The singleton core in the college admissions problem and its application to the National Resident Matching Program (NRMP)

Games and Economic Behavior 69 (2010) 150–164

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The singleton core in the college admissions problem and its application to the National Resident Matching Program (NRMP) ✩ Jinpeng Ma Department of Economics, Rutgers University-Camden, NJ 08102, United States

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 30 December 2006 Available online 4 January 2010

We show that in the marriage problem the student-optimal algorithm may in fact generate an equilibrium outcome that is college-optimal and student-pessimal in terms of the true preferences even though it is student-optimal and college-pessimal in terms of the submitted preferences. In the college admissions problem, the student-optimal algorithm generates either a matching that is not stable for the true preferences or a matching that is college-optimal and student-pessimal in terms of the true preferences. Thus, our results show that, in the absence of certain match variations, the newly designed student-optimal algorithm adopted by the NRMP since 1998 either may be bias in favor of hospitals in terms of the true preferences or fails to produce a true stable matching. We also discuss when the core is large and when the core is a singleton at a Nash equilibrium. © 2010 Elsevier Inc. All rights reserved.

JEL classification: C78 D71

1. Introduction Consider the marriage problem. As a man, which algorithm do you choose, the man-optimal algorithm or the womanoptimal algorithm? Now consider the college admissions problem. As a student, which algorithm do you choose, the studentoptimal algorithm or the college-optimal algorithm? These two questions turn out to be in the center of a controversy about algorithm choice in the National Resident Matching Program (NRMP), a clearinghouse mechanism that matches about 20,000 graduating medical students with residency programs in the USA each year since 1951. Prior to 1998, the NRMP used essentially the hospital-optimal and studentpessimal algorithm. For a given set of submitted rank-order lists (ROLs) of preferences, this preexisting algorithm generates a stable matching that is hospital-optimal and student-pessimal for each submitted rank-order lists (ROLs) (Roth, 1984a). Williams (1995a, 1995b) criticized the use of the hospital-optimal and student-pessimal algorithm of the NRMP to be biased in favor of hospitals at the expense of students and to give students the incentives for misreporting of their ROLs. The criticisms raised by Williams (1995a) received a great deal of attention around the time from the American Medical Student Association (AMSA), which was together with the Public Citizens’ Health Research Group to strongly voice a change in algorithm in the NRMP (AMSA and Public Citizen Health Research Group, 1995): The direction in which the algorithm bias should lie now becomes the obvious question. There are several compelling reasons why a student-optimal algorithm would be a fair and more reasonable choice of stable algorithms than the current hospital-optimal algorithm. First, the historical basis for choosing the current algorithm was to eliminate the ✩ I thank Vince Crawford, Fuhito Kojima, Al Roth, Tayfun Sonmez, Kevin J. Williams, and, especially, Bob Aumann and the anonymous referee for helpful comments that have greatly improved the paper. I also thank audiences of the Symposium in honor of Robert Aumann and of the 2006 SED meeting in Bodrum. E-mail address: [email protected].

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J. Ma / Games and Economic Behavior 69 (2010) 150–164

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emphasis the pre-match process placed on the students’ strategic ability as a factor in residency placement. As an orderly, centralized system, the current hospital-optimal algorithm has somewhat ameliorated this strategic component though it still retains some of the incentives for students to misrepresent their preferences; a student optimal-algorithm would remove these incentives and complete this goal. Second, since it is impossible to remove all the incentives for hospitals to misrepresent, it would be best to choose the student-optimal algorithm to remove incentives, at least for students. In other words, within the set of stable algorithms, you either have incentives for both the hospitals and the students to misrepresent their true preferences or only for the hospitals. Third, there is reason to believe that hospitals are not as particular about precise assignments as are individual students and would subsequently lose very little in a student-optimal algorithm. Finally, there is evidence that minority students are disproportionately at risk from the hospital-optimal algorithm’s bias. Minority students tend to match with less frequency to their stated first choices than other groups of students. Because of this, a significantly larger portion of them are exposed to the negative effects of the hospital optimal bias. Granted, changing the algorithm to a student-optimal one will not deter any given hospital from composing a preference list under racially biased considerations (this falls outside the algorithm’s sphere of influence). What it would do though is remove the unbalanced, added negative effects levied against minority students made possible by the hospital-optimal algorithm. In 1998, the NRMP decided to adopt the Roth–Peranson newly designed student-optimal and hospital-pessimal algorithm. The new algorithm has been in use since then in the NRMP and in more than thirty specialty matches. If without certain match variations such as married couples and supplemental rank order lists, this new algorithm produces a stable matching, for each set of submitted ROLs, that is student-optimal and hospital-pessimal in the sense that each student (hospital) likes it at least (most) as well as any other stable matchings; see, e.g., Gale and Shapley (1962), Roth (1985), Roth and Sotomayor (1990), Roth and Peranson (1999). Thus, the new algorithm has been commonly believed to be more in favor of students at the expense of hospitals. It is exactly this view we like to question in this paper. It is true that the student-optimal algorithm reaches a stable matching that all students like the most among all stable matchings in terms of submitted preferences. But is it also true that such a stable matching is the one that all students like the most in terms of their true preferences? That is, is the change from the hospital-optimal algorithm to the student-optimal algorithm more beneficial to students for the true preferences? Of course, what matters the most is the true preferences not the submitted ones. We also know that the submitted preferences may well be different from the true ones, since Roth’s impossibility theorem shows that no stable algorithm exists such that it is always the best for every agent to reveal her true preferences. If the submitted preferences are different from the true preferences, an answer to the question is urgently in demand. In this paper we study the marriage and the college admissions problems, the two theoretical models on which the NRMP market is based. We show that the student-optimal algorithm may not fulfill its goal as what has been commonly believed. Our results show that the student-optimal algorithm may in fact generate an equilibrium outcome that is hospitaloptimal and student-pessimal in terms of the true ROLs of preferences, even though the outcome is student-optimal and hospital-pessimal in terms of the submitted ROLs of preferences. Our results show that the hospital-optimal algorithm may not do worse than the student-optimal algorithm from the aspect of students’ welfare, in terms of their true preferences. As a result, we show that the criticisms made by Williams (1995a, 1995b) and the American Medical Student Association (AMSA) and the Public Citizens’ Health Research Group about the bias of the hospital-optimal algorithm, while true for the submitted preferences, may not be true for the true preferences. A truncation strategy for a student or a hospital is a ROL of preference that is order-consistent with her or its true ROL of preferences but has fewer acceptable hospitals or students. Truncation strategies exclude some other more complicated strategies such as change in orders that may be profitable for misreporting. Roth and Rothblum (1999) studied the marriage problem and showed that more complicated and profitable strategies other than truncation do exist. But they also showed that players need to know all preferences of the other players in order to exploit the benefits of such complicated strategies in manipulation. They showed why this class of simple strategies in truncation are plausible in an environment with low information about the preferences of all other players. Williams (1995a, 1995b) and Peranson and Randlett (1995) both suggested that students can use truncation to be profitable in misreporting under the hospital-optimal algorithm. The Nash equilibrium identified by this paper for the student-optimal algorithm is as follows: Each student chooses her dominant strategy by submitting her true ROLs of preferences and each hospital program chooses an equilibrium strategy that is a truncation of its true preferences (Roth and Vande Vate, 1991; Roth and Rothblum, 1999). Then we show that in the marriage problem, any outcome of such an equilibrium must be hospital-optimal and student-pessimal in terms of the true preferences. In the college admissions problem, any outcome of such an equilibrium must be either instable or hospital-optimal and student-pessimal in terms of the true preferences. In the college admissions problem, when hospitals are restricted to report just truncations, a Nash equilibrium always exists. Moreover, the equilibrium outcome, if not stable for the true profile, will give no hospital worse interns than those under the hospital-optimal and student-pessimal true stable matching. That is, with truncations alone in manipulation, the worst that a hospital can obtain is the interns under the hospital-optimal and student-pessimal true stable matching. With respect to algorithms’ bias, the student-optimal algorithm may be even worse than the hospital-optimal algorithm from the welfare aspect of students.

