Understanding Stable Matchings: A Non-Cooperative Approach KANDORI, Michihiro, KOJIMA, Fuhito, and YASUDA, Yosuke∗ March 8, 2009 Preliminary and incomplete. Do not circulate.
Abstract We present a series of non-cooperative games with monotone best replies whose set of Nash equilibria coincides with the set of stable matchings. A number of key features of stable matchings can be understood as familiar properties of games with monotone best replies. We also provide some new results on stable matchings, by making use of the techniques developed for games with monotone best replies.
∗
Kandori: Faculty of Economics, University of Tokyo (email: e-mail:
[email protected]); Kojima: Cowles Foundation, Yale University and Department of Economics, Stanford University (email:
[email protected]); Yasuda: National Graduate Institute for Policy Studies (email:
[email protected]). We are grateful to Hiroyuki Adachi, Federico Echenique, Drew Fudenberg, Hideo Konishi, Paul Milgrom, Tayfun Sonmez, Satoru Takahashi, Utku Unver and seminar participants at Boston College, Caltech, Edinburgh, Princeton and Waseda for comments and discussions.
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1
Framework
1.1
Preliminary definitions
A market is tuple Γ = (S, C, P ). S and C are finite and disjoint sets of students and colleges. We denote N = S ∪ C. For each s ∈ S, Ps is a strict preference relation over 2C . For each c ∈ C, Pc is a strict preference relation over 2S .1 The non-strict counterpart of Pi is denoted by Ri , so we write XRi X 0 if and only if XPi X 0 or X = X 0 . For each s ∈ S and C 0 ⊆ C, the chosen set Chs (C 0 ) is a set such that (1) Chs (C 0 ) ⊆ C 0 , and (2) C 00 ⊆ C 0 implies Chs (C 0 )Rs C 00 . In words, Chs (C 0 ) is the set of colleges that s would choose if she can choose partners freely from C 0 . For each c ∈ C and S 0 ⊆ S, the set Chc (S 0 ) is similarly defined. A matching µ is a (possibly empty) correspondence from C ∪ S to C ∪ S such that µ(s) ⊆ C for every s ∈ S, µ(c) ⊆ S for every c ∈ C and, for every i, j ∈ N , i ∈ µ(j) if and only if j ∈ µ(i). We abuse the notation and, for any i ∈ N , write µPi ν if µ(i)Pi ν(i) and µRi ν if µ(i)Ri ν(i). Given a matching µ, we say that it is blocked by (s, c) ∈ S × C if s ∈ / µ(c), s ∈ Chc (µ(c) ∪ s) and c ∈ Chs (µ(s) ∪ c). A matching µ is individually rational if for every i ∈ N , Chi (µ(i)) = µ(i). A matching µ is pairwise stable if it is individually rational and is not blocked. We simply refer to pairwise stability as stability when there is no confusion. For each i ∈ S (respectively i ∈ C), her preference relation Pi is substitutable if for any X, X 0 ⊆ S (respectively X, X 0 ⊆ C) with X ⊆ X 0 , we have Chi (X 0 ) ∩ X ⊆ Chi (X) (Kelso and Crawford, 1982). That is, a partner who is chosen from a larger set of potential partners is always chosen from a smaller set of potential partners. If every agent has substitutable preferences, then there exists a pairwise-stable matching (Roth, 1984).
1.2
Normal-form game
We consider the following students’ final offers game, which is a simplified version of games analyzed by Sotomayor (2004) and Echenique and Oviedo (2006). The set of players is S, while colleges in C are passive players.2 In the first stage, each s ∈ S simultaneously announces a subset of colleges Cs ⊆ C, to which she makes an offer. In the second stage, 1
In many situations, it is natural to assume a college has capacity constraints, that is, the college cannot admit more than a certain number of students. Such feasibility constraints can be accommodated by assuming preferences appropriately. For example, we can simply assume that the college disprefers any set of students of cardinality exceeding the capacity to the null set of students. 2 We can alternatively define a two-stage game with colleges being players in the second stage. The analysis is unchanged.
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each college c ∈ C chooses to match to its most preferred subset of students among those who proposed to it, i.e. to Chc (Sc ) where Sc := {s ∈ S|c ∈ Cs }. The outcome of the game is uniquely specified by µ(c) = Chc (Sc ) for each c ∈ C. We write CS 0 := (Cs )s∈S 0 for any S 0 ⊆ S, C−s = CS\s , and φ(CS ) to be the matching that results in the end of this game when students announce CS .3 A strategy profile CS is a Nash equilibrium if φ(CS )Rs φ(Cs0 , C−s ) for every s ∈ S and Cs0 ⊆ S. We introduce an order on the set of strategies by Cs0 Cs if Cs ⊆ Cs0 . For any S 0 ⊆ S, we define the order by CS0 0 CS 0 if Cs ⊆ Cs0 for every s ∈ S 0 .
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Results
Consider a students’ final offers game where preferences are not necessarily substitutable. For any C−s , let As (C−s ) := {c ∈ C|s ∈ Chc (Sc0 ∪ s)}, where Sc0 := {s0 ∈ S \ s|c ∈ Cs0 }. Given any strategy profile CS and s ∈ S, it is easy to see that Cs0 is a best response of s to C−s if and only if Chs (As (C−s )) ⊆ Cs0 ⊆ Chs (As (C−s )) ∪ (C \ As (C−s )). Lemma 1 1. Let CS be a Nash equilibrium. If Cs Cs0 and Cs0 is a best response to C−s for every s ∈ S, then φ(CS0 ) = φ(CS ) and CS0 is also a Nash equilibrium. 2. Let CS be a Nash equilibrium. If preferences of all colleges are substitutable and, Cs0 Cs and Cs0 is a best response to C−s for every s ∈ S, then φ(CS0 ) = φ(CS ) and CS0 is also a Nash equilibrium. Proof. Part 1. By definition of CS0 we have φ(CS )(c) ⊆ {s ∈ S|c ∈ Cs0 } ⊆ {s ∈ S|c ∈ Cs } for each college. This and rationality of each college imply φ(CS0 ) = φ(CS ). To show that Cs0 is a Nash equilibrium, note that for each s ∈ S we have {c ∈ C|s ∈ Chc (Sc0 ∪{s})} ⊆ {c ∈ C|s ∈ Chc (Sc ∪ {s})} where Sc = {s ∈ S|c ∈ Cs } and Sc0 = {s ∈ S|c ∈ Cs0 } by 0 rationality of college preferences. Therefore Cs0 is a best response to C−s for each s ∈ S, completing the proof. Part 2. Since colleges have substitutable preferences, φ(CS0 ) = φ(CS ). Since college preferences are substitutable and Cs ⊆ Cs0 for every s ∈ S, s ∈ / Chc (Sc ∪ {s}) implies 0 0 s∈ / Chc (Sc ∪ {s}) for every s ∈ S and c ∈ C. Therefore CS is a Nash equilibrium. As the following example shows, when college preferences are not subsitutable, the conclusion of part 2 of the above Lemma does not necessarily hold. 3
We often write x for a singleton set {x} for simplicity.
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Example 1 Let C = {c}, S = {s1 , s2 }, c s ∅ for s ∈ {s1 , s2 } and {s1 , s2 } c ∅ c {s} for s ∈ {s1 , s2 }. It is a Nash equilibrium for each student to apply to no college. On the other hand, it is not an equilibrium for s1 to apply to c while s2 applies to no college, since if s2 instead applies to c, c accepts both s1 and s2 , thus benefitting s2 . In the sequel, we assume that every agent has substitutable preferences unless stated otherwise. We say that Cs and Cs0 are outcome equivalent under C−s if φ(Cs , C−s ) = φ(Cs0 , C−s ). Given C−s , a strategy Cs of s ∈ S is a maximal strategy under C−s if Cs0 ⊆ Cs for every strategy Cs0 that is outcome equivalent to Cs under C−s . Lemma 2 Given any strategy profile CS and any s ∈ S, there exists a maximal strategy for s that is outcome equivalent to Cs under C−s . Proof. Since every college has substitutable preferences, For any C−s , As (C−s ) := {c ∈ C|s ∈ Chc (Sc0 ∪ s)}, where Sc0 := {s0 ∈ S \ s|c ∈ Cs0 }. In words, As (C−s ) is the set of colleges that, if s proposes, would accept her offer. Now given an arbitrary Cs and C−s , let C¯s := Cs ∪ (C \ As (C−s )). We shall show C¯s is the maximal strategy that is outcome equivalent to Cs . To show this, first note that all colleges that receive an offer from s under Cs receive one from s under C¯s since Cs ⊆ C¯s . This implies that all colleges that accept an offer from s under Cs accept one from s under C¯s . Second, by definition of C¯s , no colleges that do not accept an offer from s under Cs accept an offer from s under C¯s . Thus C¯s is outcome equivalent to Cs . Second, consider any Cs0 that is not a subset of C¯s . By definition of C¯s , this means that there exists a college c ∈ Cs0 \ C¯s such that s ∈ Chc (Sc0 ∪ s). Then c ∈ φ(Cs0 , C−s )(s) while c ∈ / φ(Cs , C−s )(s) by assumption, and hence φ(Cs0 , C−s )(s) 6= φ(Cs , C−s ), so Cs0 is not outcome equivalent to Cs . This completes the proof. Given s ∈ S and C−s , a strategy of s is the maximal best response of s to C−s if it is a best response of s to C−s and it is a maximal strategy under C−s . The maximal best response function br : (2C )|S| → (2C )|S| is a function such that, for each s ∈ S, brs (CS ) is the maximal best response of s to C−s . By Lemma 2, the function br is well-defined. Lemma 3 Given any strategy profile CS and s ∈ S, brs (CS ) = Chs (As (C−s )) ∪ (C \ As (C−s )). Proof. Immediate from Lemma 2.
