Matching with Interdependent Values

Matching with Interdependent Values

Fuhito Kojima1

April 15, 2015

1

Interdependent Values

Today’s topic is interdependence in matching theory and mechanism design. Main Papers today are:

1 _{“Two-Sided Matching with Interdependent Values,”}

Archishman Chakraborty, Alessandro Citanna and Michael Ostrovsky,Journal of Economic Theory.

2 _{”Efficient Assignment under Interdependent Values,”}

Interdependent Values

Interdependent values: your utility depends not only on your own private information but also on others’.

Interdependent values seem important in many economic situations.

1 _{A painting may be real or a fake, and someone might know}

the answer.

2 Oil drilling companies bidding for a tract of land, when the

amount of oil is unknown.

3 _{Hiring an assistant professor.}

4 _{Schools; information of other parents may be relevant.} 5 _{Elections: The “jury theorem”; selecting the “right”}

president.

6 _{Other examples?}

Two-Sided Matching with Interdependent Values

More specifically, interdependence seems important in the context of two-sided matching:

1 _{Top law schools don’t hire a new professor until she has}

received offers from other law schools.

2 _{Law professors submit papers to multiple journals}

One More Example

Dugatkin (1992) “Sexual selection and imitation: Females copy the mate choice of others” about mate choice among

Trinidadian guppies.

The Game

There are students and colleges (for simplicity, assume each college has one seat).

Assume that student preferences are publicly known (and so students cannot misreport preferences); a strong assumption, but seems necessary...

1 _{For each student, nature selects her quality.}

2 _{Each college receives a signal about the quality profile of} students.

3 _{All colleges report a signal to the mechanism (colleges can} misreport their signals).

4 _{Based on reported signals, the mechanism generates a} (possibly a distribution over) matching.

5 _{Each agent observes her “information” (like their own} matching; varies across models, will specify later). 6 _{Colleges simultaneously decide to do one of the following:}

1 _{Do nothing.}

2 _{Unilaterally drop its matched student.} 3 Make an offer to a student to rematch.

Stability

Definition

Given an information structure, matching mechanismµis

stableunder that information structure if

1 _{it never matches a student to an unacceptable college, and}

2 _{there exists a perfect Bayesian equilibrium of the game in}

which, on the equilibrium path, all colleges report true signals and accept their assigned match.

Note that stability concept should specify the information structure because of interdependent values.

A mechanism isstrongly stablewhen it is stable in the case

when everyone sees the entire matchings.

A mechanism isweakly stablewhen it is stable in the case

when everyone sees only their own matchings.

Information Structure

Theorem

If a mechanism is stable under some information structure, then it is also stable for every coarser information structure.

Theorem

Strong Stability

Does there exist a strongly stable mechanism?

Theorem

There does not necessarily exist a strongly stable matching mechanism.

The impossibility is proven even when students have homogeneous preferences, i.e., all students have the same ranking over colleges.

Proof (Example 1 of the paper)

Proof strategy: Find a counterexample. Consider necessary conditions for strong stability until we find a contradiction.

Two colleges, 1 and 2 with one seat each.

Three students,s1,s2,s3. Each student is either a low type ql >0 or high typeqh>ql, both with positive probability (and

independently).

Signal: each college receives (independently) signalH withph

if the student is a high type, and withpl <phif she is a low type.

For both colleges, utility for a student with qualityqisq (and 0 if unmatched).

Step 1

Claim

Suppose that college 1 observes one H (say, for s1) and two Ls (say, for s2and s3). Then the mechanism can match1with the Ls (s2or s3) only if college2’s signal about s1is L and its signal about1’s match is H.

This is because, if not, college 1 can always rematch withs1 and strictly benefit

Step 2

Claim

If college 2 is matched to a student (say s1) for whom 2 received signal L while another student (say s2) for whom2 received signal H is unmatched, then it should be the case that college1received H for s1and L for s2.

Step 3

Claim

Suppose 1 observes two H signals and one L signal,(H,H,L). Then the mechanism must match 1 to a student whose signal is HH whenever there is an HH student.

To show this, it suffices to show that college 1 can always guarantee to match with an HH student when there is one.

Consider college 1’s misreport by “inverting,”(L,L,H). Then,

1 If 1 is matched withs₁ors₂, then such a student is of HH

signals (see Step 1)

2 _{If 1 is matched withs}

Step 4 (Final Step)

Suppose 2 observes two H signals and one L signal,(H,H,L). Then it is not optimal for college 2 to report its signals truthfully and accept its assigned match.

Proof approach: A pretty exhaustive search (as far as I can tell): Use Steps 1-3 to decide all the possibilities.

A basic idea:

1 _{Suppose 2 receives signal}(H,H,L).

2 _{Then consider misreport of 2 by inverting the signal,}

(L,L,H); seems like an intuitive way to misreport!

3 _{Then it will allow 2 to mach (after rematching) withs1ors2,}

Weak Stability

Can we restore existence of stable matchings if information is more limited?

Remark: Less information makes stability easier (an intuitive result: Theorem 1 of the paper).

But Theorem

There does not necessarily exist a weakly stable matching mechanism.

Remark: Weak stability seems the weakest possible stability with respect to information.

Proof (Example 2 of the paper)

Colleges 1 and 2, with one seat each.

Studentss1,s2.

