Can we design a mechanism without money that is efﬁcient and incentive compatible Yes, if the agents know their own values of the objects; private values

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Efficient Assignment with Interdependent

Values

Yeon-Koo Che1 Jinwoo Kim2 Fuhito Kojima3

April 23, 2013

1_{Columbia University} 2_{Seoul National University} 3

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Assignments without Money

Many resources are assigned without using money

I office assignment, student placement in public schools, university housing allocation, organ allocation, task assignments in organizations

Can we design a mechanism without money that is

efficientandincentive compatible?

Yes, if the agents know their own values of the objects; private values

I _{Serial dictatorships (Svensson, 99; Abdulkadiro ˘glu &} S ¨onmez, 98), top trading cycles mechanisms

(Abdulkadiro ˘glu & S ¨onmez, 03), hierarchical exchanges (Papai, 00), trading cycles (Pycia & ¨Unver 11) ...

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Assignments without Money

(Abdulkadiro ˘glu & S ¨onmez, 03), hierarchical exchanges (Papai, 00), trading cycles (Pycia & ¨Unver 11) ...

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Assignments without Money

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Interdependent Values

Values areinterdependentwhen one’s valuation of objects

depends on the information/signals possessed by others

E.g. in school choice, many parents have insufficient

information about the fitness of schools for their children

I _{Seek advice from other parents through word-of-mouth} communication, online social networks ...

Are there incentive compatible mechanisms that implement

the efficient assignments in the interdependent values

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Interdependent Values

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Interdependent Values

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Ex Post Incentive Compatibility

Recall that a (direct) mechanism is a mapping that maps

each signal profile to an assignment (stochastic

assignments allowed).

A mechanism isex post incentive compatibleif truth-telling is a Nash equilibriumeven after all signals are revealed

I _{Best alternative to strategy-proofness in the interdependent} values environment

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Ex Post Incentive Compatibility

Recall that a (direct) mechanism is a mapping that maps

each signal profile to an assignment (stochastic

assignments allowed).

A mechanism isex post incentive compatibleif truth-telling is a Nash equilibriumeven after all signals are revealed

I _{Best alternative to strategy-proofness in the interdependent} values environment

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Our Main Results

No ex post IC mechanism can attain a Pareto efficient

assignment whenever there existsnon-trivial preference

interdependence(and a few auxiliary conditions)

Any ex post “group” incentive compatible mechanism can

only attain atrivial assignment, which is constant

irrespective of signal profiles.

In the two-agent and two-object case, there is a

mechanism that is Pareto efficient and Bayesian incentive

compatible under some reasonable conditions

→It may be important to consider mechanisms thatviolate ex

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Our Main Results

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Our Main Results

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Our Main Results

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Related Literature

Impossibility results with mechanism designwith money

I _{Jehiel & Moldovanu (01): Impossibility of efficient and} Bayesian incentive compatible mechanism

I _{Jehiel et al. (06): Impossibility of nontrivial and ex post} incentive compatible mechanism

Impossibility result reinforced by our paper but... with some substantial differences

I _{Utilitarian efficiency}_vs._{Pareto efficiency} I _{Public decision}_vs. _{private goods allocation} I _{Multidimensional}_vs._{single dimensional signals} Two-sided matching with interdependent values

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Related Literature

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Related Literature

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Illustrative Example: Inefficiency of Ex Post IC

mechanisms

We begin with a simple example (generalized later):

Two agents 1,2 are assigned two objectsa,b.

