Gale-Shapley Stable Marriage Problem Revisited: Strategic Issues and Applications

Home , Core (game theory), Perfect information, Stable marriage problem, Strategy (game theory)

Chung-Piaw Teo • Jay Sethuraman • Wee-Peng Tan Department of Decision Sciences, Faculty of Business Administration, National University of Singapore, FBA 1-15 Law Link, Singapore 117591 Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027 Department of Decision Sciences, Faculty of Business Administration, National University of Singapore, FBA 1-15 Law Link, Singapore 117591 [email protected] • [email protected]

e study strategic issues in the Gale-Shapley stable marriage model. In the first part Wof the paper, we derive the optimal cheating strategy and show that it is not always possible for a woman to recover her women-optimal stable partner from the men-optimal stable matching mechanism when she can only cheat by permuting her preferences. In fact, we show, using simulation, that the chances that a woman can benefit from cheating are slim. In the second part of the paper, we consider a two-sided matching market found in Singapore. We study the matching mechanism used by the Ministry of Education (MOE) in the placement of primary six students in secondary schools, and discuss why the current method has limited success in accommodating the preferences of the students, and the specific needs of the schools (in terms of the “mix” of admitted students). Using insights from the first part of the paper, we show that stable matching mechanisms are more appropriate in this matching market and explain why the strategic behavior of the students need not be a major concern. (Stable Marriage; Strategic Issues; Gale-Shapley Algorithm; Student Posting Exercise)

1. Introduction matching mechanism (a complete description appears This paper is motivated by a study of the mech- in §4). In fact, two well-known stable matching anism used to assign primary school students in mechanisms—the students-optimal stable mechanism, Singapore to secondary schools. The current assign- and the schools-optimal stable mechanism—appear to ment process requires that the primary school stu- be natural candidates; however, neither of these dents submit a rank-ordered list of six schools to the mechanisms (nor any other stable matching mech- Ministry of Education. Students are then assigned to anism) is capable of eliciting truthful participation secondary schools based on their preferences, with from all of the participants. Our main purpose is to priority given to those with higher examination scores show that the students have very little room to mis- in the Primary School Leaving Examination (PSLE). represent their preferences, under either mechanism. The current matching mechanism is plagued by sev- While the strategic issues facing the students under eral problems. We argue that a satisfactory resolu- the students-optimal mechanism are well understood, tion of these problems necessitates the use of a stable the insights under the schools-optimal mechanism

Management Science © 2001 INFORMS 0025-1909/01/4709/1252$5.00 Vol. 47, No. 9, September 2001 pp. 1252–1267 1526-5501 electronic ISSN TEO, SETHURAMAN, AND TAN Gale-Shapley Stable Marriage Problem Revisited appear to be new. The most suitable stable match- be complete, and no one is to be declared as unac- ing mechanism for the Singapore Posting Exercise ceptable. A matching is just a one-to-one mapping will ultimately be a variant of these two well-known between the two sexes such that a man m is mapped matching mechanisms, as local issues and features to a woman w if and only if w is mapped to m, have to be incorporated. However, our results on sim- and m and w are acceptable to each other. (In the ulation experiments under the students-optimal and economics literature, this is defined as an individually schools-optimal mechanisms suggest that, regardless rational matching.) A matching is said to be unstable of the choice of the stable matching mechanism, the (under either model) if there is a man-woman pair, students have little incentive to misrepresent their who both prefer each other to their partners in ; this preferences. This addresses one of the major concerns pair is said to block the matching , and is called of the participants in the Singapore Posting Exercise. a blocking pair for .Astable matching is a match- A distinguishing feature of our model is that the ing that is not unstable. The significance of stability participants are required to submit complete preference is best highlighted by a system where acceptance of lists. Strategic issues in the stable marriage problem the proposed matching is voluntary. In such a setting, have been explored in settings that allow incomplete an unstable matching cannot be expected to remain preference lists; but, to the best of our knowledge, intact, as the blocking pair(s) may discover that they no systematic study has been made when complete could both improve their match by joint action: The preference lists are required. As we shall see, such a man and woman involved in a blocking pair could requirement imposes a severe restriction on the strate- just “divorce” their respective partners and “elope.” gic choices available to the participants, resulting in In addition to formulating several versions of the substantially reduced incentives to cheat. To study the stable matching problem, Gale and Shapley (1962) immunity of the schools-optimal mechanism to cheat- described a simple algorithm that always finds a sta- ing by the students, we need to first determine the ble matching for any instance of the stable marriage optimal cheating strategy in the classical stable mar- problem. This elegant result sparked the interest of riage model where the participants are only allowed many researchers, resulting in a thorough investiga- to cheat by reversing the true preference order of tion of the structural properties of the stable mar- acceptable partners; They are not allowed to truncate riage model. A property that is especially relevant to the preference list, as is the case usually considered our study is that the set of all stable matchings for in the literature. The main result of this paper shows an instance of the stable marriage problem forms a that this problem can be solved in polynomial time, lattice, with the extremal elements being the so-called using a simple extension of the classical Gale-Shapley “men-optimal” and “women-optimal” stable match- algorithm. ings. In fact, the algorithm of Gale and Shapley (1962) Preliminaries. Stable matching problems were first that established the existence of a stable marriage con- studied by Gale and Shapley (1962). In a stable mar- structs a men-optimal stable matching. This algorithm riage problem there are two finite sets of participants: is commonly known as the men-propose algorithm the set of men (M) and the set of women (W ). We because it can be expressed as a sequence of “pro- assume that every member of each sex has strict pref- posals” from the men to the women. Note that the erences over the members of the opposite sex. In the Gale-Shapley algorithm can be easily adapted to yield model that allows rejection, the preference list of a a women-optimal stable matching by simply inter- participant can be “truncated” in the sense that par- changing the roles of men and women; this is com- ticipants have the option of declaring some others monly called the women-propose algorithm. All of the as unacceptable; in this model, we also include the results in this paper will be stated under the assump- possibility that a participant may be unmatched, i.e., tion that the men-optimal stable matching mechanism the participant’s assigned partner in the matching is is used; by interchanging the role of men and women, himself/herself. In contrast, in the Gale-Shapley model, analogous results can be derived for the women- the preference lists of the participants are required to optimal mechanism.

