Monotonicity and consistency in matching markets

Transcription

1 Int. J. Game Theory (2006) 34: DOI /s ORIGINAL ARTICLE Manabu Toda Monotonicity and consistency in matching markets Published online: 17 January 2006 Springer-Verlag 2006 Abstract Objective: To obtain axiomatic characterizations of the core of one-toone and one-to-many matching markets. Methods: The axioms recently applied to characterize the core of assignment games were adapted to the models of this paper. Results: The core of one-to-one matching markets is characterized by two different lists of axioms. The first one consists of weak unanimity, population monotonicity, and Maskin monotonicity. The second consists of weak unanimity, population monotonicity, and consistency. If we allow for weak preferences, the core is characterized by weak unanimity, population monotonicity, Maskin monotonicity, and consistency. For one-to-many matchings, the same lists as for the case of strict preferences characterize the core. Conclusions: The cores of the discrete matching markets are characterized by axioms that almost overlap with the axioms characterizing the core of the continuous matching markets. This provides an axiomatic explanation for the observations in the literature that almost parallel properties are obtained for the core of the two models. We observe that Maskin monotonicity is closely related to consistency in matching markets. Keywords Two-sided matchings Maskin monotonicity Population monotonicity Consistency JEL Classification C71 C78 This research is financially supported by Waseda University Grant for Special Research Projects #2000A 887, 21COE-GLOPE, and Grant-in-Aid for Scientific Research # , JSPS. This paper was presented at the 7th. International Meeting of the Society for Social Choice and Welfare held in Osaka, Japan. The comments of the participants are gratefully acknowledged. The author thanks Professors William Thomson, Eiichi Miyagawa and anonymous referees for their valuable comments and suggestions. Any remaining errors are independent. M. Toda School of Social Sciences, Waseda University, , Tokyo, Japan

2 14 M. Toda 1 Introduction Most of the literature on two-sided matchings has focused on two different models of markets. One is the discrete model of marriage and college admissions due to Gale and Shapley (1962), and the other is the continuous model of job matching due to Shapley and Shubik (1972). Although they are apparently distinct, it has been long recognized that almost parallel properties hold for the cores of the two markets, such as lattice structure, polarization, single-agent property, and others. A number of explanations for the similarities underlying the two models have been offered. Kaneko (1982) considered a class of balanced NTU games that covers both. Roth et al. (1990) observed that most of the parallel properties follow from the duality and complementary slackness of closely related linear programming problems. Eriksson and Karlander (2000) proposed a generalized framework including both markets. Sotomayor (2000) extended their setting and obtained several properties of stable matchings in a unified way. Fujishige and Tamura (2003) considered a similar hybrid model. In the present paper, we take an axiomatic approach. Related to this paper, Sasaki and Toda (1992) characterize the core of marriage markets and Sasaki (1995) characterizes the core of assignment games. The axioms in the two papers have little overlap, so that the cores of the two markets are quite dissimilar in terms of their axioms. In the following sections, we show that this is not the whole story. First, we find a new set of axioms characterizing the core of marriage markets that almost overlap with the axioms characterizing the core of assignment games recently obtained by Toda (2003). We characterize the core by Pareto optimality, population monotonicity, and consistency. Toda (2003) characterizes the core by Pareto optimality, population monotonicity, consistency, and pairwise monotonicity. Then, the only difference is pairwise monotonicity. This axiom says that the total payoff of a pair should not decrease when the worth of the pair increases. In marriage markets, a solution simply selects a subset of matchings but in assignment games, a solution selects a subset of pairs of a matching and a payoff vector. Hence, in assignment games, pairwise monotonicity is required to capture the behaviour of a solution in choosing payoff vectors, while in marriage markets, it is not needed. Then, the result of this paper is related to the one in Toda (2003), however, the two papers are logically independent. For instance, in assignment games, a solution satisfying Pareto optimality, population monotonicity, and pairwise monotonicity is a subsolution of the core. In marriage markets, pairwise monotonicity is redundant, but a solution satisfying Pareto optimality and population monotonicity may not be a subsolution of the core. Some important observations follow from our result. First, in our characterization, consistency can be replaced by Maskin monotonicity. Second, Pareto optimality can be replaced by weak unanimity, which requires that if each agent most prefers a matching at which no agent is single, then the matching is the unique recommendation. The first observation shows that consistency and Maskin monotonicity play the same role in establishing the uniqueness of the core. In principle, Maskin monotonicity has no logical relation with consistency because the former assumes a fixed population while the latter assumes a variable population. In matching markets,

3 Monotonicity and consistency in matching markets 15 the set of alternatives and the set of agents happen to be equal. This coincidence is behind the equivalence between the two axioms. In the second part of this paper, we consider an extension of the model which allows for weak orderings. In this case, there exists a proper subsolution of the core satisfying either Maskin monotonicity or consistency, but no proper subsolution satisfies both. Therefore, the core is the unique solution satisfying weak unanimity, population monotonicity, Maskin monotonicity, and consistency. In the final part of this paper, we consider college admissions. The extension to this case is non-trivial. For instance, in marriage markets, weak unanimity and population monotonicity imply individual rationality. In college admissions markets, this is not true. We also need to be careful in extending consistency and other axioms to this case. However, our main message is the same. The core is characterized by weak unanimity, population monotonicity, and consistency, in which consistency can be replaced by Maskin monotonicity. The next section describes the basic model and axioms. The third section obtains logical relations among the axioms and establishes core characterizations in marriage markets. The fourth section considers marriage markets with possibly weak preferences. The final section considers college-admissions problems. The appendices show the logical independence of the axioms and discuss some possible modifications of the axioms. 2 Model and axioms Let I and J be two disjoint sets of infinitely many potential agents. We call an agent in I a man and an agent in J a woman. Let M and W be the sets of all nonempty finite subsets of I and of J, respectively. A matching problem is a triplet γ = (M, W, P) such that (M, W ) M W and P ={P a a M W} is a preference profile. For each m M, P m is a strict ordering over the set W {φ}, and for each w W, P w is a strict ordering over the set M {φ}, where φ means remaining single. For each a M W, R a is the weak preference ordering associated with P a. That is, br a b if and only if either bp a b or b = b.ifbp a φ, b is acceptable for a. For each (M, W) M W, amatching is a mapping µ from M W into M W {φ} satisfying the following three conditions: (1) µ is one-to-one on the set {a M W µ(a) φ}. (2) For each a M W,ifµ(a) φ, then µ µ(a) = a. (3) If µ(m) φ and µ(w) φ, then µ(m) W and µ(w) M, respectively. We often write (m, w) µ to denote µ(m) = w or equivalently µ(w) = m. When µ(a) = φ, we also write a µ.ifµ(a) φ for each a M W, µ is complete. A complete matching exists if and only if M and W have the same cardinality. Let M(M, W) be the set of all matchings. For each γ = (M, W, P), we write M(γ ) to denote M(M, W) although this set does not depend on P. If µ(a)r a φ for each a, µ is individually rational. If there exists no µ µ such that µ (a)p a µ(a) for each µ (a) µ(a), µ is Pareto optimal. IfwP m µ(m) and mp w µ(w), pair (m, w) blocks µ. If µ is individually rational and has no blocking pair, it is stable. For each γ, IR(γ ), PO(γ ), and S(γ ) denote the sets of all individually rational matchings, Pareto optimal matchings, and stable matchings, respectively.