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Proposition 1 The following sets of matchings coincide: (i) The set of stable matchings, (ii) The set of Nash equilibrium outcomes, and (iii) The set of Nash equilibrium outcomes in maximal strategies. Proof. The equivalence of (i) and (ii) are a minor modification of Sotomayor (2004) and Echenique and Oviedo (2006). The equivalence between (ii) and (iii) follows from Lemma 1.
The following example shows that the set of maximal strategies may not coincide with the set of stable matchings as defined in this paper (that is, pairwise stable matchings) when college preferences are not substitutable. Example 2 Let C = {c}, S = {s1 , s2 }, c s ∅ for s ∈ {s1 , s2 } and {s1 , s2 } c ∅ c {s} for s ∈ {s1 , s2 }. This is the same market as the one in Example 1. The empty matching (a matching where every agent is unmatched) is a pairwise stable matching. However the following argument shows that the empty matching is not an outcome of any Nash equilibrium in maximal strategies. First, (Cs1 , Cs2 ) = (∅, ∅) is not a Nash equilibrium in maximal strategies, since a maximal best response of s1 to Cs2 is {c}. Second, ({c}, ∅) is not a Nash equilibrium since s2 can profitably deviate to {c} and be matched to c while she is unmatched under ({c}, ∅). Third, (∅, {c}) is not a Nash equilibrium since s1 can profitably deviate to {c} and be matched to c while she is unmatched under (∅, {c}). Finally, ({c}, {c}) is a Nash equilibrium in maximal strategies with outcome φ({c}, {c})(c) = {s1 , s2 }. A matching µ is said to be Hatfield-Milgrom stable if it is individually rational and there is no student s ∈ S and C 0 ⊆ C such that C 0 s µ(s) and s ∈ Chc (µ(c) ∪ {s}) for each c ∈ C 0 . In general, any outcome of a Nash equilibrium in students’ final offers game is Hatfield-Milgrom stable. Proposition 2 The students’ final offers game has 0 (i) Strategic complementarity: That is, for any s ∈ S, C−s C−s and any best response 0 0 Cs to C−s , there exists a best response Cs to C−s such that Cs0 Cs . Equivalently, 0 brs (Cs , C−s ) brs (CS ). 0 (ii) Negative externality for students: That is, for any s ∈ S, CS and CS0 with C−s C−s , 0 if Cs = brs (CS ) then φ(CS )Rs φ(CS ).
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(iii) Positive externality for colleges: That is, for any c ∈ C, CS and CS0 CS , φ(CS0 )Rc φ(CS ). 0 Proof. (i) Since C−s −s C−s and colleges have substitutable preferences, we have 0 As (C−s ) ⊆ As (C−s ). Then, we have
brs (CS ) = Chs (As (C−s )) ∪ (C \ As (C−s )) (Lemma 3) 0 0 )). ) ∪ (C \ As (C−s ⊂ Chs (As (C−s )) ∩ As (C−s 0 ) ∪ (C \ The second line holds because (1) Chs (As (C−s )) ⊂ Chs (As (C−s )) ∩ As (C−s 0 0 As (C−s )) and (2)(C \ As (C−s )) ⊂ (C \ As (C−s )). Now, note further that student s’s 0 0 substitutable preferences imply Chs (As (C−s )) ∩ As (C−s ) ⊆ Chs (As (C−s )). This establishes brs (CS ) = Chs (As (C−s )) ∪ (C \ As (C−s )) 0 0 ⊂ Chs (As (C−s )) ∪ (C \ As (C−s )) = brs (CS0 ). 0 0 (ii) Since college preferences are substitutable, C−s C−s implies As (C−s ) ⊆ As (C−s ). 0 0 Thus φ(CS )(s) = Chs (As (C−s ))Rs Chs (As (C−s )) = φ(CS )(s). (iii) Since CS0 CS , Sc0 ⊆ Sc , where Sc0 is the set of students that apply to c under CS0 . Thus φ(CS0 )(c) = Chc (Sc0 )Rc Chc (Sc ) = φ(CS )(c), completing the proof.
Proposition 2 (i) is our key result, which enables us to connect any matching model to a corresponding strategic complementarity (non-cooperative) game. To obtain this nice property, allowing overbooking by students, i.e., applying to more colleges than their serious ones plays a crucial role. In what follows, we will briefly explain why this overbooking is important in the context of many-to-one matching where each student can be matched at most one college. We note that it is crucial that each student can apply to any subset of colleges irrespective of their preferences. To see this point consider a many-to-one matching where, in contrast to our students’ final offers game, each student can be matched to at most one college and each student should apply to only one college simultaneously with other students. Heuristically, we assume that “overbooking” is prohibited. Therefore, the number of strategies is equal to the number of colleges plus 1 (not applying to any college). Perhaps the most natural strategy ordering one may think of is to define applying to a more preferred college as a larger strategy for each student. For example, if a student prefers college ci to cj , then strategy ci is called larger than cj for the student. Unfortunately, the following simple example shows that this seemingly natural ordering fails to establish strategic complementarity.
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Example 3 Consider a market with two students s1 , s2 , and two colleges c1 , c2 , each of which has a quota of 1 student. Assume that all students (resp. colleges) are acceptable to all colleges (resp. students), and that both students prefer c1 to c2 , and both colleges prefer s1 to s2 . Since c1 is a better college than c2 , c1 is a larger strategy than c2 for each student. Then, it is easy to see that s2 ’s best reply function is not increasing with respect to s1 ’s strategy. Note that s2 ’s best reply is c1 when s1 takes smaller strategy c2 , while it decreases to c2 , if s1 increases her strategy to c1 . Therefore, “better college” = “larger strategy” does not yield increasing best replies.
The above example suggests it is difficult to endow strategy orderings that yield strategic complementarity. Indeed this difficulty is formally stated in the next impossibility result, which shows that Proposition 2 (i) cannot hold unless overbooking is allowed.
Proposition 3 Consider a students’ final offers game in which each student is restricted to apply to one college, i.e., overbooking is prohibited. Then it is impossible to endow each student’s strategy space with a partial order such that, for all matching problems (S, C, P ) 1. the ordering induces a lattice for each student’s strategy space, 2. there exists a best reply selection (a single-valued function that is selected from the best response correspondence) that is non-decreasing with respect to the ordering. Proof. Consider a market with three students s1 , s2 , s3 , and two colleges c1 , c2 , each of which has a quota of 1 student. Fix a preference profile of students such that s1 c s2 c s3 c ∅ for c = c1 , c2 , and both colleges are acceptable to all students. By condition (1), each student has maximal and minimal strategies. This, combined with the fact that there are only three strategies (applying to c1 , c2 , and not applying to any college), implies that c1 c2 or c2 c1 according to the partial order of strategies for each student. Since we have three students, there are at least two students si and sj with i, j, ∈ {1, 2, 3}, i < j such that the ordering on their strategies over two colleges coincide, say c1 c2 without loss of generality. Consider strategy profiles where sk with k 6= i, j applies to no college. When si increases her strategy from c2 to c1 , it is uniquely optimal for sj to decrease his strategy strictly from c1 to c2 . Thus condition (2) is violated, completing the proof. Let (L, ) be a complete lattice. Function f : L → L is said to be monotone increasing if, for all x, y ∈ L, x y implies f (x) f (y).
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Result 1 (Tarski’s Fixed Point Theorem) Let (L, ) be an nonempty complete lattice and function f : L → L is monotone increasing. Then the set of all fixed points of f is a nonempty complete lattice with respect to . Lemma 4 The function br is monotone increasing with respect to . Proof. This is a direct consequence of Proposition 2-(i). By Result 1 and Lemma 4, we have the following result. Theorem 1 (Existence of a Stable Matching) In the students’ final offers game, the set of Nash equilibria in maximal strategies forms a non-empty lattice with respect to the partial order . Therefore the set of stable matchings is nonempty. A stable matching µ is a student-optimal stable matching if µRs ν for every s ∈ S and every stable matching ν. A stable matching µ is a college-optimal stable matching if µRc ν for every c ∈ C and every stable matching ν. A stable matching µ is a studentpessimal stable matching if νRs µ for every s ∈ S and every stable matching ν. A stable matching µ is a college-pessimal stable matching if νRc µ for every c ∈ C and every stable matching ν. Theorem 2 (Side-Optimal/Pessimal Stable Matchings) There exist a student-optimal stable matching and a college-optimal stable matching. The student-optimal stable matching coincides with the college-pessimal stable matching and the college-optimal stable matching coincides with the student-pessimal stable matching. Proof. By Theorem 1, there exist Nash equilibria C¯S and CS in maximal strategies, such that C¯S CS CS for every Nash equilibrium CS in maximal strategies. By items (ii) and (iii) of Proposition 2, this implies that φ(CS )Rs φ(CS )Rs φ(C¯S ) for every s ∈ S and φ(C¯S )Rc φ(CS )Rc φ(CS ) for every c ∈ C, completing the proof.
2.1
Deferred Acceptance Algorithm as a Learning Process
In this section, we investigate the student-proposing deferred acceptance algorithm, or the deferred acceptance algorithm for short (Gale and Shapley, 1962). More specifically, we consider a sequential version of the deferred acceptance algorithm originally formulated in one-to-one matching by McVitie and Wilson (1970) extended to the many-to-many setting. Let S be ordered in an arbitrary manner, by s1 , s2 , . . . , s|S| .