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Strategy. Consider a market in which men and choices, and the ministry pads each list by ranking women submit their preference lists to a centralized the remaining schools using location as the only cri- agency that matches the participants by computing terion (schools close to the student’s home are ranked the men-optimal stable matching. An important dif- higher). All assignments are done in a centralized ficulty that arises is that this matching mechanism is manner, and no student is allowed to approach the manipulable by the women: Some women can inten- schools privately for admission purposes. tionally submit false preferences and obtain better However, such passive roles for the schools partners. Such (strategic) questions have been stud- will change in the near future. In preparation ied for the stable marriage problem by mathemati- for a knowledge-based economy, the ministry has cal economists and game theorists, with the goal of reiterated its intention to shift from the current quantifying the potential gains of a deceitful par- examination-oriented system to one that focuses on ticipant. An early result in this direction is due to ability-driven education. In line with this goal, the Roth (1982), who proved that when the men-optimal ministry intends to give the schools administrative mechanism is used, none of the men benefits by and professional autonomy. Student assessments will submitting false preferences, regardless of how the also be reviewed to meet the objective of develop- other participants report their preferences. By sub- ing creative independent learners. The new university mitting a false preference list, a man can, at best, admissions system to be implemented from 2003 will obtain his (true) optimum stable partner, which he not rely solely on the GCE A-level exam results, obtains anyway by submitting his true preference list. but instead make a holistic evaluation of a student’s In game-theoretic parlance, stating true preferences potential. The current assignment process of students is a dominant strategy for the men. However, Gale to schools is not consistent with this new focus of and Sotomayor (1985b) showed that the women can the ministry. Furthermore, the design of the current still force the men-optimal mechanism to return the assignment process has also given rise to several women-optimal solution. The optimal cheating strat- operational problems (see §4), and our purpose is to egy in this case is simple: Each woman w submits a argue that a centralized stable matching mechanism preference list that is the same as her true preference of the type proposed by Gale and Shapley is supe- list, except that she declares men who rank below her rior. Our goal, in the end, is to convince the read- women-optimal partner as unacceptable. Our main ers that regardless of whether the students-optimal or goal in this paper is to study the analogous question schools-optimal mechanism is used, there is no sig- in the Gale-Shapley model, in which all the partici- nificant incentive for the students or the schools to pants are required to submit complete preference lists. strategize and exploit the system. We emphasize that we consider only a centralized There is an obvious relation between the MOE market that computes the men-optimal matching, and problem and the stable marriage problem introduced all the men and women can only manipulate the out- earlier: In the MOE problem, the students play the role of “women” and the secondary schools play the come by permuting their preference lists. We do not role of “men.” Observe that there is also a crucial dif- consider a decentralized market, where manipulation ference: In the MOE problem, many “women” (stu- possibilities are richer, nor do we consider other ways dents) can be assigned to the same “man” (secondary of manipulating the outcome. school), whereas in the stable marriage model we Motivation. The motivation for studying strategic require that the matching be one-to-one. Since the questions in the Gale-Shapley model is the Singapore students (schools) in this setting are not allowed school admissions problem (referred to hereafter as to declare any school (student) as unacceptable, the the MOE (Ministry of Education) problem) briefly MOE problem can be modeled as a many-to-one described earlier. Currently, the schools are merely matching problem with complete preference lists. seen as resources by the ministry to be allocated The issues of strategic manipulation in the sta- to students. Furthermore, for ease of implementable marriage model with rejection are well under- tion, each student is only allowed to submit six stood (cf. Roth and Sotomayor 1991 and the references

1254 Management Science/Vol. 47, No. 9, September 2001 TEO, SETHURAMAN, AND TAN Gale-Shapley Stable Marriage Problem Revisited therein). However, little is known in the case of Roth and Rothblum (1999). In a centralized many-to- stable marriage models (one-to-one and many-to- one market (with rejection allowed) operating under one) without rejection, which, as we discussed, is the schools-optimal mechanism, Roth and Rothblum a more suitable representation of the MOE prob- consider the strategic issues facing the students. In a lem; a notable exception is the recent work of low information environment, where preferences of Tadenuma and Toda (1998), in which they consider the other participants are not known with certainty, the implementation question, and show that no sta- they conclude that stating preferences that reverse the ble matching correspondence can be implemented in true preference order of two acceptable schools is not Nash equilibrium as long as M=W > 2. (Stronger beneficial to the students, while submitting a trunca- nonimplementability results hold for the rejection tion of the true preferences may be. Our simulation- model, see Kara and Sonmez 1996.) based experimental results in this paper suggest that this observation is valid even in the perfect informa- Results and Structure of the Paper. To derive tion setting, i.e., knowing the reported preferences of insights into the strategic behavior of the participants all the other participants may still not allow one to in the MOE problem, we first consider the one-to-one benefit from cheating, if one is only allowed to reverse model, and study the following question: In the sta- the order of the schools, but not allowed to submit a ble marriage model with complete preferences, with truncated list. the men-optimal mechanism, is there an incentive for Finally, we note that the strategic issues facing the the women to cheat? If so, what is an optimal cheat- schools, under the schools-optimal mechanism, is a ing strategy for a woman? Can a woman always force nontrivial problem: Roth (1985) showed that, contrary the mechanism to return her women-optimal part- to the one-to-one case, the schools-optimal match- ner? In §2 we present the main result of this paper: ing need not even be weakly-Pareto optimal for the An optimal cheating strategy can be constructed in schools, and there is no stable matching mechanism polynomial time. A related issue concerns the robust- that makes it a dominant strategy for all the schools ness of the men-optimal mechanism. In the rejection to state their true preferences. We shall see in §4 why model, it is well known that the women can easily this is not an important issue in the MOE problem. manipulate the men-optimal mechanism, and, in fact, almost all the women will submit false preferences. In sharp contrast, our results for the Gale-Shapley model 2. Optimal Cheating in the in §3 provide evidence that the men-optimal mecha- Gale-Shapley Model nism is fairly robust, and that there is very little incen- Our standing assumption in this paper is that woman tive for the women to cheat. In particular, restricting w is the only deceitful participant and she knows the strategic choices of the women drastically reduces the reported preferences of all the other partici- their benefits from cheating, thereby reducing the pos- pants, which remain fixed throughout. We shall show, sibility that a woman will cheat. Armed with this the- eventually, that cheating opportunities for woman w oretical understanding of the Gale-Shapley model, we are uncommon, in spite of the assumption that she prescribe some recommendations in §4 to improve the has perfect information. We consider only a central- current “matching mechanism” used in the Singapore ized market, although the algorithm is phrased as a MOE Secondary School Posting Exercise. In particu- sequence of proposals from the men to the women. lar, we argue that a stable matching mechanism is We visualize the deceitful woman as running this more appropriate for the MOE problem, and that algorithm in her head and submitting the optimal some of the other difficulties inherent in the present cheating strategy thus computed as her preference list system can be effectively addressed by a stable match- to the centralized market. ing mechanism. We shall begin by considering the following ques- It is instructive to compare our results to some tion: Consider a centralized market in which the men- recent results obtained in a very interesting study by propose algorithm is literally used to compute the