Signals are single-dimensional;s1,s2∈[0,1]

Letui(s)≡v_ai(s)−v_bi(s)bei0utility difference betweena andb, and assume

∂ui(s)

∂si >

∂u−i(s)

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Illustrative Example: Inefficiency of Ex Post IC

mechanisms

0 1

1

A B

C D

E

Sbb

Sab Sba

Saa

s1 s2

I2

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Illustrative Example: Inefficiency of Ex Post IC

mechanisms

0 1

1

A B

C D

E

Sbb

Sab Sba

Saa

s1 s2

I2

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Illustrative Example: Inefficiency of Ex Post IC

mechanisms

0 1

1

A B

C D

E

Sbb

Sab Sba

Saa

s1 s2

I2

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Illustrative Example: Inefficiency of Ex Post IC

mechanisms

0 1

1

A B

C

D

E

Sbb

Sab Sba

Saa

s1 s2

I2

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Illustrative Example: Inefficiency of Ex Post IC

mechanisms

0 1

1

A B

C D

E

Sbb

Sab Sba

Saa

s1 s2

I2

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Illustrative Example: Inefficiency of Ex Post IC

mechanisms

0 1

1

A B

C D

E

Sbb

Sab Sba

Saa

s1 s2

I2

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General Setup

The results can be generalized to assignment ofnagents

tonobjects, exactly one object to each agent.

Incentive requirement: (weak) Ex post IC.

Three main assumptions; Interdependence, Richness,

Connectedness.

Assumption 1(Interdependence)For any agents i,j, objects a6=b, whenever vi

a(s) =vbi(s), there is a vector zj such that ∇_zjv_ai(s)6=∇_zjv_bi(s).

Requires that agentj0s signal influences agenti’s

preferences between any object pairs, at least wheni is

indifferent (doesn’t require value interdependence to be

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General Setup

Connectedness.

Assumption 1(Interdependence)For any agents i,j, objects

a6=b, whenever vi

a(s) =vbi(s), there is a vector zj such that ∇_zjv_ai(s)=6 ∇_zjv_bi(s).

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General Setup

Connectedness.

Assumption 1(Interdependence)For any agents i,j, objects

a6=b, whenever vi

a(s) =vbi(s), there is a vector zj such that ∇_zjv_ai(s)=6 ∇_zjv_bi(s).

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Inefficiency of weakly Ex Post IC mechanisms;

Assumptions

Fix objectsa,b agentsi,j, and signal profiless−ij ∈S−ij.

Fork,k0∈ {a,b}, define

S_kkij 0(s−ij)≡the set of signal profiles(si,sj)for which

1. i ranksk first and the other object in{a,b}second, 2. j ranksk0 first and the other object in{a,b}second, and 3. all others rankaandbbelow any other objects.

Assumption 2(Rich Domain)There exist agents i,j, objects a,b, and signals of others s−ij such that S_kkij 0(s−ij)is non-empty

for all k,k0 ∈ {a,b}

The Rich Domain assumption can be satisfiedevenwhen

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Inefficiency of weakly Ex Post IC mechanisms;

Assumptions

Assumption 2(Rich Domain)There exist agents i,j, objects

a,b, and signals of others s−ij such that S_kkij 0(s−ij)is non-empty

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Inefficiency of weakly Ex Post IC mechanisms;

Assumptions

Assumption 2(Rich Domain)There exist agents i,j, objects

a,b, and signals of others s−ij such that S_kkij 0(s−ij)is non-empty

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Inefficiency of weakly Ex Post IC mechanisms;

Assumptions

Fork ∈ {a,b}, define

S_kij

_·

(s−ij)≡set of signal profiles(si_,_sj₎_{for which}

1. agentiranksk first and the other object in{a,b}second; 2. agentjranksaandbabove any other object;

3. all others ranka,bbelow any other object

Assumption 3(Connectedness)For some i,j, a,b, and s−ij that satisfy the Rich Domain assumption, and for some

k ∈ {a,b}, both S_kij

_·

(s−ij)and S

_·

ij_k(s−ij)are connected.