1 _s

1prefers 1 to 2 to∅,

2 _s

2prefers 2 to 1 to∅.→Heterogeneous preferences!

Signal structure

1 _{no uncertainty about students}

1’s quality.

2 _Students

2’s qualityqis either 2 or−2, each with probability 1/2.

3 _{College 1 receives an uninformative signal, while 2}

receives a perfectly informative signal (i.e., knowsq).

Utility for colleges are given by

u1(s1;q) =1, u2(s1;q) =3,

Proof

Proof strategy: Consider necessary conditions for weak stability until we find a contradiction.

Denote a matching by (1’s match, 2’s match), like(s1,s2).

First, note that the mechanism can never leave 1 unmatched (why?)→Should produce only

(s1,s2),(s1,∅),(s2,s1),(s2,∅)

with positive probability.

Further, the mechanism can never leave both college 2 and students1unmatched (why?) →This kills(s2,∅), and should produce only

Keep in mind that only(s1,s2),(s1,∅),(s2,s1)can be produced

with positive probability.

Further, the mechanism

1 _{cannot match 2 tos}

2ifq=−2 (why?), 2 _{cannot leave 2 unmatched ifq=2 (why?).}

→As(s1,s2),(s1,∅),(s2,s1)are the only possibility, can

produce

1 _(s

1,s2)only ifq=2, 2 _(s

1,∅)only ifq=−2.

Suppose(s1,∅)is produced with positive probability.→college 1, upon seeing it being matched tos1, can make a rematching offer tos2(this is the key point of the proof: why is this move profitable for 1?)→ifq=−2, then(s2,s1)is produced with probability one.

Ifq=2, then either(s1,s2)or(s2,s1)should be produced from the last paragraph.→But if(s1,s2)is produced with positive probability, then 2 has strict incentive to reportqto be−2 and produce(s2,s1)(from last paragraph).→So the mechanism should always produce(s2,s1), regardless ofq.

Remarks

In the proof, the combination manipulation is not used. →Impossibility holds even if the stability requirement is weakend, prohibiting combined manipulations.

Student preferences are heterogeneous (unlike the example for strong stability).

A Possibility Result

Suppose student preferences are homogeneous: college 1 best, 2 second, and so on.

Theorem

If student preferences are homogeneous, then there is a weakly stable matching mechanism in that market.

Note that strong stability was impossible even under homogeneous student preferences.

Proof: consider theserial dictatorship:

1 _{college 1 receives its favorite studentbased only on its}

signal,

2 _{college 2 receives its favorite remaining studentbased only}

on its signal and 1’s matching,

3 _{and so on.}

Serial dictatorship is a weakly stable mechanism when student preferences are homogeneous (why is that?).

A “Detail Free” Implementation of SD

A serial dictatorship can be implemented in a “detail-free” manner (c.f. Wilson 1987)

Further Questions

For weak stability, is the serial dictatorships “the only stable mechanisms”? Anything else?

Existence under strong stability seems hopeless. But is there any sufficient condition for strongly stable mechanisms?

Any condition under which a weakly stable matching exists even when student preferences are heterogeneous?

Issues with the assumption that student preferences are publicly known.

Is there any “continuity” with the complete information case?

1 _{When is such an assumption realistic? Labor markets?}

School choice (appeals process)?

2 _{What if students as well as colleges have interdependent}

values?

Group incentive compatibility (Chakraborty and Citanna 2012)

One-Sided Matching: Ex Post Incentive Compatibility

Che, Kim, and Kojima (2013); one-sided matching (object allocation).

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

Best alternative to strategy-proofness in the interdependent values environment

Main Findings

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

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)≡vi

a(s)−vbi(s)bei

0_{utility difference betweena}

andb, and assume ∂ui(s)

∂si >

∂u−i(s)

∂si >0,∀s ∈[0,1]

Illustrative Example: Inefficiency of Ex Post IC

mechanisms

0 1

1

A B

C D

E

Sbb

Sab

Sba

Saa

s1 s2

I2

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

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 ranksk}0 _{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}

Inefficiency of weakly Ex Post IC mechanisms;

Assumptions

Fork ∈ {a,b}, define

S_kij

_·

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

_·

_·

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.

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*.

Theorem 2Under the assumptions of Interdependence, Rich

Domain*, and Connectedness*, ifϕis ex post group incentive compatible, thenϕis constant across signals.

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,jk}impliesϕi(sk) =ϕi(sk+1) Lastly,ϕik_(s

Bayesian Incentive Compatible Mechanisms

A mechanism isBayesian incentive compatibleif truth-telling is a Bayesian Nash equilibrium

That is, the truth-telling is a mutual best response for each

agentiknowingsi only

Focus on the 2×2 case with single dimensional signals

For eachi=1,2,si _{is drawn from[0,1]following cdfF}i_(·)

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

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

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 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 alls}j

Related Literature and Conclusion

Implementing desirable outcomes are difficult in the interdependent values setup

Impossibility of stability (Chakraborty, Citanna, and Ostrovsky)

Impossibility results with ex post incentive compatibility (Che, Kim, and Kojima)

Some possibility results

Impossibility results with mechanism designwith money

Jehiel & Moldovanu (01): Impossibility of efficient and Bayesian incentive compatible mechanism

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

Directions for future research

Generalization of Bayesian IC mechanisms for object allocation

Comparative study of some practical assignment mechanisms