Management Science/Vol. 47, No. 9, September 2001 1255 TEO, SETHURAMAN, AND TAN Gale-Shapley Stable Marriage Problem Revisited men-optimal matching. Suppose woman w has no preference lists is the women-optimal solution (with knowledge of the preferences of any of the other partic- respect to the original preference lists). To prove our ipants, and that she is the only deceitful participant. result, however, we have to show that when a woman Suppose also that she is allowed to reject proposals. unilaterally declares all the men as unacceptable, this Is it possible for her to identify her women-optimal is enough to induce her optimal partner to propose to partner by just observing the sequence of proposals her in the course of executing the men-propose algo- she receives? Somewhat surprisingly, the answer is rithm. Furthermore, we need to show that no higher- yes! If w simply rejects all the proposals made to ranked man on her list will propose to her even after her, then the best (according to her true preference she rejects the proposal from her optimal partner. list) man among those who propose to her is her Proof of Theorem 1. Let w denote the women- women-optimal partner. Our algorithm for finding the optimal partner for w. We modify w’s preference list optimal cheating strategy in the Gale-Shapley model by inserting the option to remain single in the list, builds on this insight: Woman w rejects as many pro- immediately after w. (We declare all men that are posals as possible, while remaining engaged to some inferior to w as unacceptable to w.) Consequently, man who proposed earlier in the algorithm. Using a in the men-propose algorithm, all proposals inferior backtracking scheme, she uses the matching mecha- to w will be rejected. Nevertheless, since there is nism repeatedly to find her optimal cheating strategy. a stable matching (with respect to the original prefer- Given our standing assumption that woman w has ences) with w matched to w, our modification does complete knowledge of the reported preferences of not destroy this solution, i.e., this solution remains the other participants, and that she is the only agent stable with respect to the modified list. It is also acting strategically, it is clear what she would do: She well known that in any stable matching instance, the would run the algorithm to find her optimal cheating set of people who are single is the same for all sta- strategy privately and submit this (optimal) prefer- ble matchings (cf. Roth and Sotomayor 1990, p. 42). ence list to the centralized market. Thus, w must be matched in all stable matchings with respect to the modified preference list. The men- 2.1. Finding Your Optimal Partner optimal matching for this modified preference list We first describe Algorithm OP—an algorithm to must match w to w, since w is now the worst compute the women-optimal partner for w using the partner for w with respect to the modified list. In par- men-propose algorithm. (Recall that we do this under ticular, w must have proposed to w during the exe- the assumption that woman w is allowed to remain cution of the men-propose algorithm. Note that until single.) w proposes to w, the men-propose algorithm for the modified list runs exactly in the same manner as Algorithm OP in Step 1 of OP. The difference is that Step 1 of OP will 1. Run the men-propose algorithm, and reject all reject the proposal from w, while the men-propose proposals made to w. At the end, w and a man, say algorithm for the modified list will accept the pro- m, will remain single. posal from w,asw prefers w to being single. 2. Among all the men who proposed to w in Step 1, Hence, clearly w is among those who proposed to let the best man (according to w)bem . w in Step 1 of OP, and so m ≥w w. Suppose m >w w. Consider the modified list in Theorem 1. m is the women-optimal partner for w. which we place the option of remaining single imme- Remark. Gale and Sotomayor (1985b) showed that diately after m. We run the men-propose algorithm when each woman declares all the men inferior to with this modified list. Again, until m proposes to w, her women-optimal partner as unacceptable, then the the algorithm runs exactly the same as in Step 1 of men-optimal mechanism will be forced to return OP, after which the algorithm returns a stable partner the women-optimal stable matching. This is because for w who is at least as good as m according to w the only stable matching solution for the modified (under both the original and the modified lists, since