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The intuition of our results is quite simple. Consider the marriage problem. Once the man-optimal algorithm is used, when all women report truthfully, the algorithm indeed produces the man-optimal stable matching in terms of the true preferences. But truthful reporting by women is not rational for women. By telling a lie, women can achieve a matching that should be no worse than by telling the truth. With truncation, women can in fact achieve the true women-optimal stable matching. This intuition may be summarized by the following, quoted from the book by Harry G. Frankfurt, On Bullshit, published by Princeton University Press in 2005: Someone who lies and someone who tells the truth are playing on opposite sides, so to speak, in the same game. Each responds to the facts as he understands them, although the response of the one is guided by the authority of the truth, while the response of the other defies that authority and refuses to meet its demands. Telling a lie is an act with a sharp focus. It is designed to insert a particular falsehood at a specific point in a set or system of beliefs, in order to avoid the consequences of having that point occupied by the truth. This requires a degree of craftsmanship, in which the teller of the lie submits to objective constraints imposed by what he takes to be the truth. The liar is inescapably concerned with truth-values. In order to invent a lie at all, he must think he knows what is true. And in order to invent an effective lie, he must design his falsehood under the guidance of that truth. Gale and Sotomayor (1985) studied the marriage problem and identified a woman strong Nash equilibrium under the man-optimal algorithm whose outcome is the true woman optimal stable matching. In their woman strong Nash equilibrium, each man states his true preferences, and each woman states a strategy that is a truncation of her preferences up to her mate, if any, under the true woman optimal stable matching. Thus, to play such an equilibrium, all women must know their mates under the true woman optimal stable matching; see Roth and Sotomayor (1990) for a discussion. Our result for the marriage problem is similar to that in Gale and Sotomayor (1985) in terms of the outcome. However, examples of the marriage problem show that there are many other Nash equilibria that are covered by our result but not by Gale and Sotomayor (1985). We provide several examples to show how robust our results are. Example 5 in Section 5 is a marriage problem that has two stable matchings: the true man-optimal and the true woman-optimal stable matchings. In the game induced by the man-optimal algorithm under which each man chooses his dominant strategy and states his true preferences, there are seven Nash equilibria in pure strategy. Three Nash equilibria give rise to the true woman-optimal stable matching and four other give rise to the true man-optimal stable matching. But only the first three Nash equilibria are trembling-hand perfect. They are the only Nash equilibria in truncation. The other four are not Nash equilibria in truncation and they are not immune to Selten’s trembling-hand errors.1 Example 11 in Section 6.1 is a college admissions problem that has a unique stable matching for the true profile. Under the student-optimal algorithm, there is a unique Nash equilibrium in truncation which is also the only trembling-hand perfect Nash equilibrium. But the outcome of this equilibrium is not stable for the true preferences. There are six other Nash equilibria that give rise to the true stable matching. But none of them is Nash equilibrium in truncation and passes the test of trembling-hand errors. With Examples 5 and 11, we also show how it is possible for a college to be beneficial from overreporting its quota, which is not possible in the game studied by Sonmez (1997). Example 14 in Section 6.1 is a college admissions problem that has two stable matchings for the true profile. Under the student-optimal algorithm, any Nash equilibrium whose outcome is not the true college-optimal stable matching is not trembling-hand perfect. All trembling-hand perfect Nash equilibria give rise to the true college-optimal stable matching. In the paper, we also discuss when the core is small and when it is large. Roth and Peranson (1999) remarkably found out that the set of stable matchings (the core) in the NRMP for a given profile of reported ROLs is quite small. For the same set of reported ROLs, the switch from the original hospital-optimal to the new student-optimal algorithm only affected few applicants (approximately 0.1%). If the core is found to be small, “there are very few opportunities for participants to engage in strategic manipulation of their stated preferences when it comes to making and accepting offers” (Roth and Peranson, 1999). A natural question is whether such a remark also applies to the true preferences. An answer to the question depends on how much one may know about participants’ true preferences from their stated equilibrium preferences. We show that very little information about the true preferences can be derived from stated preferences. There are situations under which the set of stable matchings in terms of the true preferences is a singleton yet the set of stable matchings in terms of stated preferences at a Nash equilibrium is very large. There are also situations such that the set of stable matchings in terms of true preferences is very large yet the set of stable matchings in terms of stated preferences at a Nash equilibrium is a singleton. The paper is organized as follows: Section 2 introduces the marriage and the college admissions problem. Section 3 shows that the core at a Nash equilibrium can be very large. Section 4 discusses when the core at a Nash equilibrium is a singleton. Section 5 presents the main result for the marriage problem. Section 6 presents the main result for the college admissions problem. Appendix A concludes the paper.

1

Each Nash equilibrium here uses strategies that are ‘bad’ defined by Roth and Sotomayor (1990).

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2. The model We use some definitions from Roth and Sotomayor (1990) (also see Ma, 2002). The college admissions problem consists of two finite and disjoint sets, a set S = { S 1 , . . . , S n } of students and a set C = {C 1 , . . . , C r } of colleges (or hospitals), with each college C j ∈ C having a quota q C j  1 of enrollments. Each student S i ∈ S is enrolled in at most one college and has strict preferences P S i over C ∪ { S i }, leaving the possibility that student S i may prefer not to go to colleges. For example, suppose P S i is C 1 , C 2 , S i , C 3 , . . . . Then this means that C 1 is the college student S i prefers the most. He prefers college C 2 to “not go to college”. College C 3 and all others are unacceptable for S i . Each college C j ∈ C has strict preferences P C j over S ∪ {C j }. For example, suppose P C j is S 1 , S 2 , C j , S 3 , . . . . Then this means that college C j prefers to enroll student S 1 rather than student S 2 . She prefers to enroll student S 2 rather than “leave a position unfilled”. Student S 3 and all others are unacceptable for C j . Let R S i and R C j denote the weak preferences associated with P S i and P C j respectively. Let Ω S i denote the strict preferences for student S i ∈ S and ΩC j denote the set of all strict preferences for college C j .  set of all  Let Ω = S i ∈ S Ω S i × C j ∈C ΩC j denote the set of all preference profiles. The marriage problem is the college admissions problem with q C j = 1 for every college C j ∈ C . For the marriage problem, we identify S as the set of men M and C the set of women W . Define an unordered family of elements of any set X to be a collection of elements in which the order is immaterial. The set of unordered families of elements of X is denoted by X .

Definition. A matching μ is a function μ : S ∪ C → S ∪ C such that (a) |μ( S i )| = 1 for every S i ∈ S and μ( S i ) ∈ C whenever μ( S i ) = S i ; (b) |μ(C j )| = qC j for every C j ∈ C , and if | S ∩ μ(C j )| < qC j then μ(C j ) is filled to qC j by copies of C j ; (c) μ( S i ) = C j if and only if S i ∈ μ(C j ). Let M denote the set of all matchings. Definition. Let μ, λ ∈ M be two matchings. We say that a preference P¯ C j for a college C j over sets of students is responsive to its preference P C j over individual students if, whenever μ(C j ) = λ(C j ) ∪ S k \ {τ } for τ ∈ λ(C j ) and S k ∈ / λ(C j ), then

μ(C j ) P¯ C j λ(C j ) if and only if S k P C j τ .

Gale and Shapley (1962) originated the study of the college admissions problem. Roth (1985) reformulated the problem and introduced the notion of responsive preferences. Henceforth, we assume that colleges’ preferences over groups of students are complete, transitive, and responsive. We always use the notation P¯ C j to designate college C j ’s (responsive) preferences over groups of students and P C j without a bar for college C j ’s preferences over individual students. A pair of a student S i and a college C j blocks a matching μ if they are not matched under μ but student S i prefers college C j to his assignment μ( S i ) and college C j prefers student S i to some member σ ∈ μ(C j ), i.e., C j P S i μ( S i ) and S i P C j σ for some σ ∈ μ(C j ). Definition. Given a profile P ∈ Ω , a matching μ is (a) individually rational if μ( S i ) R S i S i for all S i ∈ S and σ R C j C j for every σ ∈ μ(C j ) for all C j ∈ C ; (b) pairwise stable if it is not blocked by any pairs of a student and a college; (c) stable if it is both individually rational and pairwise stable. Let I R ( P ) and S ( P ) denote the set of all individually rational matchings and the set of all stable matchings respectively with respect to a profile P ∈ Ω . Definition. A (matching) mechanism ϕ : Ω → M is a map from profiles to matchings. A mechanism if ϕ ( Q ) ∈ S ( Q ) for all Q ∈ Ω . Let Φ denote the set of all stable mechanisms.

