The Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18) Coalition Manipulation of Gale-Shapley Algorithm∗ Weiran Shen, Pingzhong Tang Yuan Deng Institute for Interdisciplinary Information Sciences Department of Computer Science Tsinghua University Duke University Beijing, China Durham, NC 27708, USA {emersonswr,kenshinping}@gmail.com [email protected] Abstract women. At each round of the procedure, each man proposes to his favorite woman among those who have not rejected It is well-known that the Gale-Shapley algorithm is not truth- ful for all agents. Previous studies in this category concentrate him yet. Then each woman rejects all men but her favorite on manipulations using incomplete preference lists by a sin- one. The algorithm terminates when no man can make any gle woman and by the set of all women. Little is known about proposal. manipulations by a subset of women. It is shown that the matching computed by the Gale- In this paper, we consider manipulations by any subset of Shapley algorithm is stable and the algorithm is guaranteed women with arbitrary preferences. We show that a strong to terminate for each legal input, which immediately implies Nash equilibrium of the induced manipulation game always that every instance of the stable matching problem has a sta- exists among the manipulators and the equilibrium outcome ble matching. There are many interesting structural results in is unique and Pareto-dominant. In addition, the set of match- the literature of stable matching theory. For example, among ings achievable by manipulations has a lattice structure. We all stable matchings, the matching computed by the Gale- also examine the super-strong Nash equilibrium in the end. Shapley algorithm is preferred by all men to other match- ings, and thus is called the M-optimal (W-pessimal) match- Introduction ing. Similarly, the W-optimal (M-pessimal) matching can be The stable matching theory was introduced by Gale and found by switching the roles of men and women. In fact, all Shapley (1962). Since then, stability has been a central men and women have opposite preferences over the set of concept in matching market design. The area has attracted stable matchings, i.e., for every two stable matchings μ1 and intensive research attention, putting theory into practice μ2, all men prefer μ1 to μ2, if and only if all women prefer through a large amount of important applications, such as μ2 to μ1. Moreover, the set of all stable matchings forms a college admissions and school matchings (Abdulkadiroglu lattice structure. and Sonmez¨ 2003; Abdulkadiroglu,˘ Pathak, and Roth 2005; Gale and Shapley 1962), hospitals-residents matchings (Irv- Incentive Issue ing and Manlove 2009; Irving, Manlove, and Scott 2000; However, the Gale-Shapley algorithm suffers from the in- Roth 1996), kidney exchange programs (Abraham, Blum, ¨ centive issue, i.e., some agents have incentive to misreport and Sandholm 2007; Roth, Sonmez,¨ and Unver 2004; 2005; their preference lists. Although it is shown that the Gale- Liu, Tang, and Fang 2014), and water right trading (Liu et Shapley algorithm is group strategy-proof for all men (Du- al. 2016; Zhan et al. 2017). bins and Freedman 1981)1, when the algorithm is adopted, We study the standard stable matching model, where two the women may have incentives to misreport their pref- set of agents, namely men and women, have preferences erences. Moreover, a well-known impossibility result by over each other. A matching is a one-to-one correspondence Roth (1982) states that no stable matching algorithm is truth- between the two sets. A pair of a man and a woman, who ful for all agents. are not matched together, but prefer each other to their des- Gale and Sotomayor (1985) shows that if all women trun- ignated partner, is said to be a blocking pair. A match- cate their preference lists properly, the Gale-Shapley algo- ing is called stable if there exists no blocking pair. The rithm will output a matching that matches each of them to Gale-Shapley algorithm, which was first proposed by Gale their partner in the W-optimal matching. Teo, Sethuraman, and Shapley (1962), takes as input the preference lists of O(n2) and Tan (2001) provide a polynomial time algorithm to find all agents, and computes a stable matching in time. the optimal single-agent truncation manipulation. However, The algorithm simulates the procedure of men proposing to ∗This work was supported in part by the National Natural Sci- 1Precisely, group strategy-proof means no coalition manipula- ence Foundation of China Grant 61561146398 and a China Youth tion can make all men in the coalition strictly better off, in this 1000-talent program. 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). 1210 little is known when only a subset of players can misreport removed in order for the W-optimal matching to be the final their preference lists. output. 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 We analyze the manipulation problem in the stable matching a stable matching mechanism and prove that it is computa- problem, where agents can report a preference list over any tionally hard to find a manipulation, even for a single ma- subset of the other sex. Contrary to most existing works, we nipulator. 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 ∈ M is a strict total order m over a subset of W , where This result generalizes the results by Teo, Sethuraman, 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 ∈ 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.