1256 Management Science/Vol. 47, No. 9, September 2001 TEO, SETHURAMAN, AND TAN Gale-Shapley Stable Marriage Problem Revisited we have not altered the order of the men before m under the Gale-Shapley model. The reason is that this on the list). The matching obtained is thus stable with greedy improvement technique does not allow for the respect to the original list. This contradicts our ear- possibility of rejecting the current best partner, in the lier assumption that w is the best stable partner hope that this rejection will trigger a proposal from a for w. better would-be partner. Our algorithm in this paper, Suppose w is allowed to modify her preference list which is described next, does precisely that. An illus- after each run of the men-propose algorithm, and the trative example appears soon after the description of algorithm is to be repeated until w concludes that the algorithm. she has found her best possible partner. Theorem 1 Pw = m m m Let 1 2 n be the true preference essentially says that knowing the set of proposals list of woman w, and let Pmw be a preference list w woman receives is enough to allow her to con- for w so that the men-propose algorithm will return struct her optimal cheating strategy, if she is allowed m as her men-optimal partner. In the case that man m to declare certain men as unacceptable; she need not cannot be obtained by w as her men-optimal partner know the preferences of any of the other participants for any preference list, we set Pmw to be the null involved, as long as they behave truthfully. In the next list, as a matter of convention. Our algorithm con- section, we shall use this insight to construct an opti- structs Pmw (if man m is attainable by woman w) w mal cheating algorithm for , under the additional or determines whether Pmw is the null list (if man restriction that she is not allowed to declare any man m is not attainable by woman w) iteratively, and con- as unacceptable. sists of the following steps: 1. Run the men-propose algorithm with the true 2.2. Cheating Your Way to a Better Marriage preference list Pw for woman w. Keep track of all Observe that the procedure of §2.1 only works when men who propose to w. Let the men-optimal partner woman w is allowed to remain single throughout the for w be m, and let Pmw be the true preference matching process. Suppose the central planner does list Pw. not give the woman an option to declare any man as 2. Suppose mj proposed to w in the men-propose unacceptable. How do we determine her best attain- algorithm. By moving mj to the front of the list able partner by manipulation? This is essentially a Pmw, we obtain a preference list for w such that restatement of our original question: Who is the best the new men-optimal partner (after running the men- stable partner woman w can have when the men- m optimal mechanism is used and when she can lie only propose algorithm on this modified list) is j . Let Pm w m by permuting her preference list. Note that the prefer- j be this altered list. We say that j is a w ences of the remaining participants (except woman w) potential partner for . are fixed throughout. 3. Repeat Step 2 to obtain a preference list for every m w A natural extension of Algorithm OP is for woman man (other than ) who proposed to woman in the w to: (i) accept a proposal first, and then reject algorithm; after this, we say that we have exhausted m all future proposals; (ii) from the list of men who man , the men-optimal partner obtained with the proposed to her but were rejected, find her most preference list Pmw. All potential partners of w preferred partner; repeat the men-propose algorithm that come from running the men-propose algorithm until the stage when this man proposes to her; using the list Pmw have been found. (iii) reverse the earlier decision and accept the pro- 4. If a potential partner for w, say man u, has not posal from this most preferred partner, and continue been exhausted, run the men-propose algorithm with the men-propose algorithm by rejecting all future Puw as the preference list of w. Identify new poten- proposals; and (iv) repeat (ii) and (iii) until w can- tial partners. Repeat Steps 2–3 with u in place of m. not obtain a better partner from all other proposals. 5. Repeat Step 4 until all potential partners of w are Unfortunately, this simple strategy does not always exhausted. Let N denote the set of all potential (and yield the best stable partner a woman can achieve hence exhausted) partners for w.

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6. Among the men in N let ma be the man woman exhausted after this. No new potential partner is w prefers most. Then Pmaw is an optimal cheating found, and so the algorithm terminates. strategy for w. Remark. Example 1 shows that woman 1 could The men in the set N at the end of the algorithm cheat and get a partner better than her men-optimal have the following crucial properties: partner. However, her women-optimal partner in this • For each man m in N , there is an associated pref- case turns out to be Man 1. Hence Example 1 also erence list for w such that the men-propose algorithm shows that Woman 1 cannot always assure herself returns m as the men-optimal partner for w with this of her women-optimal partner through cheating, in list. contrast to the case when rejection is allowed in the • All other proposals in the course of the men- cheating strategy. N propose algorithm come from other men in . (Oth- We conclude this section by stating and proving the erwise, there will be some potential partners who are main result. Recall that ma is the best man according not exhausted.) to w’s true preference list among the men in set N With each run of the men-propose algorithm, we constructed by the algorithm. exhaust a potential partner, and so this algorithm executes at most n men-propose algorithms before Theorem 2. = Pmaw is an optimal strategy for termination. woman w. Example 1. Consider the following stable marriage Proof (by contradiction). We use the convention problem: that rm = k if man m is the kth man on woman w ∗ = m m m 1 23451 1 12354 ’s true preference list. Let p1 p2 pn 2 34512 2 21453 be a preference list that gives w her best stable part- 3 51423 3 32514 ner when the men-optimal mechanism is used. Let

4 31245 4 45123 this man be denoted by mpb, and let woman w strictly

5 15234 5 51234 prefer mpb to ma (under her true preference list), i.e.,

True Preferences of True Preferences of rmpb

1258 Management Science/Vol. 47, No. 9, September 2001 TEO, SETHURAMAN, AND TAN Gale-Shapley Stable Marriage Problem Revisited

m = m m = m (ii) q1 p1 qb pb, where the position 3. Strategic Issues in the m of those men who rank higher than pb is Gale-Shapley Problem unchanged. By requiring the women to submit complete pref- rm = s + m ∈ m m m (iii) a 1 a qs+1 qs+2 qn . erence lists, we are clearly restricting their strategic m m m (iv) The men in the set qs+1 qs+2 qn options; thus many of the strong structural results retain their relative position with respect to one known for the rejection model may not hold for the ∗ another under . Gale-Shapley model. This is good news, for it reduces w ˜ Note that the men-optimal partner of under the incentive for a woman to cheat. In this section, we m m m cannot come from the set qs+1 qs+2 qn . present several examples to show that the strategic Otherwise, since the set of men who proposed behavior of the women can be very different under in the course of the algorithm must come from the models with and without rejection. m m m qs+1 qs+2 qn , and since the preference list ∗ retains the relative order of the men in this 3.1. The Best Man (Obtained by Cheating) ∗ set, the same partner would be obtained under . May Not Be Women-Optimal ∗ This leads to a contradiction as is supposed to In the two-sided matching model with rejection, it is w return a better partner for . Hence, we can see not difficult to see that the women can always force ˜ that under , we already get a better partner than the men-optimal mechanism to return the women- under . optimal solution (for instance, each woman declares m Now, since the preference list returns a with as unacceptable those men who are inferior to her true rm = s + N a 1, we may conclude that the set women-optimal partner). In the Gale-Shapley model, (obtained from the final stage of the algorithm) which forbids rejection, the influence of the women does not contain any man of rank lower than is far less, even if they collude. A simple example is s + N ⊆ m m m 1. Thus qs+1 qs+2 qn . Suppose when each woman is ranked first by exactly one man. m m m N qs+1 qs+2 qw do not belong to the set , In this case, there is no “conflict” among the men, m m and qw+1 is the first man after qs who belongs to and in the men-optimal solution, each man is matched the set N . By construction of N , there exists a per- to the woman he ranks first. In this case, the algo- ˆ m w mutation with qw+1 as the stable partner for rithm will terminate with the men-optimal matching, under the men-optimal mechanism. Furthermore, all regardless of how the women rank the men in their of those who propose to w in the course of the lists. So ruling out the strategic option of remaining algorithm are in N , and hence they are no better single for the women significantly affects their ability m w than a to . Furthermore, all proposals come from to change the outcome of the game by cheating. m m m N ⊆ m men in qw+1 qw+2 qn , since qs+1, By repeating the above analysis for all the other m m qs+2 qn . women in Example 1, we conclude that, by cheating By altering the order of those who did not pro- unilaterally, the best possible partners for Women 1, pose to w, we may assume that ˆ takes the follow- 2, 3, 4, and 5 are, respectively, Men 3, 1, 2, 4, and 3. m m m m m m ing form: q1 q2 qs−1 qs qw qw+1 An interesting observation is that Woman 5 cannot , where the first qw+ 1 men in the list are identi- benefit by cheating alone (she can only get her men- cal to those in ˜ . But, the men-optimal stable solution optimal partner no matter how she cheats). However, obtained using ˆ must also be stable under ˜ , since if Woman 1 cheats using the preference list (3, 4, 1, w m is match to qw+1, and the set of men she strictly 2, 5), Woman 5 will also benefit by being matched to m ˆ ˜ prefers to qw+1 is identical in both and . This Man 5, who is first in her list. is a contradiction as ˜ is supposed to return a men- optimal solution better than ma. 3.2. Does Cheating Pay? This implies that ∗ does not exist, and so is opti- Roth (1982) shows that under the men-optimal mech- mum and ma is the best stable partner w can get by anism, the men have no incentive to alter their true permuting her preference list. preference lists. In the rejection model, however, Gale