ϕ : Ω → M is stable

It follows from Lemma 5.6 in Roth and Sotomayor (1990) that S ( P ) is nonempty for each profile P ∈ Ω . Therefore, the set of stable mechanisms Φ is nonempty. A stable  mechanism  ϕ ∈ Φ and an underlying true profile P ∈ Ω induce a normal form game Γ (ϕ , P ) as follows. The set Ω = S i ∈ S Ω S i × C j ∈C ΩC j is the set of strategies of the game Γ (ϕ , P ) and the outcome function is ϕ . Our definitions of strictly and weakly dominated strategies follow from standard textbooks of game theory (see e.g. Mas-Colell et al., 1995). Definition. Given a game Γ (ϕ , P ), a strategy Q i for a player i dominates another strategy Q i for the same player if player i likes ϕi ( Q i , Q −i ) at least as well as ϕi ( Q i , Q −i ) for all possible profiles Q −i of strategies from other players. A dominant strategy for a player i is a strategy Q i∗ such that Q i∗ dominates his/her all other strategies Q i . Definition. Given a game Γ (ϕ , P ), a strategy Q i for player i weakly dominates another Q i if Q i dominates Q i and if, there exists at least one Q −i such that player i prefers ϕi ( Q i , Q −i ) to ϕi ( Q i , Q −i ). A strategy Q i for player i strictly dominates another Q i if, for all Q −i , player i prefers ϕi ( Q i , Q −i ) to ϕi ( Q i , Q −i ). Definition. A profile Q ∈ Ω is a Nash equilibrium of a game Γ (ϕ , P ) if



ϕ S i ( Q − S i , Q S i ) R S i ϕ S i Q − S i , Q S i



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for all S i ∈ S, Q S ∈ Ω S i and i



ϕC j ( Q −C j , Q C j ) R¯ C j ϕC j Q −C j , Q C j



for all C j ∈ C , Q C ∈ ΩC j . j

Let N (ϕ , P ) the set of all Nash equilibria in pure strategy of the game Γ (ϕ , P ). We follow Roth and Vande Vate (1991) and Roth and Rothblum (1999) to introduce a class of simple strategies called “truncation”. This class of strategies are introduced for the marriage problem in these papers. A college C j is acceptable to a student S i if C j R S i S i . A student S i is acceptable for a college C j if S i R C j C j . A truncation strategy Q S i (with respect to P S i ) for a student S i contains k (0  k) acceptable colleges such that the first k elements of Q S i are the first k elements, in the same order, in her true preference P S i , and the (k + 1)th element in Q S i is S i . The ordering after the (k + 1)th element S i in Q S i does not matter in our study. Similarly, a truncation strategy Q C j for a college C j (with respect to P C j ) contains k (0  k) acceptable students such that the first k elements of Q C j are the first k elements, in the same order, in her true preference P C j , and the (k + 1)th element in Q C j is C j . Again the ranking after the (k + 1)th element C j does not matter. Truncation strategies exclude some other more complicated strategies such as changes in orders that may be profitable for misrepresentation. Roth and Rothblum (1999) showed that more complicated and profitable strategies other than truncation do exist. But they also show that players need to know all preferences of the other players in order to exploit the benefits of such complicated strategies in manipulation. They convincingly showed why this class of simple strategies in truncation are plausible for the marriage problem in an environment with low information about the preferences of all other players. In the Sorority rush market Mongell and Roth (1991) found that truncation strategies are in fact used by players in practice: Players often truncate after their first choice. A Nash equilibrium Q of a game Γ (ϕ , P ) is a Nash equilibrium in truncation if all equilibrium strategies are in truncation. 3. Large core at Nash equilibrium Consider a marriage problem ( M , W , P n ) with n men and n women, where P n is a profile of preferences with every pair ( w , m) mutually acceptable (Roth and Sotomayor, 1990). We will first provide an example that has a very large core at a Nash equilibrium. Theorem 1. Let ϕ M be the man-optimal algorithm. For each i  0 there exist a marriage problem ( M , W , P n ) of size n = 2i and a Nash equilibrium Q n of the game induced by ϕ M and P n such that the number of stable matchings in terms of the Nash equilibrium Q n is at least 2n−1 . The main idea in the proof of this result is to use a result in Irving and Leather (1986) to construct a marriage problem and a Nash equilibrium to have the property stated in the theorem. Proof. Irving and Leather (1986) or Theorem 3.19 in Roth and Sotomayor (1990) has shown that there exists a marriage problem ( M , W , Q n ) such that the number of stable matchings in terms of profile Q n is at least 2n−1 . Let μ be the man-optimal stable matching for ( M , W , Q n ). We now construct a marriage problem ( M , W , P n ) such that Q n is a Nash equilibrium for the game induced by ϕ M and P n . For each man m ∈ M, his true preference P n (m) is the same as his equilibrium preference Q n (m). For each woman w ∈ W , since μ( w ) = w, let P n ( w ) = (μ( w ), . . .). Under the man-optimal algorithm ϕ M , it is known from Roth (1982) or Roth and Sotomayor (1990) that the true preference P n (m) is a dominant strategy for each man m. Clearly, Q n ( w ) is a Nash equilibrium strategy for each woman w since each women w is matched at Q n under ϕ M with a mate μ( w ) she likes the most under her true preference P n ( w ). This completes the proof. 2 Theorem 1 has shown that the maximum number of stable matchings in terms of submitted preferences at certain Nash equilibria can grow exponentially as the size n of the market grows. From the above proof, one can also see that the set of stable matchings in terms of submitted preferences at the equilibrium Q n is not necessarily small, even though there is a unique stable matching in terms of the true profile P . That is, even though the core for the true profile P n is very small, the core for a Nash equilibrium Q n can be very large. For a marriage problem in which not every pair (m, w ) is mutually acceptable, one can construct the profile P in the proof of Theorem 1 for the women by P ( w ) = (μ( w ), . . .) for all w such that μ( w ) = w and P n ( w ) = ( w , . . .) otherwise. With such a minor modification in the construction of the true profile P , we obtain an even stronger result. Corollary 2 below shows that given any profile of preferences Q , one can find a marriage problem ( M , W , P ) such that Q is a Nash equilibrium for the game induced by ϕ M and P . Since Q is arbitrary, this result implies that there are many Nash equilibria Q such that the set of stable matchings in terms of Q is not small.

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Corollary 2. Let ( M , W , Q ) be any marriage problem, not necessarily with an equal number of men and women, and some pairs ( w , m) may not be mutually acceptable. Then there exists a marriage problem ( M , W , P ) such that Q is a Nash equilibrium of the game induced by ϕ M and P . Similar results to Corollary 2 can be shown for the college admissions problem for both the college-optimal and the student-optimal algorithm, with some modifications in the construction of the true profile P . 4. Singleton core at Nash equilibrium In the NRMP, Roth and Peranson (1999) found out that the set of stable matchings (the core) in the NRMP for a given profile of reported ROLs is quite small, though not singleton. For the same set of reported ROLs, the switch from the original hospital-optimal to the new student-optimal algorithm only affected few applicants (approximately 0.1%). In the course of designing, testing, and evaluating the new clearinghouse algorithm [adopted by the NRMP since 1998], some surprising properties of large labor markets emerged. The high transaction costs involved in interviewing place a practical limit on how many interviews are conducted, and one consequence of this is that the set of stable outcomes is very small, and there are very few opportunities for participants to engage in strategic manipulation of their stated preferences when it comes to making and accepting offers. (Neither of these would be the case in the absence of transaction costs.) (Roth and Peranson, 1999) In many specialty matches where there are no match variations as in the NRMP, the core in terms of submitted preferences is not just small. It is essentially singleton. For examples, Elliott Peranson (see Peranson and Randlett, 1995) had conducted some preliminary experiments on the dental residencies in the United States, with approximately 2000 students, and the first-year positions with law firms in Canada, with approximately 800 students, and found that the set of stable matchings in terms of submitted preferences for each of these two entry-level labor markets is indeed a singleton: For the same set of submitted preferences, the student-optimal and the hospital-optimal algorithm produced an identical matching. Roth and Peranson (1999) had studied the matches of Thoracic Surgery from year 1991 to 1996 and found that the matches of 1991, 1994 and 1996 all have a singleton core. Only two students were affected in the matches of 1992 and 1993 by the change in algorithm, which may be the consequence of some minor deviations from the equilibrium strategies.2 Dr. August Colenbrander, Coordinator of the Ophthalmology Matching Program, had studied 24 past Ophthalmology matches and found that 20 out of these 24 matches have a singleton core. The cores of the other four matches are essentially singleton. In three matches, two students are affected and in one match three students are affected by the change in algorithm.3 Theorem 1, the large core at a Nash equilibrium, provides a sharp contrast to the empirical findings of the small core in the NRMP and of the singleton cores in the specialty matches. Under what circumstances is the core singleton, not just small? Ma (2002) has provided an answer with Nash equilibrium in truncation. This result will be needed in our proofs below. Let ϕ be any stable algorithm. The set of all Nash equilibria of the game Γ (ϕ , P ) can be decomposed into two sets, N 1 (ϕ , P ) and N 2 (ϕ , P ), where N 1 (ϕ , P ) is the set of all Nash equilibria Q such that S ( P ) ∩ S ( Q ) = ∅ and N 2 (ϕ , P ) is the ˜ (ϕ , P ) be the set of all Nash equilibria in truncation. set of all Nash equilibria Q such that S ( P ) ∩ S ( Q ) = ∅. Let N

˜ (ϕ , P ) in truncation, the set of stable matchings S ( Q ) contains a single Theorem 3. (See Ma, 2002.) (a) For all Nash equilibria Q ∈ N matching ϕ ( Q ). (b) For all Q ∈ N 2 (ϕ , P ), the equilibrium outcome ϕ ( Q ) is a true stable matching, no matter how many stable matchings are there in terms of Q . Theorem 3(a) shows that the core at a Nash equilibrium profile in truncation must be a singleton. Theorem 3(b) shows that although there may be many matchings that are stable in terms of a submitted equilibrium profile Q , as shown in Theorem 1 and Corollary 2, the equilibrium outcome ϕ ( Q ) must be stable for the true profile P , as long as Q does admit a true stable matching. 5. Main result: the marriage problem Theorem 4. Consider the marriage problem with strict preferences. Let ϕ M be the man-optimal algorithm. Suppose that Q is a reported profile of preferences such that each man chooses his dominant strategy and reports his true preferences and each woman chooses a strategy in truncation in the matching game Γ (ϕ M , P ) induced by ϕ M and the underlying true profile of preferences P . If the reported profile Q is a Nash equilibrium, then the equilibrium outcome ϕ M ( Q ) is the true woman optimal stable matching. Moreover, the set

2 As seen above, Roth and Peranson (1999) used mainly the transaction costs to explain the small core in the NRMP markets. The transaction costs may explain why the core is small. But it is not ideal to explain why the core is in fact a singleton for so many specialty matches. 3 See February 1996 press release: “Ophthalmology Match Needs Students’ Concerns”.