4 16 M. Toda Because S(γ ) coincides with the core in the usual sense, we simply call it the core of γ. For each γ, there exists a unique matching µ M S(γ ), called the men-optimal stable matching, satisfying µ M (m)r m µ(m) for each m M and each µ S(γ ). The women-optimal stable matching µ W S(γ ) is analogously defined and exists uniquely. (See, e.g., Roth and Sotomayor 1990.) A solution is a correspondence ϕ which associates with each γ a non-empty subset ϕ(γ) M(γ ). We now introduce axioms. Individual rationality (I.R): for each γ, ϕ(γ) IR(γ ). Pareto optimality (P.O): for each γ, ϕ(γ) PO(γ ). Weak unanimity (W.U): for each γ = (M, W, P), if there exists a complete matching µ such that µ(a) is most preferred by each a M W, then ϕ(γ) ={µ}. Mutually best (M.B): for each γ = (M, W, P) and (m, w) M W, ifm and w most prefer each other, then (m, w) µ for each µ ϕ(γ). For γ = (M, W, P), γ = (M,W, P ) is an extension of γ if (1) M M and W W, (2) P m agrees with P m on W {φ} for each m M, (3) P w agrees with P w on M {φ} for each w W. In particular, if M M and W = W, γ is an M-extension of γ and if W W and M = M, γ is a W-extension of γ. Population monotonicity (P.MON): for each γ = (M, W, P) and each M- extension γ of γ,ifµ ϕ(γ), then there exists µ ϕ(γ ) such that µ(m)r m µ (m) for each m M. Ifγ is a W-extension of γ, there exists µ ϕ(γ ) satisfying a symmetric requirement for each w W. Let γ = (M, W, P) and µ M(γ ). For each m M, we define L(µ, R m ) {a W {φ} µ(m) R m a}. For each w W, we define L(µ, R w ) analogously. We say that γ = (M, W, P ) is obtained from γ by a monotonic transformation at µ if L(µ, R a ) L(µ, R a ) for each a M W. Maskin monotonicity (M.MON): for each γ and µ ϕ(γ), ifγ is obtained from γ by a monotonic transformation at µ, then µ ϕ(γ ). For each γ = (M, W, P) and each µ M(γ ), let M M and W W.Ifµ(M W ) M W {φ} and γ is an extension of γ = (M,W, P ), then γ is a reduced problem of γ at µ. We denote by µ M W the restriction of µ to M W. Consistency (CONS): for each γ and each µ ϕ(γ), ifγ = (M,W, P ) is a reduced problem of γ at µ, then µ M W ϕ(γ ). Individual rationality and Pareto optimality are standard. Weak unanimity says that if a complete matching exists and is most preferred by every agent, it should be the unique recommendation. Mutually best is stronger than weak unanimity, requiring a pair of mutually best agents to be matched at every solution outcome. This axiom plays important instrumental roles in the subsequent discussions but does not explicitly appear in the main theorems. Note that weak unanimity also follows from Pareto optimality. Maskin monotonicity says that if a preference profile changes in such a way that the ranking of a social choice does not decrease in everyone s preferences,

5 Monotonicity and consistency in matching markets 17 then the outcome is still recommended. In other words, social decision positively depends on individual evaluations. It is a necessary condition for a social choice correspondence to be Nash implementable (see Maskin 1999). Population monotonicity, which originates in Thomson (1983), 1 requires that if a new entrant enters into a market, no incumbent should gain. This kind of solidarity axiom usually applies to a single-valued function. In this paper, however, we need to consider multi-valued functions and take into account the gender of the entrant. In our definition, population monotonicity requires that if the number of agents on one side of a market increases while the opposite side remains fixed, no incumbent on the same side as the entrants is made better off at the worst possible solution outcome. In order to distinguish it from other concepts, let us call this property lower population monotonicity. On the other hand, we may require no incumbent on the same side as the entrants to be made better off at the best possible solution outcome. More precisely, for each γ = (M, W, P) and each M-extension γ of γ, if µ ϕ(γ ), there exists µ ϕ(γ) such that µ(m)r m µ (m) for each m M and a symmetric condition is satisfied by a W-extension γ of γ. Let us call this upper population monotonicity. In Appendix B, we show that the main results of the next section can not be obtained under upper population monotonicity. Needless to say, if we strengthen our definition by requiring both upper and lower population monotonicity, our uniqueness theorems remain valid. Consistency requires that for each solution outcome, the same outcome should be recommended for each reduced problem that results when some agents leave with what they received. It is one of the central principles in recent axiomatic work. For good surveys of this axiom, we refer the reader to Peleg (1992) and Thomson (2003). It is obvious that the core satisfies I.R, P.O, M.B and CONS. Tadenuma (1993) first observed that the core satisfies M.MON. Proposition 2.1 The core satisfies P.MON. Proof For each γ = (M, W, P) and each W -extension γ = (M, W, P ) of γ, let µ M and µ M be the men-optimal stable matchings in γ and in γ, respectively. Theorem 2.25 in Roth and Sotomayor (1990) states that for each w W, µ M (w)r w µ M (w). Corollary 2.14 in Roth and Sotomayor (1990) states that for each µ S(γ ) and each w W, µ(w)r w µ M (w). Therefore, for each w W, µ(w)r w µ M (w). For an M-extension γ = (M,W,P ) of γ, a symmetric argument applies. 3 Characterizations and some implications In this section, we explore some logical relations between the axioms and establish characterization theorems. Lemma 3.1 If a solution ϕ satisfies W.U and P.MON, then it satisfies M.B. 1 For a comprehensive survey of this principle, see Thomson (1995).