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Initialization: Let t = 0 and µ0 be the empty matching, that is, µ0 (s) = ∅ for every s ∈ S. Let A0 (s) = C for every s ∈ S. Iteration: For each t, Step t proceeds as follows. 1. If µt (s) = Chs (At (s)) for all s ∈ S, then µt+1 = µt and At+1 (s) = At (s) for all s ∈ S. 2-1. Otherwise, let st be a student with the smallest index such that µt (s) 6= Chs (At (s)). At+1 (s) = At (s) for all s 6= st . Let u = 0 and At,0 (st ) = At (st ). 2-2. For each u, Student st applies to the set of colleges Chs (At,u (s)). Each college c ∈ Chs (At,u (s)) chooses its most preferred group of students among those who applied, that is, Chc (µt (c) ∪ st ). If no college rejects st , then let the resulting matching be µt+1 and At+1 (st ) = At,u (st ). If some college rejects st , then delete the set of colleges that rejected st from At,u (st ) and let it be At,u+1 (st ). We say that the algorithm terminates in step T if µT (s) = Chs (AT (s)) for all s ∈ S and µt (s) 6= Chs (At (s)) for some s ∈ S for all t < T . Note that we have not shown that the deferred acceptance algorithm terminates at this point: We will derive termination as a consequence of Theorem 3 below. We will relate the deferred acceptance algorithm to a learning process on our students’ final offers game. For that purpose, we first define the following learning process, called the best response dynamics. Initialization: Let Cs0 = ∅ for each s ∈ S. Iteration: For each t: If CSt is a Nash equilibrium, then µt+1 = µt . If not, let st be the student with the smallest index whose current strategy is not a best response. Let st change her strategy to Cst+1 = brs (CSt ) while every other student s keeps taking Cst+1 = Cst . We will say that the best response dynamics converges at a finite step T if C T is a Nash equilibrium and C t is not a Nash equilibrium for any t < T . Theorem 3 φ(CSt ) = µt for each t ∈ {0, 1, 2, . . . }. t Proof. We will show, by induction, that φ(CSt ) = µt and At (s) ⊇ As (C−s ) for all 0 0 s ∈ S and t = 1, 2, . . . . For t = 0, φ(CS ) = µ since both are the empty matching and 0 A0 (s) ⊇ As (C−s ) since A0 (s) = C for all s ∈ S by definition. t Suppose φ(CSt ) = µt and At (s) ⊇ As (C−s ) for all s ∈ S. If µt (s) = Chs (At (s)) for all s ∈ S, then for each s ∈ S, µt (s) = µt+1 (s) and At (s) = At+1 (s) by definition and since µt
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is a stable matching, by Theorem 1, C t is a Nash equilibrium and hence by definition of the t+1 best response dynamics, CSt+1 = CSt so φ(CSt+1 ) = µt+1 and At+1 (s) = At (s) ⊇ As (C−s ). So t t t suppose µ (s) 6= Chs (A (s)) for some s ∈ S and let s be such a student with the smallest t index. Then, by construction of the deferred acceptance algorithm, At,u (st ) ⊇ Ast (C−s t) = t+1 t+1 t+1 t Ast (C−st ) for all u and hence A (s ) ⊇ As (C−st ). Also by substitutability of preferences t+1 of colleges and st , At+1 (s) = At (s) ⊇ As (C−s ) for all s 6= st . Moreover µt+1 (st ) = Chst (As (µt )) and for any s 6= st , µt+1 (s) = µt (s) \ {c ∈ µt+1 (st )|s ∈ / Chc (µt (c) ∪ st )}. In the best response dynamics, φ(CSt+1 )(st ) = Chst (A(C−s )) = Chst (As (µt )) = µt+1 (st ), and for any s 6= st , by substitutability, φ(CSt+1 )(s) = φ(CSt )(s) \ {c ∈ φ(CSt+1 )(st )|s ∈ / t t t+1 Chc (µ (c) ∪ s )} = µ (s). This completes the proof. Corollary 1 The student-proposing deferred acceptance algorithm terminates in a finite number T of steps. The resulting matching µT is the student-optimal stable matching. Proof. Since the students’ final offers game is a game with a finite set of strategies and strategic complementarity, the best-response dynamics as described above converges in a finite time T at the smallest Nash equilibrium (note that the dynamics starts at the smallest strategy profile). Since the smallest Nash equilibrium results in the studentoptimal stable matching by Theorem 2, by Theorem 3 the student-proposing deferred acceptance algorithm terminates in a finite number of steps T and the resulting matching µT is the student-optimal stable matching.
2.2
Understanding Properties of Stable Matchings
Recall a market is tuple Γ = (S, C, P ). Let Γ−c = (S, C \c, P−c ) be a market in which college c is not present but otherwise the same as Γ. Formally, we define Γ−c from Γ simply by changing preferences of c in such a way that Chc (S 0 ) = ∅ for every S 0 ∈ S. This definition is convenient since we need to re-define neither preferences of students nor strategy sets in the new market Γ−c with this definition. Let br−c be the maximal best response function in Γ−c . Since preferences are substitutable, br(CS ) ⊆ br−c (CS ) for any CS ⊆ C S . Theorem 4 (Comparative Statics) Let C¯S and C¯S−c be the largest Nash equilibria in maximal strategies in Γ and Γ−c , respectively. Then φ(C¯S )Rs φ(C¯S−c ) for every s ∈ S and φ(C¯S−c )Rc0 φ(C¯S ) for every c0 ∈ C \ c. Let CS and C−c S be the smallest Nash equilibria in −c maximal strategies in Γ and Γ , respectively. Then φ(CS )Rs φ(C−c S ) for every s ∈ S and −c 0 φ(CS )Rc0 φ(CS ) for every c ∈ C \ c.
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Proof. Since the students’ final offers game has strategic complementarity, we have C¯S = sup{CS ⊆ C S |CS br(CS )}, C¯S−c = sup{CS ⊆ C S |CS br−c (CS )}. Since br(CS ) br−c (CS ), CS br(CS ) implies CS br−c (CS ). Thus C¯S = sup{CS ⊆ C S |CS br(CS )} sup{CS ⊆ C S |CS br−c (CS )} = C¯S−c . Moreover c is not matched to any s ∈ S in Γ−c . Therefore φ(C¯S )Rs φ(C¯S−c ) for every s ∈ S and φ(C¯S−c )Rc0 φ(C¯S ) for every c0 ∈ C \ c. Similarly, CS = inf{CS ⊆ C S |CS br(CS )} inf{CS ⊇ C S |CS br−c (CS )} = C¯S−c , −c 0 and hence φ(CS )Rs φ(C−c S ) for every s ∈ S and φ(CS )Rc0 φ(CS ) for every c ∈ C \ c, completing the proof.
Let CS∗ be any Nash equilibrium in maximal strategies in Γ, and now assume c becomes unavailable to be matched to any student, and students try to make new offers and so on. This story suggests the following equilibration process, called the vacancy chain dynamics associated with CS∗ and c (Blum, Roth, and Rothblum, 1997). Initialization: Let CS (0) = CS∗ . Iteration: For each Step t ∈ {1, 2, . . . }, let CS (t) = br−c (CS (t)). We say that the vacancy chain dynamics terminates in finite steps if CS (t + 1) = CS (t) for some t ∈ {0, 1, . . . }. Theorem 5 (Vacancy Chain Dynamics) For any Nash equilibrium CS∗ in maximal strategies in Γ and c ∈ C, the vacancy chain dynamics associated with CS∗ and c terminates in finite steps at a Nash equilibrium CS0 in maximal strategies in Γ−c , and φ(CS∗ )Rs φ(CS0 ) for every s ∈ S and φ(CS0 )Rc0 φ(CS∗ ) for every c0 ∈ C \ c. Proof. Recall that br−c (CS ) br(CS ) for any CS ⊆ C S , and br(CS∗ ) = CS∗ since CS∗ is a Nash equilibrium in maximal strategies in Γ. These properties imply that CS (1) = br−c (CS∗ ) br(CS∗ ) = CS∗ .
(1)
By an inductive argument, property (1) implies CS (t) ⊇ CS (t − 1) for all t ∈ {1, 2, . . . }. Since the set of strategies is a finite set, this implies that the algorithm terminates to a Nash equilibrium CS0 in Γ−c satisfying CS0 CS∗ . This set inclusion and the fact c rejects every s in Γ−c imply φ(CS∗ )Rs φ(CS0 ) for every s ∈ S and φ(CS0 )Rc0 φ(CS∗ ) for every c0 ∈ C \ c, completing the proof.
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Remark 1 For Theorems 4 and 5, we considered only situations in which one college c ∈ C exits from the market for simplicity. It is tedious but easy to extend the analysis to the case in which a group of colleges or students enter or exit the market simultaneously. Consider the following property due to Alkan (2002), Alkan and Gale (2003) and Hatfield and Milgrom (2005). Definition 1 (The Law of Aggregate Demand, Size Monotonicity) Preference relation Pc satisfies the law of aggregate demand (size monotonicity) if |Chc (S 0 )| ≥ |Chc (S 00 )| for every S 00 ⊆ S 0 ⊆ S. Similarly, preference relation Ps satisfies the law of aggregate demand if |Chs (C 0 )| ≥ |Chs (C 00 )| for every C 00 ⊆ C 0 ⊆ C. With substitutability and the law of aggregate demand, we show a version of the socalled “rural hospitals theorem.” Roth (1986) shows one version of the rural hospitals theorem for many-to-one matching with responsive preferences. More specifically, he shows that every hospital that has unfilled positions at some stable matching is assigned exactly the same doctors at every stable matching. Martinez, Masso, Neme, and Oviedo (2000) generalize the theorem for substitutable and q-separable preferences. Although there is no obvious notion of “unfilled positions” under substitutability4 , Theorem 6 below shows that a weaker version of the rural hospitals theorem still holds, as formulated by McVitie and Wilson (1970), Gale and Sotomayor (1985a; 1985b) and Roth (1984). More specifically, every hospital signs exactly the same number of contracts at every stable allocation, although the doctors assigned and the terms of contract can vary. Theorem 6 (The Rural Hospital Theorem) If preference relations of all agents satisfy substitutability and the law of aggregate demand, then every student and college are matched with the same number of partners at every stable matching. Proof. Let C¯S and CS be the largest and smallest Nash equilibria in maximal strategies. For any Nash equilibrium CS in maximal strategies, by Theorem 1 we have C¯S CS CS . Since college preferences satisfy the law of aggregate demand, this implies |φ(C¯S )(c)| ≥ |φ(CS )(c)| ≥ |φ(CS )(c)| for every c ∈ C.