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of stable matchings S ( Q ) with respect to the reported equilibrium profile Q is a singleton, no matter how many true stable matchings are there in S ( P ). That is, S ( Q ) contains nothing but the true woman optimal stable matching. Kara and Sönmez (1997) has showed that no selection of the core is implementable in Nash equilibrium. Theorem 4 shows that the woman (man) optimal stable matching is implemented by the man (woman) optimal algorithm in a subset of Nash equilibria that are in truncation,4 a result that is quite peculiar in nature. It shows that the man-optimal algorithm ϕ M does not generate the true woman optimal stable matching at a Nash equilibrium Q under which all men state their true preferences only if some women do not use strategies in truncation, resulting in a true stable matching (Roth, 1984b) that all women prefer less than otherwise. But Roth and Rothblum (1999) showed that strategies in truncation are the class of profitable strategies in manipulation that require the least information about others’ preferences (thus the simplest class in comparison with changes in orders); also see Roth and Peranson (1999), Roth and Vande Vate (1991). For the game induced by the man-optimal algorithm, Gale and Sotomayor (1985) identified a woman strong Nash equilibrium whose outcome is the true woman optimal stable matching. In their woman strong Nash equilibrium, each man states his true preferences, and each woman states a strategy that is a truncation of her preferences up to her mate, if any, under the true woman optimal stable matching. This woman strong Nash equilibrium may deserve coordination in behavior among all women; see Roth and Sotomayor (1990) for a discussion. The class of equilibria covered by Theorem 4 is different in idea and only depends on the unilateral behavior from the women side, no group coordination among women is in fact needed. Example 5 shows that there are many equilibria that are covered by Theorem 4 but not by Gale and Sotomayor (1985). In this example, there are two true stable matchings. All Nash equilibria that are also trembling-hand perfect under the man-optimal algorithm give rise to the true woman-optimal stable matching.5 Those equilibria whose outcomes are the true man-optimal stable matching are the equilibria that are not trembling-hand perfect. Whether a Nash equilibrium passes the test of trembling-hand errors is important in practice, since players eventually have to make some good guessing (subject to small errors) about others’ strategies before submitting their own ones. Example 5. There are two men: m1 , m2 , and two women: w 1 , w 2 . The true profile of preferences P is given as follows:

P m1 = ( w 1 , w 2 , m 1 )

P m2 = ( w 2 , w 1 , m 2 )

P w 1 = (m2 , m1 , w 1 )

P w 2 = (m1 , m2 , w 2 )

There are two stable matchings with respect to P . One is the woman optimal stable matching





μW = ( w 1 , m2 ); ( w 2 , m1 )

and the other the man optimal stable matching





μM = ( w 1 , m1 ); ( w 2 , m2 )

When the man-optimal algorithm is used and both women report their true preferences in a straightforward manner, the man optimal stable matching μ M is obtained. However, the women can do better via manipulation of their true preferences. Consider the matching game Γ (ϕ M , P ) induced by the man-optimal algorithm ϕ M and P , under which all men choose their dominant strategies and report their true preferences. There are six strategies for each woman w ∈ { w 1 , w 2 }: (m2 , m1 , w ), (m1 , m2 , w ), (m1 , w , m2 ), (m2 , w , m1 ), ( w , m1 , m2 ) and ( w , m2 , m1 ). But the last two strategies are in an equivalent class, which can be represented by one strategy. Since each man states his true preferences, the game Γ (ϕ M , P ) can be represented in a strategic form as follows, where w 1 is the row player and w 2 is the column player. Nash equilibria and strategies in truncation are highlighted. Note that if a woman is matched with the man she likes the most in her true preferences, she is assigned a payoff 2. If she is matched with the man she likes the second, she is assigned a payoff 1. If she is unmatched with any man, she is assigned a payoff 0. For example, if w 1 is matched with m2 , w 1 has a payoff 2. Consider the following three Nash equilibria in truncation:

Q = ( P m1 , P m2 , Q w 1 , Q w 2 ) where

a) Q w 1 = (m2 , w 1 , m1 ), Q w 2 = (m1 , w 2 , m2 ); b) Q w 1 = (m2 , m1 , w 1 ), Q w 2 = (m1 , w 2 , m2 ); c) Q w 1 = (m2 , w 1 , m1 ), Q w 2 = (m1 , m2 , w 2 ). 4

The existence of such an equilibrium follows from Gale and Sotomayor (1985). Our trembling-hand perfect Nash equilibrium of the game induced by the man or student optimal algorithm does not allow the men or students to make mistakes. This makes sense since all men or students know their true preferences without errors and they state their true preferences with no errors. 5

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157

Table 1 Matching game: w 1 the row player and w 2 the column player.

(m2 , m1 , w 1 ) (m2 , w 1 ) (m1 , m2 , w 1 ) (m1 , w 1 ) (w1)

(m2 , m1 , w 2 )

(m1 , m2 , w 2 )

(m1 , w 2 )

(m2 , w 2 )

(w2)

(1, 1) (0, 1) (1, 1) (1, 1) (0, 1)

(1, 1) (2, 2) (1, 1) (1, 1) (0, 2)

(2, 2) (2, 2) (1, 0) (1, 0) (0, 2)

(1, 1) (0, 1) (1, 1) (1, 1) (0, 1)

(2, 0) (2, 0) (1, 0) (1, 0) (0, 0)

Case a is the woman strong Nash equilibrium identified by Gale and Sotomayor (1985). Cases b and c are the two new equilibria covered by Theorem 1 but not by Gale and Sotomayor (1985). All these equilibrium outcomes are the same: the true woman-optimal stable matching. Note that there are four more Nash equilibria in pure strategy:

Q = ( P m1 , P m2 , Q w 1 , Q w 2 ) where

d) Q w 1 = (m1 , w 1 , m2 ), Q w 2 = (m2 , w 2 , m1 ); e) Q w 1 = (m1 , w 1 , m2 ), Q w 2 = (m2 , m1 , w 2 ); f) Q w 1 = (m1 , m2 , w 1 ), Q w 2 = (m2 , w 2 , m1 ); g) Q w 1 = (m1 , m2 , w 1 ), Q w 2 = (m2 , m1 , w 2 ). Each of them from d to g gives rise to the true man-optimal stable matching μ M . One may observe that the equilibrium strategies for women w 1 and w 2 in cases d to g are all weakly dominated. Therefore, none of them is trembling-hand perfect. In contrast, the equilibria in cases a to c are trembling-hand perfect. Thus, the Nash equilibria that are plausible in this example are those three from cases a to c, each of which gives rise to the true woman optimal stable matching μ W , even if the man-optimal algorithm is in use. Let us summarize our analysis in Example 5 as follows: Consider the matching game Γ (ϕ M , P ) induced by the manoptimal algorithm ϕ M and the true profile P of preferences under which each man chooses his dominant strategy and states his true preferences. Then the only three trembling-hand Nash equilibria that give rise to the (true) woman-optimal stable matching μ W are the Nash equilibria in truncation. Note that the core S ( Q ) for each misreported equilibrium profile Q from equilibria a to g is a singleton, even if the core S ( P ) for the true profile P is not. This example shows that the small core S ( Q ) at each equilibrium from a to g does not deliver the information about how many opportunities the two women have in their strategic manipulation of their true preferences. Strategies in truncation for women are important for Theorem 4. Without truncation, then the set S ( Q ) at a Nash equilibrium is not necessarily a singleton, as shown in Corollary 2. Thus Theorem 4 fails without truncation. 6. Main result: the college admissions problem Theorem 6. Assume n  2, r  2, and q C j  2 for at least one C j . Consider the college admissions problem with strict preferences. Assume that colleges’ preferences over groups of students are responsive, complete, and transitive. Let ϕ S be the student-optimal algorithm. Suppose that Q is a reported profile of preferences such that each student chooses his dominant strategy and reports his true preferences and each college chooses a strategy in truncation in the matching game Γ (ϕ S , P ) induced by ϕ S and the underlying true profile of preferences P . Suppose that the reported profile Q is a Nash equilibrium. Then either ϕ S ( Q ) is not stable for P or ϕ S ( Q ) is the true college optimal stable matching. Moreover, the set of stable matchings S ( Q ) with respect to the reported equilibrium profile Q is a singleton. Theorem 6 shows that there is a situation under which not only the core is small but also it contains only the true college-optimal stable matching under the student-optimal algorithm. This result is in contrast to the situation when all agents choose to report their preferences in a straightforward manner. It suggests that the switch from the hospital-optimal to the student-optimal algorithm in the NRMP may in fact result in a matching that is the worst for all students in terms of the true preferences, even though the new algorithm generates a matching that is optimal for students in the submitted preferences. Moreover, hospitals can profit for the use of the new algorithm through a very simple class of strategies in truncation. The next corollary of Theorems 4 and 6 shows that the student-optimal algorithm does not produce the true college optimal stable matching only under two conditions: Either the reported profile Q is not a Nash equilibrium in truncation or no matching in S ( Q ) is stable for P .

