optimal allocations with capacity constrained...

Optimal Allocations with Capacity ConstrainedVerification

Albin Erlanson∗ and Andreas Kleiner∗∗

∗University of Essex and ∗∗Arizona State University

University of MannheimOctober 29, 2019

Introduction: Mechanism design & verifiable information

I How to design institutions and rules such that a desirableoutcome can be reached, although information is privatelyheld?

I Standard model assumes soft information, e.g. aboutpreferences

I In some applications, private information is based on hardfacts

I For example, “experts” with private information– Information is based on hard evidence

I Designer can verify experts’ private informationI Examples: work permits, research grants, . . .

1

Introduction: assigning working permits in Canada

I Recruit skilled workers:“Canada is changing its economic immigration programs toprovide more opportunities to prospective skilled immigrants”

I Policy in place from 2015, called “express entry”I Applicants are ordered in the pool according to a scoreI Score is based on factors that are known to contribute to

economic success

2

Canada is changing its economic immigration programs to provide more opportunities to prospective skilled immigrants.As of January 2015, skilled foreign workers have access to Express Entry, which covers Canada’s key economic immigration programs:

• the Federal Skilled Worker Program;• the Federal Skilled Trades Program; • the Canadian Experience Class; and• a portion of the Provincial Nominee Program.

Candidates who are invited to apply for permanent residence under the Express Entry system will bene� t from fast processing times of six months or less. Express Entry also provides a pathway for skilled workers to connect with potential job opportunities in Canada prior to arrival. Express Entry ensures that the candidates who are most likely to succeed economically – not simply those � rst in line – are able to immigrate to Canada.

How Express Entry Works

WHAT PROSPECTIVE CANDIDATES NEED TO KNOWWHAT PROSPECTIVE CANDIDATES NEED TO KNOW

ITA

Express Entry Pool

Create your profi le

Apply for permanent

residence online

Invitation to Apply

ENTRÉE EXPRESSEXPRESS ENTRY

STEP 1 STEP 2

Job Bank

WHAT PROSPECTIVE CANDIDATES NEED TO KNOW

registration and self-promotion to

employers

www.canada.ca/ExpressEntry

Express your interest in coming to Canada as a skilled foreign worker. Create an online Express Entry pro� le and tell us about your skills, work experience, language ability, education and other details. Before doing this, you will need to take a language test in English or French. If you were educated outside of Canada, you may also need to have your education assessed against Canadian standards. More information on language and education assessments is available online.If you meet the criteria of one of the federal economic immigration programs subject to Express Entry, you will be placed in a pool of pre-screened candidates. If you do not already have a Canadian job o� er or a nomination from a province/territory, you must register with the Government of Canada’s Job Bank. Job Bank is an easy, online search tool that will help you get matched with jobs in Canada based on your skills, knowledge and experience.

Express Entry Pool You will be given a score to determine your place in the Express Entry pool using a Comprehensive Ranking System that includes factors known to contribute to economic success (such as language, education, and work experience).

� ere will be regular rounds of invitations issued to candidates from the Express Entry pool, inviting them to apply for permanent residence. Candidates with the highest scores, including those who have a valid job o� er or a provincial/territorial nomination, will be invited to apply.

Your Express Entry pro� le will be valid for 12 months. During that time, you will need to update your pro� le if circumstances change such as your level of education or language test results.

Important: Filling out an online Express Entry pro� le is not a guarantee that you will qualify for permanent residence. If you are invited to apply for permanent residence, information provided in your Express Entry pro� le will be veri� ed at that time.

You will receive an Invitation to Apply for permanent residence if you:• have a valid job off er from an employer in Canada

(subject to the Labour Market Impact Assessment process in place at that time);• have been nominated by a province or territory; or• are among the top ranked in the pool based on your skills and experience.

Citizenship and Immigration Canada will process the majority of complete permanent residence applications received within six months or less. Candidates in the Express Entry pool who do not receive an Invitation to Apply for permanent residence after 12 months can resubmit their pro� le and re-enter the pool if they still meet the criteria.

For more details on Express Entry, see www.canada.ca/ExpressEntry.

Step 1Potential candidates create an online Express Entry profi le

Step 2Selected candidates are invited to submit an electronic application for permanent residence

Introduction: verification and assigning working permits

I m working permits to assign among n applicants, m < nI Allocate them to applicants with highest expected earningsI An application consists of qualifications, degrees, . . .I Private information but it can be verifiedI At most k applicants can be verified, k < m

What is the optimal way to allocate these permits, given thepossibility of verifying agents?

