Understanding Stable Matchings: a Non$Cooperative Approach

Understanding Stable Matchings: A Non-Cooperative Approach

KANDORI, Michihiro, KOJIMA, Fuhito, and YASUDA, Yosuke

March 9, 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]); Ko- jima: Cowles Foundation, Yale University and Department of Economics, Stanford University (email: [email protected]); Yasuda: National Graduate Institute for Policy Studies (email: yya- [email protected]). We are grateful to Hiroyuki Adachi, Federico Echenique, Drew Fudenberg, Hideo Kon- ishi, Paul Milgrom, Tayfun Sonmez, Satoru Takahashi, Utku Unver and seminar participants at Boston College, Caltech, Edinburgh, Princeton and Waseda for comments and discussions.

1 1 Framework

1.1 Preliminary de…nitions A market is tuple = (S;C;P ). S and C are …nite and disjoint sets of students and C colleges. We denote N = S C. For each s S, Ps is a strict preference relation over 2 . [ 2 S 1 For each c C, Pc is a strict preference relation over 2 . The non-strict counterpart of Pi 2 is denoted by Ri; so we write XRiX0 if and only if XPiX0 or X = X0.

For each s S and C0 C, the chosen set Chs(C0) is a set such that (1) Chs(C0) C0, 2 and (2) C00 C0 implies Chs(C0)RsC00: In words, Chs(C0) is the set of colleges that s would choose if she can choose partners freely from C0. For each c C and S0 S, the set Chc(S0) 2 is similarly de…ned. A matching is a (possibly empty) correspondence from C S to C S such that [ [ (s) C for every s S, (c) S for every c C and, for every i; j N, i (j) if and 2 2 2 2 only if j (i). We abuse the notation and, for any i N, write Pi if (i)Pi(i) and 2 2 Ri if (i)Ri(i). Given a matching , we say that it is blocked by (s; c) S C if s = (c), s 2 2 2 Chc((c) s) and c Chs((s) c). A matching is individually rational if for every [ 2 [ i N, Chi((i)) = (i). A matching is pairwise stable if it is individually rational 2 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 2 2 any X;X0 S (respectively X;X0 C) with X X0, we have Chi(X0) 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’…nal o¤ers game, which is a simpli…ed 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 …rst stage, each s S simultaneously 2 announces a subset of colleges Cs C, to which she makes an o¤er. 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 de…ne a two-stage game with colleges being players in the second stage. The analysis is unchanged.

2 each college c C chooses to match to its most preferred subset of students among those 2 who proposed to it, i.e. to Chc(Sc) where Sc := s S c Cs . The outcome of the game f 2 j 2 g is uniquely speci…ed by (c) = Chc(Sc) for each c C. We write CS := (Cs)s S for any 2 0 2 0 S0 S, C s = CS s, and (CS) to be the matching that results in the end of this game n 3 when students announce CS.

A strategy pro…le CS is a Nash equilibrium if (CS)Rs(Cs0 ;C s) for every s S and 2 C0 S. We introduce an order on the set of strategies by C0 Cs if Cs C0 . For any s s s S0 S, we de…ne the order by C0 CS if Cs C0 for every s S0. S0 0 s 2

2 Results

Consider a students’…nal o¤ers 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 Cs . f 2 j 2 [ g f 2 n j 2 0 g Given any strategy pro…le CS and s S, it is easy to see that C0 is a best response of s to 2 s C s if and only if

Chs(As(C s)) Cs0 Chs(As(C s)) (C As(C s)): [ n

Lemma 1 1. Let CS be a Nash equilibrium. If Cs C0 and C0 is a best response to s s C s for every s S, then (CS0 ) = (CS) and CS0 is also a Nash equilibrium. 2

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 (CS0 ) = (CS) and 2 CS0 is also a Nash equilibrium.

Proof. Part 1. By de…nition of C0 we have (CS)(c) s S c C0 s S c S f 2 j 2 sg f 2 j 2 Cs for each college. This and rationality of each college imply (C0 ) = (CS). To show g S that C0 is a Nash equilibrium, note that for each s S we have c C s Chc(S0 s ) s 2 f 2 j 2 c[f g g c C s Chc(Sc s ) where Sc = s S c Cs and S0 = s S c C0 by f 2 j 2 [ f g g f 2 j 2 g c f 2 j 2 sg rationality of college preferences. Therefore Cs0 is a best response to C0 s for each s S, 2 completing the proof.