Management Science/Vol. 47, No. 9, September 2001 1259 TEO, SETHURAMAN, AND TAN Gale-Shapley Stable Marriage Problem Revisited and Sotomayor (1985a) show that a woman has an represents the men-optimal and women-optimal part- incentive to cheat as long as she has at least two dis- ners of each participant. For instance, Man 1 and tinct stable partners. From Pittel (1989), we know that Woman 1 have preference lists “64782351” (i.e., Man 1 the average number of stable matchings is asymptot- prefers Woman 6 to Woman 4 to Woman 7, etc., and ically n logn/e, and with high probability, the rank similarly Woman 1 prefers Man 6 to Man 4 to Man 7, of the women-optimal and men-optimal partners for etc.). Under the men-optimal mechanism, Man 1 is the women are, respectively, logn and n/ logn. matched to Woman 6, and Woman 1 is matched to Thus, in typical instances of the stable marriage game Man 7. Thus the women-optimal partner for Woman 1 under the rejection model, most of the women will is Man 6 (from symmetry), and so she has an incen- not reveal their true preferences. tive to lie about her preferences. In fact, this exam- Many researchers have argued that the troubling ple shows that 6 out of the 8 women benefit from implications from these studies are not relevant in cheating in the rejection model. The situation for the practice, as the model assumes that the women have Gale-Shapley model is very different: given that the complete knowledge of all the other participants, as women can only misrepresent their preferences by well as their preference lists. For the Gale-Shapley order reversals, no woman can benefit from cheating. model we consider, it is natural to ask whether it We ran the algorithm on 1,000 instances, with pref- pays (as in the rejection model) for a typical woman erences of each individual generated uniformly at to solicit information about the preferences of the random, for n = 8. The number of women who benefit remaining participants in the game. from cheating is tabulated in Table 1. Example 2. We have designed a Visual Basic pack- Interestingly, the number of women who benefit age for the stable marriage problem with complete from cheating is surprisingly low. In fact, in 74% preference lists. The following example, in one of the of the instances, the men-optimal solution is their simulation experiments, illustrates the dramatic dif- only option, no matter how they cheat. The average ference in the behavior of the women in the models percentage of women who benefit from cheating is with and without rejection: merely 5.06. In the example, the men and women are labelled To look at the typical strategic behavior on larger from 1 to 8. Both man i and woman i have identical instances of the stable marriage problem, we ran the n = preference lists. The number next to the list in Figure 1 heuristic on 1,000 random instances for 100. The cumulative plot is shown in Figure 2. In particular, in more than 60% of the instances at most 10 women Figure 1 Stable Marriage Model with Rejection Versus One Without Rejection (out of 100) benefited from cheating, and in more than 96% of the instances at most 20 women benefited from cheating. The average number of women who benefited from cheating is 9.515%. Thus, the chances that a typical woman can benefit from acquiring perfect information (i.e., knowing the preferences of the other participants) is slim in the Gale-Shapley model. We have performed similar experiments for large instances of the Gale-Shapley model. Because of com- putational requirements, we can only run the experiment on 100 random instances of the problem with

Table 1 Number of women who beneﬁted 0 1 2 3 45678

Number of observations 740 151 82 1971000

1260 Management Science/Vol. 47, No. 9, September 2001 TEO, SETHURAMAN, AND TAN Gale-Shapley Stable Marriage Problem Revisited

Figure 2 Beneﬁts of Cheating

500 men and women. The insights obtained from the solution is more appropriate, and how some of the 100 by 100 cases carry over: the number of women problems under the current system can be addressed who benefit from cheating is again not more than 10% using a stable matching mechanism. of the total number of the women involved. In fact, the average was close to 6% of the women popula- 4.1. Current Assignment Process tion in the problem. This suggests that the number Prior to taking their Primary School Leaving Exam- of women who benefit from cheating in the Gale- ination (PSLE), the students are required to submit Shapley model with n women grows at a rate slower (around August) their rank ordering of schools to than a linear function of n. A detailed probabilistic the MOE, which oversees the posting exercise. In the analysis of this phenomenon is a challenging problem, option form, each student is required to list six sec- which we leave for future research. ondary schools she would like to attend, in order of decreasing desirability. The students (rather, their par- 4. Singapore MOE Posting Exercise ents) are advised by the MOE to make realistic choices The admission to secondary schools in Singapore is because the preference list, once submitted, cannot centrally controlled. A distinguishing feature of the be changed. After the PSLE results are announced, Singapore school admissions problem (MOE problem) the top ten percent performers are given a second is that the students are not allowed to remain unas- option to indicate their preferences for the Indepen- signed at the end of the posting exercise, and that the dent schools and the Special Assistance Plan (SAP) schools are not allowed to reject any students if there schools. These are the elite schools in Singapore that are still places available. In what follows, we first admit only the top students. present a summary of the current matching mecha- The posting of students to secondary schools is nism, along with a brief description of its shortcom- (partially) computerized. All students are ranked ings in §4.1. In §4.2, we explain why a stable matching according to their aggregate PSLE scores regardless of

