marriage, honesty, & stability

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Marriage, Honesty, & Stability. Nicole Immorlica , Northwestern University. Math & Romance. Love, marriage, & bipartite graphs. t he boys. Twiggy. Jake. Jake > Elwood > Curtis > Ray. Holly > Claire > Twiggy > Jill. Claire. Elwood. Marriage. Love. - PowerPoint PPT Presentation

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Marriage, Honesty, & Stability Nicole Immorlica, Northwestern UniversityMath &RomanceLove, marriage, & bipartite graphsthe boysand girlsJakeElwoodCurtisRayTwiggyClaireJillHollyHolly > Claire > Twiggy > JillClaire > Jill > Twiggy > HollyTwiggy > Jill > Holly > ClaireHolly > Claire > Twiggy > JillJake > Elwood > Curtis > RayJake > Curtis > Elwood > RayRay > Curtis > Elwood > JakeRay > Jake > Elwood > CurtisLoveMarriage

3A pair (m,w) is a blocking pair for a matching if m prefers w to his match and w prefers m to her match.When is marriage stable?blocking pairJakeElwoodCurtisRayTwiggyClaireJillHollyHolly > Claire > Twiggy > JillClaire > Jill > Twiggy > HollyTwiggy > Jill > Holly > ClaireHolly > Claire > Twiggy > JillJake > Elwood > Curtis > RayJake > Curtis > Elwood > RayRay > Curtis > Elwood > JakeRay > Jake > Elwood > Curtis4When is marriage stable?blocking pairA pair (m,w) is a blocking pair for a matching if m prefers w to his match and w prefers m to her match.stabilityA matching is stable if there is no blocking pair.The courtship ritualJakeElwoodCurtisRayHolly > Claire > Twiggy > JillClaire > Jill > Twiggy > HollyTwiggy > Jill > Holly > ClaireHolly > Claire > Twiggy > JillTwiggyClaireJillHollyJake > Elwood > Curtis > RayJake > Curtis > Elwood > RayRay > Curtis > Elwood > JakeRay > Jake > Elwood > CurtisWhats math say about romance?Question 1. Do stable matchings always exist?Question 2. Are they easy to find?Question 3. Does the courtship ritual work?Question 4. Are stable matchings unique?Question 5. If not, who benefits?What would Cupid prove?Theorem. Courtship algorithm terminates.Theorem. Resulting marriage is stable.Corollary. Stable marriages always exist and are easy to find.

Well prove this using potential functions.Step 1: proof of terminationTheorem.The courtship ritual terminates.Proof.Define a potential function: the number of names on boys lists.Note this is strictly decreasing and always non-negative.Step 2a: defining invariantThe girls options only ever improve!Invariant.For every girl G and boy B, if G is crossed off Bs list it is because shes married to him or has a suitor she prefers. because B only crosses off G after proposing, and G only rejects proposals when she finds someone better.Step 2b: proof of stabilityTheorem.The resulting matching is stable.Proof.Consider boy B and girl G that are not married to each other.Suppose G was crossed off Bs list. Then G prefers husband to B, so wont elope with B.Suppose G is on Bs list. Then B didnt propose to G yet, so B prefers wife to G, so wont elope with G.What can math say on romance?Question 1. Do stable matchings always exist?Question 2. Are they easy to find?Question 3. Does the courtship ritual work?

Question 4. Are stable matchings unique????Question 5. If not, who benefits????Yes, yes, yes.Recall JakeElwoodCurtisRayHolly > Claire > Twiggy > JillClaire > Jill > Twiggy > HollyTwiggy > Jill > Holly > ClaireHolly > Claire > Twiggy > JillTwiggyClaireJillHollyJake > Elwood > Curtis > RayJake > Curtis > Elwood > RayRay > Curtis > Elwood > JakeRay > Jake > Elwood > CurtisIn an alternate universeJakeElwoodCurtisRayHolly > Claire > Twiggy > JillClaire > Jill > Twiggy > HollyTwiggy > Jill > Holly > ClaireHolly > Claire > Twiggy > JillTwiggyClaireJillHollyJake > Elwood > Curtis > RayJake > Curtis > Elwood > RayRay > Curtis > Elwood > JakeRay > Jake > Elwood > CurtisConclusion: not uniqueJakeElwoodCurtisRayTwiggyClaireJillHollyHolly > Claire > Twiggy > JillClaire > Jill > Twiggy > HollyTwiggy > Jill > Holly > ClaireHolly > Claire > Twiggy > JillJake > Elwood > Curtis > RayJake > Curtis > Elwood > RayRay > Curtis > Elwood > JakeRay > Jake > Elwood > CurtisStable spousesIn general, there are many stable marriages.Boy BGirl GGirl GA person P is a stable spouse of a person P if P is married to P in some stable matching.stable spouseBoys are happier than girlsTheorem.Boys marry their most preferred stable wife. and girls get their least preferred stable husband!wayProof.Proof.Of 1st claim, by contradiction. Suppose some boy isnt married to favorite stable spouse.He must have proposed to her and been refused.Let B be 1st boy to lose his favorite stable wife G.Then G must have had a proposal from a boy B she preferred to B.Since B has not yet crossed off his favorite stable wife, B must love G more than any stable wife.But then B and G will elope in marriage which matches B to G, contradicting stability of wife G.Proof.Of 2nd claim, by contradiction. Suppose there is a stable matching in which girl G gets worse husband.Let B be her husband when boys propose and B be her husband in worse matching.Then G prefers B to B by assumption.Furthermore, by 1st claim, G is favorite stable wife of B, so B prefers G to wife in worse matching.But then B and G will elope in worse matching.But of course symmetryIf girls propose, then they will get their favorite stable husbands.Applications of Stable MatchingCentralized two-sided markets.