Weaker than convexity; satisfied if value functions are

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Inefficiency of weakly Ex Post IC mechanisms;

Assumptions

S

_·

ij_k(s−ij)≡set of signal profiles(si_,_sj₎_{for which}

1. agentiranksaandbabove any other object;

2. agentjranksk first and the other object in{a,b}second; 3. all others ranka,bbelow any other object

_·

(s−ij)and S

_·

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Inefficiency of weakly Ex Post IC mechanisms;

Assumptions

S

_·

(s−ij)and S

_·

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Inefficiency of weakly Ex Post IC mechanisms;

Assumptions

S

_·

(s−ij)and S

_·

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Inefficiency of weakly Ex Post IC mechanisms;

The Result

Theorem 1Under the assumptions of Interdependence, Rich

Domain, and Connectedness, there exists no mechanism that

is both Pareto efficient and weakly ex post incentive compatible.

The proof is done by finding a pair of agents and signals

such that a contradiction as in the previous 2-agent

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Inefficiency of weakly Ex Post IC mechanisms;

The Result

Domain, and Connectedness, there exists no mechanism that

is both Pareto efficient and weakly ex post incentive compatible.

The proof is done by finding a pair of agents and signals

such that a contradiction as in the previous 2-agent

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Impossibility of Ex Post Group Incentive Compatibility

A mechanismϕisex post group incentive compatibleif, for

any signal profiles, there exist no group of agentsN0and

their reported signals such that everyone in the group can

be made weakly better off, with at least one agent strictly.

We slightly modify some of the previous assumptions

(skipped): Rich Domain*, Connectedness*.

Domain*, and Connectedness*, ifϕis ex post group incentive

compatible, thenϕis constant across signals.

The (individual) ex post incentive compatibility is not

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Impossibility of Ex Post Group Incentive Compatibility

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Impossibility of Ex Post Group Incentive Compatibility

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Impossibility of Ex Post Group Incentive Compatibility

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Intuition for the result

Given any two signal profilessands˜, we construct a step-wise path,s=s0→s1→ · · · →sm = ˜s, such that each signal profile is associated with strict preferences and for eachk,

(1) sk andsk+1differ in the signal ofonly one agent, sayjk;

(2) betweensk andsk+1, ordinal preferences differ forat most

one agent, sayik 6=jk

Given (1) & (2), the ex post IC forjk implies

ϕjk₍_s

k) =ϕjk(sk+1)

Given this & (2), for anyi∈ {/ ik,jk}, the ex post group IC applied to{i,j_k}impliesϕi(s_k) =ϕi(s_k₊₁)

Lastly,ϕik₍_s

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Intuition for the result

ϕjk_(s

k) =ϕjk(sk+1)

Given this & (2), for anyi∈ {/ ik,jk}, the ex post group IC applied to{i,j_k}impliesϕi(s_k) =ϕi(s_k₊₁)

Lastly,ϕik₍_s

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Intuition for the result

ϕjk_(s

k) =ϕjk(sk+1)

Given this & (2), for anyi∈ {/ ik,jk}, the ex post group IC

applied to{i,j_k}impliesϕi(s_k) =ϕi(s_k₊₁)

Lastly,ϕik₍_s

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Intuition for the result

ϕjk_(s

k) =ϕjk(sk+1)

Given this & (2), for anyi∈ {/ ik,jk}, the ex post group IC

applied to{i,j_k}impliesϕi(s_k) =ϕi(s_k₊₁)

Lastly,ϕik_(s

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Bayesian Incentive Compatible Mechanisms

A mechanism isBayesian incentive compatibleif

truth-telling is a Bayesian Nash equilibrium

I _{That is, the truth-telling is a mutual best response for each} agentiknowingsi _only

Focus on the 2×2 case with single dimensional signals

I _{For each}_i=1,2,si _{is drawn from}_[0_,_1]_{following cdf}_Fi₍_·₎

I _{Assume that signals are independently distributed (though} correlated signals are fine as long as they are not too negatively correlated)

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Bayesian Incentive Compatible Mechanisms

A mechanism isBayesian incentive compatibleif

truth-telling is a Bayesian Nash equilibrium

I _{That is, the truth-telling is a mutual best response for each} agentiknowingsi _only

Focus on the 2×2 case with single dimensional signals

I _{For each}_i=1,2,si _{is drawn from}_[₀_,₁_]_{following cdf}_Fi₍_·₎

I _{Assume that signals are independently distributed (though} correlated signals are fine as long as they are not too negatively correlated)

