Coalition Manipulation of the Gale-Shapley Algorithm Weiran Shen and Pingzhong Tang Yuan Deng Institute for Interdisciplinary Information Sciences Department of Computer Science Tsinghua University Duke University Beijing, China Durham, NC 27708, USA femersonswr,[email protected] [email protected] Abstract him yet. Then each woman rejects all men but her favorite one. The algorithm terminates when no man can make any It is well-known that the Gale-Shapley algorithm is not truth- proposal. ful for all agents. Previous studies in this category concentrate on manipulations using incomplete preference lists by a sin- It is shown that the matching computed by the Gale- gle woman and by the set of all women. Little is known about Shapley algorithm is stable and the algorithm is guaranteed manipulations by a subset of women. to terminate for each legal input, which immediately implies In this paper, we consider manipulations by any subset of that every instance of the stable matching problem has a sta- women with arbitrary preferences. We show that a strong ble matching. There are many interesting structural results in Nash equilibrium of the induced manipulation game always the literature of stable matching theory. For example, among exists among the manipulators and the equilibrium outcome all stable matchings, the matching computed by the Gale- is unique and Pareto-dominant. In addition, the set of match- Shapley algorithm is preferred by all men to other match- ings achievable by manipulations has a lattice structure. We ings, and thus is called the M-optimal (W-pessimal) match- also examine the super-strong Nash equilibrium in the end. ing. Similarly, the W-optimal (M-pessimal) matching can be found by switching the roles of men and women. In fact, all Introduction men and women have opposite preferences over the set of stable matchings, i.e., for every two stable matchings µ1 and The stable matching theory was introduced by Gale and µ2, all men prefer µ1 to µ2, if and only if all women prefer Shapley (1962). Since then, stability has been a central µ2 to µ1. Moreover, the set of all stable matchings forms a concept in matching market design. The area has attracted lattice structure. intensive research attention, putting theory into practice through a large amount of important applications, such as Incentive Issue college admissions and school matchings (Abdulkadiroglu and Sonmez¨ 2003; Abdulkadiroglu,˘ Pathak, and Roth 2005; However, the Gale-Shapley algorithm suffers from the in- Gale and Shapley 1962), hospitals-residents matchings (Irv- centive issue, i.e., some agents have incentive to misreport ing and Manlove 2009; Irving, Manlove, and Scott 2000; their preference lists. Although it is shown that the Gale- Roth 1996), kidney exchange programs (Abraham, Blum, Shapley algorithm is group strategy-proof for all men (Du- and Sandholm 2007; Roth, Sonmez,¨ and Unver¨ 2004; 2005; bins and Freedman 1981)1, when the algorithm is adopted, Liu, Tang, and Fang 2014), and water right trading (Liu et the women may have incentives to misreport their pref- al. 2016; Zhan et al. 2017). erences. Moreover, a well-known impossibility result by We study the standard stable matching model, where two Roth (1982) states that no stable matching algorithm is truth- set of agents, namely men and women, have preferences ful for all agents. over each other. A matching is a one-to-one correspondence Gale and Sotomayor (1985) shows that if all women trun- between the two sets. A pair of a man and a woman, who cate their preference lists properly, the Gale-Shapley algo- are not matched together, but prefer each other to their des- rithm will output a matching that matches each of them to ignated partner, is said to be a blocking pair. A match- their partner in the W-optimal matching. Teo, Sethuraman, ing is called stable if there exists no blocking pair. The and Tan (2001) provide a polynomial time algorithm to find Gale-Shapley algorithm, which was first proposed by Gale the optimal single-agent truncation manipulation. However, and Shapley (1962), takes as input the preference lists of little is known when only a subset of players can misreport all agents, and computes a stable matching in O(n2) time. their preference lists. The algorithm simulates the procedure of men proposing to women. At each round of the procedure, each man proposes 1Precisely, group strategy-proof means no coalition manipula- to his favorite woman among those who have not rejected tion can make all men in the coalition strictly better off, in this context. If considering the case where no man is worse off and at Copyright c 2018, Association for the Advancement of Artificial least one man is strictly better off, the