3

Introduction: allocations with capacity constrainedverification

There is a group of n agents and a principal with m objects

The principal determines an allocation, and m < n

I Agents have information that is relevant for the principalI Agents would like to have an objectI Principal can learn agents’ typesI Learning is capacity constrained:

at most k agents’ information can be learned, where k < mI Monetary transfers are not possible

4

Possible mechanisms

I Mechanism 1:

0 t̄

Allocate k objects efficiently

Verify k agents

Improve by allocating remaining objects randomly:I Mechanism 2:

0 t̄

Allocate k objects efficiently and remaining

Verify k agents

m − k objects randomly with an equal probability

Two possible mechanisms, both incentive compatible

5

Possible mechanisms

I Mechanism 3:

0 t̄c

Allocate randomlywith constant probability

Never verify

Allocate up to m objects efficiently

Verify k agents

I Not ex-post incentive compatible:for example, if ti < c and m agents to the right of c,agent i could deviate and report t ′i > c

I Bayesian incentive compatible for sufficiently high cutoff c

6

Aside: Incentive constraintsBayesian incentive compatibility:

Pr(Getting an object; ti , ti )︸︷︷︸Truthtelling

≥ Pr(Getting an object; t ′i , ti )︸︷︷︸Lying

I In many settings local incentive constraints are sufficient

ti t ′iI In our model local constraints are not sufficient

P

ti t ′i

7





ti t ′i

I In our model local constraints are not sufficientP

ti t ′i

7





ti t ′iI In our model local constraints are not sufficient

P

ti t ′i

7

Incentive compatibility

Local incentive constraints are not sufficient!

Let P(ti ) = Pr(Getting an object; ti , ti )

Incentive compatibility must hold for “worst-off” types

8

Incentive compatibility

A“worst-off” type t ′i ∈ Ti is such that

P = P(t ′i ) ≤ P(ti ) for all ti ∈ Ti

For a mechanism to be IC it follows thatI Types with higher probability of getting an object must be

verified more often, but how often?I Worst-off types do not want to mimic any other type

9

Possible mechanisms

I Mechanism 3:

0 t̄c

Allocate randomlywith constant probability

Never verify

Allocate up to m objects efficiently

Verify k agents

I Bayesian incentive compatible for sufficiently high cutoff c

10

Possible mechanisms

There are many other mechanisms1. How many objects to allocate efficiently?2. More than two regions, if so how to allocate?3. Where should the cutoffs be?4. How to verify such that the mechanism is BIC?

11

Plan for rest of the talk

I Related literatureI ModelI Optimal mechanismI Characterization of BIC mechanismsI Feasible reduced forms

12

Related Literature

I Costly verification with transfersTownsend JET 1979; Gale & Hellwig RES 1985; Border & Sobel RES1987

I Private good allocation with verificationBen-Porath, Dekel & Lipman AER 2014; Mylovanov & ZapechelnyukAER 2017

I Mechanism design with evidenceGreen & Laffont RES 1986; Glazer and Rubinstein ECMA 2004;Ben-Porath, Dekel & Lipman ECMA 2019; ...

I Interim allocation rulesBorder ECMA 1991; Che et al. ECMA 2013

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Model

• n agents with:

Types (symmetric) ti ∈ Ti = [0, 1], private informationDistribution F , independent, and density fPreferences ui (ti ) > 0 objects are desirable, private values

and each agent need at most one object

• m homogeneous objects, where n > m

• One principal with:

Payoff∑

i∈I′ ti , where I ′ is the set of agents that get an objectAction whom to verify and then decide an allocation

• No monetary transfers

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Verification

I Possible to randomly verify agentsI If agent i is verified the principal learns agent i ’s true typeI The penalty available is not to give an object to a lying agent

Verification Technology:I At most k agents can be verified, and k < m

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Optimal mechanism

0 1BIC Bk Bm

Allocaterandomly

Never verify

Allocate to k highest& allocate randomly

Verify sometimes

Allocate efficiently

Verify sometimes

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Optimal mechanism

I Case 1: At least k reports in region Bm

0 t̄BIC Bk Bm


Verify k agents

Allocaterandomly

Never verify

Allocaterandomly

Never verify

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Optimal mechanism

I Case 2: Less than k reports in region Bm

0 t̄BIC Bk Bm

Allocate to k highest

Verify k highest


Verify k highest

Allocaterandomly

Never verify

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Optimal mechanism: Reduced forms

0 1BIC Bk Bm

Allocaterandomly

Never verify


Verify sometimes


Verify sometimes

I Induces an expected probability of winning an object P∗(ti )

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Main result: optimal mechanism

Let A∗(ti ) = P∗(ti )− inft′i P∗(t ′i ).

TheoremThe mechanism (P∗,A∗) maximizes the expected utility of theprincipal and is therefore optimal among all BIC mechanisms.