Part 2. Since colleges have substitutable preferences, (CS0 ) = (CS). Since college preferences are substitutable and Cs C0 for every s S, s = Chc(Sc s ) implies s 2 2 [ f g s = Chc(S0 s ) for every s S and c C. Therefore C0 is a Nash equilibrium. 2 c [ f g 2 2 S 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. f g

3 Example 1 Let C = c ;S = s1; s2 , c s for s s1; s2 and s1; s2 c c s f g f g ; 2 f g f g ; f g for s s1; s2 . It is a Nash equilibrium for each student to apply to no college. On the 2 f g 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 bene…tting 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 (Cs;C s) = (Cs0 ;C s). Given C s, a strategy Cs of s S is a maximal strategy under C s if 2 Cs0 Cs for every strategy Cs0 that is outcome equivalent to Cs under C s.

Lemma 2 Given any strategy pro…le CS and any s S, there exists a maximal strategy 2 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 f 2 C s Chc(Sc0 s) , where Sc0 := s0 S s c Cs . In words, As(C s) is the set of j 2 [ g f 2 n j 2 0 g colleges that, if s proposes, would accept her o¤er.

Now given an arbitrary Cs and C s, let Cs := Cs (C As(C s)). We shall show Cs is [ n the maximal strategy that is outcome equivalent to Cs.

To show this, …rst note that all colleges that receive an o¤er from s under Cs receive

one from s under Cs since Cs Cs. This implies that all colleges that accept an o¤er from s under Cs accept one from s under Cs. Second, by de…nition of Cs, no colleges that do not accept an o¤er from s under Cs accept an o¤er from s under Cs. Thus Cs is outcome equivalent to Cs. Second, consider any Cs0 that is not a subset of Cs. By de…nition of Cs, this means that there exists a college c Cs0 Cs such that s Chc(Sc0 s). Then c (Cs0 ;C s)(s) 2 n 2 [ 2 while c = (Cs;C s)(s) by assumption, and hence (Cs0 ;C s)(s) = (Cs;C s), so Cs0 is not 2 6 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 2 is a best response of s to C s and it is a maximal strategy under C s. The maximal best C S C S response function br : (2 )j j (2 )j j is a function such that, for each s S, brs(CS) ! 2 is the maximal best response of s to C s. By Lemma 2, the function br is well-de…ned.

Lemma 3 Given any strategy pro…le CS and s S, 2

brs(CS) = Chs(As(C s)) (C As(C s)): [ n Proof. Immediate from Lemma 2.

4 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 modi…cation 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 de…ned 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 f g f g ; 2 f g f g ; f g for s s1; s2 . This is the same market as the one in Example 1. The empty matching 2 f g (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, (Cs ;Cs ) = ( ; ) is not a Nash equilibrium in 1 2 ; ; maximal strategies, since a maximal best response of s1 to Cs is c . Second, ( c ; ) is not 2 f g f g ; a Nash equilibrium since s2 can pro…tably deviate to c and be matched to c while she is f g unmatched under ( c ; ). Third, ( ; c ) is not a Nash equilibrium since s1 can pro…tably f g ; ; f g deviate to c and be matched to c while she is unmatched under ( ; c ). Finally, ( c ; c ) f g ; f g f g f g is a Nash equilibrium in maximal strategies with outcome ( c ; c )(c) = s1; s2 . f g f g f g A matching is said to be Hat…eld-Milgrom stable if it is individually rational and there is no student s S and C0 C such that C0 s (s) and s Chc((c) s ) for 2 2 [ f g each c C0. In general, any outcome of a Nash equilibrium in students’…nal o¤ers game 2 is Hat…eld-Milgrom stable.

Proposition 2 The students’…nal o¤ers game has

(i) Strategic complementarity: That is, for any s S, C0 s C s and any best response 2 Cs to C s, there exists a best response Cs0 to C0 s such that Cs0 Cs. Equivalently, brs(Cs;C0 s) brs(CS).

(ii) Negative externality for students: That is, for any s S, CS and CS0 with C0 s C s, 2 if Cs = brs(CS) then (CS)Rs(CS0 ).