(2)
On the other hand, since college preferences are substitutable, C¯S CS CS implies that, for every s ∈ S, As (C¯−s ) ⊆ As (C−s ) ⊆ As (C−s ). Since student preferences satisfy the law of aggregate demand, this implies |φ(CS )(s)| ≥ |φ(CS )(s)| ≥ |φ(C¯S )(s)| for every s ∈ S. 4
(3)
A discussion on this point is found in Hatfield and Kojima (2008), where a rural hospital theorem similar to Theorem 6 is shown in the context of many-to-one matching with contracts as in Hatfield and Milgrom (2005).
12
Inequalities (2) and (3) can both hold if and only if all weak inequalities hold with equality, completing the proof. Theorem 6 is applicable to the previous domains of McVitie and Wilson (1970), Gale and Sotomayor (1985a; 1985b), Roth (1984; 1986) and Martinez, Masso, Neme, and Oviedo (2000). Hatfield and Milgrom (2005) show the theorem in the context of many-to-one matching with contracts under substitutes and the law of aggregate demand, and Hatfield and Kojima (2008) establish the same conclusion while relaxing the substitute to a condition called unilateral substitutes. Both the substitutes and unilateral substitutes conditions reduce to the substitutability condition when terms of contracts are exogenously given as in our paper.
3
Generalized Threshold Strategies
In this section we consider an alternative class of strategies. Given s ∈ S and C−s , the generalized threshold best response of s to C−s is defined as {c ∈ C|c ∈ Chs (As (C−s ) ∪ c)}. Generalized threshold best response function br : (2C )|S| → (2C )|S| is a function such that, for each s ∈ S, brs (CS ) is the generalized best response of s to C−s , that is, brs (CS ) = {c ∈ C|c ∈ Chs (As (C−s ) ∪ c)}, for every s ∈ S. A generalized threshold best response is clearly a best response. Moreover, as in Proposition 2, the set of stable matchings can be shown to be equivalent to the set of Nash equilibrium outcomes in generalized threshold best responses, and further that, as in Lemma 4, the generalized threshold best response function is monotone increasing. Here we show the monotonicity (other proofs are essentially the same as before). Proposition 4 When students and colleges have substitutable preferences, generalized threshold best response function is monotone increasing. 0 Proof. Suppose C−s −s C−s . Since colleges have substitutable preferences, we have 0 0 ), and by definition this is As (C−s ) ⊆ As (C−s ). We need to show brs (C−s ) ⊂ brs (C−s 0 equivalent to c ∈ Chs (As (C−s ) ∪ c) ⇒ c ∈ Chs (As (C−s ) ∪ c). This directly follows from the substitutability of s’s preferences. (Note that c is available both in the larger set As (C−s )∪c 0 and the smaller set As (C−s ) ∪ c, and c is chosen in the larger set. Then c must also be chosen in the smaller set.)
13
An important point to know is that this is the smallest (in the sense of set inclusion) selection of best responses which is “always” monotonic. Our “always” requirement is important because in a specific game, there can be a monotone selection of best replies that is smaller than the generalized threshold best response. For example, suppose that (i) there are two colleges C = {c, c0 }, (ii) c0 would not accept any student, and (iii) c would accept student s only. Assume further that c0 Ps c and cPs ∅. Then, xs (C−s ) = {c} is monotone increasing (because it is a constant function), and this is smaller than the generalized threshold best response brs (C−s ) = {c, c0 } (again a constant function). To formally define the smallest (in the sense of set inclusion) selection of best replies which is “always” monotonic, we look at a generally applicable selection criterion of a best reply which relies only on the essential features of the game. Denote by As (C−s ; PC ) the set of colleges that would accept s, given other students’ offers C−s and college preferences PC . The set of best responses of s can be determined by As (C−s ; PC ) and s’s own preferences Ps .5 We restrict our attention to a selection of best response that only depends on those two essential data. For each matching problem Γ = (S, C, P ), a best response selection specifies a best reply (of the students’ final offer game) for each student s as brs (C−s | Γ). A best response selection is said to be essential when brs (C−s | Γ) depends only on the essential aspects of the game, As (C−s ; PC ) and Ps . In other words, a best response selection is essential if there is one function ρs such that, for all matching problem Γ, brs (C−s | Γ) = ρs (As (C−s ; PC ), Ps ). Proposition 5 Assume |S| ≥ 2. The generalized threshold best response function is the smallest (in the sense of set inclusion) essential best response selection, which specifies monotone increasing best replies in all students’ final offer games where all agents in both sides have substitutable preferences. Proof. The generalized threshold best response function is associated with essential best response selection ρ∗s (As (C−s ; PC ), Ps ) ≡ {c ∈ C|c ∈ Chs (As (C−s ; PC ) ∪ c)}, and Proposition 4 establishes that it specifies a monotone increasing best reply. Hence we only need to show that, if an essential best response selection ρs (As (C−s ; PC ), Ps ) specifies a monotone increasing best reply, it is larger than the generalized threshold best response selection: {c ∈ C|c ∈ Chs (As (C−s ; PC ) ∪ c)} ⊆ ρs (As (C−s ; PC ), Ps ). Suppose this is not true and there are C−s , Ps , PC and c0 ∈ C such that c0 ∈ Chs (As (C−s ; PC ) ∪ c0 ) 5
Note that Chs (As (C−s ; PC )) is the optimal set of colleges for s, given other students’ offers C−s and college preferences PC . Obviously this is a best reply. Since s does not mind adding any offer that is going to be rejected to this optimal set, adding such colleges provides another best reply. The set of all best replies, or the best reply correspondence is given by BRs (C−s ) = {Chs (As (C−s ); PC ) ∪ X | X ⊂ C As (C−s ; PC )}.
14
and c0 ∈ / ρs (As (C−s ; PC ), Ps ). Note that c0 cannot be s’s choice in the available colleges for him As : c0 ∈ / Chs (As (C−s ; PC )). This is because any best response (in particular ρs (As (C−s ; PC ), Ps )) must contain the best attainable colleges for s, Chs (As (C−s ; PC )). Hence c0 must be a college which rejects s, or c0 ∈ / As (C−s ; PC ). Then we show that the essential best response selection ρs does not specify a monotone increasing best reply in some game. 0 defined by Cs0 0 = Cs0 \ {c0 } for Case 1: Assume s ∈ Chc0 ({s}). In that case consider C−s 0 0 every s0 6= s. Then C−s −s C−s and As (C−s ; PC ) = As (C−s ; PC ) ∪ c0 . Our premise that c0 is in the generalized threshold best reply, namely c0 ∈ Chs (As (C−s ; PC ) ∪ c0 ), implies 0 0 , and in par; PC )). Hence c0 must be in any best response against C−s c0 ∈ Chs (As (C−s 0 0 0 ticular c ∈ ρs (As (C−s ; PC ), Ps ). This contradicts our premise c ∈ / ρs (As (C−s ; PC ), Ps ) and monotonicity of the best reply. Case 2: Assume s ∈ / Chc0 ({s}). Let Pc00 be a responsive preference relation with quota one 0 by Cs0 0 = such that s0 Pc00 sPc00 ∅ for every s0 6= s, and let PC0 = (Pc00 , PC\{c0 } ). Also define C−s 0 0 Cs0 ∪{c0 } for every s0 6= s. Then As (C−s ; PC ) = As (C−s ; PC0 ), so c0 ∈ / ρs ((As (C−s ; PC0 ), Ps ) = 00 0 00 −s C−s by Cs000 = Cs0 \{c0 } for every s0 6= s. Then C−s ρs ((As (C−s ; PC ), Ps ). Now define C−s 0 00 ; PC0 ) ∪ c0 . Our premise that c0 is in the generalized threshold ; PC0 ) = As (C−s and As (C−s 0 00 best reply, namely c0 ∈ Chs (As (C−s ; PC0 ) ∪ c0 ), implies c0 ∈ Chs (As (C−s ; PC0 )). Hence c0 00 00 ; PC0 ), Ps0 ). This , and in particular c0 ∈ ρs (As (C−s must be in any best response against C−s 0 ; PC0 ), Ps ) and monotonicity of the best reply. contradicts c0 ∈ / ρs (As (C−s We also consider the smallest best response selection when we fix the game, that is, players and their preferences. Let C(s) be the set of colleges that never accept student s under any strategy profile, that is, C(s) = {c ∈ C|s ∈ / Chc (S 0 ), ∀S 0 ⊆ S}. For each s ∈ S, the modified threshold best response function br∗s (·) is defined by br∗s (CS ) = brs (CS ) \ C(s). The modified threshold best response simply deletes from the generalized best response all colleges that do not choose s in any instance. When a college has substitutable preferences (as assumed in this paper), s is never chosen from any set of available students if and only if s ∈ / Chc ({s}). Therefore, br∗s (CS ) = brs (CS ) \ {c ∈ C|s ∈ / Chc ({s})}.