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Bayesian incentive compatibility

A direct mechanism is a pair (p, a)I pi (ti , t−i ) is the probability that agent i gets an object

conditional on not failing the auditI ai (ti , t−i ) is the probability that agent i is verified

DefinitionA mechanism (p, a) is Bayesian incentive compatible (BIC) if,for all ti , t ′i ∈ [0, 1],

ui (ti ) · Et−i

[pi (ti , t−i )

]≥

ui (ti ) · Et−i

[pi (t ′i , t−i )(1− ai (t ′i , t−i ))︸︷︷︸

no verification

+ 0 · ai (t ′i , t−i , s)︸︷︷︸lie is detected

].

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Revelation principle

PropositionIt is without loss of generality to consider direct mechanisms of theform (p, a), where

p :T → [0, 1]n Allocation rulea :T → [0, 1]n Auditing rule

in which truth-telling is a Bayesian equilibrium.T denotes the product of types spaces ×n

i=1Ti

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Objective of the principal

maxp,a

Et[∑

ipi (ti , t−i )ti

](P)

s.t. (p, a) is Bayesian incentive compatibility (BIC)at most k agents are verified (k-feasibility)at most m objects are allocated (m-feasibility)

I We allow for stochastic allocation and verification rules

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BIC implies that for all ti ∈ Ti , t ′i ∈ Ti

ui (ti ) · Et−i

[pi (ti , t−i )

]≥

ui (ti ) · Et−i

[pi (t ′i , t−i )(1− ai (t ′i , t−i ))︸︷︷︸

no verification

+ 0 · ai (t ′i , t−i , s)︸︷︷︸lie is detected

]

We assume from now on that ai (t) > 0 only if pi (t) = 1.

infti∈Ti

Pi (ti ) ≥ Pi (t ′i )− Ai (t ′i )

24



Et−i

[pi (ti , t−i )

]≥ Et−i

[pi (t ′i , t−i )(1− ai (t ′i , t−i ))︸︷︷︸

no verification

]


infti∈Ti

Pi (ti ) ≥ Pi (t ′i )− Ai (t ′i )

24



infti∈Ti

Et−i

[pi (ti , t−i )

]≥ Et−i

[pi (t ′i , t−i )(1− ai (t ′i , t−i ))︸︷︷︸

no verification

]


infti∈Ti

Pi (ti ) ≥ Pi (t ′i )− Ai (t ′i )

24


BIC implies that for all t ′i ∈ Ti

infti∈Ti

Et−i

[pi (ti , t−i )

]≥ Et−i

[pi (t ′i , t−i )(1− ai (t ′i , t−i ))︸︷︷︸

no verification

]


infti∈Ti

Pi (ti ) ≥ Pi (t ′i )− Ai (t ′i )

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Characterization of BIC mechanisms

Let Pi (ti ) = Et−i [pi (ti , t−i )] and Ai (ti ) = Et−i [ai (ti , t−i )].

LemmaA mechanism (p, a) is Bayesian incentive compatible (BIC) if andonly if, for all i ∈ I and all t ′i ∈ Ti

infti∈Ti

Pi (ti ) ≥ Pi (t ′i )− Ai (t ′i ) (BIC)

I A reduced form allocation rule is a mapping Pi : Ti → [0, 1].Thus, Pi from above is a reduced form allocation rule.

I A reduced form verification rule is a mappingAi : Ti → [0, 1]. Thus, Ai from above is a reduced formverification rule.

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Proof sketch: Principal’s maximization problem

maxp,a

∑iEti

[Pi (ti )ti

](P)

s.t.inf

ti∈TiPi (ti ) ≥ Pi (t ′i )− Ai (t ′i ) for all t ′i ∈ Ti (BIC)∑

ai (t) ≤ k (k-feasibility)∑pi (t) ≤ m (m-feasibility)

I A maximization problem in reduced forms, or interim rulesI Need to ensure that Pi and Ai are feasible

26

Aside: Feasible reduced forms

There is one object, i.e., m = 1. An allocation rule is feasible ifn∑

i=1pi (t) ≤ 1, for all t ∈ T

A reduced form P : Ti → [0, 1] is implementable if there exists afeasible allocation rule p such that

P(ti ) =∫

T−ip(ti , t−i )dF n−1(t−i )

I Which reduced forms are implementable?

27

Excursion: Feasible reduced forms

I N = {1, 2} and T = {t`, th}, Pr(t`) = Pr(th) = 1/2I Given P, find p(t`, t`), p(t`, th), p(th, t`) and p(th, th) s.t.:

P(t`) = 12p(t`, t`) + 1

2p(t`, th) (t`)

P(th) = 12p(th, t`) + 1

2p(th, th) (th)

I Symmetry: p(t`, t`) ≤ 1/2, p(th, th) ≤ 1/2 andp(t`, th) + p(th, t`) ≤ 1

I What about (P(t`),P(th)) = ( 15 ,

45 ), is that feasible?