National Residency Matching Program (NRMP) since 1950s Dental residencies and medical specialties in the US, Canada, and parts of the UKNational university entrance exam in Iran Placement of Canadian lawyers in Ontario and Alberta Sorority rush Matching of new reform rabbis to their first congregation Assignment of students to high-schools in NYC

Theorem 1. The matching produced by the men-proposing algorithm is the best stable matching for men and the worst stable matching for women. This matching is called the men-optimal matching.

Theorem 2. The order of proposals does not affect the stable matching produced by the men-proposing algorithm.

Theorem 3. In all stable matchings, the set of people who remain single is the same. Classical Results NRMPNational Residency Matching ProgramAfter finishing med school in the US, students must complete internships at hospitals prior to receiving their degreeStudents interview at hospitals and rank their preferencesHospitals rank students they interviewed A hospital-optimal deferred acceptance algorithm determines placementQuestion. Do participants have an incentive to announce a list other than their real preference lists?

Answer. Yes! In the men-proposing algorithm, sometimes women have an incentive to be dishonest about their preferences.Incentive Compatibility Recall JakeElwoodCurtisRayHolly > Claire > Twiggy > JillClaire > Jill > Twiggy > HollyTwiggy > Jill > Holly > ClaireHolly > Claire > Twiggy > JillTwiggyClaireJillHollyJake > Elwood > Curtis > RayJake > Curtis > Elwood > RayRay > Curtis > Elwood > JakeRay > Jake > Elwood > CurtisA small white lieJakeElwoodCurtisRayHolly > Claire > Twiggy > JillClaire > Jill > Twiggy > HollyTwiggy > Jill > Holly > ClaireHolly > Claire > Twiggy > JillTwiggyClaireJillHollyJake > Elwood > Curtis > RayJake > Curtis > Elwood > RayRay > Curtis > Elwood > JakeRay > Jake > Elwood > CurtisNext Question. Is there any truthful mechanism for the stable matching problem?

Answer. No! Roth (1982) proved that there is no mechanism for the stable marriage problem in which truth-telling is the dominant strategy for both men and women. Incentive Compatibility Next Question. When can people benefit from lying?

Theorem. The best match a woman can receive from a stable mechanism is her optimal stable husband with respect to her true preference list and others announced preference lists.

In particular, a woman can benefit from lying only if she has more than one stable husband. Incentive Compatibility Data from NRMP show that the chance that a participant can benefit from lying is slim.1993 1994 1995 1996 # applicants 20916 22353 22937 24749 # positions 22737 22801 2280622578 # applicants who could lie 1620 14 21 Explanations (Roth and Peranson, 1999)The following limit the number of stable husbands of women:

Preference lists are correlated. Applicants agree on which hospitals are most prestigious; hospitals agree on which applicants are most promising. If all men have the same preference list, then everybody has a unique stable partner, whereas if preference lists are independent random permutations almost every person has more than one stable partner. (Knuth et al., 1990)

Preference lists are short.Applicants typically list around 15 hospitals. A Probabilistic Model Men. Choose preference lists uniformly at random from lists of at most k women.Women. Randomly rank men that list them.

Conjecture (Roth and Peranson, 1999). Holding k constant as n tends to infinity, the fraction of women who have more than one stable husband tends to zero. Our Results Theorem. Even allowing women arbitrary preference lists in the probabilistic model, the expected fraction of women who have more than one stable husband tends to zero.Economic Implications Corollary 1. When other players are truthful, almost surely a random players best strategy is to tell the truth.

Corollary 2. The stable marriage game has an equilibrium in which in expectation a (1-o(1)) fraction of the players are truthful.

Corollary 3. In stable marriage game with incomplete information there is a (1+o(1))-approximate Bayesian Nash equilibrium in which everybody tells the truth.

Structure of proof Step 1. An algorithm that counts the number of stable husbands of a