Incentive compatibility in 2-sided matching markets

1. Incentive compatibility in 2-sided matching markets Mohammad Mahdian Yahoo! Research Based on joint work with Nicole Immorlica

2. Centralized matching markets • Many examples: • certain job markets • match-making markets • auction houses • kidney exchange markets • Netflix DVD rental market • … • The objective of the “center” is to find a matching that is optimal from individuals’ perspective.

3. Stable Marriage • Consider a set of nwomen and nmen. • Each person has an ordered list of some members of the opposite sex as his or her preference list. • Let µ be a matching between women and men. • A pair (m, w) is a blocking pair if both m and w prefer being together to their assignments under µ. Also, (x, x) is a blocking pair, if x prefers being single to his/her assignment under µ. • A matching is stable if it does not have any blocking pair.

4. Example Lucy Peppermint Marcie Sally Charlie Linus Schroeder Franklin Stable!

5. Deferred Acceptance Algorithms (Gale and Shapley, 1962) • In each iteration, an unmarried man proposes to the first woman on his list that he hasn’t proposed to yet. • A woman who receives a proposal that she prefers to her current assignment accepts it and rejects her current assignment. This is called the men-proposing algorithm.

6. Example Lucy Peppermint Marcie Sally Charlie Linus Schroeder Franklin Stable!

7. Classical Results • Theorem 1. The order of proposals does not affect the stable matching produced by the men-proposing algorithm. • Theorem 2. 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 3. In all stable matchings, the set of people who remain single is the same.

8. Applications of stable matching • Stable marriage algorithm has applications in the design of centralized two-sided markets. For example: • National Residency Matching Program (NRMP) since 1950’s • Dental residencies and medical specialties in the US, Canada, and parts of the UK. • New York school match • National university entrance exam in Iran • Placement of Canadian lawyers in Ontario and Alberta • Sorority rush • Matching of new reform rabbis to their first congregation • …

9. Incentive Compatibility • Question: 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.

10. Example Lucy Peppermint Marcie Sally Charlie Linus Schroeder Franklin Stable!

11. Incentive Compatibility • Next 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.

12. However, data from NRMP show that the chance that a participant can benefit from lying is slim.

13. Number of applicants who could lie can be computed using the following theorem. 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.

14. 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.

15. A Probabilistic Model • Men choose preference lists uniformly at random from lists of at most kwomen. • 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.

16. Our Results • Theorem. Even allowing women arbitrarypreference lists in the probabilistic model, the expected fraction of women who have more than one stable husband tends to zero.

17. Economic Implications • Corollary 1. When other players are truthful, almost surely a given player’s 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.

18. Structure of proof • Step 1: An algorithm that counts the number of stable husbands of a given woman. • Step 2: Bounding the probability of having more than one stable husband in terms of the number of singles • Step 3: Bounding the number of singles by the solution of the occupancy problem.

19. Step 1:Finding stable husbands ofg • Use men-proposing algorithm to find a stable matching. • Whenever the algorithm finds a stable matching, have g divorce her husband and continue the men-proposing algorithm (but now g has a higher standard for accepting new proposals). • Terminate when either • a man who is married in the men-optimal matching runs through his list, or • a woman who is single in the men-optimal matching receives a proposal.

20. Question: If each woman has an arbitrary complete preference list, and each man has a random list of k women, what is the probability that this algorithm returns more than one stable husband for g? • The main tool that we will use to answer this question is the principle of deferred decisions: Men do not pick the list of their favorite women in advance; Instead, every time a man needs to propose, she picks a woman at random and proposes to her. A man remains single if he gets rejected by k different women.

21. Stable! End! Lucy Peppermint Marcie Sally Charlie Linus Schroeder Franklin

22. Step 2: Bounding the probability • Consider the moment when the algorithm finds the first (i.e., men-optimal) matching. Call this matching μ. • Let A denote the set of women who are single in μ, and X denote |A| . • Fix random choices before the algorithm finds μ, and let probabilities be over random choices that are made after that.

23. Step 2, cont’d. • Look at the sequence of women who receive a proposal. • The probability that the algorithm finds another stable husband for g is bounded by the probability that g comes before all members of A in this sequence. This probability is 1/(X+1). • Therefore, the probability that g has more than one stable husband is at most

24. Step 3:Number of singles • We need to compute E[1/(X+1)], where X is the number of singles in the men-optimal matching. • Simple Observation: The probability that a woman remains single is at least the probability that she is never named by men.

25. Step 3, cont’d. • Let Ym,n denote the number of empty bins in an experiment where m balls are dropped independently and uniformly at random in n bins. • Lemma. • Proof Sketch: Assume (without loss of generality!) that men are amnesiacs and might propose to a woman twice. The total number of proposals (bins) is at most (k+1)n w.h.p.

26. The occupancy problem • Lemma. • Proof sketch: • Use the principle of inclusion and exclusion to compute E[1/(Ym,n+1)] as a summation. • Compare this summation to another (known) summation term-by-term.

27. Putting it all together… Theorem. In the model where women have arbitrary complete preference lists and men have random lists of size k, the probability that a fixed woman has more than one stable husband is at most

28. Generalizations • More general classes of distributions: • Arbitrary non-uniform distribution instead of the uniform distribution: still we can prove that the probability tends to zero. • Many-to-one matchings: • [Kojima & Pathak]: result generalizes.

29. Open Questions • Stable matching with couples: Why has the NRMP algorithm found a matching every year? • Restricting to complete preference lists: There are similar observations about the probability that a participant can benefit from lying. (Teo, Sethuraman, Tan, 2001)