28

Excursion: Feasible reduced forms

P(t`)

P(th)

0

( 12 ,

12 )

( 14 ,

34 )

( 15 ,

45 )

( 34 ,

14 )Feasible P’s

Increasing P’s

29

Aside: Feasible reduced forms

I For each α ∈ [0, 1], set Eα = {ti ∈ Ti |P(ti ) ≥ α}

Theorem (Border, 1991)Let P : Ti → [0, 1]. Then P is implementable if and only if foreach α ∈ [0, 1]

n∫

EαP(ti )dF ≤ 1− F n(E c

α) (Border)

30

Feasibility constraints for PThere is a feasibility constraint on P from

∑pi (t) ≤ m:

LemmaP is implementable if and only if

∫Eα P(ti )dF ≤ CAPm(Eα).

I Follows from Che et al. (2011)

There is also a feasibility constraint on A from∑

ai (t) ≤ k:LemmaA is feasible if and only if

∫E A(ti )dF ≤ CAPk(E , p).

Since P(ti )− inft′i ∈Ti P(t ′i ) = A(ti ), we get∫E

P(ti )dF (ti ) ≤ CAPm(E )∫E

P(ti )− inf P dF (ti ) ≤ CAPk(E , p)

31

Intuition for optimal mechanism: Constraints on P

t0 1

CAPm

CAPk

IC : “Incentive-Constraint”

BIC Bk Bm

32


t0 1

CAPm ∫E

P(ti )dF (ti ) ≤ CAPm(E)

CAPk


BIC Bk Bm

32


t0 1

CAPm

CAPk ∫E

P(ti )− inf PdF (ti ) ≤ CAPk (E , 1)


BIC Bk Bm

32


t0 1

CAPm

CAPk


BIC Bk Bm

32

Main result: optimal mechanism

TheoremThere is ϕ ∈ (0,m/n) such that the mechanism (P∗,A∗) with thereduced form allocation rule

P∗(ti ) =

− 1

n f (ti ) CAP ′m(ti ) for ti ∈ Bm

− 1n f (ti ) CAP ′k(ti ) + ϕ for ti ∈ Bk

ϕ else.

and verification rule A∗ = P∗ − ϕ is optimal.

33

Optimal mechanism: Allocation rule (ex-post)

Given t ∈ T let S be the set of agents such that(1) ti ∈ Bm and among the m-highest reports or(2) ti ∈ Bk and ti among the k-highest reports

The optimal ex-post allocation rule p∗ given a profile t ∈ T is:I all agents in S gets an objectI any remaining objects are allocated in a scramble randomly

among agents that didn’t get an object such that each type inBk ∪ BIC has the same interim probability of getting an objectin the scramble.

34

Optimal cutoffs

Optimal mechanism is parametrized by one parameter:

0 1BIC Bk Bma(ϕ) b(ϕ)

I Increasing ϕ increases the allocation probability for lowesttypes

I Fewer audits necessary, CAPk constraint relaxesI But leaves fewer objects for efficient allocationI Hence, a′(ϕ) > 0 and b′(ϕ) < 0

I First-order condition:∫ b(ϕ)

0 tdF + [1− F (b(ϕ))]b(ϕ) = a(ϕ)

35

Optimal cutoffs

Optimal mechanism is parametrized by one parameter:

0 1BIC Bk Bma(ϕ) b(ϕ)

I Increasing ϕ increases the allocation probability for lowesttypes

I Fewer audits necessary, CAPk constraint relaxesI But leaves fewer objects for efficient allocationI Hence, a′(ϕ) > 0 and b′(ϕ) < 0I First-order condition:

∫ b(ϕ)0 tdF + [1− F (b(ϕ))]b(ϕ) = a(ϕ)

35

Optimal cutoffsMechanism 2: Allocate to k highest types, remainder randomly

t0 1

CAPm

CAPk

IC : “Incentive–Constraint”

36

Optimal cutoffsMechanism 3: Allocate efficient above some cutoff, remainderrandomly

t0 1

CAPm

CAPk

IC : “Incentive–Constraint”

37

Optimal mechanism is not EPIC

Proof.1. P is the essentially unique interim allocation rule. In particular,on Bm we allocate m objects efficiently.2. Consider agent i and a type profile t such that m other agentshave a type in Bm above ti .3. If agent i is truthful, he won’t get an object.4. If he claims to be a high type, he will get an object whenever heis not verified.5. Not all m agents can get verified.

What is the optimal EPIC mechanism?

38

Summary

I Provide a model for the allocation of homogeneous objectswith verifiable private information

I Characterize the optimal allocation ruleI Further questions:

How does the optimal mechanism change when there is a shiftin the distribution?

I How does the optimal mechanism change as we increase thenumber of objects?

I If there is a cost of increasing k: what is the optimal k∗?

39

Thank you for your attention!

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