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Consider the mechanism (denotedϕ∗) as in the figure: s1 s2 0 1 1 ¯ s2 ¯ s1 (1,0) (1,0) (1,0) (0,1) (0,1) (0,1)

(p,1−p)

(p0,1−p0)

I2

I1

E.g. (p,1−p)means that agent 1 (resp. 2) receives object

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Possibility of Efficiency with Bayesian IC

ϕ∗ is clearly Pareto efficient (if implemented as described)

Sufficient and necessary condition forϕ∗ to be Bayesian

IC:

For eachi=1,2, the “threshold type”¯si is indifferent

between reportingsi >¯si andˆsi <¯si

Theorem 3There exists a pairp,p0 ∈[0,1]that makesϕ∗

Bayesian incentive compatible, if and only if either

Z 1

0

u1(¯s1,s2)dF2(s2)≥0≥ Z 1

0

u2(s1,¯s2)dF1(s1), or

Z 1

0

u1(¯s1,s2)dF2(s2)≤0≤ Z 1

0

u2(s1,¯s2)dF1(s1)

This condition requires that the preferences of the two

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Possibility of Efficiency with Bayesian IC

Sufficient and necessary condition forϕ∗ to be Bayesian IC:

Z 1

0

u1(¯s1,s2)dF2(s2)≥0≥ Z 1

0

u2(s1,¯s2)dF1(s1), or

Z 1

0

u1(¯s1,s2)dF2(s2)≤0≤ Z 1

0

u2(s1,¯s2)dF1(s1)

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Possibility of Efficiency with Bayesian IC

Z 1

0

u1(¯s1,s2)dF2(s2)≥0≥

Z 1

0

u2(s1,¯s2)dF1(s1), or

Z 1

0

u1(¯s1,s2)dF2(s2)≤0≤

Z 1

0

u2(s1,¯s2)dF1(s1)

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Possibility of Efficiency with Bayesian IC

Z 1

0

u1(¯s1,s2)dF2(s2)≥0≥

Z 1

0

u2(s1,¯s2)dF1(s1), or

Z 1

0

u1(¯s1,s2)dF2(s2)≤0≤

Z 1

0

u2(s1,¯s2)dF1(s1)

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This condition is sufficient andnecessaryfor there to be a

Bayesian IC and Pareto efficient mechanism (not

necessarily of the form ofϕ∗ before) if we require the mechanism to beex post monotonicin the sense that

ϕi_a(·,sj)is non-decreasing for allsj

Open question: Generalization of the possibility result to

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This condition is sufficient andnecessaryfor there to be a

Bayesian IC and Pareto efficient mechanism (not

necessarily of the form ofϕ∗ before) if we require the mechanism to beex post monotonicin the sense that

ϕi_a(·,sj)is non-decreasing for allsj

Open question: Generalization of the possibility result to

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Concluding Remarks

This paper is (to our knowledge) the first to study the NTU

assignment problem in the interdependent values setup

I _{Impossibility results with ex post incentive compatibility} I _{Some possibility result with Bayesian incentive compatibility} I _{It may be important to pay attention to mechanisms that}

violate ex post IC but satisfy Bayesian IC if

interdependence of valuations exists; a sharp contrast to private-values setting.

Directions for future research

I _{Generalization of Bayesian IC mechanisms} I _{Comparative study of some practical assignment}

mechanisms

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Concluding Remarks

This paper is (to our knowledge) the first to study the NTU

assignment problem in the interdependent values setup

I _{Impossibility results with ex post incentive compatibility} I _{Some possibility result with Bayesian incentive compatibility} I _{It may be important to pay attention to mechanisms that}

violate ex post IC but satisfy Bayesian IC if

interdependence of valuations exists; a sharp contrast to private-values setting.

Directions for future research

I _{Generalization of Bayesian IC mechanisms} I _{Comparative study of some practical assignment}

mechanisms