5 (iii) Positive externality for colleges: That is, for any c C, CS and C0 CS, (C0 )Rc(CS). 2 S S

Proof. (i) Since C0 s s C s and colleges have substitutable preferences, we have As(C0 s) As(C s): Then, we have

brs(CS) = Chs(As(C s)) (C As(C s)) (Lemma 3) [ n Chs(As(C s)) As(C0 s) (C As(C0 s)). \ [ n The second line holds because (1) Chs(As(C s)) Chs(As(C s)) As(C0 s) (C \ [ n As(C0 s)) and (2)(C As(C s)) (C As(C0 s)). Now, note further that student s’s n n substitutable preferences imply Chs(As(C s)) As(C0 s) Chs(As(C0 s)). This establishes \

brs(CS) = Chs(As(C s)) (C As(C s)) [ n Chs(As(C0 s)) (C As(C0 s)) = brs(CS0 ). [ n

(ii) Since college preferences are substitutable, C0 s C s implies As(C0 s) As(C s). Thus (CS)(s) = Chs(As(C s))RsChs(As(C0 s)) = (CS0 )(s). (iii) Since C0 CS, S0 Sc, where S0 is the set of students that apply to c under C0 . S c c S Thus (CS0 )(c) = Chc(Sc0 )RcChc(Sc) = (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 brie‡y 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 irre- spective of their preferences. To see this point consider a many-to-one matching where, in contrast to our students’…nal o¤ers 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). Per- haps the most natural strategy ordering one may think of is to de…ne 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.

6 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’sbest 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 di¢ cult to endow strategy orderings that yield strategic complementarity. Indeed this di¢ culty 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’…nal o¤ers 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 pro…le 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, 2 f g say c1 c2 without loss of generality. Consider strategy pro…les where sk with k = i; j 6 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 increas- ! ing if, for all x; y L, x y implies f(x) f(y). 2

7 Result 1 (Tarski’sFixed Point Theorem) Let (L; ) be an nonempty complete lattice and function f : L L is monotone increasing. Then the set of all …xed 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’…nal o¤ers 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 is a student-optimal stable matching if Rs for every s S 2 and every stable matching . A stable matching is a college-optimal stable matching

if Rc for every c C and every stable matching . A stable matching is a student- 2 pessimal stable matching if Rs for every s S and every stable matching . A stable 2 matching is a college-pessimal stable matching if Rc for every c C and every 2 stable matching .

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 CS and CS in maximal strategies, such that CS CS C for every Nash equilibrium CS in maximal strategies. By items S (ii) and (iii) of Proposition 2, this implies that (C )Rs(CS)Rs(CS) for every s S and S 2 (CS)Rc(CS)Rc(C ) for every c C, completing the proof. S 2

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 speci…cally, 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 . j j

8 Initialization: Let t = 0 and 0 be the empty matching, that is, 0(s) = for every ; s S. Let A0(s) = C for every s S. 2 2 Iteration: For each t, Step t proceeds as follows.

t t t+1 t t+1 t 1. If (s) = Chs(A (s)) for all s S, then = and A (s) = A (s) for all 2 s S. 2 t t t 2-1. Otherwise, let s be a student with the smallest index such that (s) = Chs(A (s)). 6 At+1(s) = At(s) for all s = st. Let u = 0 and At;0(st) = At(st). 6 t t;u 2-2. For each u, Student s applies to the set of colleges Chs(A (s)). Each college t;u c Chs(A (s)) chooses its most preferred group of students among those who 2 t t t applied, that is, Chc( (c) s ). If no college rejects s , then let the resulting [ matching be t+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).

T T We say that the algorithm terminates in step T if (s) = Chs(A (s)) for all s S t t 2 and (s) = Chs(A (s)) for some s S for all t < T . Note that we have not shown that 6 2 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’ …nal o¤ers game. For that purpose, we …rst de…ne the following learning process, called the best response dynamics.

Initialization: Let C0 = for each s S. s ; 2 t t+1 t t Iteration: For each t: If CS is a Nash equilibrium, then = . If not, let s be the student with the smallest index whose current strategy is not a best response. Let t t+1 t s change her strategy to Cs = brs(CS) while every other student s keeps taking t+1 t Cs = Cs.

We will say that the best response dynamics converges at a …nite step T if CT is a Nash equilibrium and Ct is not a Nash equilibrium for any t < T .