6 18 M. Toda Proof Let γ = (M, W, P) and µ ϕ(γ). We denote M ={m 1,...,m n } and W ={w 1,...,w k }. Suppose that m 1 and w 1 most prefer each other. We introduce new men M ={m 2,...,m k } and new women W ={w 2,...,w n } such that m i and w i most prefer each other for i = 2,...,n and m j and w j most prefer each other for j = 2,...,k. Let γ = (M M,W W, P ) be the extension of γ so obtained. By W.U, ϕ(γ ) ={µ }, where (m 1,w 1 ) µ, (m i,w i ) µ for i = 2,...,n and (m j,w j ) µ for j = 2,...,k. Let γ = (M M,W,P ) be the problem obtained by deleting W from γ. Because γ is a W-extension of γ and ϕ(γ ) ={µ }, by P.MON, for each µ ϕ(γ ), µ (w 1 )R w1 µ (w 1 ) = m 1. Since m 1 is most preferred by w 1, µ (w 1 ) = m 1 for each µ ϕ(γ ). Since γ is an M-extension of γ, by P.MON again, there exists µ ϕ(γ ) such that µ(m 1 )R m1 µ (m 1 ) = w 1. Because w 1 is most preferred by m 1, µ(m 1 ) = w 1. Lemma 3.2 If a solution ϕ satisfies M.B and P.MON, then it satisfies I.R. Proof Let γ = (M, W, P) and µ ϕ(γ). We denote W ={w 1,...,w k } and introduce new men M ={m 1,m 2,...,m k } such that m j and w j most prefer each other for j = 1,...,k. Let γ = (M M,W,P ) be the extension of γ so obtained. By M.B, ϕ(γ ) ={µ }, where (m j,w j ) µ for j = 1,...,k and m µ for each m M. Because γ is an M-extension of γ, by P.MON, µ(m)r m µ (m) = φ for each m M. By a symmetric argument, µ(w)r w φ for each w W. By Lemmas 3.1 and 3.2, W.U and P.MON imply I.R. However, the following example shows that even if we assume P.O and P.MON, we may not obtain a subsolution of the core. Example 3.1 For each γ = (M, W, P), let ϕ(γ) be the set of all µ PO(γ ) satisfying the following two conditions: µ M (m)r m µ(m)r m µ W (m) for each m M, (1) µ W (w)r w µ(w)r w µ M (w) for each w W. (2) It is easy to see that S(γ ) ϕ(γ) and hence ϕ is well-defined. Let ˆγ = ( ˆM,Ŵ, ˆP) be the problem such that ˆM ={m 1,m 2,m 3 }, Ŵ ={w 1,w 2,w 3 } and ˆP is determined by m 1 : w 2 >w 1 >w 3 w 1 : m 2 >m 1 >m 3, m 2 : w 3 >w 2 >w 1 w 2 : m 3 >m 2 >m 1, m 3 : w 1 >w 3 >w 2 w 3 : m 1 >m 3 >m 2. It is obvious that in ˆγ, µ M ={(m 1,w 2 ), (m 2,w 3 ), (m 3,w 1 )}, µ W ={(m 1,w 3 ), (m 2,w 1 ), (m 3,w 2 )}. Let µ ={(m 1,w 1 ), (m 2,w 3 ), (m 3,w 2 )}. Since µ is Pareto optimal and satisfies conditions (1) and (2), µ ϕ( ˆγ). But, (m 3,w 3 ) blocks µ so that µ / S( ˆγ). Therefore, ϕ(γ) S(γ ) in general. Note that the best elements of S(γ ) and of ϕ(γ) coincide and the worst elements of both sets also coincide. Since S satisfies upper and lower population monotonicity, ϕ obviously satisfies both.

7 Monotonicity and consistency in matching markets 19 If either M.MON or CONS is additionally assumed, we obtain a subsolution of the core. The next two lemmas show this. Lemma 3.3 If a solution ϕ satisfies I.R, M.B and M.MON, then it is a subsolution of the core. Proof Let γ = (M, W, P) and µ ϕ(γ). Since ϕ(γ) IR(γ ), it suffices to prove that µ has no blocking pair. By way of contradiction, suppose that pair (m, w) blocks µ. Let P be a preference profile such that m and w most prefer each other but their preference orderings over the agents other than m and w remain the same as in γ, and the preference of every other agent is the same as in γ. Then, γ = (M, W, P ) is obtained from a monotonic transformation of γ at µ.by M.MON, µ ϕ(γ ). Because (m, w) / µ, this contradicts M.B. Lemma 3.4 If a solution ϕ satisfies I.R, M.B and CONS, then it is a subsolution of the core. Proof Let γ = (M, W, P) and µ ϕ(γ). Since ϕ(γ) IR(γ ), it suffices to prove that µ has no blocking pair. By way of contradiction, suppose that pair (m, w) blocks µ. Let γ = (M,W, P ) be the reduced problem of γ at µ such that M {m, µ(w)} M and W {w, µ(m)} W. Note that m and w most prefer each other in γ. By CONS, µ M W ϕ(γ ). Because (m, w) / µ M W, this contradicts M.B. We also obtain the following results. Lemma 3.5 No proper subsolution of the core satisfies M.MON. Proof Theorem 1 in Kara and Sönmez (1996) states that if a solution ϕ satisfies I.R, P.O and M.MON, the core is a subsolution of ϕ. Because a subsolution of the core satisfies I.R and P.O, no proper subsolution of the core satisfies M.MON. Lemma 3.6 No proper subsolution of the core satisfies CONS. We defer a proof of this lemma to Section 5 in which a more general result will be proved. Now, we are ready to state the main theorems of this section. Theorem 3.1 The core is the unique solution satisfying W.U, P.MON, and M.MON. Theorem 3.2 The core is the unique solution satisfying W.U, P.MON, and CONS. In Theorems 3.1 and 3.2, W.U can be replaced by P.O. In assignment games, Toda (2003) recently characterizes the core by Pareto optimality, population monotonicity, consistency, and pairwise monotonicity. Pairwise monotonicity is a weak form of coalitional monotonicity which originates in Zhou (1991). It requires that if the worth of a pair increases, the sum of their payoffs should not decrease. In assignment games, a solution specifies a subset of pairs of a matching and a payoff vector. This is a reason why we need the additional axiom in the continuous model which refers to payoff distribution. Moreover, it is obvious from the lemmas of this section that the core is the unique solution satisfying I.R, M.B, and M.MON, in which M.MON is replaced by CONS.

