Background Monogamous Case Sisterhood Polygamy, Blacklists, and Mismatched in the Gale-Shapley Matching Algorithm Quotas Yannai A. Gonczarowski Einstein Institute of Mathematics and Center for the Study of Rationality The Hebrew University of Jerusalem June 3, 2013 Joint work with Ehud Friedgut The Electronic Journal of Combinatorics 20(2) (2013), #P12 Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 1 / 13
Background The Stable Matching Problem Monogamous Case Polygamy, Blacklists, and Mismatched • Two disjoint finite sets to be Quotas matched: women W and men M . • Assume 1-to-1 for now. • Assume | W | = | M | for now. Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 2 / 13
Background The Stable Matching Problem Monogamous Case Polygamy, Blacklists, and Mismatched • Two disjoint finite sets to be Quotas matched: women W and men M . • Assume 1-to-1 for now. • Assume | W | = | M | for now. • Preferences for each woman and for each man. • Assume a strict order of preference for each woman over all men and vice versa. • Assume no blacklists for now. Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 2 / 13
Background The Stable Matching Problem Monogamous Case Polygamy, Blacklists, and Mismatched • Two disjoint finite sets to be Quotas matched: women W and men M . • Assume 1-to-1 for now. • Assume | W | = | M | for now. • Preferences for each woman and for each man. • Assume a strict order of preference for each woman over all men and vice versa. • Assume no blacklists for now. • The goal: a stable matching. • If w and m are matched, and if w ′ and m ′ are matched, then w and m ′ should not both prefer each other over their spouses. Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 2 / 13
Background The Gale-Shapley Deferred-Acceptance Algorithm Monogamous Case Polygamy, Blacklists, and Mismatched Quotas Gale and Shapley (1962) The following algorithm yields a stable matching. 1 On each night, every man serenades under the window of the woman he prefers most out of those who have not yet rejected him. Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 3 / 13
Background The Gale-Shapley Deferred-Acceptance Algorithm Monogamous Case Polygamy, Blacklists, and Mismatched Quotas Gale and Shapley (1962) The following algorithm yields a stable matching. 1 On each night, every man serenades under the window of the woman he prefers most out of those who have not yet rejected him. 2 On each night, every woman rejects all the men serenading under her window, except for the one she prefers most among them. Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 3 / 13
Background The Gale-Shapley Deferred-Acceptance Algorithm Monogamous Case Polygamy, Blacklists, and Mismatched Quotas Gale and Shapley (1962) The following algorithm yields a stable matching. 1 On each night, every man serenades under the window of the woman he prefers most out of those who have not yet rejected him. 2 On each night, every woman rejects all the men serenading under her window, except for the one she prefers most among them. 3 When no more rejections occur, each woman is matched with the man serenading under her window. Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 3 / 13
Background Gender Duality and Manipulation Incentives Monogamous Case Gale and Shapley (1962) Polygamy, Blacklists, and Mismatched No stable matching is better for any man. Quotas McVitie and Wilson (1971) No stable matching is worse for any woman. Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 4 / 13
Background Gender Duality and Manipulation Incentives Monogamous Case Gale and Shapley (1962) Polygamy, Blacklists, and Mismatched No stable matching is better for any man. Quotas McVitie and Wilson (1971) No stable matching is worse for any woman. Dubins and Freedman (1981) No subset of men can lie in a way that would make them all better off lying. Gale and Sotomayor (1985) Generally, there exists a woman who would be better off lying. Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 4 / 13