Theorem 3 (Ct ) = t for each t 0; 1; 2;::: . S 2 f g t t t t Proof. We will show, by induction, that (CS) = and A (s) As(C s) for all 0 0 s S and t = 1; 2;::: . For t = 0, (CS) = since both are the empty matching and 02 0 0 A (s) As(C s) since A (s) = C for all s S by de…nition. t t t 2t t t Suppose (CS) = and A (s) As(C s) for all s S. If (s) = Chs(A (s)) for all 2 s S, then for each s S, t(s) = t+1(s) and At(s) = At+1(s) by de…nition and since t 2 2 9 is a stable matching, by Theorem 1, Ct is a Nash equilibrium and hence by de…nition of the t+1 t t+1 t+1 t+1 t t+1 best response dynamics, CS = CS so (CS ) = and A (s) = A (s) As(C s ). So t t t suppose (s) = Chs(A (s)) for some s S and let s be such a student with the smallest 6 2 t;u t t index. Then, by construction of the deferred acceptance algorithm, A (s ) Ast (C st ) = t+1 t+1 t t+1 Ast (C st ) for all u and hence A (s ) As(C st ). Also by substitutability of preferences t t+1 t t+1 t t+1 t of colleges and s , A (s) = A (s) As(C s ) for all s = s . Moreover (s ) = t t t+1 t t6 +1 t t t Chst (As( )) and for any s = s , (s) = (s) c (s ) s = Chc( (c) s ) . 6 t+1 t n f 2 j 2 t t[+1 tg In the best response dynamics, (CS )(s ) = Chst (A(C s)) = Chst (As( )) = (s ), t t+1 t t+1 t and for any s = s , by substitutability, (CS )(s) = (CS)(s) c (CS )(s ) s = t t 6 t+1 n f 2 j 2 Chc( (c) s ) = (s). This completes the proof. [ g

Corollary 1 The student-proposing deferred acceptance algorithm terminates in a …nite number T of steps. The resulting matching T is the student-optimal stable matching.

Proof. Since the students’…nal o¤ers game is a game with a …nite set of strategies and strategic complementarity, the best-response dynamics as described above converges in a …nite time T at the smallest Nash equilibrium (note that the dynamics starts at the smallest strategy pro…le). Since the smallest Nash equilibrium results in the student- optimal stable matching by Theorem 2, by Theorem 3 the student-proposing deferred acceptance algorithm terminates in a …nite number of steps T and the resulting matching T is the student-optimal stable matching.

2.2 Understanding Properties of Stable Matchings

c Recall a market is tuple = (S;C;P ). Let = (S;C c; P c) be a market in which college n c c is not present but otherwise the same as . Formally, we de…ne from simply by

changing preferences of c in such a way that Chc(S0) = for every S0 S. This de…nition ; 2 is convenient since we need to re-de…ne neither preferences of students nor strategy sets in c c the new market with this de…nition. Let br be the maximal best response function c c S in . Since preferences are substitutable, br(CS) br (CS) for any CS C . c Theorem 4 (Comparative Statics) Let CS and CS be the largest Nash equilibria in c c maximal strategies in and , respectively. Then (CS)Rs(CS ) for every s S and c c 2 (CS )Rc0 (CS) for every c0 C c. Let CS and CS be the smallest Nash equilibria in 2 c n c maximal strategies in and , respectively. Then (CS)Rs(CS ) for every s S and c 2 (C )Rc (C ) for every c0 C c. S 0 S 2 n

10 Proof. Since the students’…nal o¤ers game has strategic complementarity, we have

S CS = sup CS C CS br(CS) ; f j g c S c C = sup CS C CS br (CS) : S f j g c c Since br(CS) br (CS), CS br(CS) implies CS br (CS). Thus S S c c CS = sup CS C CS br(CS) sup CS C CS br (CS) = C : f j g f j g S c c Moreover c is not matched to any s S in . Therefore (CS)Rs(CS ) for every s S c 2 2 and (C )Rc (CS) for every c0 C c. S 0 2 n Similarly,

S S c c C = inf CS C CS br(CS) inf CS C CS br (CS) = C ; S f j g f j g S c c and hence (C )Rs(C ) for every s S and (C )Rc (C ) for every c0 C c, S S 2 S 0 S 2 n completing the proof.

