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Random m atching markets

Random m atching markets. Itai Ashlagi. Stable Matchings = Core Allocations. A matching is stable if there is no man and woman who both prefers each other over their current match.

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Random m atching markets

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  1. Random matching markets Itai Ashlagi

  2. Stable Matchings = Core Allocations • A matching is stable if there is no man and woman who both prefers each other over their current match. • The set of stable matchings is a non-empty lattice, whose extreme points are the Men Optimal Stable Match (MOSM) and the Women Optimal Stable Match (WOSM) • Men are matched to their most preferred stable woman under the MOSM and their least preferred stable woman under the WOSM

  3. Multiplicity • Core has a lattice structure and can be large (Knuth) • Roth, Peranson (1999) – small core in the NRMP • Hitsch, Hortascu, Ariely (2010) – small core in online dating • Banerjee et al. (2009) – small core in Indian marriage markets

  4. Random matching markets • Random Matching Market: A set of men and a set of women • Man has complete preferences over women, drawn randomly at uniform iid. • Woman has complete preferences over men, drawn randomly at uniform iid.

  5. Questions • Average rank, or to whom should agents expect to match? • Define men’s average rank of their wives under excluding unmatched men from the average • Average rank is if all men got their most preferred wife, higher rank is worse. • How many agents have multiple stable assignments? • Agents can manipulate iff they have multiple stable partners

  6. Large core in random balanced markets Theorem[Pittel 1998, Knuth, Motwani, Pittel 1990] Consider a random market with men and women. 1. With high probability, the average ranks in the MOSM are and 2. The fraction of agents that have multiple stable partners tends to 1 as tends to infinity. Suppose there are n+1 men and n women and the

  7. Random markets with short preference lists have a small core Roth, Peranson (1999) – find a small core in the NRMP Theorem[Immorlica, Mahdian 2005]: Consider a market with n men and n women where men have short(constant length) uniform at random preference lists and women have arbitrary preferences. Then of men and women have multiple stable matches. Kojima, Pathak 2009 show that there is a limited scope for manipulating a stable mechanism in such many-to-one large markets (each hospital has a constant number of positions).

  8. Literature (multiplicity) • Pittel(1989), Knuth, Motwani, Pittel (1990), Roth, Peranson (1999) – when there are equal number of men and women • Under MOSM men’s average rank of wives is , but it is under the WOSM • The core is large – most agents have multiple stable partners • Roth, Peranson (1999) – small core in the NRMP • Immorlica, Mahdian (2005) and Kojima, Pathak (2009) show that if one side has short random preference lists the core is small • Many (popular) agents are unmatched • Hitsch, Hortascu, Ariely (2010) – small core in online dating • Banerjee et al. (2009) – small core in Indian marriage markets

  9. Literature • Hitsch, Hortascu, Ariely(2010) – online dating • Banerjee, Duflo, Ghatak, Lafortune (2009) - Indian marriage markets • Abdulkadiroglu, Pathak, Roth (2005) – NYC school choice All use Deferred Acceptance (Gale & Shapely) to make predictions… • Crawford (1991) – comparative statics on adding men (women), but only in a given stable matching.

  10. Large core in balanced markets Theorem[Pittel 1998, Knuth, Motwani, Pittel 1990] Consider a random market with men and women. 1. With high probability, the average ranks in the MOSM are and 2. The fraction of agents that have multiple stable partners tends to 1 as tends to infinity. Question: Suppose there are n+1 men and n women and men are proposing. What is the likely average men’s rank of their wives?” [Gil Kalai’s blog]

  11. Matching markets are very competitive When there are unequal number of men and women: • The short side is much better off under all stable matchings;roughly, the short side “chooses” and the long side gets “chosen” • sharp effect of competition despite heterogeneity • The core is small;there is little difference between the MOSM and the WOSM • Small core despite long lists and uncorrelated preferences Question: Are there any real/natural matching markets with large cores?

  12. Men’s average rank of wives,

  13. Men’s average rank of wives,

  14. Percent of matched men with multiple stable partners

  15. Theorem [Ashlagi, Kanoria, Leshno 2013] Consider a random market with men and women,for . With high probability in any stable matching and Moreover, And of men and women have multiple stable matches.

  16. That is, with high probability under all stable matchings • Men do almost as well as they would if they choose in a random order, ignoring women’s preferences. • Women are either unmatched or roughly getting a randomly assigned man. • The core is small

  17. Corollary 1: One women makes a difference In a random market with men and women, with high probability and in all stable matches, and a vanishing fraction of agents have multiple stable partners. (n=1000, log n= 6.9, n/logn = 145) (n=100000,log n =11.5, n/logn =8695)

  18. Corollary 2: Large Unbalance Consider a random market with men and women for . Then for we have that, with high probability, in all stable matchings and Ashlagi, Braverman, Hassidim (2011) also establish a small core in this setting.

  19. Strategic implications • Men proposing DA (MPDA) is strategyproof for men, but no stable mechanism is strategyproof for all agents. • A woman can manipulate MPDA only if she has multiple stable husbands • Misreport truncated preferences. • In unbalanced matching market a diminishing number of woman have multiple stable husbands • Under full information, a diminishing fraction can manipulate • At the interim, under mild assumptions on utilities, all agents reporting truthfully is an -Nash

  20. Intuition • In a competitive assignment market with homogenous buyers and homogenous sellers the core is large, but the core shrinks when there is one extra seller. • In a matching market the addition of an extra woman makes all the men better off • Every man has the option of matching with the single woman • But only some men like the single woman • Changing the allocation of some men requires changing the allocation of many men. If some men are made betteroff, some women are made worse off creating more options for men. All man benefit, and the core is small.