15
Proposition 6 Fix a game. The modified threshold best response function is the smallest (in the sense of set inclusion) essential best response selection, which specifies monotone increasing best replies in all students’ final offer games where all agents in both sides have substitutable preferences. Proof. The modified threshold best response function is associated with the essential / Chc ({s})}, best response selection ρ∗s (As , Ps ) ≡ {c ∈ C|c ∈ Chs (As ∪ c)} \ {c ∈ C|s ∈ and Proposition 4 establishes that it specifies a monotone increasing best reply. Hence we only need to show that, if an essential best response selection ρs (As , Ps ) specifies a monotone increasing best reply, it is larger than the modified threshold best response selection: {c ∈ C|c ∈ Chs (As ∪ c)} \ {c ∈ C|s ∈ / Chc ({s})} ⊆ ρs (As , Ps ). Suppose this 0 0 is not true and there is c ∈ Chs (As ∪ c ) such that s ∈ Chc0 ({s}) while c0 ∈ / ρs (As , Ps ). 0 0 Note that c cannot be s’s choice in the available colleges As for him: c ∈ / Chs (As ). This is because any best response (in particular ρs (As , Ps )) must contain the best attainable colleges for s, Chs (As ). Hence c0 must be a college which rejects s, or c0 ∈ / As . Then we show that the essential best response selection ρs does not specify a monotone increasing best reply. Recall that, when colleges have substitutable preferences, As (C−s ) is monotone 0 such that decreasing. Since s ∈ Chc0 ({s}) by assumption, we can find C−s −s C−s 0 0 0 As (C−s ) = As ∪ c and As (C−s ) = As . Our premise that c is in the modified threshold best 0 reply, namely c0 ∈ Chs (As ∪ c0 ), implies c0 ∈ Chs (As (C−s )). Hence c0 must be in any best 0 , and in particular c0 ∈ ρs (As (C−s ), Ps ) = ρs (As , Ps ). This contradicts response against C−s our premise c0 ∈ / ρs (As , Ps ). More generally we present, without a proof, a characterization of the set of essential best response selections in a fixed game. An essential best response selection is monotone if and only if br∗s (CS ) ∪ D(C−s ), where D(C−s ) is a non-decreasing selection (with respect to C−s ) from C(CS ). We can show all previous conclusions, such as Theorems 1, 2 and 3 using generalized threshold best responses instead of maximal best responses. One potential advantage of generalized threshold best responses is its simplicity in some special cases. Suppose that student s has responsive preferences: Student s’s preference relation Ps is responsive with quota qs if for any T ⊂ C with |T | < qs , and any c, c0 ∈ C\T , we have (i) [T ∪ c] Ps [T ∪ c0 ] ⇔ c Ps c0 and (ii) [T ∪ c] Ps T ⇔ c Pi ∅, 16
and for any T ⊆ C with |T | > qs , we have ∅ Ps T . Symmetric definition applies to colleges. Then any generalized threshold strategy is of the form {c ∈ C|cRs c} for some “threshold college” c ∈ C (this is why we use the term “generalized threshold strategy.”). Moreover, any two strategies of the form can be compared since {c ∈ C|cRs c} {c ∈ C|cRs c0 } ⇐⇒ c0 Rs c, for two strategies of s. Thus when students have responsive preferences, one can restrict attention to one dimensional strategies of (generalized) threshold best responses. Having a one-dimensional strategy set for each player makes the analysis simple, and enables us to show further results. For example, adopting the argument by the working paper version of Kukushkin, Takahashi, and Yamamori (2008) (Takahashi and Yamamori, 2008), with one-dimensional strategy space and strategic complementarity (in the sense of increasing best response), one can show that the best response dynamics with any initial strategy profile converges in finite time to a Nash equilibrium.
3.1
Structure of the set of stable matchings
We can further obtain the upper bound and uniqueness conditions for stable matchings by restricting our attention to many-to-one cases. Suppose there are N students and L colleges, and each student can be matched with only one college. For simplicity, assume that all colleges are acceptable to all students.6 Then, let us assign an integer to each generalized threshold strategy for every student in the way that a strategy is assigned an integer k if its threshold college is the k’th best college for the student. That is, each student is endowed with (generalized threshold) strategies {1, 2, ..., L} such that strategy k is to apply to all the colleges weakly preferred to her k’th best college. Let x∗∗ and x∗ be the largest and smallest pure strategy Nash equilibria (x∗∗ ≥ x∗ ). Remember that each Nash equilibrium corresponds to a stable matching. Now we are ready to show the following results giving the upper bound of the size of stable matchings. Denote a best response function (of the form of a generalized threshold strategy) of player i by BRi . Theorem 7
(i) Suppose there is an integer K such that BRi (x∗−i + (k, ..., k)) − BRi (x∗−i ) < k
(4)
for all k ≥ K, and for all player i, at the smallest Nash equilibrium x∗ . Then, each student is matched with at most K different colleges in the set of all stable matchings. 6
It is easy to incorporate unacceptable colleges, which would not change any of the following results.
17
(ii) Suppose there is an integer K such that, 0
for all x ≥ x with
N X
(x0i
N X
− xi ) ≥ K,
i=1
Then
PN
0 i=1 (xi
BRi (x0−i )
N X − BRi (x−i ) < (x0i − xi ).
i=1
i=1
− xi ) < K must hold for any equilibria x, x0 .
∗ Proof. (i) It is sufficient to show that x∗∗ i − xi < K for all student i. Let n be a student ∗ ∗∗ ∗ ∗∗ ∗ ∗∗ ∗ where x∗∗ i − xi is the largest, i.e., xn − xn ≥ xi − xi for all i. Suppose that xn − xn ≥ K. ∗ Let us denote k ≡ x∗∗ n − xn . Then, we have a contradiction: ∗ ∗∗ ∗ x∗∗ n − xn = BRn (x−n ) − BRn (x−n ) ∗ ≤ BRn (x∗−n + (k, ..., k)) − BRn (x∗−n ) < k = x∗∗ n − xn .
The first equality comes from the fact that x∗∗ and x∗ are Nash equilibria. The weak ∗ ∗∗ ∗ inequality comes from non-decreasing best replies and k ≡ x∗∗ n − xn ≥ xi − xi for all i (the definition of player n). The strict inequality is the condition (4). P (x0 −x ) ≥ K, for some equilibria x, x0 . (ii) Suppose, contrary to our claim, we had N P PN i=1 i 0 i 0 Then, the above condition would apply and i=1 BRi (x−i ) − BRi (x−i ) < N i=1 (xi − xi ), which would contradict the equilibrium conditions BR(x0 ) = x0 and BR(x) = x. The insight of the above two results comes from the well-known sufficient conditions for the unique equilibrium point. Theorem 7-(i) is a discrete analogue of the slope condition for the unique pure strategy equilibrium, when a strategy is one-dimensional continuous variable xi ∈ [0, x] and the best replies are increasing. The slope condition is for any i and any x−i ,
X ∂BRi j6=i
∂xj
(x−i ) < 1.
Another well-known condition for uniqueness is the contraction mapping property. Let BR(x) = (BR1 (x−1 ), ..., BRN (x−N )). The contraction mapping condition is ∃β ∈ (0, 1) ∀x, x0 |BR(x0 ) − BR(x)| ≤ β |x0 − x| , where | | can be any norm in RN , but consider the norm |x| = |x1 | + · · · + |xN |. Then, the contraction mapping condition is stated as 0
∃β ∈ (0, 1) ∀x, x
N N X X BRi (x0−i ) − BRi (x−i ) ≤ β |x0i − xi | . i=1
i=1
18
When best replies are non-decreasing, the following relaxation of this contraction mapping condition suffices for the uniqueness. 0
∀x ≥ x, x 6= x
0
N X
BRi (x0−i )
N X − BRi (x−i ) < (x0i − xi ).
i=1
i=1
Theorem 7-(ii) states the discrete analogue of this contraction condition. The above theorem also seems to be similar to the small core theorem by Immorlica and Mahdian (2005) and Kojima and Pathak (2009), although the exact relation is yet to be found. A slightly weaker sufficient condition than Theorem 7-(ii) is obtained as follows. Theorem 8 Suppose there is an integer K such that, N X ∀x ≥ x with (x0i − xi ) ≥ K ∃n BRn (x0−n ) − BRn (x−n ) < x0n − xn . 0
i=1
Then
PN
0 i=1 (xi
− xi ) < K must hold for any equilibria x, x0 .
P 0 Proof. Suppose, contrary to our claim, we had N i=1 (xi − xi ) ≥ K, for some equilibria x, x0 . Then, the above would apply and ∃n BRn (x0−n ) − BRn (x−n ) < x0n − xn . This would contradict the equilibrium conditions BRn (x0−n ) = x0n and BR(x−n ) = xn . Substituting K = 1 into Theorem 8, we obtain the following uniqueness result. Corollary 2 There exists a unique Nash equilibrium if the following condition is satisfied. ∀x0 ≥ x, x 6= x0 ∃n BRn (x0−n ) − BRn (x−n ) < x0n − xn .
(5)
It is easy to verify that condition (5) is satisfied when all colleges have the same preferences over students. Thus, Corollary 2 gives us a weaker uniqueness condition than the well-known one that players on one side all have the identical preferences by Gusfield and Irving (1989).
4
Aggregating Men and Women: A Two-Player Game with Strategic Substitutes
A remarkable property of stable matchings is partial alignment of interests among the players on the same side. For example, in the marriage problem there is a stable outcome 19
which is unanimously best for all men (among all stable outcomes), and similarly there is the women-optimal stable outcome. In addition, the men-optimal outcome is the unanimously worst for women. Our Theorem 1 provides some intuition in terms of the non-cooperative game with strategic complementarities that implements stable outcome as Nash equilibria. In this section, we demonstrate the alignment of interests among men and among women, together with the conflict of interest between men and women, in a more pronounced way. In particular, we construct a non-cooperative game that implements stable outcomes as Nash equilibria, in such a way that (i) all men are aggregated into one player and all women are aggregated into another player, and (ii) the resulting 2-player game has strategic substitutes, where a larger strategy is more aggressive. The first point highlights the alignment of interests among men and among women, while the second point demonstrates the stark conflict of interests between men and women. In particular, the latter implies that the game has the “top dog” property, according to the well-known taxonomy of business strategies by Fudenberg and Tirole (1984). Also, this 2-player game looks very much like the Nash demand game about the bargaining over the division of $1.