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Corollary 7. Consider the college admissions problem with strict preferences. Assume that colleges’ preferences over groups of students are responsive, complete and transitive. Let ϕ S be the student-optimal algorithm. Let Q be a Nash equilibrium in truncation of the matching game Γ (ϕ S , P ) such that each student chooses his dominant strategy and states his true preferences. Then ϕ S ( Q ) is not the true college optimal stable matching only if there is no true stable matching in S ( Q ). The proofs of Theorem 6 and Theorem 4 will be given in Appendix A. Our results for both the marriage problem and the college admissions problem show that the comments made around 1995 by the American Medical Student Association (AMSA) and the Public Citizens’ Health Research Group cited in the introduction of this paper can be very wrong. For example, the comment The direction in which the algorithm bias should lie now becomes the obvious question. . . . [A] student-optimal algorithm would be a fair and more reasonable choice of stable algorithms than the current hospital-optimal algorithm. may not be right at all. Unfortunately, our results presented here were not known around the time. They may be quite relevant for the NRMP with flexible-salaries match which is currently under a study by Crawford (2005). It should be aware that there may not exist a set of truncated strategies for colleges that forms a Nash equilibrium in the game Γ (ϕ S , P ) under which each student chooses his dominant strategy and states his true preferences, as shown by the following example. In the following game, we use S i jk for the strategy ( S i , S j , S k , C ) for a college C . In the payoff matrix, S i j means that a college is matched with students S i and S j ; And S i means that college C is matched with one student S i and fills its class with one copy C . Example 8.6 There are three students S = { S 1 , S 2 , S 3 , S 4 } and two colleges C = {C 1 , C 2 }. Each college has a quota of 2. Their true preferences are given as follows:

P S 1 = (C 2 , C 1 , S 1 ) P S 2 = (C 1 , C 2 , S 2 ) P S 3 = (C 2 , C 1 , S 3 ) P S 4 = (C 1 , C 2 , S 4 ) P C1 = ( S 1, S 3, S 2, S 4, C1) P C2 = ( S 2, S 1, S 4, S 3, C2) There are two stable matchings with respect to the true preference profile P :









μC = (C 1 ; S 2 , S 3 ); (C 2 ; S 1 , S 4 ) and

μ S = (C 1 ; S 2 , S 4 ); (C 2 ; S 1 , S 3 ) The matching game of Example 8:

S 1324 S 132 S 13 S1 C1

S 2143

S 214

S 21

S2

C2

S 24

( S 24 , S 13 ) ( S 23 , S 14 ) ( S 3 , S 12 ) (C 1 , S 12 ) (C 1 , S 12 )

( S 23 , S 14 ) ( S 23 , S 14 ) ( S 3 , S 12 ) (C 1 , S 12 ) (C 1 , S 12 )

( S 23 , S 1 ) ( S 23 , S 1 ) ( S 3 , S 12 ) (C 1 , S 12 ) (C 1 , S 12 )

( S 13 , S 2 ) ( S 13 , S 2 ) ( S 13 , S 2 ) (S1, S2) (C 1 , S 2 )

( S 13 , C 2 ) ( S 13 , C 2 ) ( S 13 , C 2 ) (S1, C2) (C 1 , C 2 )

( S 13 , S 24 ) ( S 13 , S 24 ) ( S 13 , S 24 ) ( S 1 , S 24 ) (C 1 , S 24 )

Note that college C 2 has incentive to deviate to strategy S 24 from any of its truncated strategies. So no set of truncated strategies from colleges forms a Nash equilibrium in this example. However, if the strategy space for each college contains its truncations only, then the strategy profile ( S 132 , S 214 ) is a Nash equilibrium in truncations if college C 2 prefers { S 1 , S 4 } to { S 2 , C 2 }. If college C 2 prefers { S 2 , C 2 } to { S 1 , S 4 }, otherwise, the Nash equilibrium in truncations is the strategy profile ( S 132 , S 2 ). The former equilibrium outcome is the true college optimal stable matching μC and the later is unstable for the true profile P . It is of interest to note that each college obtains a class under the equilibrium strategic profile ( S 132 , S 2 ) that is even better than its class under μC . We will show this result holds in general. The study of Nash equilibrium in the college admissions problems is quite subtle because the responsiveness provides a family of distinct preferences over groups of students. When the number of students is large, such a family of preferences can be huge and quite complicated.

6

We thank the anonymous referee for providing this example, which also motivates the results in Theorem 9 below.

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Theorem 9. Assume n  2, r  2, and q C j  2 for at least one C j . Consider the college admissions problem with strict preferences. Assume that colleges’ preferences over groups of students are responsive, complete, and transitive. Let ϕ S be the student-optimal algorithm. If each college is restricted to report its truncation strategies in the matching game Γ (ϕ S , P ) induced by ϕ S and the underlying true profile of preferences P , then there always exists a Nash equilibrium Q ∗ under which each student chooses his dominant strategy and states his true preferences. Moreover, this equilibrium matching ϕ S ( Q ∗ ) satisfies the following: Either ϕ S ( Q ∗ ) is the true college optimal stable matching or no college gets worse off under ϕ S ( Q ∗ ) than under the true college optimal stable matching. The proof of Theorem 9 is given in Appendix A. It mainly depends on the decomposition lemma in the marriage problem. Corollary 10 follows from Theorems 6 and 9 and the lattice structure of the set of stable matchings. Corollary 10. Assume n  2, r  2, and q C j  2 for at least one C j . Consider the college admissions problem with strict preferences. Assume that colleges’ preferences over groups of students are responsive, complete, and transitive. Let ϕ S be the student-optimal algorithm and Q be a Nash equilibrium in the game Γ (ϕ S , P ) under which each student chooses his dominant strategy and states his true preferences and each college is restricted to report truncations. Then the following holds: Each college obtains a class under ϕ S ( Q ) that is no worse than its class under the true college optimal stable matching. 6.1. Examples In this subsection we will provide examples to show how robust our result in Theorem 6 is. First note that the condition that ϕ S ( Q ) is stable for P in Theorem 6 is not dispensable, unlike Theorem 4 in the marriage problem. This can be shown by the following example, due to Tayfun Sonmez (1997). In fact, it is Example 5 with an exception that one man is willing to have two wives. Our analysis of this example shows that the only trembling-hand Nash equilibrium of the game induced by the student-optimal algorithm is the Nash equilibrium in truncation identified in Theorem 6. Moreover, its outcome is unstable for the true preferences. Example 11. (See Sonmez, 1997.) There are two colleges C = {C 1 , C 2 } and two students S = { S 1 , S 2 }. College C 1 has a quota q C 1 = 2 and college C 2 has quota q C 2 = 1. Let

P C1 = ( S 1, S 2, C1)

P C2 = ( S 2, S 1, C2)

P S 1 = (C 2 , C 1 , S 1 )

P S 2 = (C 1 , C 2 , S 2 )

be the true preferences. There is a unique stable matching





μ = (C 1 ; S 2 , C 1 ); (C 2 ; S 1 )

for the profile P = ( P C 1 , P C 2 , P S 1 , P S 2 ) of true preferences. The matching

  λ = (C 1 ; S 1 , C 1 ); (C 2 ; S 2 )

is not stable for P . When q C 1 = 1, this college admissions problem becomes the marriage problem in Example 5 (colleges are men and students are women). With the same profile P of preferences, μ is the woman-optimal stable matching μ W and λ is the man-optimal stable matching μ M . We present the analysis of this example with two claims whose proofs are given in Appendix A. Claim 12. For Example 11, consider the matching game Γ (ϕ S , P ) induced by ϕ S and P under which each student chooses his dominant strategy and states his true preferences. Then the only Nash equilibrium in truncation of the game Γ (ϕ S , P ) is the unique tremblinghand perfect Nash equilibrium whose outcome is the matching λ, which is not stable for P . Claim 13. For Example 11, consider the matching game Γ (ϕ C , P ) induced by ϕ C and P under which college C 2 chooses its dominant strategy and states its true preferences. Then there exists a unique trembling-hand perfect Nash equilibrium of the game Γ (ϕ C , P ) whose outcome is the matching λ, which is not stable for P . Claims 12 and 13 show that both the student-optimal algorithm and the college-optimal algorithm can fail to produce a matching that is stable for the true profile even though both are stable algorithms for submitted profiles. This is in the contrast to the marriage problem. Indeed, a result in Roth (1984b) showed that the outcome of any trembling-hand perfect Nash equilibrium in the marriage problem of the game induced by the man-optimal (or woman-optimal) algorithm must be a matching that is stable for the true profile. Consider a situation where colleges publicly announce their quotas first which are then fixed as in the NRMP, then the social planner announces a matching mechanism under which players submit their ROLs as strategies. Claims 12 and 13 and