8 20 M. Toda Then, we observe the followings: First, Maskin monotonicity and consistency play the same role. Second, individual rationality and population monotonicity also play the same role in pinning down the core. Maskin monotonicity and individual rationality are very different notions from consistency and population monotonicity. The former two assume a fixed population, while the latter two assume a variable population. In matching markets, however, the set of agents and the set of alternatives happen to be equal. These equivalences between the axioms go part of the way towards explaining this coincidence. 4 Weak preferences Up to now, we assume that each agent has a strict ordering and has the option to be single. In this section, we allow possibly weak orderings but the single status is unacceptable. This is exemplified by labour markets, in which workers may be indifferent between some of the jobs but losing a job is the worst choice. In the previous section, the core has no proper subsolution satisfying Maskin monotonicity or consistency. On the domain of this section, this is no longer true. 2 We show by examples that there exists a proper subsolution of the core satisfying Maskin monotonicity or consistency. However, no proper subsolution of the core satisfies both. Then, we characterize the core by weak unanimity, population monotonicity, Maskin monotonicity, and consistency. In this section, both I and J are assumed to be countably infinite. A matching problem (with weak preferences) is a triplet γ = (M, W, R) such that (M, W ) M W and R {R a a M W} is a preference profile. For each m M, R m is a weak ordering on W {φ} and for each w W, R w is a weak ordering on M {φ}. For each a M W, P a and I a denote the strict and indifference relations derived from R a, respectively. We assume that wp m φ and mp w φ for each m M and each w W.Amatching is analogously defined as in Section 2 and M(γ ) denotes the set of all matchings. A stable matching is analogously defined as in Section 2 and the set of all stable matchings denoted S(γ ) is called the core 3 of γ. The axioms in Section 2 are easily adapted to the domain of this section. In the following, we show that the core satisfies P.MON but the other axioms are obviously satisfied by the core. In order to show P.MON, we need an additional definition. A problem ˇγ = (M, W, ˇP) with strict preferences is a tie-breaker of γ = (M, W, R) if Pˇ a does not contradict the strict ordering P a derived from R a for each a M W. Proposition 4.1 The core satisfies P.MON. Proof Let µ S(γ ) for γ = (M, W, R) and let γ = (M, W, R ) be a W -extension of γ. Let ˇγ = (M, W, ˇP) be a tie-breaker of γ such that µ(a) is 2 Kara (1996) and Tadenuma and Toda (1998) independently show that there exists a proper subsolution of the core satisfying Maskin monotonicity when all preferences are strict in the setting of this section. 3 Since we allow weak preferences, the core and the strong core may be different. Recently, Ehlers (2004) shows that the core is the minimal Maskin monotonic supersolution of the strong core.

9 Monotonicity and consistency in matching markets 21 most preferred in the indifference class of R a at µ(a) for each a M W.Itis obvious that µ S( ˇγ). Let ˇγ = (M, W, ˇP ) be a tie-breaker of γ such that the tie-breaking rule in ˇγ does not contradict the one in ˇγ. Because ˇγ is a W-extension of ˇγ, by Proposition 2.1, there exists µ S( ˇγ ) such that µ(w) Pˇ w µ (w) for each w W with µ(w) µ (w). Therefore, µ(w)r w µ (w) for each w W.Itis obvious that µ S(γ ). For an M-extension of γ, a symmetric argument applies. By the same argument as in Section 3, we see that Lemmas 3.1, 3.2, 3.3, and 3.4 are valid on the domain of this section, however, the following propositions show that Lemmas 3.5 and 3.6 are not. Proposition 4.2 The core has a proper subsolution satisfying W.U, P.MON, and M.MON. Proof Let ˆγ = ( ˆM,Ŵ, ˆP) be the problem considered in Example 3.1. Note that S( ˆγ)={µ M,µ W, ˆµ}, in which µ M ={(m 1,w 2 ), (m 2,w 3 ), (m 3,w 1 )}, µ W ={(m 1,w 3 ), (m 2,w 1 ), (m 3,w 2 )}, ˆµ ={(m 1,w 1 ), (m 2,w 2 ), (m 3,w 3 )}. Let ˆƔ be the set of all problems γ = ( ˆM,Ŵ,R) such that L( ˆµ, R a ) L( ˆµ, ˆR a ) for each a ˆM Ŵ. For each γ ˆƔ, let ϕ(γ) = S(γ ) \{ˆµ} and each γ / ˆƔ, let ϕ(γ) = S(γ ). It is easily seen that for each γ ˆƔ, µ M (m)p m w and µ W (w)p w m for each m M and each w W. Then, µ M,µ W S(γ ) and hence ϕ(γ). In order to show that ϕ satisfies W.U, 4 let µ M(γ ) be the matching most preferred by every agent in γ = ( ˆM,Ŵ,R). Ifµ ˆµ, it is obvious that ϕ(γ) ={µ} since ϕ is a subsolution of S. Ifµ = ˆµ, then L( ˆµ, R a ) L( ˆµ, ˆR a ) for each a ˆM Ŵ. Therefore, γ / ˆƔ and hence ϕ(γ) = S(γ ) ={µ}. By the same argument as in Example 3.1, ϕ satisfies P.MON. In order to show that ϕ satisfies M.MON, let µ ϕ(γ) for γ = ( ˆM,Ŵ,R) and γ = ( ˆM,Ŵ,R ) be a problem obtained from γ by a monotonic transformation at µ.ifγ ˆƔ, then µ ˆµ. Because ϕ is a subsolution of S, it follows from Maskin monotonicity of S that µ ϕ(γ ).Ifγ / ˆƔ, we distinguish two cases; Case 1: µ ˆµ. In this case, Maskin monotonicity of S obviously implies µ ϕ(γ ). Case 2: µ =ˆµ. Because γ is obtained from γ by a monotonic transformation at ˆµ, L( ˆµ, R a ) L( ˆµ, R a ) for each a ˆM Ŵ. Since γ / ˆƔ, there exists a ˆM Ŵ such that b / L( ˆµ, ˆR a ) for some b L( ˆµ, R a ). Since b L( ˆµ, R a ), γ / ˆƔ and ϕ(γ ) = S(γ ). By Maskin monotonicity of S, µ =ˆµ ϕ(γ ). Proposition 4.3 The core has a proper subsolution satisfying W.U, P.MON, and CONS. 4 We should note that W.U only applies if µ(a) is uniquely most preferred under P a for each a M W.

10 22 M. Toda Proof Since I and J are countably infinite, there exist bijections σ I : I N and σ J : J N, where N is the set of all positive integers. For each γ = (M, W, R), let ˇγ be the tie-breaker of γ in accordance with the decreasing orders of the numbers in {σ I (m) m M} and in {σ J (w) w W}. Then, we define ϕ(γ) = S( ˇγ). By W.U and P.MON of S, it is obvious that ϕ satisfies both axioms. In order to show that ϕ satisfies CONS, for µ ϕ(γ), let γ = (M,W, R ) be a reduced problem of γ at µ. Since ϕ(γ) = S( ˇγ)and ˇγ is a reduced problem of ˇγ at µ, by the consistency of S, µ M W S( ˇγ ) = ϕ(γ ). It is easy to see that ϕ is a proper subsolution of S. Proposition 4.4 No proper subsolution of the core satisfies both M.MON and CONS. Proof Let ϕ be a subsolution of S satisfying both M.MON and CONS and let µ S(γ ) for γ = (M, W, R). Let ˇγ = (M, W, ˇP) be a tie-breaker of γ such that for each a M W, µ(a) is most preferred in the indifference class of R a at µ(a). It is clear that µ S( ˇγ). By the same arguments as in the proof of Lemma 5.8 which will be proved in the next section, there exists a problem γ = (M,W, P ) such that S(γ ) ={µ }, µ M W = µ, and ˇγ is a reduced problem of γ at µ. Since ϕ(γ ) ={µ } and ϕ satisfies CONS, µ ϕ( ˇγ). Because γ is obtained from ˇγ by a monotonic transformation at µ, by M.MON, µ ϕ(γ). This proves that ϕ S. The main result of this section is as follows. Theorem 4.1 The core is the unique solution satisfying W.U, P.MON, M.MON, and CONS. Proof Suppose that a solution ϕ satisfies the axioms. By the same argument as in Section 3, ϕ is a subsolution of S. Then, the conclusion follows from Proposition College admissions problems In this section, we consider college admissions. The formal setting is the same as the one in Chapter 5 of Roth and Sotomayor (1990) except that we have a variable population. Let I and J be disjoint sets of infinitely many potential agents. In the following, we call an agent in I a college and an agent in J a student. Let C and S be the sets of all non-empty and finite subsets of I and of J, respectively. A college admissions problem is a list γ (C,S,q,R) such that (C, S) C S, q = (q c ) c C is a vector of positive integers representing quotas of each college, and R ={R c c C} {P s s S} is a preference profile. We assume that each s S has a strict ordering P s over the set C {φ}, where φ means attending no college. For each c C, R c is a complete and transitive preordering over the set c (S) ={G S 1 G q c } {φ}, where φ means admitting no student. When no confusion arises, c (S) is simply denoted by c. By abuse of notation, each a C S {φ} is identified with the singleton set {a}. We assume that R c defines a strict ordering P c over the set S {φ} of individual students. The same notation P c denotes the strict preference relation on c (S) derived from R c.for

