN, x) ji and xi 2 Bi implement xi

4. Otherwise implement xi for i = (j lj ) modN + 1• x 2 F(R) all agents announcing the same equilibrium (B1,

…, BN, x) 2 in state R is a NE• Case-1 NE: (B1,…, BN, x) verifies x, is a budget eqm in state

R x 2 F(R) • Other cases: all but (perhaps) one agent could deviate to

Case 4 to get his best outcome by NVP, x 2 F(R)

• May further reduce cost e.g. by letting Bi be announced only by agent i modN+1– McKelvey, Reiter-Reichelstein

• Key: each agent’s budget set must be determined by others’ messages – prevent “price manipulation”

• With private information, incentives require constructing Bi without using i’s info extra communication cost

Communication Cost of Selfishness (with Ron Fadel)

Incentive-Compatibility with private information:

1. Ex Post IC: Each agent’s strategy is optimal given others’ even if he knew other agents’ types

– Weaker than Dominant-Strategy IC: in an extensive form, don’t consider others’ unused strategies

2. Bayesian IC: Each agent’s strategy is optimal given others’ on expectation over their types

Communication Cost of Selfishness• CCS = (Minimal Communication Cost of

an Incentive-Compatible mechanism) - Communication Complexity

• Related work: Reichelstein 1984, Green and Laffont 1987, Lahaie and Parkes 2004, Feigenbaum el al. 2004, Johari 2004

Setup• Quasilinear payoffs: ui(y)+ti, with y 2 Y

• ui 2 Ui ½ Y - private type of agent i

• Types drawn independently (basic model)

• Decision function f: U1£…£ UI ! Y

• Don’t care about transfers

Communication Protocols

1

2

2

1

• Binary extensive-form message game (tree)• Agents’ (type-contingent) strategies• Outcomes assigned to leaves

y1

y2

y3

y4

y5

Communication cost = 3 bits (worst-case)

u1 u1

u1'u1'

, t14, t2

4

Incentivizing a Protocol

• Assign incentivizing payments to the leaves• Hide history (create Information Sets) to

prevent contingent deviations • A given protocol may not be incentivizable

– may need more communication

1

2

2

1

y1

y2

y3

y4

y5

, t11, t2

1 , t1

2 , t22

, t13, t2

3

, t15, t2

5

Sources of CCS

• EPIC: No need to hide info; CCS comes from the need to compute payments

• BIC: CCS comes from need to hide info; (computing payments does not require extra bits)

• Restrict to f that is implementable in some IC mechanism (e.g. full revelation)

– for others, CCS = 1

A Protocol that can’t be EPIC incentivized

• Allocate one object between two agents efficiently; values v1 2 {1,2,3,4}, v2 2 [0,5].

• Protocol:– Agent 1 announces v1 (2 bits)– Agent 2 “takes” iff v2 ¸ v1 (1 bit)

• Intuition: for EPIC, agent 1 must be charged a price within 1 of v2, but it is not revealed

• Can show that any EPIC protocol must take ¸ 3 bits; EPIC CCS = 1

• A similar example for BIC

An Upper Bound on CCS(both BIC and EPIC)

• Take a protocol P computing f (which is implementable)• Can incentivize agents if observe their strategies in P

– Transfers need only depend on the strategies– Punish strategies that are not consistent with any type

• Protocol in which all agents simultaneously announce their strategies in P is incentivizable

1

y4

1

2

2

y1

y2

y3

y5

An Upper Bound on CCS(both BIC and EPIC)

• A d-bit protocol P has · 2d decision nodes agents’ strategies can be described with · 2d bits

Comm. Complexity with IC · 2Comm. Complexity

• Is the bound tight? For EPIC, open question

1

2

2

1

y1

y2

y3

y4

y5

Example: Low CCS for Efficiency

• f(u) solves maxy i ui(y) • Let agents announce final utilities wi = ui(y), pay

each agent ti = jiwj

Agents become a “team” Efficient protocol is EPIC Even if can’t reach full efficiency, strategies

maximizing expected surplus form a BNE

• Also, any EPIC implementable f (e.g., efficiency) has BIC CCS = 0

Exponential Bound reached for BIC

• Expert privately knows 1-to-1 mapping between K decisions and K consequences– Also has a private utility over decisions

• Boss privately knows desired consequence

Decisions

Consequences

25413Expert

Boss

Bound is reached for BIC

• A simple protocol: – Boss announces desired consequence (logK bits),

– Expert decides (logK bits)

• Not BIC – Expert will maximize own utility

Decisions

Consequences

3 1 4 5 2Expert

Boss

Bound is reached for BIC

• A BIC protocol: – Expert reveals mapping

( log(K!)~KlogK bits – exponentially longer)

– Boss decides

• We show that any BIC protocol takes ¸ K/2 bits - exponentially longer than the simple protocol

Decisions

Consequences

25413Expert

Boss

Extensions: CCS is unbounded for• Average-case communication complexity

– Example: allocate object efficiently between 2 agents with values uniformly drawn from [0,1]

• Bisection takes on expectation 4 bits (Arrow et al.)• EPIC for Agent 1 requires essentially charging him agent 2’s

value, which has unbounded entropy

• BIC CCS with Correlated Types – Example: Agent 1 has “large” type that determines

binary outcome and correlates with Agent 2’s type • BIC must punish Agent 1 when “caught lying” by Agent 2

• EPIC CCS with Interdependent Valuations: vi(x,si,s-i), where si is agent i’s type– Example: Agent 1 knows whether he should get the

object but his value for it is known only to Agent 2• EPIC requires charging Agent 1 this value

Open Questions

• EPIC CCS – how high can it be?

• In what practical problems is CCS low?

• Can CCS be reduced substantially if ICs only need to be satisfied approximately (equivalently, utilities given with finite precision)?

Ideas for understanding firms

• Coase (1937), Simon (1951): firms are islands of authority where “discovering what the relevant prices are” is too costly– Have example where authority achieves

efficiency with probability 1, but verifying it (finding prices) takes exponentially more bits

• Communication may be distributed to economize on individual costs– Individual costs may be reduced by hiring extra

agents (“managers”) to relay prices (cf. computational models of Radner - van Zandt)