4.1
Threshold demand game
Consider the many-to-many case, where all players have responsive preferences. In the previous section we showed that threshold strategies can be defined in this case. We exploit this fact to construct a game with Properties (i) and (ii) above. We first construct a game with the original set of players. We then go on to show that all players on the same side (all students, for example) can be aggregated into a single player. To this end, we first introduce a generalized version of threshold strategies. We have defined generalized threshold best responses, but more generally generalized threshold strategies are defined as follows. We confine our attention to a student s ∈ S, but symmetric definitions apply to a college c ∈ C. For any C 0 ⊆ C the generalized threshold strategy of s associated with C 0 is defined as ts (C 0 ) = {c ∈ C|c ∈ Chs (C 0 ∪ c)}. We say that C 0 is a threshold of ts (C 0 ). There are several properties of generalized threshold strategies. Let IRs = {C 0 ⊆ C|C 0 = Chs (C 0 )} 20
be the set of groups of colleges that are individually rational for student s. (i) C 0 ⊂ ts (C 0 ) if C 0 ∈ IRs . (ii) When s has substitutable preferences, ts (C 0 ) = ∪C 00 ⊂C Chs (C 0 ∪ C 00 ). (iii) There can be distinct thresholds C 0 , C 00 ∈ IRs (C 0 6= C 00 ) resulting in the same set of offers ts (C 0 ) = ts (C 00 ). 00
(iv) “Rising threshold” from C 0 to C 00 , if defined as C Ps C 0 , does not generally mean that 00 the student is making a smaller set of offers: C 0 , C 00 ∈ IRs and C Ps C 0 do not imply ts (C 0 ) ⊃ ts (C 00 ) (v) However, a modified version holds: C 0 , C 00 ∈ IRs and C 00 = Chs (C 0 ∪ C 00 ) imply ts (C 0 ) ⊃ ts (C 00 ). Both (i) and (ii) directly follow from the definitions. Item (ii) motivates our definition of a threshold strategy. It is the set of offers accepted by the player, if offered together with the threshold (and possibly something else). Item (iii) can be shown by an example. Suppose s has capacity or quota qs = 2, a and b are individually rational, and {a, b}Ps a, and {a, b}Ps b. Then ts (a) = ts (b) = {a, b}. Item (iv) can also be shown by an example. Suppose s has quota qs = 2 and additive utility function u(w) = 10, u(x) = 3, u(y) = 2, u(z) = 1, and u(∅) = 0. Then, {w, z}Ps {x, y} (because u(w) + u(z) > u(x) + u(y)) but ts ({x, y}) + ts ({w, z}). This is because z ∈ ts ({w, z}) (Item (i)) but z ∈ / ts ({x, y}) (because both x and y are better than z). Basically threshold {w, z} is better than threshold {x, y}, because w is very good (while z is a lousy choice). Item (v) is shown as follows. We need to show that C 00 = Chs (C 0 ∪ C 00 ) and c ∈ ts (C 00 ) imply c ∈ ts (C 0 ). Equivalently, we need to show C 00 = Chs (C 0 ∪ C 00 ) and c ∈ Chs (C 00 ∪ c) imply c ∈ Chs (C 0 ∪ c). C 00 = Chs (C 0 ∪ C 00 ) means that C 0 is rejected in C 0 ∪ C 00 , and the substitutable preferences of s implies that C 0 is also rejected in a larger set C 0 ∪ C 00 ∪ c (as substitutability implies the rejected set is monotone increasing in the opportunity set). Then a revealed preference argument shows that Chs (C 00 ∪ c) = Chs (C 0 ∪ C 00 ∪ c). By the substitutable preferences of s, c ∈ (Chs (C 00 ∪c) =) Chs (C 0 ∪C 00 ∪c) implies c ∈ Chs (C 0 ∪c). Consider a game where all players are required to take a generalized threshold strategy. Hence, a strategy of a student s is represented by a threshold xs = C 0 ⊂ C. This means that s is “making offers” to colleges ts (C 0 ) = {c ∈ C|c ∈ Chs (C 0 ∪ c)}.
21
Student s’s strategy space is given by Xs = 2C , and this includes an empty set (i.e., s can choose threshold xs = ∅, and in that case he makes offers to {c ∈ C|c ∈ Chs (c)} = {c ∈ C|cPs ∅}). We order a student’s strategy in terms of the set of offers he is making: xs s x0s ⇔ ts (xs ) ⊂ ts (x0s ). This definition matches to our everyday language: a “higher threshold” means a smaller set of offers. Similarly, a college c’s strategy space is given by Xc = 2S , and the same ordering is applied to colleges’ strategies (xc c x0c ⇔ tc (xc ) ⊂ tc (x0c ).). Given a strategy profile x, students and colleges are matched in the following way. Student s is matched to college c if and only if they make offers to each other. Let ms be the set of colleges matched to student s under threshold strategy profile x. Note that ms depends only on colleges’ offers xC and the student’s own offers xs . Hence we denote it by ms (xs , xC ). Similarly, we define the set of students accepted to college c by mc (xc , xS ).7 This is an essential feature of our model. In our original game in Section 2, where students make offers first and then colleges respond, there is direct conflict of interests between students. For example, whether student s is admitted to college c crucially depends on who else also applies to college c. In the current game, colleges and students move simultaneously, and colleges are required to accommodate all accepted offers. This means that student s can be matched to college c as long as c is making an offer to s, and this is independent of what offers other students are making. Hence, in our simultaneous offers game, we can effectively eliminate strategic interaction, or competition, among students (and among colleges, by a symmetric argument). We assume that players have lexicographic preferences: they prefer a better matching (first priority) and then they prefer to choose a threshold that is indeed realized (second priority). In other words, other things being equal, each player is inclined to choose a “serious threshold” that is going to be realized. This assumption is employed to ensure that a Nash equilibrium always provides a stable outcome. Without this, an unstable outcome with the following feature can be a Nash equilibrium. In the equilibrium outcome, there are a student s and a college c who are not matched, and they would rather be matched to each other. In such a situation, a player can profitably deviate to the threshold which matches the realized outcome (∅), under the lexicographic preferences. If one player, say s, is using threshold ∅, however, he is making an offer to a valuable partner c, and c can unilaterally deviate to make an offer to s. Hence the situation where c and s are not matched cannot be an equilibrium. 7
Hence, the matching function µ resulting from strategy profile x is given by µ(s) = ms (xs , xC ) for s ∈ S and µ(c) = mc (xc , xS ) for c ∈ C.
22
Formally, we specify student s’s payoff function us in the current game in such a way that us (x) > us (x0 ) if ms (xs , xC )Ps ms (x0s , x0C ), us (x) > us (x0 ) if ms (xs , xC ) = ms (x0s , x0C ) = xs and x0s 6= xs . The second line is our key assumption. Other things being equal, a player would like to choose a threshold that is actually realized. The specification of us for the remaining case ( ms (xs , xC ) = ms (x0s , x0C ) = ms , xs 6= ms and x0s 6= ms ) is arbitrary. Note that the absence of strategic interaction between students in the outcome (the resulting matching) carries over to the payoffs: us only depends on xs and xC . With an abuse of notation, we sometimes write us = us (xs , xC ). A college’s lexicographic payoff uc is similarly defined. Let us motivate the assumption of lexicographic preferences. We do not argue that this is a particularly compelling assumption about people’s preferences. Rather, our point is that this rather weak assumption helps us to obtain a better understanding of stable matchings, in terms of a simple non-cooperative game. Alternatively, we can design the rules of the game in such a way that players end up having the lexicographic payoffs. For example, we can pay a small amount of money (say 1 cent) to a player when his threshold is equal to his realized match. If 1 cent is lower than the marginal disutility of any change in the realized match, then we do get the lexicographic payoffs. As before, when a college c has a quota qc (the maximum number of students it can accept) and the number of students assigned to c in our game exceeds its quota qc , we specify a sufficiently low payoff to c (so that c is worse off than in the situation where it finds no match). The same is true for students. Hence, “overbooking” is allowed in our game and players seek to avoid overbooking at their own discretion. By specifying quotas, our many-to-many matching model incorporates, as special cases, the one-to-many and one-to-one cases. This is another essential ingredient of our model. Hence we have completely specified strategy spaces and payoffs of our game. We call this a threshold demand game. (Named after the famed simultaneous offer bargaining game, the Nash demand game.) Suppose for the moment we do not assume substitutability or responsiveness, and consider the stability as defined below. The stability concept here requires no blocking by any subset of players. Formally, a set of students S 0 and a set of colleges C 0 block a given matching µ, if (i) S 0 and C 0 are not matched under µ (∀s ∈ S 0 µ(s) ∩ C 0 = ∅ and ∀c ∈ C 0 µ(c) ∩ S 0 = ∅), and (ii) ∀s ∈ S 0 C 0 ⊂ Chs (µ(s) ∪ C 0 ) and ∀c ∈ C 0 S 0 ⊂ Chc (µ(c) ∪ S 0 ). (S 0 , C 0 ) is called a blocking coalition. A matching µ is stable if it is individually rational (same definition as before) and there is not blocked by any set of players. Stability is equivalent to pairwise stability when substitutability (or a stronger notion of responsiveness)
23
holds. (By substitutability, if an outcome is blocked by a set of players T , it can be blocked by s, c ∈ T .) Now we are ready to show that there is one-to-one correspondence between the pure strategy Nash equilibria in this game and the set of stable matchings. Theorem 9 Consider many-to-many matching and suppose that all players have substitutable preferences. In the threshold demand game, a pure strategy Nash equilibrium achieves a stable matching. Conversely, any stable matching can be achieved by a pure strategy Nash equilibrium in the threshold demand game. Proof. The proof is omitted, as it is an adaptation of Echenique and Oviedo (2006). Now we show that the players on the same side can be aggregated into a single player in the threshold demand game defined above. In this game all players simultaneously make offers to each other, and players are required to accommodate all accepted offers. Then, a student’s payoff only depends on his own offers and colleges’ offers, and it is independent of other students’ offers. Hence there is no direct strategic interaction between students (and similarly between colleges). Theorem 10 In a threshold demand game, define the representative student whose strategy consists of students’ thresholds xS = (xs )s∈S ∈ XS and whose payoff is any function strictly increasing in students’ payoffs: WS ((us )s∈S ) and WS is strictly increasing in us for all s ∈ S. Similarly define the representative college who has strategy space XC and payoff WC ((uc )c∈C ) which is strictly increasing in uc for all c ∈ C. The resulting two-player game, the aggregated threshold demand game, has the same set of pure strategy Nash equilibria as the threshold demand game. Proof. Note that the payoff functions of the representative players are given by πS (x) = WS ((us (xs , xC )s∈S )) and πC (x) = WC ((uc (xc , xS )c∈C )). Let x∗ be a pure strategy Nash equilibrium of the threshold demand game. We show that x∗ is also a Nash equilibrium of the aggregated threshold demand game. Consider the representative student’s gain from deviation πS (xS , x∗C )−πS (x∗S , x∗C ) (a symmetric argument applies to the representative college). Since x∗ is a Nash equilibrium in the threshold demand game, we have us (xs , x∗C ) ≤ us (x∗s , x∗C ) for all s. This shows that πS (xS , x∗C ) − πS (x∗S , x∗C ) ≤ 0, because πS = WS ((us )s∈S ) is increasing in its arguments. Hence x∗ is also a Nash equilibrium of the aggregated threshold demand game. Let us now assume 24
that x∗ is not a pure strategy Nash equilibrium of the threshold demand game. We show that x∗ is not a Nash equilibrium of the aggregated threshold demand game. Since x∗ is not a Nash equilibrium of the threshold demand game, there is a player who can gain by a unilateral deviation. Without loss of generality, suppose that student s can gain by deviating to xs . When representative student deviates to (xs , x∗−s ), student s can benefit (us (xs , x∗C ) > us (x∗s , x∗C )), while the payoff to any other student s0 6= s remains unchanged (us0 (x∗s0 , x∗C )). By definition, representative student gains from deviation to (xs , x∗−s ), and we conclude that x∗ is not a Nash equilibrium of the aggregated threshold demand game.