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Example 5 provide another interesting result. A college can benefit from overreporting its quota. Example 5 shows that it is the man-optimal matching μ M that is obtained under the woman-optimal algorithm and it is the woman-optimal matching μW that is obtained under the man-optimal algorithm. Thus, there is no chance in the marriage problem of Example 5 for a man to marry a woman he likes the most under both algorithms. However, if one man just misreports that he is willing to marry with two women, then he will marry with the woman he likes the most under both algorithms, as shown in Claims 12 and 13, even though he still marries with just one woman. Now we provide an example under which there is no trembling-hand Nash equilibrium in the game Γ (ϕ S , P ) whose outcome is the true student-optimal stable matching. In fact, the outcomes of all trembling-hand Nash equilibria in this example are the true college-optimal stable matching. Example 14. There are three students S = { S 1 , S 2 , S 3 } and two colleges C = {C 1 , C 2 }. College C 1 has a quota of 2 and college C 2 has a quota of 1. Their true preferences are given as follows:

P S 1 = (C 1 , C 2 , S 1 ) P S 2 = (C 2 , C 1 , S 2 ) P S 3 = (C 1 , C 2 , S 3 ) P C1 = ( S 2, S 1, S 3, C1) P C2 = ( S 3, S 1, S 2, C2) There are two stable matchings in terms of the true preference profile P :









μC = (C 1 ; S 1 , S 2 ); (C 2 ; S 3 ) and

μ S = (C 1 ; S 1 , S 3 ); (C 2 ; S 2 )

Now consider the game Γ (ϕ S , P ) under which each student chooses his dominant strategy and states his true preferences. There are sixteen strategies for each college c = C 1 , C 2 . It is clear that strategies ( S 1 , c , S 2 , S 3 ) and ( S 1 , c , S 3 , S 2 ) are in an equivalent class for a stable algorithm. For simplicity, we just use ( S 1 , c ) to denote a strategy in this equivalent class. The others are similar:

Q c1 = ( S 1 , c ) Q c2 = ( S 2 , c ) Q c3 = ( S 3 , c ) Q c4 = ( S 1 , S 2 , c ) Q c5 = ( S 2 , S 1 , c ) Q c6 = ( S 1 , S 3 , c ) Q c7 = ( S 3 , S 1 , c ) Q c8 = ( S 2 , S 3 , c ) Q c9 = ( S 3 , S 2 , c ) Q c10 = ( S 1 , S 2 , S 3 , c ) Q c11 = ( S 2 , S 1 , S 3 , c ) Q c12 = ( S 1 , S 3 , S 2 , c ) Q c13 = ( S 3 , S 1 , S 2 , c ) Q c14 = ( S 2 , S 3 , S 1 , c ) Q c15 = ( S 3 , S 2 , S 1 , c ) Q c16 = (c ) By responsiveness, the preferences over groups of students for C 1 may be given by ({ S 1 , S 2 }, { S 2 , S 3 }, { S 1 , S 3 }, { S 2 }, { S 1 }, { S 3 }, {C 1 }), which are assigned payoffs (6, 5, 4, 3, 2, 1, 0), respectively. For college C 2 , we assign payoffs 3, 2, 1, 0 for S 3 , S 1 , S 2 , C 2 , respectively. The payoff vector for the true college-optimal stable matching μC is (6, 3) and the payoff vector for the true student-optimal stable matching μ S is (4, 1).

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161

Table 2 Matching game: C 1 the row player and C 2 the column player.

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q 10 Q 11 Q 12 Q 13 Q 14 Q 15 Q 16