11 Monotonicity and consistency in matching markets 23 each c C, let us denote by S c c the most preferred entering class of c. We additionally assume a regularity condition on R c ; for any two groups of students that differ in only one student, the one containing the more desirable student is preferred to the other. Formally, R c is responsive if for each G S with G q c 1, the following two conditions are satisfied. (1) For each s, s S \ G, G {s}p c G {s } if and only if sp c s. (2) For each s S \ G, G {s}p c G if and only if sp c φ. For each γ = (C,S,q,R) and each (c, s) C S, ifcp s φ, c is acceptable for s and if sp c φ, s is acceptable for c. For each (C, S) C S and each q = (q c ) c C,amatching is a correspondence µ from C S into C S {φ} satisfying the following three conditions: (1) for each s S, µ(s) is a singleton set and µ(s) φ implies µ(s) C, (2) for each c C, 1 µ(c) q c and µ(c) φ implies µ(c) S, (3) for each (c, s) C S, µ(s) = c if and only if s µ(c). If µ(s) = φ and µ(c) = φ, then s is enrolled in no college and c admits no student, respectively. In each case, we write s µ and c µ. Ifµ(s) = c or equivalently s µ(c), we write (c, s) µ. A matching µ is complete if µ(a) φ for each a C S. Let M(C,S,q) denote the set of all matchings. For each γ = (C,S,q,R), we write M(γ ) to denote M(C,S,q) although this set does not depend on R. If µ(a)r a φ for each a C S, µ is weakly individually rational.ifµ(s)r s φ for each s S and sr c φ for each s µ(c) and each c C, µ is individually rational. For a weakly individually rational matching, the entering class of each college is more preferable than admitting no student, but unacceptable students may be admitted. On the other hand, for an individually rational matching, each college admits acceptable students only. If there exists no µ such that µ (a)r a µ(a) for each a C S and µ (a)p a µ(a) for some a C S, µ is Pareto optimal. For µ M(γ ) and (c, s) C S, if either one of the following two conditions is satisfied, (1) cp s µ(s), sp c φ and µ(c) <q c, (2) cp s µ(s) and sp c j for some j µ(c), then pair (c, s) blocks µ. Ifµ is individually rational and has no blocking pair, it is stable. For each γ, IR w (γ ), IR(γ ), PO(γ ), and S(γ ) denote the sets of all weakly individually rational matchings, individually rational matchings, Pareto optimal matchings, and stable matchings, respectively. For the same reason as in Section 2, we call S(γ ) the core of γ. The C-optimal matching denoted µ C S(γ ) and the S-optimal matching denoted µ S S(γ ) are analogously defined as in Section 2. For each γ, µ C and µ S exist uniquely. (See, Chapter 5 of Roth and Sotomayor (1990)). A solution is a correspondence ϕ which associates with each γ a non-empty subset ϕ(γ) M(γ ). Now, we introduce axioms. Weak individual rationality (W.I.R): for each γ, ϕ(γ) IR w (γ ). Weak unanimity (W.U): for each γ = (C,S,q,R), if there exists a complete matching µ such that µ(a) is most preferred by each a C S, then ϕ(γ) ={µ}.

12 24 M. Toda Mutually best (M.B): for each γ = (C,S,q,R) and (c, s) C S, if the most preferred entering class S c of c is non-empty and each s S c most prefers c, then µ(c) = S c for each µ ϕ(γ). The axioms of individual rationality (I.R), Pareto optimality (P.O), population monotonicity (P.MON) and Maskin monotonicity (M.MON) are analogously defined as in Section 2. The core satisfies these axioms. 5 Lemma 5.1 If a solution ϕ satisfies W.U and P.MON, then it satisfies M.B. Proof Let γ = (C,S,q,R) and µ ϕ(γ). Suppose that φ S 1 c1 is the most preferred entering class of c 1 C and each s 1 S 1 most prefers c 1.For each c c 1, we introduce a set S c of q c new students such that each s S c most prefers c and S c is the most preferred entering class of c. For each s / S 1, we also introduce a new college c with quota one such that c and s most prefer each other. Let γ be the extension of γ so obtained. By W.U, ϕ(γ ) ={µ } where µ (c 1 ) = S 1, µ (c) = S c for each c c 1 and µ (s ) = c for each s / S 1. Let γ be the problem obtained by deleting all new students from γ. By P.MON, for each µ ϕ(γ ) and each s S, µ (s) = µ (s) because µ (s) is the best choice of each s S. Hence, µ (c 1 ) = S 1. Because γ is a C-extension of γ and S 1 is the best choice of c 1, by P.MON again, µ(c 1 ) = S 1. Lemma 5.2 If a solution ϕ satisfies M.B and P.MON, then it satisfies W.I.R. Proof Let γ = (C,S,q,R) and µ ϕ(γ). For each c C, introduce a set S c of q c new students such that each s S c most prefers c and S c is the most preferred entering class of c. Let γ be the extension of γ so obtained. By M.B, ϕ(γ ) ={µ } where µ (c) = S c for each c C and µ (s) = φ for each s S. Then, by P.MON, µ(s)r s φ for each s S. Next, for each s S, we introduce a new college c with quota equal to one such that c and s most prefer each other. Let γ be the extension of γ so obtained. By M.B, ϕ(γ ) ={µ }, where µ (s) = c for each s S and µ (c) = φ for each c C. Then, by P.MON, µ(c)r c φ for each c C. In Section 3, Lemma 3.2 shows that W.U and P.MON imply I.R. The following example shows that this is no longer true in college admissions problems. Example 5.1 We extend the solution in Example 3.1 to the domain of this section. For each γ = (C,S,q,R), let ϕ(γ) be the set of all µ PO(γ ) satisfying the following two conditions: µ C (c)r c µ(c)r c µ S (c) for each c C, (3) µ S (s)r s µ(s)r s µ C (s) for each s S. (4) Since it is easy to show that ϕ satisfies P.MON and P.O, it also satisfies W.U. and M.B. In order to show that ϕ violates I.R, let γ = (C,S,q,R) be defined as follows; C ={c 1,c 2,c 3 }, q c = 2 for each c C, and S ={s 1,s 2,s 3,s 4,s 5,s 6 }. The preferences of colleges over individual students are given by c 1 : s 1 >s 2 >s 5 >s 3 >s 4 >φ>s 6, c 2 : s 3 >s 4 >s 1 >s 2 >s 5 >s 6 >φ, c 3 : s 5 >s 6 >s 3 >s 4 >s 1 >s 2 >φ, 5 Sonn (1994) first observed that the core satisfies M.MON on the domain of this section.