  21. Proof overview Calculate the WOSM using: • Algorithm 1: Men-proposing Deferred Acceptance gives MOSM • Algorithm 2: MOSM→WOSM Both algorithms use a sequence of proposals by men Stochastic analysis by sequential revelation of preferences.

  22. Algorithm 1: Men-proposing DA (Gale & Shapley) Everyone starts unmatched. Each man one at a time, begins a ‘chain’. Chain beginning with : • Set to be the proposer. • The proposer proposes to his (next) most preferred woman • If is unmatched, end chain and start a new one with • Otherwise, rejects her less preferred man between and her current partner. Repeat, with rejected man proposing.

  23. Algorithm 2: MOSM → WOSM We look for stable improvement cycles for women. We iterate: Phase:For candidate woman , reject her match starting a chain. Two possibilities for how the chain ends: • (Improvement phase) Chain reaches new stable match. • (Terminal phase) Chain ends with unmatched woman is ’s best stable match. is no longer a candidate.

  24. Algorithm 2: MOSM → WOSM Initialize • Choose in if non-empty. • Phase: Record the current match as . Woman rejects her partner, man , starting a chain where accept a proposal only if the proposal is preferred to . • Two possibilities for how the chain ends: • (Improvement phase) If the chain ends with acceptance by , we have found a new stable match. Return to Step 2. • (Terminal phase) Else the chain ends with acceptance by . Woman has found her best stable partner. Roll the match back to . Add to S and return to Step1.

  25. Illustration of Algorithm 2: MOSM → WOSM prefers to rejects to start a chain

  26. Illustration of Algorithm 2: MOSM → WOSM prefers to

  27. Illustration of Algorithm 2: MOSM → WOSM

  28. Illustration of Algorithm 2: MOSM → WOSM prefers to

  29. Illustration of Algorithm 2: MOSM → WOSM New stable match found (Convince yourself that there is no blocking pair), Update match and continue.

  30. Illustration of Algorithm 2: MOSM → WOSM

  31. Illustration of Algorithm 2: MOSM → WOSM

  32. Illustration of Algorithm 2: MOSM → WOSM prefers to rejects to start a chain

  33. Illustration of Algorithm 2: MOSM → WOSM prefers to

  34. Illustration of Algorithm 2: MOSM → WOSM ends with a proposal to unmatched woman is ’s best stable partnerW and

  35. Illustration of Algorithm 2: MOSM → WOSM ends with a proposal to unmatched woman is ’s best stable partnerand similarly and already had their best stable partner

  36. Illustration of Algorithm 2: MOSM → WOSM ends with a proposal to unmatched woman is ’s best stable partner and similarly and already had their best stable partner

  37. Illustration of Algorithm 2: MOSM → WOSM

  38. Illustration of Algorithm 2: MOSM → WOSM rejects to start a new chain

  39. Illustration of Algorithm 2: MOSM → WOSM prefers to But is already matched to her best stable partner

  40. Illustration of Algorithm 2: MOSM → WOSM prefers to But is already matched to her best stable partner is ’s best stable partner

  41. Illustration of Algorithm 2: MOSM → WOSM prefers to But is already matched to her best stable partner is ’s best stable partner

  42. Algorithm 2: MOSM → WOSM Initialize • Choose in if non-empty. • Phase: Record the current match as . Woman rejects her partner, man , starting a chainwhere accepts a proposal only if the proposal is preferred to . • Two possibilities for how the chain ends: • (Improvement phase) If the chain ends with acceptance by , we have found a new stable match. Return to Step 2. • (Terminal phase) Else the chain ends with acceptance by . Woman has found her best stable partner. Roll the match back to . Add to S and return to Step1.

  43. Algorithm 2: MOSM → WOSM Initialize (S set of women matched to best stable partners) • Choose in if non-empty. • Phase: Record the current match as . Woman rejects her partner, man , starting a chain. A proposal is accepted if the woman prefers the proposer to her match under . • Two possibilities: • (Improvement cycle) If there is an acceptance by a woman in the chain, we have found a new stable match. Update , erase the women with new stable matches from the chain and continue. • (Terminal phase) Else the chain ends with acceptance by . Woman has found her best stable partner. Roll the match back to . Add and all women who accepted proposals in this chain to S and return to Step 1.

  44. Proof idea: • Analysis of MPDA is similar to that of Pittel (1989) • Coupon collectors problem • Analysis of Algorithm 2: MOSM → WOSM more involved. • S grows quickly (set of woman that are already matched to best stable partner) • Once S is large improvement phases are rare • Together, in a typical market, very few agents participate in improvement cycles.

  45. Why does the set S (women matched to best stable partners) grows quickly? • After finding the MOSM, men haven’t made and women haven’t received many proposals. • Proposals to or with probability ~1/n • => After eliminating cycles first terminal phase will include women!

  46. Further questions • Can we allow correlation in preferences? • Perfect correlation leads to a unique core. • But “short side” depends on more than • Tiered market: 15 men, 20 women: 10 top, 10 mid • What is a general balance condition? • Are there any real/natural matching markets with large cores?

  47. Correlation -

  48. Correlation -

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