Management Science/Vol. 47, No. 9, September 2001 1261 TEO, SETHURAMAN, AND TAN Gale-Shapley Stable Marriage Problem Revisited their choices. Currently, the assignment of students to release of the PSLE scores. Knowing that many good schools is conducted in multiple phases, which are as students will be applying to the good schools, parents follows: cannot afford to reveal their true preferences. If they Phase 1 Consider the top students who have were to list down just the ideal top six schools as their applied to the independent schools in the second six choices, they run the risk of being rejected by all option, and assign them to these schools. these six schools—this happens if their child performs Phase 2 Consider the top students who have below their expectations in the PSLE and fails to qual- applied to SAP schools in the second option, and ify for these schools. Worse, this may actually result assign them to SAP schools. in their child getting into a school that is completely Phase 3 For historical and administrative reasons, out of their consideration, as manual posting would certain secondary schools have affiliated primary then be carried out. The parents are thus forced to be schools. This is to foster closer collaboration between more “realistic” when choosing the schools for their the primary and secondary schools. Students who child. Consequently, the parents are faced with the opt to go to the affiliated secondary schools will difficult problem of optimally utilizing the six options be given higher priority. Consider all students who provided by the MOE. have selected their affiliated secondary school as their The MOE has tried to simplify matters for the par- top choice, and assign them to the affiliated schools ents by providing them with information such as the (subject to availability of places). schools’ cut-off points for the previous year to help Phase 4 Consider the remaining students one by them make rational decisions. In recent years, the one, and match them to the schools according to the MOE has also provided a rating of all the schools in choices they made in August. Singapore so that the parents will be better informed Note that in the process of assigning the students of the schools’ strengths and weaknesses. However, to schools, a student who ranks his affiliated sec- this still does not address the parents’ major concern ondary school as his first choice is given priority for in formulating their choices: they have to submit their that school. In this way, the current assignment pro- choices before their children take the examination! In cess has managed to incorporate some of the schools’ fact, one parent lamented on a national newspaper: selection criterion, which actually place more impor- “How do we choose when we do not know how they tance on “affiliation” than on performance in the will do at the national level?” (The Straits Times,17 PSLE; not all students, however, can gain admission to their affiliated secondary schools because such August 1996). A good matching mechanism should allow admissions are subject to vacancies in these schools. the students to submit their true ranking list, without the When a student fails to get admission to any of need to strategize based on their PSLE results. the schools of her choice, she is assigned (manually) We note that it is not possible for the Ministry to a school (in nearby postal districts) that still has to solicit the students’ options after the release of vacancies. If there is no such school available, the the examination results, due to the tight time span student is assigned (manually) to a school in some between the release of results and the start of the new other postal district that still has vacancies. The stu- school term. dents are notified of the schools they are assigned Problem 2 Another problem with the current assign- to by late December, just in time for the new school ment process is that other than affiliation, the gen- semester, which starts early January. The current sys- uine preferences of the schools are not considered. tem is plagued with several problems, some of which The schools have to accept the students assigned are as follows. to them under the process. Furthermore, the num- Problem 1 Decision making by the parents is com- ber of places available in the school is also largely plicated by the fact that (i) parents are only given determined by the MOE. To attract the desired stu- limited options to express their preferences, and dents, many schools, especially the newer ones, resort (ii) parents have to make their decisions prior to the to advertising campaigns such as organizing open

1262 Management Science/Vol. 47, No. 9, September 2001 TEO, SETHURAMAN, AND TAN Gale-Shapley Stable Marriage Problem Revisited houses, sending brochures to parents, and conduct- Although a substantial number (30%) of respon- ing talks at primary schools. Though such market- dents think that distance to school is important, and ing efforts help, schools still depend signiﬁcantly on hence they have no reason to worry about this cri- the ST School 100 ranking (provided by the MOE) terion under the current assignment process, it is to attract their desired students. However, since the interesting to note that 70% of respondents do not ST ranking is decided mainly by the schools’ per- consider distance to school as an important factor in formance in the GCE “O”-level examination, over- their ranking of schools. The current system are not reliance on the ranking could induce the schools to able to cater to these people in its assignment. A good overemphasize the importance of academic results. assignment process should match all students to schools This is clearly an undesirable repercussion as it actu- according to the true preferences of the students. ally impedes the Singapore government from achiev- Problem 4 Finally, since the examination results are ing the “Thinking Schools, Learning Nation” vision. released only in late November, the MOE can only A good matching mechanism should allow the schools to obtain the second option form not earlier than early rank the students using different criteria, depending on December. As such, the entire assignment process is the strength and areas of excellence the schools wish to conducted only in a span of two weeks as the assign- emphasize. ments have to be made known to the students by Problem 3 A third problem lies in the logic of the the third week of December. An unfortunate event current assignment process at Phase 4 of the exercise. developed in December 1997 when the Ministry had When all the six choices are exhausted, the students to publicly apologize for an administrative error: are then assigned to schools in nearby postal districts. It has been discovered that there were some dis- However, according to a 1992 survey (The Straits crepancies in the posting of pupils within the Nor- Times, 19 August 1992), only 30% of the respondents mal (academic) course; 3,278 pupils out of a total of think that the distance to school is an important factor 9,417 are affected. The ministry is re-posting the 3,278 in their selection of schools. While this is a nonnegli- pupils (The Straits Times, December 1997). gible fraction of the respondents, it is distressing that The ministry apologized for the error and said that several other factors, considered more important by the the error occurred when wrong data was entered respondents, were completely ignored in the current into its computers during the assignment process. The assignment process. The major factors identiﬁed in human error was discovered only when “the min- the survey are: istry checked again and the parents gave feedback.” Average Score This ultimately led to some resentment and confu- Which Factors Matter Most (out of 10) sion among the students and parents, as some of them had already bought the books and met the teachers in Quality of O-level passes 75 Percentage of O-level passes 74 their previously assigned secondary school. The fact Entry cut-off points 66 that the re-assignment was made barely a week before Value-added 66 the commencement of the new school term aggra- Percentage of students accepted 61 vated the seriousness of the human errors. As far as ECA performance 4 9 possible, the matching mechanism should be fully automated to mitigate the chances of human errors and reduce Other factors considered Percentage of the time taken to post students to schools. very important decision makers

