Matching with Interdependent Values

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Matching with Interdependent Values

Fuhito Kojima1

April 23, 2013

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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 Elections: The “jury theorem”; selecting the “right” president. 5 _{Other examples?}

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Two-Sided Matching with Interdependent Values

Based on Chakraborty et al. (2009 JET).

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

simultaneously (don’t do it in economics!) → An acceptance of a paper for publication by a journal may make it easier for the author to publish it in a better one:

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One More Example

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

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The Game

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

Assume (a strong assumption) that student preferences are publicly known (and so students cannot misreport preferences); discussed later.

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.

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Stability

Definition

Given an information structure, matching mechanismµ isstable

under 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.

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Strong Stability

Does there exist a strongly stable mechanism?

Theorem

There does not necessarily exist a strongly stable matching mechanism.

As we will see, the impossibility is shown even when students have homogeneous preferences, i.e., all students have the same ranking over colleges.

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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 with ph if the student is a high type, and withpl <ph if she is a low type. For both colleges, utility for a student with qualityq isq (and 0 if unmatched).

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Step 1

Claim

Suppose that college 1 observes one H (say, for s1) and two Ls (say, for s2 and s3). Then the mechanism can match1 with the Ls (s2 or s3) only if college2’s signal about s1 is L and its signal about1’s match is H.

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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 whom 2received signal H is unmatched, then it should be the case that college1received H for s1 and L for s2.

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Step 3

Claim

Suppose 1 observes two H signals and one L signals,(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 with s₁ or s₂, then such a student is of HH

signals (see Step 1)

2 If 1 is matched with s3, then college 1 can rematch with the

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Step 4 (Final Step)

Claim

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) with}_s₁_or_s₂_,

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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.

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Proof (Example 2 of the paper)

Colleges 1 and 2, with one seat each.

Studentss1,s2.

1 _s₁ _{prefers 1 to 2 to}∅_,

2 s₂ prefers 2 to 1 to∅.→Heterogeneous preferences!

Signal structure

1 _{no uncertainty about student}_s₁_{’s quality.}

2 Students₂’s qualityq is 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,

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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 student s1 unmatched (why?) → This kills (s2,∅), and should produce only

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

with positive probability.

Further, the mechanism

1 _{cannot match 2 to}_s₂ _if_q₌−_{2 (why?),} 2 _{cannot leave 2 unmatched if}_q_{= 2 (why?).}

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

1 ₍_s₁_,_s₂_{) only if}_q_{= 2,} 2 ₍_s₁_,∅_{) only if}_q₌−_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.

Then college 1 has strict incentives to rematch withs1(why?), a

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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).

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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 student based only on

its signal and 1’s matching,

3 and so on.

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Different Versions of Serial Dictatorships

There are multiple weakly stable mechanisms: different versions of serial dictatorship. For example,

1 college 1 receives its favorite student based only on its signal, 2 college 2 receives its favorite remaining studentbased only on

1’s and its own signal,

3 and so on.

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A “Detail Free” Implementation of SD

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

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Revelation Principle

The analysis has been restricted to direct revelation mechanisms (except for some of the serial dictatorship implementations). Is it without loss of generality?

Myerson’s original result is not applicable.

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Further Questions

For weak stability, is the serial dictatorships “the only stable mechanisms”? There are versions of serial dictatorship mechanisms under interdependent values, with differing outcomes. 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.

1 When is such an assumption realistic? Labor markets? School

choice (appeals process)?

2 What if students as well as colleges have interdependent