13 Monotonicity and consistency in matching markets 25 and the preference profile of students is given by s 1 : c 3 >c 2 >c 1 >φ, s 2 : c 3 >c 2 >c 1 >φ, s 3 : c 1 >c 3 >c 2 >φ, s 4 : c 1 >c 3 >c 2 >φ, s 5 : c 2 >c 1 >c 3 >φ, s 6 : c 2 >c 1 >c 3 >φ. We assume that college c 1 has a responsive preference such that {s 5,s 6 } > {s 3,s 4 }. It is obvious that µ C (c 1 ) ={s 1,s 2 }, µ C (c 2 ) ={s 3,s 4 }, µ C (c 3 ) ={s 5,s 6 }, µ S (c 1 ) ={s 3,s 4 }, µ S (c 2 ) ={s 5,s 6 }, µ S (c 3 ) ={s 1,s 2 }. If µ is given by µ(c 1 ) ={s 5,s 6 }, µ(c 2 ) ={s 1,s 2 }, µ(c 3 ) ={s 3,s 4 }, we can easily check that µ ϕ(γ). Since s 6 µ(c 1 ) is not acceptable for college c 1, ϕ violates I.R. Lemma 5.3 Under M.B and M.MON, W.I.R is equivalent to I.R. Proof Let ϕ be a solution satisfying M.B, M.MON, andw.i.r. Let γ = (C,S,q,R) and µ ϕ(γ). For each c C with µ(c) φ, let µ (c) = {s µ(c) sp c φ}. Because of W.I.R and responsiveness, µ (c). Consider a problem γ = (C,S,q,R ) such that µ (c) is the set of all acceptable students of c C with µ(c) φ, no student is acceptable for each c C with µ(c) = φ and µ(s) is the best choice of each s S. Because γ is obtained by a monotonic transformation of γ at µ, by M.MON, µ ϕ(γ ). On the other hand, by M.B, for each ν ϕ(γ ), ν(c) = µ (c) for each c C with µ(c) φ. By W.I.R, for each ν ϕ(γ ), ν(c) = φ for each c C with µ(c) = φ. Therefore, ϕ(γ ) ={µ }. In other words, µ(c) = µ (c) for each c C with µ(c) φ, which implies that every s µ(c) is acceptable for c C with µ(c) φ. Lemma 5.4 If a solution ϕ satisfies I.R, M.B and M.MON, then it is a subsolution of the core. Proof Let γ = (C,S,q,R) and µ ϕ(γ). By way of contradiction, suppose that µ has a blocking pair (c, s). Let s µ(c) be the least preferred student in µ(c) when µ(c). Note that if µ(c) <q c, µ(c) {s} is the best choice of c within µ(c) {s} and if µ(c) =q c, ( µ(c) \{s } ) {s} is the best choice of c within µ(c) {s}. We replace the preference of college c by another preference R c such that every student in µ(c) {s} is acceptable but no other student is acceptable. For each ŝ µ(c), we replace his or her preference by another preference P ŝ such that c is uniquely acceptable. We also replace P s by P s such that c and µ(s) are only acceptable colleges for s. Let γ = (C,S,q,R ) be the problem so obtained. It is obvious that γ is obtained by a monotonic transformation of γ at µ. Then, by M.MON, µ ϕ(γ ). However, the most preferred entering class of college c in γ is either µ(c) {s} or ( µ(c) \{s } ) {s} and every student in µ(c) {s} most prefers college c. Therefore, by M.B, for every µ ϕ(γ ), either µ (c) = µ(c) {s}

14 26 M. Toda or µ (c) = ( µ(c) \{s } ) {s}. In each case, µ (c) µ(c), which contradicts µ ϕ(γ ). Base on the result of Kara and Sönmez (1997), by the same argument as in Section 3, Lemma 3.5 is extended as follows. Lemma 5.5 No proper subsolution of the core satisfies M.MON. Then, the first main result of this section is as follows. Theorem 5.1 The core of college admissions problems is the unique solution satisfying W.U, P.MON, and M.MON. Proof Let ϕ be a solution satisfying the axioms. By Lemma 5.1, ϕ satisfies M.B. By Lemmas 5.2 and 5.3, ϕ satisfies I.R. Then, by Lemma 5.4, ϕ S. Lemma 5.5 implies that ϕ S. Next, we extend the definition of consistency to the domain of this section in the same way as Ergin (2002). For a given γ = (C,S,q,R) and µ M(γ ), let C C, S S. Suppose that all agents outside C S leave the market with what they receive at µ. Then, what is left for C S? If c leaves γ, every s µ(c) also leaves the market. Hence, for each s S, µ(s ) C unless µ(s ) = φ. Similarly, if µ(c) =q c and every s µ(c) leaves γ, then c also leaves the market. Hence, for each c C with µ(c ) =q c, µ(c ) S. On the other hand, for each c C with µ(c ) <q c,itmay be the case that µ(c ) S =, however, regardless of whether µ(c ) is quotafilling or not, the number of positions offered by c C should be reduced to q c µ(c ) \ S. Formally, for each γ = (C,S,q,R) and µ M(γ ), let C C and S S satisfy µ(c ) S for each c C with µ(c ) =q c and µ(s ) C {φ}. 6 Let q c = q c µ(c ) \ S for each c C and q = (q c ) c C. Let R be the collection of the restrictions of the preferences of agents in C S to c (S ) or C {φ}. Then, γ = (C,S,q, R ) is a reduced problem of γ at µ. For a reduced problem γ of γ at µ, we define µ C S M(γ ) as follows; For each c C, { µ µ(c ) (S {φ}) C S (c ) = φ if µ(c ) (S {φ}), otherwise, and each s S, µ C S (s ) = µ(s ). Consistency (CONS): for each γ and each µ ϕ(γ), ifγ = (C,S,q, R ) is a reduced problem of γ at µ, then µ C S ϕ(γ ). We have shown that under M.B and M.MON, W.I.R is equivalent to I.R. If M.MON is replaced by CONS, the same equivalence is obtained. Lemma 5.6 Under M.B and CONS, W.I.R is equivalent to I.R. 6 If µ is a one-to-one matching, these conditions are equivalent to µ(c S ) C S {φ}.