Background Gender Duality and Manipulation Incentives Monogamous Case Gale and Shapley (1962) Polygamy, Blacklists, and Mismatched No stable matching is better for any man. Quotas McVitie and Wilson (1971) No stable matching is worse for any woman. Dubins and Freedman (1981) No subset of men can lie in a way that would make them all better off lying. Gale and Sotomayor (1985) Generally, there exists a woman who would be better off lying. Note: the latter two do not follow from the former two. Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 4 / 13
Background Example: Manipulation by Women Monogamous Case Men’s Preferences Women’s Preferences Polygamy, m 1 w 2 w 1 w 3 w 1 m 1 > m 2 > m 3 Blacklists, and Mismatched m 2 w 1 w 2 w 3 w 2 m 2 > m 1 Quotas m 3 w 1 w 3 w 2 w 3 any Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 5 / 13
Background Example: Manipulation by Women Monogamous Case Men’s Preferences Women’s Preferences Polygamy, m 1 w 2 w 1 w 3 w 1 m 1 > m 2 > m 3 Blacklists, and Mismatched m 2 w 1 w 2 w 3 w 2 m 2 > m 1 Quotas m 3 w 1 w 3 w 2 w 3 any w 1 w 2 w 3 1 m 2 , m 3 m 1 Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 5 / 13
Background Example: Manipulation by Women Monogamous Case Men’s Preferences Women’s Preferences Polygamy, m 1 w 2 w 1 w 3 w 1 m 1 > m 2 > m 3 Blacklists, and Mismatched m 2 w 1 w 2 w 3 w 2 m 2 > m 1 Quotas m 3 w 1 w 3 w 2 w 3 any w 1 w 2 w 3 1 m 2 , m 3 m 1 2 m 2 m 1 m 3 Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 5 / 13
Background Example: Manipulation by Women Monogamous Case Men’s Preferences Women’s Preferences Polygamy, m 1 w 2 w 1 w 3 w 1 m 1 > m 23 > m 32 Blacklists, and Mismatched m 2 w 1 w 2 w 3 w 2 m 2 > m 1 Quotas m 3 w 1 w 3 w 2 w 3 any w 1 w 2 w 3 1 m 2 , m 3 m 1 2 m 2 m 1 m 3 Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 5 / 13
Background Example: Manipulation by Women Monogamous Case Men’s Preferences Women’s Preferences Polygamy, m 1 w 2 w 1 w 3 w 1 m 1 > m 23 > m 32 Blacklists, and Mismatched m 2 w 1 w 2 w 3 w 2 m 2 > m 1 Quotas m 3 w 1 w 3 w 2 w 3 any w 1 w 2 w 3 1 m 2 , m 3 m 1 2 m 2 m 1 m 3 2 m 3 m 1 , m 2 Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 5 / 13
Background Example: Manipulation by Women Monogamous Case Men’s Preferences Women’s Preferences Polygamy, m 1 w 2 w 1 w 3 w 1 m 1 > m 23 > m 32 Blacklists, and Mismatched m 2 w 1 w 2 w 3 w 2 m 2 > m 1 Quotas m 3 w 1 w 3 w 2 w 3 any w 1 w 2 w 3 1 m 2 , m 3 m 1 2 m 2 m 1 m 3 2 m 3 m 1 , m 2 3 m 1 , m 3 m 2 Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 5 / 13
Background Example: Manipulation by Women Monogamous Case Men’s Preferences Women’s Preferences Polygamy, m 1 w 2 w 1 w 3 w 1 m 1 > m 23 > m 32 Blacklists, and Mismatched m 2 w 1 w 2 w 3 w 2 m 2 > m 1 Quotas m 3 w 1 w 3 w 2 w 3 any w 1 w 2 w 3 1 m 2 , m 3 m 1 2 m 2 m 1 m 3 2 m 3 m 1 , m 2 3 m 1 , m 3 m 2 4 m 1 m 2 m 3 Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 5 / 13
Background Example: Manipulation by Women Monogamous Case Men’s Preferences Women’s Preferences Polygamy, m 1 w 2 w 1 w 3 w 1 m 1 > m 23 > m 32 Blacklists, and Mismatched m 2 w 1 w 2 w 3 w 2 m 2 > m 1 Quotas m 3 w 1 w 3 w 2 w 3 any w 1 w 2 w 3 1 m 2 , m 3 m 1 2 m 2 m 1 m 3 2 m 3 m 1 , m 2 3 m 1 , m 3 m 2 4 m 1 m 2 m 3 • w 1 improved her match, but so did w 2 ; and w 3 is unharmed. Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 5 / 13
Background Example: Manipulation by Women Monogamous Case Men’s Preferences Women’s Preferences Polygamy, m 1 w 2 w 1 w 3 w 1 m 1 > m 23 > m 32 Blacklists, and Mismatched m 2 w 1 w 2 w 3 w 2 m 2 > m 1 Quotas m 3 w 1 w 3 w 2 w 3 any w 1 w 2 w 3 1 m 2 , m 3 m 1 2 m 2 m 1 m 3 2 m 3 m 1 , m 2 3 m 1 , m 3 m 2 4 m 1 m 2 m 3 • w 1 improved her match, but so did w 2 ; and w 3 is unharmed. • w 1 made w 2 “give up” m 1 by making sure w 2 is approached by someone w 2 prefers better. Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 5 / 13
Background Sisterhood Theorem Monogamous Assume that a subset of the women declare false orders of Case preference for themselves. Polygamy, Blacklists, and Mismatched We examines two runs of the Gale-Shapley algorithm: Quotas • OA — according to everyone’s true preferences; yields the matching O . • NA — according to the liars’ false preferences, and everyone else’s true preferences; yields the matching N . Yannai A. Gonczarowski (HUJI) Sisterhood in the Gale-Shapley Matching Algorithm June 3, 2013 6 / 13
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