Let CS be any Nash equilibrium in maximal strategies in , and now assume c becomes unavailable to be matched to any student, and students try to make new o¤ers 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.

c Iteration: For each Step t 1; 2;::: , let CS(t) = br (CS(t)). 2 f g We say that the vacancy chain dynamics terminates in …nite steps if CS(t + 1) = CS(t) for some t 0; 1;::: . 2 f g

Theorem 5 (Vacancy Chain Dynamics) For any Nash equilibrium CS in maximal strate-

gies in and c C, the vacancy chain dynamics associated with CS and c terminates in 2 c …nite steps at a Nash equilibrium CS0 in maximal strategies in , and (CS)Rs(CS0 ) for every s S and (C0 )Rc (C) for every c0 C c. 2 S 0 S 2 n c S Proof. Recall that br (CS) br(CS) for any CS C , and br(C) = C since C is S S S a Nash equilibrium in maximal strategies in . These properties imply that

c CS(1) = br (C) br(C) = C: (1) S S S By an inductive argument, property (1) implies CS(t) CS(t 1) for all t 1; 2;::: : 2 f g Since the set of strategies is a …nite set, this implies that the algorithm terminates to a c Nash equilibrium CS0 in satisfying CS0 CS. This set inclusion and the fact c rejects c every s in imply (C)Rs(C0 ) for every s S and (C0 )Rc (C) for every c0 C c, S S 2 S 0 S 2 n completing the proof.

11 Remark 1 For Theorems 4 and 5, we considered only situations in which one college c C 2 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 Hat…eld and Milgrom (2005).

De…nition 1 (The Law of Aggregate Demand, Size Monotonicity) Preference re-

lation Pc satis…es the law of aggregate demand (size monotonicity) if Chc(S0) j j Chc(S00) for every S00 S0 S. Similarly, preference relation Ps satis…es the law of j j aggregate demand if Chs(C0) Chs(C00) for every C00 C0 C. j j j j With substitutability and the law of aggregate demand, we show a version of the so- called “rural hospitals theorem.”Roth (1986) shows one version of the rural hospitals theorem for many-to-one matching with responsive preferences. More speci…cally, he shows that every hospital that has un…lled 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 “un…lled 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 speci…- cally, 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 CS 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 CS CS C . S Since college preferences satisfy the law of aggregate demand, this implies

(CS)(c) (CS)(c) (C )(c) for every c C: (2) j j j j j S j 2 On the other hand, since college preferences are substitutable, CS CS CS implies that, for every s S, As(C s) As(C s) As(C s): Since student preferences satisfy the law 2 of aggregate demand, this implies

(C )(s) (CS)(s) (CS)(s) for every s S: (3) j S j j j j j 2 4 A discussion on this point is found in Hat…eld 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 Hat…eld 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). Hat…eld 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 Hat…eld 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 2 generalized threshold best response of s to C s is de…ned as

c C c Chs(As(C s) c) : f 2 j 2 [ g C S C S Generalized threshold best response function br : (2 )j j (2 )j j is a function ! such that, for each s S, brs(CS) is the generalized best response of s to C s, that is, 2

brs(CS) = c C c Chs(As(C s) c) ; f 2 j 2 [ g for every s S. A generalized threshold best response is clearly a best response. Moreover, 2 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.

Proof. Suppose C0 s s C s. Since colleges have substitutable preferences, we have As(C0 s) As(C s). We need to show brs(C s) brs(C0 s), and by de…nition this is equivalent to c Chs(As(C s) c) c Chs(As(C0 s) c). This directly follows from the 2 [ ) 2 [ substitutability of s’spreferences. (Note that c is available both in the larger set As(C s) c [ and the smaller set As(C0 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 speci…c 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 f g would accept student s only. Assume further that c0Psc and cPs?. Then, xs(C s) = c f g 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). f g To formally de…ne 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’o¤ers C s and college preferences PC . The set of best responses of s can be determined by As(C s; PC ) and s’sown preferences 5 Ps. We restrict our attention to a selection of best response that only depends on those two essential data. For each matching problem = (S;C;P ), a best response selection

speci…es a best reply (of the students’ …nal o¤er game) for each student s as brs(C s j ). A best response selection is said to be essential when brs(C s ) depends only on j 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 s such that, for all matching problem ,

brs(C s ) = s(As(C s; PC );Ps). j Proposition 5 Assume S 2. The generalized threshold best response function is the j j smallest (in the sense of set inclusion) essential best response selection, which speci…es monotone increasing best replies in all students’…nal o¤er games where all agents in both sides have substitutable preferences.