Q1

Q2

Q3

Q4

Q5

Q6

Q7

Q8

Q9

Q 10

Q 11

Q 12

Q 13

Q 14

Q 15

Q 16

2, 0 3, 2 1, 2 6, 0 6, 0 4, 0 4, 0 5, 2 5, 2 6, 0 6, 0 4, 0 4, 0 5, 2 5, 2 0, 2

2, 1 0, 1 1, 1 2, 1 2, 1 4, 1 4, 1 1, 1 1, 1 4, 1 4, 1 4, 1 4, 1 4, 1 4, 1 0, 1

2, 3 3, 3 1, 0 6, 3 6, 3 4, 0 4, 0 5, 0 5, 0 6, 3 6, 3 4, 0 4, 0 5, 0 5, 0 0, 3

2, 1 3, 2 1, 2 2, 1 2, 1 4, 1 4, 1 5, 2 5, 2 4, 1 4, 1 4, 1 4, 1 4, 1 4, 1 0, 2

2, 1 0, 1 1, 1 2, 1 2, 1 4, 1 4, 1 1, 1 1, 1 4, 1 4, 1 4, 1 4, 1 4, 1 4, 1 0, 1

2, 3 3, 2 1, 2 6, 3 6, 3 4, 0 4, 0 5, 2 5, 2 6, 3 6, 3 4, 1 4, 1 5, 2 5, 2 0, 2

2, 3 3, 3 1, 2 6, 3 6, 3 4, 0 4, 0 5, 2 5, 2 6, 3 6, 3 4, 0 4, 0 5, 2 5, 2 0, 3

2, 1 0, 1 1, 1 2, 1 2, 1 4, 1 4, 1 1, 1 1, 1 4, 1 4, 1 4, 0 4, 0 4, 1 4, 1 0, 1

2, 3 3, 3 1, 1 6, 3 6, 3 4, 1 4, 1 1, 1 1, 1 4, 1 4, 1 4, 1 4, 1 4, 1 4, 1 0, 3

2, 1 3, 2 1, 2 2, 1 2, 1 4, 1 4, 1 5, 2 5, 2 4, 1 4, 1 4, 1 4, 1 4, 1 4, 1 0, 2

2, 1 0, 1 1, 1 2, 1 2, 1 4, 1 4, 1 1, 1 1, 1 4, 1 4, 1 4, 1 4, 1 4, 1 4, 1 0, 1

2, 3 3, 2 1, 2 6, 3 6, 3 4, 1 4, 1 5, 2 5, 2 4, 1 4, 1 4, 1 4, 1 4, 1 4, 1 0, 2

2, 3 3, 3 1, 2 6, 3 6, 3 4, 1 4, 1 5, 2 5, 2 4, 1 4, 1 4, 1 4, 1 4, 1 4, 1 0, 3

2, 1 0, 1 1, 1 2, 1 2, 1 4, 1 4, 1 1, 1 1, 1 4, 1 4, 1 4, 1 4, 1 4, 1 4, 1 0, 1

2, 3 3, 3 1, 1 6, 3 6, 3 4, 1 4, 1 1, 1 1, 1 4, 1 4, 1 4, 1 4, 1 4, 1 4, 1 0, 3

2, 0 3, 0 1, 0 6, 0 6, 0 4, 0 4, 0 5, 0 5, 0 6, 0 6, 0 4, 0 4, 0 5, 0 5, 0 0, 0

In the strategic form game given below, college C 1 is the row player and college C 2 is the column player. For example, the strategy Q 10 for row player C 1 is ( S 1 , S 2 , S 3 , C 1 ) and the strategy Q 10 for column player C 2 is ( S 1 , S 2 , S 3 , C 2 ). We now provide a claim about this matching game. Claim 15. Any Nash equilibrium of the matching game in Table 2 whose outcome is not the college-optimal stable matching μC is not a trembling-hand perfect Nash equilibrium. All Nash equilibria in truncation give rise to the college-optimal stable matching μC . The Nash equilibria in truncation, ( Q 11 , Q 7 ), ( Q 5 , Q 7 ), and ( Q 5 , Q 13 ) are all trembling-hand perfect. 7. Conclusion Economists are eagerly seeking a mechanism that is strategy proof so that truth-telling is a dominant strategy for every player. Unfortunately, a mechanism that is also economically efficient often fails to be strategy proof for all agents who play the game. But many efficient mechanisms do exist that do give certain, though not all, players the incentives for revealing their true preferences. The known Vickrey auction, the Gale and Shapley (1962) proposing algorithms discussed in this paper and its generalization versions in Crawford and Knoer (1981), Kelso and Crawford (1982), Roth (1984c), Crawford (2005), and the Ausubel-Milgrom proxy auction, to mention a few, are all such an example. The results in our paper pose a natural question that is needed to be answered in any economic design of a mechanism: In a strategic game such that not every player has a dominant strategy to play, should those players who do have choose their dominant strategies? Indeed, should those who have dominant strategies in truth-telling to choose from state their true preferences once such a mechanism is put in use? We provide examples in the marriage and the college admissions problems such that players in one side of the market who do have incentives to misreport can take advantage of the truthtelling given by the other side of the market so that the final outcome of the mechanism is in favor of the agents who do misreport. We believe that such a situation is not necessarily alone for the matching or auction markets. Appendix A. Proofs Next we provide a proof for Theorem 6 and Theorem 4. We need two important results from Roth and Sotomayor (1989) in our proofs. In the proof of Theorem 4, we need a result from Roth (1984b), which does not apply to the college admissions problem (see Examples 8 and 11). Theorem A (Roth and Sotomayor). Assume that colleges and students have strict preferences over individuals. Let μ and μ be stable matchings for ( S , C , P ). Then a college C j is indifferent between μ(C j ) and μ (C j ) only if μ(C j ) = μ (C j ). Theorem B (Roth and Sotomayor). Assume that colleges and students have strict preferences over individuals. Let μ and μ be stable matchings for ( S , C , P ). If μ(C j ) P¯ C j μ (C j ), then s P C j s for all s ∈ μ(C j ) and s in μ (C j ) \ μ(C j ). Note that Theorem B is quite powerful. Suppose a college C j , with quota of 2, has preferences over individual students P C j = ( S 1 , S 2 , S 3 , S 4 , C j , . . .). Without knowing any detail about the rest of the market, Theorem B shows that if C j is matched with { S 1 , S 3 } at a stable matching, then a matching such that C j is matched with { S 2 , S 4 } will not be stable for P , and vice versa. It also shows that if C j is matched with { S 1 , S 4 } at a stable matching, then a matching such that C j is matched with { S 2 , S 3 } will not be stable for P , and vice versa. This property is exactly what we want in our proof below. See Roth and Sotomayor (1989, 1990) for detail.

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Proof of Theorem 6. Suppose that ϕ S ( Q ) is not the college-optimal stable matching μC in S ( P ). Assume that ϕ S ( Q ) ∈ S ( P ). Since μC is the college optimal stable matching, it follows that μC (C j ) R¯ C j ϕCS ( Q ) for all colleges C j and j

μC (C j ) P¯ C j ϕCS j ( Q ) for some college C j , by the lattice structure. Let T = {C j : μC (C j ) P¯ C j ϕCS j ( Q )}, which is nonempty. Theorem A of Roth and Sotomayor shows that for every college C j that is not in T , we have μC (C j ) = ϕCS ( Q ). Theorem B of j Roth and Sotomayor shows that for every C j ∈ T , college C j prefers every student in μC (C j ) to every student who is in ϕCS j ( Q ) but not in μC (C j ). Since Q is a truncation and ϕ S ( Q ) is individually rational with respect to Q and P , it follows that μC ∈ S ( Q ), which is a contradiction to Theorem 3. The singleton core follows from Theorem 3 directly. 2 Proof of Theorem 4. Theorem 4.16 in Roth and Sotomayor (1990) or Roth (1984b) showed that rem 4 follows from Theorems 3 and 6 directly. 2

ϕ M ( Q ) ∈ S ( P ). Now Theo-

Proof of Theorem 9. Theorem 6 provides guidance how to construct such a Nash equilibrium. Let μC be the true college optimal stable matching. For a truncation strategy Q C j for college C j , let #( Q C j ) denote the number of students S k such that S k Q C j C j . Let Q be a profile such that each student chooses his dominant strategy and states his true preferences and each college reports a truncation at a match point with respect to μC . Thus, ϕ S ( Q ) = μC . If Q is a Nash equilibrium, then we are done. Otherwise, there exist a college C i and a truncation strategy Q C such that i

 S



ϕC i Q −C i , Q C i P¯ C i ϕCSi ( Q ). Theorem 5.34 in Roth and Sotomayor (1990) implies that #( Q C ) < #( Q C i ). Thus, we can eliminate the least preferred i student for C i in the truncation strategy Q C i by moving forward C i one position so that the least preferred student is no longer acceptable. Denote such a truncation strategy by Q˜ C i . We now replace Q C i by Q˜ C i . By Theorem 5.34 again, no college is worse off under ϕ S ( Q −C i , Q˜ C i ) than under ϕ S ( Q ). If ϕ S ( Q −C i , Q˜ C i ) is not equal to ϕ S ( Q ), then at least one college is strictly better off at the new strategy profile. Check if the new constructed profile ( Q −C i , Q˜ C i ) is a Nash equilibrium. If yes, we are done. Otherwise, one can continue this process. Because there are a finite number of students available to eliminate and no college gets worse off at each elimination, this process must end within a finite number of steps at a Nash equilibrium. Since the process starts with the college optimal stable matching μC , no college will get worse off at Q ∗ than the initial matching μC for Q ∗ = Q . 2 Table 3 Matching game Γ (ϕ S , P ): C 1 the row player and C 2 the column player.

(S1, S2, C1) (S1, C1) (S2, C2) (S2, S1, C1) (C 1 )

(S1, S2, C2)

(S1, C2)

(S2, C2)

(S2, S1, C2)

(C 2 )

(1, 1) (0, 1) (1, 1) (1, 1) (0, 1)

(1, 1) (0, 1) (1, 1) (1, 1) (0, 1)

(3, 0) (2, 2) (1, 0) (3, 0) (0, 2)

(1, 1) (2, 2) (1, 1) (1, 1) (0, 2)

(3, 0) (2, 0) (1, 0) (3, 0) (0, 0)

Proof of Claim 12. Consider the strategic form game in Table 3 where college C 1 is the row player and C 2 is the column player. College C 1 is assigned payoffs 3, 2, 1, 0 if it is matched with { S 1 , S 2 }, S 1 , S 2 , C 1 , respectively. College C 2 is assigned payoffs 2, 1, 0 if it is matched with S 2 , S 1 , C 2 , respectively. It is easy to see that the only trembling-hand perfect Nash equilibrium of the game in Table 3 is that college C 1 chooses strategy ( S 1 , C 1 ) and college C 2 chooses strategy ( S 2 , S 1 , C 2 ), resulting in a matching λ, which is different from the true stable matching μ. This completes the proof. 2 Proof of Claim 13. We first write down the strategic form game of three players, S 1 , S 2 and C 1 as follows, where C 1 is assigned payoffs 3, 2, 1 and 0 if it is matched with { S 1 , S 2 }, S 1 , S 2 , and C 1 , respectively; S 1 is assigned payoffs 2, 1 and 0 if he is matched with C 2 , C 1 and S 1 , respectively; S 2 is assigned payoffs 2, 1 and 0 if he is matched with C 1 , C 2 and S 2 , respectively: (a) College C 1 chooses strategy ( S 1 , S 2 , C 1 ) (see Table 4). (b) Player C 1 chooses strategy ( S 1 , C 1 ) (see Table 5). (c) Player C 1 chooses strategy ( S 2 , S 1 , C 1 ) (see Table 6). (d) Player C 1 chooses strategy ( S 2 , C 1 ) (see Table 7). (e) Player C 1 chooses strategy (C 1 ) (see Table 8). Now one can check that the two strategies ( S 2 , C 1 ) and (C 1 ) are weakly dominated by ( S 1 , S 2 , C 1 ) or ( S 2 , S 1 , C 1 ) for college C 1 . Moreover, the strategy (C 2 , C 1 , S 1 ) weakly dominates strategies (C 2 , S 1 ), (C 1 , C 2 , S 1 ), (C 1 , S 1 ) and ( S 1 ) for student S 1 and strategy (C 1 , C 2 , S 2 ) weakly dominates strategies (C 1 , S 2 ), (C 2 , C 1 , S 2 ), (C 2 , S 2 ) and ( S 2 ) for student S 2 . Proposition 8.F.2 in Mas-Colell et al. (1995) shows that no weakly pure strategy can be played with positive probability in any

J. Ma / Games and Economic Behavior 69 (2010) 150–164

163

Table 4 Matching game: S 1 the row player and S 2 the column player.

(C 2 , C 1 , S 1 ) (C 2 , S 1 ) (C 1 , C 2 , S 1 ) (C 1 , S 1 ) (S1)

(C 1 , C 2 , S 2 )

(C 1 , S 2 )

(C 2 , C 1 , S 2 )

(C 2 , S 2 )

(S2)

(2, 2, 1) (2, 2, 1) (1, 2, 3) (1, 2, 3) (0, 2, 1)

(2, 2, 1) (2, 2, 1) (1, 2, 3) (1, 2, 3) (0, 2, 1)

(1, 1, 2) (0, 1, 0) (1, 1, 2) (1, 1, 2) (0, 1, 0)

(1, 1, 2) (0, 1, 0) (1, 1, 2) (1, 1, 2) (0, 1, 0)

(2, 0, 0) (2, 0, 0) (1, 0, 2) (1, 0, 2) (0, 0, 0)

Table 5 Matching game: S 1 the row player and S 2 the column player.