15 Monotonicity and consistency in matching markets 27 Proof Let ϕ be a solution satisfying M.B, CONS, and W.I.R. Let γ = (C,S,q,R) and µ ϕ(γ). For each c C with µ(c) φ, let µ (c) be the same as in the proof of Lemma 5.3. By the same reasoning, µ (c) φ. By W.I.R, c is acceptable for each s µ (c). Suppose that µ (c ) µ(c ) for some c C. Consider the reduced problem γ = (C,S,q, R ) of γ at µ where C ={c } and S = µ(c ). By CONS, µ C S ϕ(γ ). Note that µ C S (c ) = µ(c ). However, in γ, µ (c ) is the most preferred entering class of c and every s µ (c ) most prefers c. Then, by M.B, ν(c ) = µ (c ) for every ν ϕ(γ ), which contradicts µ(c ) µ (c ). Lemma 5.7 If a solution ϕ satisfies I.R, M.B, and CONS, then it is a subsolution of the core. Proof Let γ = (C,S,q,R) and µ ϕ(γ). By way of contradiction, suppose that µ has a blocking pair (c, s). Case 1: µ(s) = φ. If µ(c) <q c, let γ = (C,S,q, R ) be the reduced problem of γ at µ such that C ={c} and S ={s}. On the other hand, if µ(c) =q c, there exists s µ(c) such that sp c s. Then, let γ = (C,S,q, R ) be the reduced problem of γ at µ such that C ={c} and S ={s, s }. In each case, by CONS, µ C S ϕ(γ ).Itis obvious that {s} is the most preferred entering class of c and s most prefers c in γ but µ C S (s) = φ. This contradicts M.B. Case 2: µ(s) = c for some c C. If µ(c) <q c, let γ = (C,S,q, R ) be the reduced problem of γ at µ such that C ={c, c } and S ={s}. On the other hand, if µ(c) =q c, there exists s S such that sp c s. Then, let γ = (C,S,q, R ) be the reduced problem of γ at µ such that C ={c, c } and S ={s, s }. In each case, by CONS, µ C S ϕ(γ ).It is obvious that {s} is the most preferred entering class of c and s most prefers c in γ but µ C S (s) = c. This contradicts M.B. The next lemma is so-called bracing construction (Thomson 2004). For a given problem γ, the core may have some degree of freedom, however, if we augment γ by adding some agents in a particular way, the degree of freedom can be eliminated. The following physical metaphor is useful. A bookshelf made by an unskilled person may be unstable but a craftsman can add braces to the shelf in an elaborated way to make it stable. Lemma 5.8 For each γ = (C,S,q,R) and each µ S(γ ), there exists ˆγ such that S( ˆγ)={ˆµ}, γ is a reduced problem of ˆγ at ˆµ and ˆµ C S = µ. Proof If S(γ ) =1, the conclusion is obvious, by considering ˆγ = γ and ˆµ = µ. Then, it suffices to consider the case S(γ ) 2. Let µ, µ S(γ ) for µ µ. By Theorem 5.29 in Roth and Sotomayor (1990), we distinguish two cases. Case 1: µ(ŝ)pŝµ (ŝ) for some ŝ S. In this case, we introduce a new college ĉ with qĉ = 1 such that ŝ is uniquely acceptable for ĉ. Let Pŝ ˆ be the preference ordering over C {ĉ} {φ} that coincides with P s on C {φ} satisfying µ(ŝ) Pŝ ˆ ĉ Pŝ ˆ µ (ŝ). For each s ŝ, let Pˆ s be the preference ordering over C {ĉ} {φ} that coincides with P s on C {φ} satisfying φpˆ s ĉ. For ˆq = (q, qĉ), let ˆγ = (C {ĉ},s, ˆq, ˆR) be the extension of γ so obtained. Let ˆµ, ˆµ M( ˆγ)be such that ˆµ and ˆµ coincide with µ and µ on C S, respectively, and ˆµ(ĉ) =ˆµ (ĉ) = φ. By the construction of ˆγ, ˆµ S( ˆγ)but ˆµ / S( ˆγ)

16 28 M. Toda because (ĉ, ŝ) blocks ˆµ. Theorem 5.12 in Roth and Sotomayor (1990) states that the set of all colleges that do not fill their quota is the same at every stable matching. Then, S( ˆγ) {ν M( ˆγ) ν(ĉ) = φ}. Since ν C S S(γ ) for each ν S( ˆγ),it follows that S( ˆγ) S(γ ) 1. Case 2: µ(ĉ)pĉµ (ĉ) for some ĉ C but µ (s)p s µ(s) for each s S with µ (s) µ(s). Because µ(ĉ) µ (ĉ), by Theorem 5.13 in Roth and Sotomayor (1990), µ(ĉ) = µ (ĉ) =qĉ. Hence, µ (ĉ)\µ(ĉ). Suppose that there exists s µ(ĉ) such that s Pĉ s for some s µ (ĉ) \ µ(ĉ). Because ĉ = µ (s )P s µ(s ), (ĉ, s ) blocks µ, a contradiction. Therefore, for each s µ(ĉ) and each s µ (ĉ)\ µ(ĉ), spĉ s. We introduce a new student ŝ such that ĉ is uniquely acceptable for ŝ. Let s µ (ĉ) \ µ(ĉ) be such that s Pĉs for each s µ (ĉ) \ µ(ĉ) with s s. Let Pĉ ˆ be the preference ordering over c (S {ŝ}) that coincides with Pĉ on c (S) satisfying s Pĉ ˆ ŝ Pĉ ˆ s for each s µ(ĉ). For each c ĉ, let Pˆ c be the preference ordering over c (S {s }) that coincides with P c on c (S) satisfying φpˆ c ŝ. Let ˆγ = (C, S {ŝ},q, ˆR) be the extension of γ so obtained. Let ˆµ, ˆµ M( ˆγ)be such that ˆµ and ˆµ coincide with µ and µ on C S, respectively, and ˆµ(ŝ) = ˆµ (ŝ) = φ. By the construction of ˆγ, ˆµ S( ˆγ) but ˆµ / S( ˆγ) because (ĉ, ŝ) blocks ˆµ. Again by Theorem 5.12 in Roth and Sotomayor (1990), S( ˆγ) { ν M( ˆγ) ν(ŝ) = φ }.As in the first case, it follows that S( ˆγ) S(γ ) 1. By repeating these arguments finitely many times, the desired conclusion is obtained. Then, we have the following result generalizing Lemma 3.6 in Section 3. Lemma 5.9 No proper subsolution of the core satisfies CONS. Proof Let ϕ be a subsolution of the core satisfying CONS. Let µ S(γ ). Then, by Lemma 5.8, there exists ˆγ such that S( ˆγ) ={ˆµ}, γ is a reduced problem of ˆγ at ˆµ and ˆµ C S = µ. Because ϕ is a subsolution of the core, ϕ( ˆγ)={ˆµ}. Because ϕ satisfies CONS, µ ϕ(γ). Therefore, ϕ S. Remark 5.1 In the proof of Lemma 5.8, each college added to γ has quota equal to 1. Then, our argument is valid to prove Lemma 3.6. The final conclusion of this section is as follows. Theorem 5.2 The core of college admissions problems is the unique solution satisfying W.U, P.MON, and CONS. Proof Let ϕ be a solution satisfying the axioms. By Lemma 5.1, ϕ satisfies M.B. By Lemmas 5.2 and 5.6, ϕ satisfies I.R. Then, by Lemma 5.7, ϕ S. Lemma 5.9 implies that ϕ S. Appendix A In this appendix, we show the logical independence of the axioms in the characterization theorems. We give examples of solutions different from the core, each of