Quality of teachers 88% 4.2. Beneﬁts of Stable Matching Mechanisms School discipline 82% The current posting method is appropriate for the old Quality of principal 58% paradigm in which the schools are all alike and, thus, School reputation 50% Range of school facilities 35% are viewed as resources to be allocated to the stu- Distance from school 30% dents. Also, assigning the students to schools accord- Friends in the school 30% ing to merit is a simple and convenient way to solve

Management Science/Vol. 47, No. 9, September 2001 1263 TEO, SETHURAMAN, AND TAN Gale-Shapley Stable Marriage Problem Revisited the matching problem. However, in recent years, this the MOE to prevent the schools from misrepresenting problem has assumed a new dimension as it is gen- their preferences. For instance, by working with the erally recognized that examination results alone are schools to determine their strengths and core compe- not a good way of assessing the merit of the students. tence, the MOE can specify the profile of the prospec- Thus, elaborate mechanisms are now being put in tive students that would fit into the schools’ curricu- place to assess the students in an all-round fashion. In lum. This can be used to rank the students according the future, it will no longer be possible to have a sin- to the various criteria listed down by the schools. This gle list that ranks all the students according to merit. eliminates the need for the schools to submit a rank- Also, given that the schools are being encouraged to ing list of individual students, thus giving them less develop their own identity and strength, it is highly scope for strategic manipulation. debatable whether the schools will all rank the stu- Schools-Optimal Mechanism. As in the case of the dents in the same way. Currently, besides affiliation, students-optimal matching mechanism, the danger of the assignment process cannot handle school-specific schools manipulating the assignment process can be criteria. By moving to a new paradigm in which eliminated by appropriate management control. Note schools are allowed to devise their own criteria to that even with the schools-optimal mechanism, it is rank the students, and assigning the students to known that truth revelation is not a dominant strategy schools via a stable matching mechanism, we can for the schools, and so proper management control is address some of the problems outlined in §4.1. Under still required to ensure that the schools do not mis- this new mechanism, the limit on the number of represent their preferences. However, it is still possi- choices each student is entitled to should be removed, ble that some of the students will benefit from lying. since this is one of the root causes for the problems In the rest of this section, we examine the extent to faced by the students in the current system. Although which the students benefit by misrepresenting their this will lead to an additional burden on the sys- preferences. We use the algorithm described in §2.2 to tem (data collection and processing), the benefits that obtain some insights into this problem. accrue will be well worth the effort. We designed an experiment with n = 60 students, With these changes, we can have a fully automated and k schools each with 60/k places, for k = 1, matching mechanism that can capture the true pref- 3 5 10 20. The preferences of the schools and the erences of the students, and allow the schools to turn students are generated randomly in the many-to-one from passive to active participants in the assignment model. We then convert the problem into a one-to-one process. This immediately addresses Problems 2–4 model, with each school represented as k individuals outlined in §4.1. We argue next that in a stable match- in the new model. All individuals corresponding to ing mechanism implementation, it is unlikely that the the same school will have identical preference lists. students will benefit significantly from misrepresent- Also, for the students in the new model, the prefer- ing their preference lists. In particular, we analyze ence list is generated so that the individuals corre- the strategic issues under the students-optimal and sponding to the same school are clustered together in schools-optimal mechanisms. the students’ preference list, capturing the fact that Students-Optimal Mechanism. It follows from in the many-to-one model, the students basically pro- classical results in stable matching (cf. Roth 1982) vide the preference lists for the schools only. In this that, in this model, the students have no incentive to new one-to-one model (where students correspond to misrepresent their preferences. However, the strategic women and schools correspond to men), the prefer- behavior of the schools is not well understood. In fact, ence lists are thus generated in a special manner. We to the best of our knowledge, the question of finding next report the extent to which the women (i.e., stu- the optimal cheating strategy for the schools is still dents) can cheat in this new one-to-one model, under open and appears to be a difficult problem. For the the men-optimal mechanism. Note that the number of MOE problem, however, this is less of a concern, since women who can benefit by cheating in this case is not proper management mechanisms can be instituted by directly equivalent to the number of students who can

1264 Management Science/Vol. 47, No. 9, September 2001 TEO, SETHURAMAN, AND TAN Gale-Shapley Stable Marriage Problem Revisited benefit by cheating in the many-to-one model, since 5. Concluding Remarks the one-to-one problem contains artificial strict pref- Our study here focused on two main areas: In the erences over slots of the same school. Nevertheless, first part, we studied the theoretical aspects of the the numbers reported consititue an upperbound on stable marriage problem. Specifically, we addressed the number of students who can benefit by cheating the strategic issues in the Gale-Shapley model (i.e., in the many-to-one model. a one-to-one matching model in which each partic- For each set of parameters, we ran 1,000 experi- ipant submits complete preference lists)—very little ments. The following table shows the average number was known about the strategic issues for this model of women who benefited by cheating. prior to our work. Assuming that the men-optimal mechanism is used, we derived an optimal cheating k 13 51020 strategy for the women. We also showed that the Percentage of strategic behavior of the women can be very different women who benefit by cheating 8.77 11.00 11.09 10.22 4.675 under the models with and without rejection. We saw that in our model, an optimal cheating strategy for a The cumulative distribution of the number of woman did not always guarantee her women-optimal women who benefited by cheating is shown in partner. Moreover, in sharp contrast to the rejection Figure 3. model, we showed that the chances of a woman ben- For each value of k, the y axis shows the cumula- efitting by cheating are small. tive number of women who can benefit from cheat- The second part of our study emphasized the ing. For instance, for k = 1, in close to 900 out of 1,000 importance of adopting stable matching mechanisms experiments, at least one woman benefits from cheat- in the context of the Singapore Secondary School posting. Note that in the many-to-one model with only ing exercise. There is an enormous amount of litera- one school, clearly the students cannot benefit from ture on the college admissions problem, which sheds cheating. This number drops to slightly over 800 for light on the viability of the matching mechanisms k = 3, slightly over 700 for k = 5, slightly over 500 for we proposed. For instance, a variant of the student- k = 10, and finally to around 200 for k = 20. Interest- propose mechanism suggested here has recently been recommended to be the official posting method to ingly, the number of times that no women benefit by assign interns to hospitals in the United States (Roth cheating (i.e., optimal strategy is for all the women to and Peranson 1997). The high participation rate in this tell the truth) seems to increase as k increases (From interns/hospitals market, despite the fact that partic- 100 for k = 1 to 800 for k = 20). Translated to the ipation is voluntary, shows the relevance and impor- many-to-one model, it indicates that, for instance, in tance of ensuring stability. For the Singapore posting the case k = 20 (i.e. 20 schools and 60 students), the exercise, however, participation rate is not the cen- chances that no students can benefit from cheating tral theme of the problem, since the students do not is at least 80%. It indicates that if there are relatively have a choice in participation. Interestingly, exploiting many schools in the many-to-one model compared this unique feature of the problem (that the student to the total number of students, cheating becomes cannot say “no”), we showed that a stable match- increasingly impossible. ing mechanism is still appropriate in this context. The simulation experiments also showed a interest- The current assignment process used by the MOE ing feature: The proportion of women who can benefit seems to place undue emphasis on the parents mak- from cheating hovers around 10% for all choices of ing intelligent, “realistic” choices; this is troublesome k, even though the preference lists are generated in for the parents, especially because they have to make a special manner. The numbers are similar to those their decisions before their children appear for the in the earlier experiment when the preference lists primary school leaving examination. We argued that for the one-to-one model are generated in a random adopting a stable matching mechanism results in a fashion. better system. We showed further that regardless of