Proof. The generalized threshold best response function is associated with essential

best response selection s(As(C s; PC );Ps) c C c Chs(As(C s; PC ) c) , and f 2 j 2 [ g Proposition 4 establishes that it speci…es a monotone increasing best reply. Hence we only

need to show that, if an essential best response selection s(As(C s; PC );Ps) speci…es a monotone increasing best reply, it is larger than the generalized threshold best response

selection: c C c Chs(As(C s; PC ) c) s(As(C s; PC );Ps). Suppose this is f 2 j 2 [ g not true and there are C s, Ps, PC and c0 C such that c0 Chs(As(C s; PC ) c0) 2 2 [ 5 Note that Chs(As(C s; PC )) is the optimal set of colleges for s, given other students’ o¤ers C s and college preferences PC . Obviously this is a best reply. Since s does not mind adding any o¤er 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 ) . f [ j g

14 and c0 = s(As(C s; PC );Ps). Note that c0 cannot be s’s choice in the available colleges 2 for him As: c0 = Chs(As(C s; PC )). This is because any best response (in particular 2 s(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 2 essential best response selection s does not specify a monotone increasing best reply in some game.

Case 1: Assume s Chc0 ( s ). In that case consider C0 s de…ned by Cs0 = Cs0 c0 for 2 f g 0 n f g every s0 = s. Then C s s C0 s and As(C0 s; PC ) = As(C s; PC ) c0. Our premise that 6 [ c0 is in the generalized threshold best reply, namely c0 Chs(As(C s; PC ) c0), implies 2 [ c0 Chs(As(C0 s; PC )). Hence c0 must be in any best response against C0 s, and in par- 2 ticular c0 s(As(C0 s; PC );Ps). This contradicts our premise c0 = s(As(C s; PC );Ps) and 2 2 monotonicity of the best reply.

Case 2: Assume s = Chc ( s ). Let P 0 be a responsive preference relation with quota one 2 0 f g c0 such that s0Pc0 sPc0 for every s0 = s, and let PC0 = (Pc0 ;PC c0 ). Also de…ne C0 s by Cs0 = 0 0 ; 6 0 nf g 0 Cs0 c0 for every s0 = s. Then As(C s; PC ) = As(C0 s; PC0 ), so c0 = s((As(C0 s; PC0 );Ps) = [f g 6 2 s((As(C s; PC );Ps): Now de…ne C00 s by Cs00 = Cs0 c0 for every s0 = s. Then C0 s s C00 s 0 nf g 6 and As(C00 s; PC0 ) = As(C0 s; PC0 ) c0. Our premise that c0 is in the generalized threshold [ best reply, namely c0 Chs(As(C0 s; PC0 ) c0), implies c0 Chs(As(C00 s; PC0 )). Hence c0 2 [ 2 must be in any best response against C00 s, and in particular c0 s(As(C00 s; PC0 );Ps0). This 2 contradicts c0 = s(As(C0 s; PC0 );Ps) and monotonicity of the best reply. 2 We also consider the smallest best response selection when we …x the game, that is, players and their preferences. Let C(s) be the set of colleges that never accept student s under any strategy pro…le, that is,

C(s) = c C s = Chc(S0); S0 S : f 2 j 2 8 g

For each s S, the modi…ed threshold best response function br( ) is de…ned by 2 s

br(CS) = br (CS) C(s): s s n The modi…ed 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, 2 f g

br(CS) = br (CS) c C s = Chc( s ) : s s n f 2 j 2 f g g Proposition 6 Fix a game. The modi…ed threshold best response function is the smallest (in the sense of set inclusion) essential best response selection, which speci…es monotone

15 increasing best replies in all students’…nal o¤er games where all agents in both sides have substitutable preferences.

Proof. The modi…ed threshold best response function is associated with the essential