17 Monotonicity and consistency in matching markets 29 Table 1 The independence of the axioms in Sects. 3 and 5 W.U P.MON M.MON CONS ϕ 0 No Yes Yes Yes ϕ 1 Yes No Yes Yes ϕ 2 Yes Yes No No Table 2 The independence of the axioms in Sect. 4 W.U P.MON M.MON CONS ϕ 0 No Yes Yes Yes ϕ 1 Yes No Yes Yes ϕ 3 Yes Yes No Yes ϕ 4 Yes Yes Yes No which violates exactly one axiom in each theorem. Let us start from the theorems in Section 3. Let ϕ 0 (γ ) = M(γ ), ϕ 1 (γ ) = PO(γ ), and ϕ 2 (γ ) = {µ M,µ W } for each γ = (M, W, P). Then, Table 1 summarizes the properties of these solutions, showing the logical independence of the axioms in Theorems 3.1 and 3.2 Solutions ϕ 0 and ϕ 1 can be readily adapted to the domain of Section 4. Let ϕ 3 and ϕ 4 be the solutions defined in the proofs of Propositions 4.2 and 4.3, respectively. Table 2 shows the logical independence of the axioms in Theorem 4.1. The solutions in Table 1 can be extended to the domain of Section 5. Then, Table 2 also shows the logical independence of the axioms in Theorems 5.1 and 5.2. Appendix B In this appendix, we show that (lower) population monotonicity in Theorems 3.1 and 3.2 can not be replaced by upper population monotonicity. At first, for each γ = (M, W, P), the set of agents who are single is the same at all stable matchings. (See, e.g., Theorem 2.22 in Roth and Sotomayor 1990.) We denote by M φ W φ the set of those agents. Let γ φ = (M, W, ˆP) be the problem such that for each a M φ W φ, Pˆ a = P a and for each a/ M φ W φ, Pˆ a determines the same ordering over the agents on the opposite side as P a but φ has the lowest rank in Pˆ a. Then, we define ϕ(γ) = S(γ φ ). For each γ, S(γ ) ϕ(γ). Indeed, suppose that µ S(γ ) but µ/ ϕ(γ) = S(γ φ ). Each µ S(γ ) is individually rational in γ φ so that µ has a blocking pair (m, w) in γ φ.ifµ(m) = µ(w) = φ, then m, w M φ W φ, and thus P m = P m and P w = P w. Then, (m, w) block µ in γ, which is a contradiction. If µ(m) φ and µ(w) φ, since µ IR(γ ), (m, w) block µ in γ, which is also a contradiction. Hence, without loss of generality, we assume that µ(m) φ and µ(w) = φ. Then, w W φ so that P w = P w. Since µ IR(γ ), µ(m)r m φ. Therefore, (m, w) block µ in γ, which is again a contradiction. It is obvious that S(γ ) ϕ(γ) in general. In order to show that ϕ satisfies upper population monotonicity, let γ be a W - extension of γ and µ ϕ(γ ) = S(γ φ ). Let µ W and µ W be the women-optimal

18 30 M. Toda stable matchings in γ φ and γ φ, respectively. Note that µ W ϕ(γ) and γ φ is a W-extension of γ φ. Theorem 2.25 in Roth and Sotomayor (1990) states that µ W (w) ˆR w µ W (w) for each w W. From the definition of µ W, it follows that µ W (w) ˆR w µ W (w) ˆR w µ (w) for each w W. Since S(γ ) ϕ(γ ) = S(γ φ ), µ W (w)r wφ and hence µ W (w)r w φ for each w W. Therefore, µ W (w)r w µ (w) for each w W. A symmetric argument for an M-extension shows that ϕ satisfies upper population monotonicity. For each γ and µ M(γ ), ifγ is obtained from γ by a monotonic transformation at µ, then γ φ is also obtained from γ φ by a monotonic transformation at µ. Because the core satisfies Maskin monotonicity, ϕ satisfies the same axiom. For each γ and µ M(γ ), ifγ is a reduced problem of γ at µ, then γ φ is a reduced problem of γ φ at µ. Because the core satisfies consistency, ϕ is consistent. Appendix C The mutually best axiom (M.B) in section 5 is not a unique extension of the original form in Section 2 to college admissions problems. For instance, the following two different formulations are possible. Mutually best in the first sense (MB 1 ): for each γ,ifc and s most prefer each other, then s µ(c) for each µ ϕ(γ). Mutually best in the second sense (M.B 2 ): for each γ, if there exists s S c who most prefers c, then s µ(c) for each µ ϕ(γ). Obviously, M.B 2 implies M.B in Section 5 and under responsive preferences, M.B 2 also implies M.B 1. The core satisfies M.B 2 and hence M.B 1 as well. In general, there is no logical relation between M.B 1 and M.B, however, under P.MON, M.B implies M.B 1. Lemma 5.1 states that W.U and P.MON imply M.B. If we strengthen W.U to P.O, we obtain M.B 2. 7 Lemma 5.2 shows that M.B and P.MON imply W.I.R. The proof of Lemma 5.2 suggests that if a solution ϕ satisfies M.B 1 and P.MON, µ(c)r c φ for each c and each µ ϕ(γ). However, we can not ensure that µ(s)r s φ for each s. Hence, M.B 1 and P.MON do not imply W.I.R, which indicates that M.B 1 is strictly weaker than M.B even if all preferences are responsive. References Ehlers L (2004) Monotonic and implementable solutions in generalized matching problems. J Econ Theory 114: Ergin H I (2002) Efficient resource allocation on the basis of priorities. Econometrica 70: Eriksson K, Karlander J (2000) Stable matching in a common generalization of the marriage and assignment models. Discrete Math 217: Fujishige S, Tamura A (2003) A general two-sided matching market with discrete concave utility functions. RIMS Preprint 1401, Kyoto University, Kyoto Gale D, Shapley L (1962) College admission and the stability of marriage. Am Math Monthly 69:9 15 Kaneko M (1982) The central assignment games and the assignment markets. J Math Econ 10: It is open whether or not W.U and P.MON imply M.B 2.