(C 2 , C 1 , S 1 ) (C 2 , S 1 ) (C 1 , C 2 , S 1 ) (C 1 , S 1 ) (S1)

(C 1 , C 2 , S 2 )

(C 1 , S 2 )

(C 2 , C 1 , S 2 )

(C 2 , S 2 )

(S2)

(1, 1, 2) (0, 1, 0) (1, 1, 2) (1, 1, 2) (0, 1, 0)

(2, 0, 0) (2, 0, 0) (1, 0, 2) (1, 0, 2) (0, 0, 0)

(1, 1, 2) (0, 1, 0) (1, 1, 2) (1, 1, 2) (0, 1, 0)

(1, 1, 2) (0, 1, 0) (1, 1, 2) (1, 1, 2) (0, 1, 0)

(2, 0, 0) (2, 0, 0) (1, 0, 2) (1, 0, 2) (0, 0, 0)

Table 6 Matching game: S 1 the row player and S 2 the column player.

(C 2 , C 1 , S 1 ) (C 2 , S 1 ) (C 1 , C 2 , S 1 ) (C 1 , S 1 ) (S1)

(C 1 , C 2 , S 2 )

(C 1 , S 2 )

(C 2 , C 1 , S 2 )

(C 2 , S 2 )

(S2)

(2, 2, 1) (2, 2, 1) (1, 2, 3) (1, 2, 3) (0, 2, 1)

(2, 2, 1) (2, 2, 1) (1, 2, 3) (1, 2, 3) (0, 2, 1)

(1, 1, 2) (0, 1, 0) (1, 1, 2) (1, 1, 2) (0, 1, 0)

(1, 1, 2) (0, 1, 0) (1, 1, 2) (1, 1, 2) (0, 1, 0)

(2, 0, 0) (2, 0, 0) (1, 0, 2) (1, 0, 2) (0, 0, 0)

Table 7 Matching game: S 1 the row player and S 2 the column player.

(C 2 , C 1 , S 1 ) (C 2 , S 1 ) (C 1 , C 2 , S 1 ) (C 1 , S 1 ) (S1)

(C 1 , C 2 , S 2 )

(C 1 , S 2 )

(C 2 , C 1 , S 2 )

(C 2 , S 2 )

(S2)

(2, 2, 1) (2, 2, 1) (2, 2, 1) (0, 2, 1) (0, 2, 1)

(2, 2, 1) (2, 2, 1) (2, 2, 1) (0, 2, 1) (0, 2, 1)

(0, 1, 0) (0, 1, 0) (0, 1, 0) (0, 1, 0) (0, 1, 0)

(0, 1, 0) (0, 1, 0) (0, 1, 0) (0, 1, 0) (0, 1, 0)

(2, 0, 0) (2, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0)

Table 8 Matching game: S 1 the row player and S 2 the column player.

(C 2 , C 1 , S 1 ) (C 2 , S 1 ) (C 1 , C 2 , S 1 ) (C 1 , S 1 ) (S1)

(C 1 , C 2 , S 2 )

(C 1 , S 2 )

(C 2 , C 1 , S 2 )

(C 2 , S 2 )

(S2)

(0, 1, 0) (0, 1, 0) (0, 1, 0) (0, 1, 0) (0, 1, 0)

(2, 0, 0) (2, 0, 0) (2, 0, 0) (0, 0, 0) (0, 0, 0)

(0, 1, 0) (0, 1, 0) (0, 1, 0) (0, 1, 0) (0, 1, 0)

(0, 1, 0) (0, 1, 0) (0, 1, 0) (0, 1, 0) (0, 1, 0)

(2, 0, 0) (2, 0, 0) (2, 0, 0) (0, 0, 0) (0, 0, 0)

(normal form) trembling-hand perfect Nash equilibrium. The only Nash equilibrium that survives this standard is the equilibrium where student S 1 chooses strategy (C 2 , C 1 , S 1 ), student S 2 chooses strategy (C 1 , C 2 , S 2 ), and college C 1 chooses strategies ( S 1 , S 2 , C 1 ), ( S 1 , C 1 ) or ( S 2 , S 1 , C 1 ). The only Nash equilibrium that satisfies these conditions is that student S 1 chooses strategy (C 2 , C 1 , S 1 ), student S 2 chooses strategy (C 1 , C 2 , S 2 ), and college C 1 chooses strategy ( S 1 , C 1 ), resulting in a matching λ, which is unstable in terms of P . Next we show that the identified Nash equilibrium above is indeed a trembling-hand perfect Nash equilibrium. To show this, let (1 − 4 , , , , ) be a totally mixed strategy for both S 1 and S 2 , where 0 < < 0.25. Now it is sufficient to show that there exists some small 0 such that the strategy ( S 1 , C 1 ) is a best reply for all 0 <  0 . Such an 0 can be found since the payoff of playing strategy ( S 1 , C 1 ) for college C 1 equals

2(1 − 4 )2 + 8 , which is greater than its payoff

(1 − 4 )2 + 13 + 7 2 of playing strategy ( S 1 , S 2 , C 1 ) for some sufficiently small

. This completes the proof. 2

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Proof of Claim 12. There are three types of Nash equilibria in Table 2 with equilibrium payoffs of (6, 3), (5, 2) and (4, 1). Those equilibria with payoffs (5, 2) and (4, 1) are in weakly dominated strategies either for C 1 or C 2 . This completes the proof (see Proposition 8.F.2 in Mas-Colell et al., 1995) that any Nash equilibrium whose outcome is not the college-optimal stable matching μC is not trembling-hand perfect. The fact that the Nash equilibria in truncation, ( Q 11 , Q 7 ), ( Q 5 , Q 7 ), and ( Q 5 , Q 13 ), are all trembling-hand perfect follows the converse of Proposition 8.F.2 in Mas-Colell et al. (1995). This completes the proof. 2 References AMSA and Public Citizen Health Research Group, 1995. Report on Hospital Bias in the NRMP. Crawford, V., 2005. The flexible-salary match: a proposal to increase the salary flexibility of the National Resident Matching Program, UCSD. Crawford, V., Knoer, E.M., 1981. Job matching with heterogeneous firms and workers. Econometrica 49, 437–450. Gale, D., Shapley, L.S., 1962. College admissions and the stability of marriage. Amer. Math. Mon. 69, 9–15. Gale, D., Sotomayor, M., 1985. Some remarks on the stable matching problem. Discrete Appl. Math. 11, 223–232. Irving, Robert W., Leather, Paul, 1986. The complexity of counting stable marriages. SIAM J. Computing 15, 655–667. Kara, T., Sönmez, T., 1997. Implementation of college admission rules. Econ. Theory 9, 197–218. Kelso, A.S. Jr., Crawford, V., 1982. Job matching, coalition formation, and gross substitutes. Econometrica 50, 1483–1504. Ma, J., 2002. Stable matchings and the small core in Nash equilibrium in the college admissions problem. Rev. Econ. Design 7, 117–134. Mas-Colell, A., Whinston, M., Green, J., 1995. Microeconomic Theory. Oxford University Press. Mongell, S., Roth, A., 1991. Sorority rush as a two-sided matching mechanism. Amer. Econ. Rev. 81, 441–464. Peranson, E., Randlett, R.R., 1995. The NRMP matching algorithm revisited: theory versus practice. Acad. Med. 70 (6), 477–484. Roth, A.E., 1982. The economics of matching: stability and incentives. Mathematics Operations Res. 7, 617–628. Roth, A.E., 1984a. The evolution of the labor market for medical interns and residents: a case study in game theory. J. Polit. Economy 92, 991–1016. Roth, A.E., 1984b. Misrepresentation and stability in the marriage problem. J. Econ. Theory 34, 383–387. Roth, A.E., 1984c. Stability and polarization of interests in job matching. Econometrica 52, 47–57. Roth, A.E., 1985. The college admissions problem is not equivalent to the marriage problem. J. Econ. Theory 36, 277–288. Roth, A.E., Rothblum, U., 1999. Truncation strategies in matching markets—in search of advice for participants. Econometrica 67, 21–44. Roth, A.E., Peranson, E., 1999. The redesign of the matching market for American physicians: some engineering aspects of economic design. Amer. Econ. Rev. 89, 748–780. Roth, A.E., Sotomayor, M., 1989. The college admissions problem revisited. Econometrica 57, 559–570. Roth, A.E., Sotomayor, M., 1990. Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis. Cambridge University Press, Cambridge. Roth, A.E., Vande Vate, J.H., 1991. Incentives in two-sided matching with random stable mechanisms. Econ. Theory 1, 31–44. Sonmez, Tayfun, 1997. Manipulation via capacities in two-sided matching markets. J. Econ. Theory 77, 197–204. Williams, K.J., 1995a. A reexamination of the NRMP matching algorithm. Acad. Med. 70 (6), 470–476. Williams, K.J., 1995b. Comments on Peranson and Randlett’s ‘The NRMP matching algorithm revisited: theory versus practice’. Acad. Med. 70 (6), 485–489.