best response selection (As;Ps) c C c Chs(As c) c C s = Chc( s ) , s f 2 j 2 [ g n f 2 j 2 f g g and Proposition 4 establishes that it speci…es a monotone increasing best reply. Hence we only need to show that, if an essential best response selection s(As;Ps) speci…es a monotone increasing best reply, it is larger than the modi…ed threshold best response selection: c C c Chs(As c) c C s = Chc( s ) s(As;Ps). Suppose this f 2 j 2 [ g n f 2 j 2 f g g is not true and there is c0 Chs(As c0) such that s Chc ( s ) while c0 = s(As;Ps). 2 [ 2 0 f g 2 Note that c0 cannot be s’schoice in the available colleges As for him: c0 = Chs(As). This 2 is because any best response (in particular s(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 2 show that the essential best response selection s does not specify a monotone increasing best reply. Recall that, when colleges have substitutable preferences, As(C s) is monotone decreasing. Since s Chc0 ( s ) by assumption, we can …nd C s s C0 s such that 2 f g As(C0 s) = As c0 and As(C s) = As. Our premise that c0 is in the modi…ed threshold best [ reply, namely c0 Chs(As c0), implies c0 Chs(As(C0 s)). Hence c0 must be in any best 2 [ 2 response against C0 s, and in particular c0 s(As(C s);Ps) = s(As;Ps). This contradicts 2 our premise c0 = s(As;Ps): 2 More generally we present, without a proof, a characterization of the set of essential best response selections in a …xed game. An essential best response selection is monotone if and only if

brs(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 , j j 2 n we have

(i) [T c] Ps [T c0] c Ps c0 and [ [ ,

(ii) [T c] Ps T c Pi ?, [ ,

and for any T C with T > qs, we have ? Ps T . Symmetric de…nition applies to j j colleges. Then any generalized threshold strategy is of the form c C cRsc for some f 2 j g 16 “threshold college”c C (this is why we use the term “generalized threshold strategy.”). 2 Moreover, any two strategies of the form can be compared since

c C cRsc c C cRsc0 c0Rsc; f 2 j g f 2 j g () 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 pro…le converges in …nite 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 f g 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 di¤erent 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,

N N N

for all x0 x with (xi0 xi) K; BRi(x0 i) BRi(x i) < (xi0 xi): i=1 i=1 i=1 X X X N Then (x0 xi) < K must hold for any equilibria x, x0. i=1 i P Proof. (i) It is su¢ cient to show that x x < K for all student i. Let n be a student i i where x x is the largest, i.e., x x x x for all i. Suppose that x x K. i i n n i i n n Let us denote k x x . Then, we have a contradiction: n n

xn xn = BRn(xn) BRn(x n) BRn(x n + (k; :::; k)) BRn(x n) < k = xn xn :

The …rst equality comes from the fact that x and x are Nash equilibria. The weak

inequality comes from non-decreasing best replies and k x x x x for all i (the n n i i de…nition of player n). The strict inequality is the condition (4). N (ii) Suppose, contrary to our claim, we had i=1(xi0 xi) K, for some equilibria x, x0. N N Then, the above condition would apply and i=1 BRi(x0 i) BRi(x i) < i=1(xi0 xi), P which would contradict the equilibrium conditions BR(x ) = x and BR(x) = x. P 0 0 P The insight of the above two results comes from the well-known su¢ cient 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 2

@BRi for any i and any x i, (x i) < 1: @xj j=i X6 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

(0; 1) x; x0 BR(x0) BR(x) x0 x ; 9 2 8 j j j j N where can be any norm in R , but consider the norm x = x1 + + xN . Then, j j j j j j j j the contraction mapping condition is stated as

N N

(0; 1) x; x0 BRi(x0 i) BRi(x i) xi0 xi : 9 2 8 i=1 i=1 j j X X

18 When best replies are non-decreasing, the following relaxation of this contraction mapping condition su¢ ces for the uniqueness.

N N

x0 x; x = x0 BRi(x0 i) BRi(x i) < (xi0 xi): 8 6 i=1 i=1 X X 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 su¢ cient condition than Theorem 7-(ii) is obtained as follows.

Theorem 8 Suppose there is an integer K such that,

x0 x with (xi0 xi) K n BRn(x0 n) BRn(x n) < xn0 xn: 8 i=1 9 X N Then (x0 xi) < K must hold for any equilibria x, x0. i=1 i P N Proof. Suppose, contrary to our claim, we had (x0 xi) K, for some equilibria i=1 i x, x0. Then, the above would apply and n BRn(x0 n) BRn(x n) < xn0 xn. This would 9 P contradict the equilibrium conditions BRn(x0 n) = xn0 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 satis…ed.