Note that aggregated threshold demand game is a game with strategic substitutes, when we order the representative student’s strategy by order x0S xS ⇔ ∀s x0s xs and the representative college’s strategies are ordered by x0C xC ⇔ ∀c x0c xc . It corresponds to the “top dog” case in Fudenberg and Tirole (1984), where a larger strategy of a player harms the opponent.
4.2
General offer demand game
We required the lexicographic payoff and threshold strategies, but now allow any offers and assume another version of lexicographic preferences. Specifically, we may assume that (i) players can make offers to any subset of partners (ii) if player i has two strategies (sets of offers) xi and x0i that yields the same match for him and one is larger than the other (xi x0i ), then i prefers the larger strategy. Let us call this general offer demand game. Now consider the following conjecture: Conjecture 1 The set of outcomes achieved by the pure strategy Nash equilibria of general offer demand game coincides with the set of stable matchings. We show that this conjecture is false. A counterexample is a many-to-many situation with two students s, s0 and two colleges c, c0 . Each player wants to be matched to all of the potential partners. If a player is matched to one partner, he is worse off than being unmatched. (Note that there are complementarities among the potential partners. The preferences here violate substitutability.) Consider the following strategy profile in general offer demand game: xs = {c}, xs0 = {c0 }, xc = {s0 }, xc = {s0 }. These “criss-cross” offers result in no matching. If s changes his offer to either {c, c0 } or {c0 }, he is matched only to c0 , and hence he is worse off. Symmetric argument applies to all other players, and hence this is a Nash equilibrium. Obviously, however, this outcome is not stable, because everyone would be better off if all players are matched to each other. We now employ some assumptions to establish (a par of) the conjecture. 25
Proposition 7 In a one-to-many matching problem, the set of outcomes achieved by the pure strategy Nash equilibria of general offer demand game is stable. Proof. Let µ be the matching achieved by a pure strategy Nash equilibrium of general offer demand game. It is individually rational, because otherwise a player is better off by making offers to a smaller subset. Suppose µ is not stable. Then, in the one-to-many case, the outcome must be blocked by one college c and the set of students S 0 (in the one-tomany case, any blocking coalition contains such a subset). By definition of blocking, c and S 0 are not matched in the Nash equilibrium. By the lexicographic payoff assumption, there is no s ∈ S 0 such that no offer is being made between c or s. (If such a case exists, c can make an offer to s. This does not change c’s match, and c is strictly better off by making a larger set of offers.) Let us suppose that c is making offers to T ⊂ S 0 and T are making no offer to c. Then, any student s ∈ T is better off by deviating to a new set of offers Chs (µ(s) ∪ c). This is a contradiction, so such a set T does not exist. Hence it must be the case that all students in S 0 are making offers to c and c is not making any offer to S 0 in the equilibrium. This is also a contradiction, because c can profitably deviates to a new offers Chc (µ(c) ∪ S 0 ). Now assume substitutability. Then, stability is equivalent to pairwise stability. (By substitutability, if a blocking coalition exists, any pair (c, s) in the blocking coalition also blocks the given allocation.) In this case, the conjecture is true: Proposition 8 If players’ preferences satisfy substitutability, the set of outcomes achieved by the pure strategy Nash equilibria of general offer demand game coincides with the pairwise stable matchings. Proof. Step 1 (Nash ⇒ stable): Let µ be the matching achieved by a pure strategy Nash equilibrium of general offer demand game. It is individually rational, because otherwise a player is better off by making offers to a smaller subset. Suppose µ is not pairwise stable. Then, the outcome must be blocked by a pair (c, s). By definition of blocking, c and s are not matched in the Nash equilibrium. By the lexicographic payoff assumption, it must not be the case that no offer is being made between c or s. (In such a case, c could make an offer to s. This would not change c’s match, and c could be strictly better off by making a larger set of offers.) Let us suppose that c is making an offer to s but s is not making an offer to c. Then, s is better off by deviating to a new set of offers Chs (µ(s) ∪ c), a contradiction. The remaining case, where c is not making an offer to s but s is making an offer to c, also leads to a contradiction. Hence any Nash outcome is pairwise stable. Step 2 (Stable ⇒ Nash): Let µ be a stable matching and define a strategy profile in general offer demand game x0 by x0i = µ(i) for all i. This may not be a Nash equilibrium, 26
because a player can gain by making an additional offer that is rejected (by the lexicographic payoff assumption, a player is better of making the larger set of offers.) We modify x0 to obtain a Nash equilibrium x∗ that also achieves µ. Let x∗s = {c ∈ C | c ∈ Chs (µ(s) ∪ c)}. By the individual rationality of µ (µ(s) = Chs (µ(s))), x∗s contains µ(s). Let x∗c be the union of µ(c) and all offers that are rejected under x∗S : x∗c = µ(c) ∪ {s ∈ S | c ∈ / x∗s }. By construction x∗ achieves outcome µ. Also by construction, there is no pair of players s and c between whom no offer is made under x∗ . This implies that, if any player i deviates from x∗i and makes any additional offer, it is accepted. It is easy to see that any player i is strictly worse off by offering a strict subset of x∗i (it either reduces the set of offers without changing i’s matching partners, or i would lose a valuable partner in µ(i).) We now show that player i is worse off by deviating to xi " x∗i . First, consider the case i ∈ S. Then, after the deviation, i is matched to the union of xi ∩ µ(i) and T ≡ xi ∩ {c ∈ C | c ∈ / Chi (µ(i) ∪ c)}. If i prefers this to µ(i), Chi (µ(i) ∪ T ) must contain at least one element in T . Otherwise, Chi (µ(i) ∪ T ) = Chi (µ(i)) = µ(i) (the last equality follows from the individual rationality of µ), contradicting our premise that [xi ∩ µ(i)] ∪ T (a subset of µ(i) ∪ T ) is better than µ(i) for i. However, by responsiveness Chi (µ(i) ∪ T ) should contain no elements in T , because c ∈ T satisfies c ∈ / Chi (µ(i) ∪ c). Hence we have established that a student cannot gain by a unilateral deviation at x∗ . It remains to show that any college i ∈ C cannot profitably deviate to xi " x∗i . After the deviation, i is matched to the union of xi ∩ µ(i) and K ≡ xi ∩{s ∈ S | i ∈ / µ(s) and i ∈ Chs (µ(s) ∪ i)}. Note that i is matched to K because ∗ under x , any student s is making an offer to any college c that satisfy c ∈ Chs (µ(s) ∪ c). By the stability of µ, any student s in K satisfies s ∈ / Chi (µ(i) ∪ s) (otherwise, s and i could block µ). Then, we can employ the same argument as in the previous case (i ∈ S) to show that any deviation to xi " x∗i is unprofitable for i ∈ C.