Gale-Shapley algorithm is Intelligence (www.aaai.org). All rights reserved. not group strategy-proof (Huang 2006). This paper is directly motivated by the recent reform of Dworczak (2016) put forward a new matching algorithm the college admissions process in China. In China, all stu- to find stable matchings, where all agents are allowed to dents are required to take the National College Entrance make proposals. Their algorithm is a natural generalization Exam before applying to the universities. The applications of the Gale-Shapley algorithm and they also characterize the are settled by the Ministry of Education using the Gale- set of stable matchings by showing that a matching is stable Shapley algorithm. However, besides the entrance exam, the if and only if it is a possible output of their algorithm. Teo, Ministry also has the independent admission program (aka Sethuraman, and Tan (2001) study a different type of ma- the university initiative admission plan). This program al- nipulation, where a woman can only permute her true pref- lows the universities to conduct independent exams to deter- erence list2. This is a natural constraint when all agents are mine their own ordering of the students. Starting from 2010, only allowed to report a complete preference list. They fo- these universities began to form leagues and determine their cus on the case where there is only a single manipulator and orderings together. Such leagues are widely believed to be give an algorithm to find the optimal manipulation that runs beneficial to their members, since they can cooperatively in polynomial time. Gupta et al. (2015) extends the algo- manipulate the admission results. However, these universi- rithm to the so-called P -stable (stable w.r.t preferences P ) ties are also faced with the problem of competition, since Nash equilibrium setting. they target for a similar set of students. Such leagues are With the impossibility result by Roth, it is clear that there urged to dissolve by the Ministry for the belief of unfairness. always exist some agents who have the incentive to manip- ulate the matching result, no matter what stable matching Our Results algorithm is applied. Nevertheless, Pini et al. (2009) design a stable matching mechanism and prove that it is computa- We analyze the manipulation problem in the stable matching tionally hard to find a manipulation, even for a single ma- problem, where agents can report a preference list over any nipulator. subset of the other sex. Contrary to most existing works, we allow any subset of women to be the manipulators. We show that a strong Nash equilibrium (i.e., no subset of manipula- Preliminaries tors can deviate and get strictly better off) always exists for In the standard stable matching problem, there are two sets any subset of women. Moreover, in the strong Nash equi- of agents: the men (denoted by M) and the women (denoted librium, each manipulator removes every man below her W- by W ). A preference profile P is the collection of the pref- optimal partner on her list and in the induced matching, all erence lists of all agents. The preference list P (m) of a man manipulators can be matched to their W-optimal partners. m 2 M is a strict total order over a subset of W , where This result generalizes the results by Teo, Sethuraman, m w1 m w2 denotes that m prefers w1 to w2. Similarly, the and Tan (2001) and Gale and Sotomayor (1985), which con- preference list P (w) of a woman w 2 W is a strict total sider manipulations by a single woman and the set of all P P order w over a subset of M. We will use m and w to women, respectively. Moreover, the equilibrium outcome is explicitly refer to the preference lists of m and w in profile unique and Pareto-dominant for all manipulators, i.e., all P , if multiple preference profiles are considered. However, manipulators reach a consensus on a single manipulation for simplicity, we always use m and m to denote the true profile. Furthermore, the set of all stable matchings attain- preferences of m and w. We slightly abuse notation and use able from general manipulations forms a join-semilattice. P (X) to denote the preference profile for a set of agents Finally, we show how to check whether such an unique X ⊂ M [ W . strong Nash equilibrium is a super-strong Nash equilibrium. A matching between men and women is a function µ : M [W ! M [W , that maps each agent to his or her partner Related Works in the matching. For example, µ(m) = w means that m is Knuth, Motwani, and Pittel (1990) show that the number of matched to w. Thus µ(m) = w if and only if µ(w) = m. In different partner that a woman can have in all stable match- any matching, a man should be matched with a woman and 1 vice versa.