Management Science/Vol. 47, No. 9, September 2001 1265 TEO, SETHURAMAN, AND TAN Gale-Shapley Stable Marriage Problem Revisited

Figure 3 Cheating in the Many-to-One Model

the particular stable matching mechanism used (the To summarize, our analysis of the strategic issues students-optimal or the schools-optimal mechanism), suggests that stable matching mechanisms can be there is no significant incentive for the students to used to address most of the shortcomings of the cur- cheat, under the perfect information scenario. Since rent assignment process. In particular, it can be used the students have little reason to cheat in the two to address the key concern of most parents: “How extremal stable matching mechanisms, it is natural to do I choose the schools if I do not know my child’s suspect that this result carries over to all the “inter- standing at the national level?” The answer to that, mediate” stable matchings as well; we leave this as a regardless of the stable matching mechanism used, is: challenging problem for future research. “Just tell the truth.” As for the strategic behavior of the schools, the adoption of any stable matching mechanism would Acknowledgments give incentives for some schools to falsify their prefer- The authors thank the editor and the two anonymous referees ences or even to falsify the number of places available for pointing many errors and omissions in earlier versions of this (see Sonmez 1997 on how the schools can manipu- paper. In particular, the referees made several suggestions that late capacities to obtain a better match under both improved the paper significantly, both in form and substance. This research was supported by the National University of Singapore students-optimal and schools-optimal mechanisms). under grant RP 3970021 and a fellowship from the Singapore-MIT In the Singapore context, however, the latter is rela- Alliance program. tively unimportant as the schools are tightly regulated by the government, and the number of places available are determined by the Ministry. Furthermore, the ranking of the students, according to the schools, is References usually done by some broad selection criteria (e.g., Dubins, L.E., D.A. Freedman. 198l. Machiavelli and the Gale- Shapley algorithm. Amer. Math. Monthly 88 485–494. examination results, affiliation, etc.) rather than at an Gale, D., L.S. Shapley. 1962. College admissions and the stability of individual level. Hence the issues of manipulation marriage. Amer. Math. Monthly 69 9–15. (both rank and capacity) on the schools’ part is less , M. Sotomayor. 1985a. Some remarks on the stable matching of a concern in this context. problem. Discrete Appl. Math. 11 223–232.

1266 Management Science/Vol. 47, No. 9, September 2001 TEO, SETHURAMAN, AND TAN Gale-Shapley Stable Marriage Problem Revisited

, . 1985b. Ms Machiavelli and the stable matching . 1984b. Stability and polarization of interests in job match- problem. Amer. Math. Monthly 92 261–268. ing. Econometrica 52 47–57. Gardenfors, P. 1975. Match making: Assignments based on bilateral . 1985. The college admissions problem is not equivalent to preferences. Behavioural Sci. 20 166–173. the marriage problem. J. Econom. Theory 36 277–288. Gusﬁeld, D., R.W. Irving. 1989. The Stable Marriage Problem: Struc- . 1990. New physicians: A natural experiment in market ture and Algorithms. MIT Press, Cambridge, MA. organization. Science 250 1524–1528. Irving, R.W. 1994. Stable marriage and indifference. Discrete App. , E. Peranson. 1997. The effects of the change in the Math. 48 261–272. NRMP matching algorithm. J. Amer. Medical Assoc. 278 Kara, T., T. Sonmez. 1996. Nash implementation of matching rules. 729–732. J. Econom. Theory 68 425–439. , U. Rothblum. 1999. Truncation strategies in matching mar- Knuth, D.E. 1976. Marriages Stables. Les Presses de l’Unversite’de kets: In search of advice for participants. Econometrica 67 Montreal, Montreal, Canada. (January), 21–43. McVitie, D.G., L.B. Wilson. 1970. Stable marriage assignments for , M. Sotomayor. 1991. Two-sided Matching: A Study in Game- unequal sets. BIT 10. theoretic Modeling and Analysis. Cambridge University Press, , . 1971. The stable marriage problem. Comm. ACM 10 Cambridge, MA. 486–490. Sonmez, Tayfun. 1997. Manipulation via capacities in two-sided Pittel, B. 1989. The average number of stable matchings. SIAM matching markets. J. Econom. Theory 77 197–204. J. Discrete Math. 530–549. Tadenuma, K., M. Toda. 1998. Implementable stable solu- Roth, A.E. 1982. The economics of matching: Stability and incen- tions to pure matching problems. Math. Social Sci. 35(2) tives. Math. Oper. Res. 7 617–628. 121–132. . 1984a. The evolution of the labor market for medical Teo, C.P., J. Sethuraman. 1998. The geometry of fractional interns and residents: A case study in game theory. J. Political stable matching and its applications. Math. Oper. Res. 23(4) Econom. 92 991–1016. 874–891.

Accepted by Bruce L. Golden; received January 1999. This paper was with the authors 10 months for 2 revisions.

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