5 5.1
Discussion Monotone methods for stable matching
The current paper is the first work that establishes the connection between two-sided matching and games with strategic complementarity. However, the mathematical structure similar to ours is recognized and investigated in recent works. A pioneering work by Adachi (2000) investigates one-to-one matching markets and finds that the space of “pre-matchings” allows for an increasing function, and that the set of fixed points of that increasing function corresponds to the set of stable matching in his domain. His method has been extended to many-to-one matching (Echenique and Oviedo, 2004), many-to-many matching (Echenique and Oviedo, 2006), many-to-one matching with contracts (Hatfield 27
and Milgrom, 2005) and supply-chain networks (Ostrovsky, 2007).8 This section studies the relationship between our method and these algorithmic approaches in the literature. To study the connection with these alternative approaches in more detail, let us first consider the solutions to the system of two equations, G1 : X2 → X1 and G2 : X1 → X2 : ( x1 = G1 (x2 ) , x2 = G2 (x1 ) or, the fixed points of mapping G(x1 , x2 ) = (G1 (x2 ), G2 (x1 )). The next (trivial) lemma shows that we can focus on one of the elements, x1 or x2 , to find out the fixed point. Lemma 5 (Composition Lemma): Define a composition mapping associated with G by Hi (xi ) ≡ Gi (Gj (xi )), i 6= j. The fixed points of Hi coincide with the fixed points of G, for i = 1, 2. More precisely, (x∗1 , x∗2 ) = G(x∗1 , x∗2 ) ⇔ x∗i = Hi (x∗i ) (and x∗j = Gj (x∗i )). Proof. Consider the case i = 1. Obviously, ( ( x∗1 = G1 (G2 (x∗1 )) x∗1 = G1 (x∗2 ) . ⇔ x∗2 = G2 (x∗1 ) x∗2 = G2 (x∗1 ) Now we present the mappings employed by the existing literature, whose fixed points coincide with stable matchings. We consider many-to-many matching (without contract). We call one side “students” and the other side “colleges”, and the set of students and colleges are denoted by S and C. A student s’s offer is a (possibly empty) subset of colleges, xs ⊂ C, and let xS = (xs )s∈S be the profile of students offers. Similar definitions apply for colleges’ offers. 1. Hatfield-Milgrom mapping HM: ( HMs (xC ) ≡ C \ rs (xC ) for all s ∈ S, HMc (xS ) ≡ S \ rc (xS ) for all c ∈ C, where rs (xC ) is the set of rejected offers by student s. More precisely, let ACs (xC ) be the set of colleges which would accept s (i.e., which are making an offer to s) at college offer profile xC . Recall that Rs (·) denotes the rejected set function of student s (Rs (Y ) = Y Chs (Y ), where Chs is student s’s chosen set function). 8
See also Fleiner (2003).
28
Then, rs (xC ) ≡ Rs (ACs (xC )). (Similar definition applies to rc .) Let HMS (xC ) = (HMs (xC ))s∈S and HMC (xS ) = (HMc (xS ))c∈C and HM = (HMS , HMC ).9 2. The mapping T (Ostrovsky (2007); see also Adachi (2000), Echenique and Oviedo (2004), and Echenique and Oviedo (2006)):10 ( Ts (xC ) ≡ {c ∈ C|c ∈ Chs (ACs (xC ) ∪ c)} for all s ∈ S, Tc (xS ) ≡ {s ∈ S|s ∈ Chc (ACc (xS ) ∪ s)} for all c ∈ C, where ACi is defined in item 1 above. TS (xC ) = (Ts (xC ))s∈S and TC (xS ) = (Tc (xS ))c∈C and T = (TS , TC ). Hatfield and Milgrom (2005) and Echenique and Oviedo (2006) showed that the set of all fixed points of those mappings coincide with the set of all stable matchings. More precisely, if (x∗S , x∗C ) is a fixed point of HM or T , then, a stable matching is obtained by matching any student and any college who are making an offer to each other at (x∗S , x∗C ). Conversely, any stable matching can be obtained by a fixed point of HM or T .11 Now, we show that those mappings can be interpreted as the best reply functions in the games we have constructed. We first consider our demand games, where students and colleges simultaneously make offers. Theorem 11 Hatfield-Milgrom mapping HM is the best reply mapping of the general offer demand game. When preferences are substitutable, mapping T is the best reply mapping of the threshold demand game, that is, Ts (xC ) = ts (brs (x)) for every s ∈ S and Tc (xS ) = tc (brc (x)) for every c ∈ C where bri is the best reply function (in the target partners) of agent i in the threshold demand game. 9
To be precise, their model is slightly different in two ways from ours: first, they allow for more than one possible contract for each doctor-hospital pair. Second, they assume that each doctor can sign at most one contract. Since the adaptation of their model to our situation seems to be straightforward, however, we ignore these differences and refer to the Hatfield-Milgrom model as the adaptation of their model to manyto-many matching without contracts. also, Hatfield and Milgrom (2005) consider mapping (HMC (xS ), HMS (HMC (xS ))) (which corresponds to their mapping F (XD , XH )), but this has the same set of fixed points as HM , as the composition lemma shows. 10 Ostrovsky (2007) considers a model in which there is a finite partially ordered set of agents and contracts are incorporated, and his function reduces to the mapping T here when we focus on many-tomany matching without contracts. 11 A comment on a finer point: The fixed points of HM and T are generally different (the fixed points of the former have more offers by definition), but the matchings obtained by the fixed point offer profiles are the same for HM and T .
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Proof. Consider the general offer demand game and let x = (xS , xC ) be a given strategy profile. Then, by the lexicographic preferences as imposed in the definition of the game, a best response of an arbitrary student s ∈ S is Chs (ACs (xC )) ∪ [C \ ACs (xC )] = C \ rs (xC ). Similarly, the best response of an arbitrary college c ∈ C is Chc (ACc (xS )) ∪ [S \ ACc (xS )] = S \ rc (xS ). Therefore the best reply mapping of the general offer game is identical to mapping HM . Next, consider the threshold demand game and let x = (xS , xC ) be a given strategy profile. Then, by the lexicographic preferences as imposed in the definition of the game, a best response of an arbitrary student s ∈ S is brs (x) = Chs (ACs (xc )). Therefore ts (brs (x)) = {c ∈ C|c ∈ Chs (Chs (ACs (xC )) ∪ c)}. Since preferences of s is substitutable by assumption, Chs (Chs (ACs (xC ))∪c) = Chs (ACs (xC )∪ c) for any c ∈ C. Therefore we obtain ts (brs (x)) = {c ∈ C|c ∈ Chs (ACs (xC ) ∪ c)} = Ts (xC ). A symmetric argument establishes tc (brc (x)) = Tc (xS ), establishing that mapping T coincides with the best response mapping in the threshold demand game. Next we consider our final offers game, where students make offers first and then colleges respond. Theorem 12 The largest best reply functions of students’ final offers game is the composition mapping associated with the mixture of Hatfield-Milgrom and T mappings (6). More precisely: Let brS (xS ) = (brs (x−s )) be the largest best reply function in the students’ final offers game. Then, brS (xS ) = HMS (TC (xS )). By Theorem 12, together with Proposition 1 before, we obtain the following corollary.
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Corollary 3 The set of matchings obtained by the fixed points of the following mapping coincides with the set of stable matchings12 : ( x0S = HMS (xC ) ≡ (C rs (xC ))s∈S . (6) x0C = TC (xS ) ≡ ({s ∈ S|s ∈ Chc (ACc (xS ) ∪ s})c∈C Remark 2 By the composition lemma, the fixed points of the largest best reply mapping of students’ final offers game corresponds to the set of fixed points of the mixed mapping (6). Intuitively, in this game, student s’s largest best reply to other students’ offers are calculated in two steps. The first step is to determine which college accepts s, given other students’ offers. A college would accept s if and only if s is a welcome addition to the given offers by other students. Hence this part is quite similar to mapping TC . (There is, however, a minor subtlety here. See the proof below.) The second step is to calculate the largest best reply. The largest best reply of student s consists of (i) the best subset of colleges that accept s and (ii) all colleges that reject s. This part is exactly equal to mapping HMS . The proof below makes this statement in a rigorous way. Proof of Theorem 12. Recall that brs (x−s ) = C Rs (As (x−s )), where Rs is the rejected set function of student s and As (x−s ) is the set of colleges which would accept s, given other students’ offers x−s in the students’ final offers game. Since HMs (TC (xS )) = C rs (TC (xS )) = C Rs (ACs (TC (xS ))), we need to show As (x−s ) = ACs (TC (xS )).
(7)
By the definition of As (x−s ), college c is in As (x−s ) if and only if s ∈ Chc (ACc (∅, x−s ) ∪ s),
(8)
where (∅, x−s ) denote a profile where student s is making an empty set offer and others are making offers x−s . Note that ACc (∅, x−s ) is simply the set of students who are making an offer to c under x−s . In contrast, the right hand side of the desired equality (7) is the set of colleges which are making offers under TC (xS ). By definition, college c is making an offer to s under TC (xS ) if and only if s ∈ Chc (ACc (xS ) ∪ s). (9) 12
The same claim also applies to mapping (
x0S = TS (xC ) . x0C = HMC (xS )
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Hence, we are done if we show (8) and (9) are equivalent. When s ∈ ACc (xS ), the right hand sides of (8) and (9) are both equal to Chc (ACc (xS )). When s ∈ / ACc (xS ), the right hand sides of (8) and (9) are both equal to Chc (ACc (xS ) ∪ s). Hence, (8) and (9) are equivalent and the proof is complete. We now examine the threshold best reply mapping in the students’ final offers game. This corresponds to mapping T , as in the threshold demand game. Theorem 13 The threshold best reply functions of students’ final offers game is the composition mapping associated with mapping T. More precisely: Let brS (xS ) = (brs (x−s )) be the threshold best reply function in the students’ final offers game. Then, brS (xS ) = TS (TC (xS )). The proof is similar to the previous one and therefore we only provide a sketch. The first step to determine the best reply is the same as in the previous proof. That is, the set of colleges which accept s given other students’ offers is given by mapping TC . Then students apply threshold best reply TS , and the end result is the threshold best reply in the students’ final offers game. Remark 3 By the discussion of this section, we have shown that the mathematical structures of the threshold demand game and general offer demand game share critical features with existing monotonicity-based approaches to stable matching. The final offer game does not have the exact counterpart in the literature, but it also shares the basic methodology with these studies. In that sense, these games provides “microfoundations” on the algorithmic approaches employed in the previous literature.
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