x0 x; x = x0 n BRn(x0 n) BRn(x n) < xn0 xn: (5) 8 6 9

It is easy to verify that condition (5) is satis…ed 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 Gus…eld 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 con‡ict 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 …rst point highlights the alignment of interests among men and among women, while the second point demonstrates the stark con‡ict 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 de…ned in this case. We exploit this fact to construct a game with Properties (i) and (ii) above. We …rst 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 …rst introduce a generalized version of threshold strategies. We have de- …ned generalized threshold best responses, but more generally generalized threshold strategies are de…ned as follows. We con…ne our attention to a student s S, but symmetric 2 de…nitions apply to a college c C. For any C0 C the generalized threshold strategy 2 of s associated with C0 is de…ned as

ts(C0) = c C c Chs(C0 c) : f 2 j 2 [ g

We say that C0 is a threshold of ts(C0). There are several properties of generalized threshold strategies. Let

IRs = C0 C C0 = Chs(C0) f j g 20 be the set of groups of colleges that are individually rational for student s.

(i) C0 ts(C0) if C0 IRs. 2

(ii) When s has substitutable preferences, ts(C0) = C C Chs(C0 C00): [ 00 [

(iii) There can be distinct thresholds C0;C00 IRs (C0 = C00) resulting in the same set of 2 6 o¤ers ts(C0) = ts(C00):

(iv) “Rising threshold”from C0 to C00, if de…ned as C 00 PsC0, does not generally mean that

the student is making a smaller set of o¤ers: C0;C00 IRs and C 00 PsC0 do not imply 2 ts(C0) ts(C00)

(v) However, a modi…ed version holds: C0;C00 IRs and C00 = Chs(C0 C00) imply 2 [ ts(C0) ts(C00): Both (i) and (ii) directly follow from the de…nitions. Item (ii) motivates our de…nition of a threshold strategy. It is the set of o¤ers accepted by the player, if o¤ered 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 Psa, f g and a; b Psb. Then ts(a) = ts(b) = a; b . Item (iv) can also be shown by an example. f g f g 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 f g f g ts( x; y ) + ts( w; z ). This is because z ts( w; z ) (Item (i)) but z = ts( x; y ) (because f g f g 2 f g 2 f g both x and y are better than z). Basically threshold w; z is better than threshold x; y , f g f g because w is very good (while z is a lousy choice).

Item (v) is shown as follows. We need to show that C00 = Chs(C0 C00) and c ts(C00) [ 2 imply c ts(C0). Equivalently, we need to show C00 = Chs(C0 C00) and c Chs(C00 c) 2 [ 2 [ imply c Chs(C0 c). C00 = Chs(C0 C00) means that C0 is rejected in C0 C00, and the 2 [ [ [ substitutable preferences of s implies that C0 is also rejected in a larger set C0 C00 c [ [ (as substitutability implies the rejected set is monotone increasing in the opportunity set).

Then a revealed preference argument shows that Chs(C00 c) = Chs(C0 C00 c). By the [ [ [ substitutable preferences of s, c (Chs(C00 c) =) Chs(C0 C00 c) implies c Chs(C0 c). 2 [ [ [ 2 [ 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 = C0 C. This means that s is “making o¤ers”to colleges

ts(C0) = c C c Chs(C0 c) : f 2 j 2 [ g

21 C Student s’sstrategy space is given by Xs = 2 , and this includes an empty set (i.e., s can choose threshold xs = ?, and in that case he makes o¤ers to c C c Chs(c) = c f 2 j 2 g f 2 C cPs? ). We order a student’sstrategy in terms of the set of o¤ers he is making: j g

xs s x0 ts(xs) ts(x0 ): s , s This de…nition matches to our everyday language: a “higher threshold”means a smaller set S of o¤ers. Similarly, a college c’sstrategy space is given by Xc = 2 , and the same ordering is applied to colleges’strategies (xc c x0 tc(xc) tc(x0 ):). c , c Given a strategy pro…le x, students and colleges are matched in the following way. Student s is matched to college c if and only if they make o¤ers to each other. Let ms be the set of colleges matched to student s under threshold strategy pro…le x. Note that ms depends only on colleges’ o¤ers xC and the student’s own o¤ers xs. Hence we denote it by ms(xs; xC ). Similarly, we de…ne the set of students accepted to college c by 7 mc(xc; xS). This is an essential feature of our model. In our original game in Section 2, where students make o¤ers …rst and then colleges respond, there is direct con‡ict 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 o¤ers. This means that student s can be matched to college c as long as c is making an o¤er to s, and this is independent of what o¤ers other students are making. Hence, in our simultaneous o¤ers game, we can e¤ectively 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 (…rst 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 pro…tably 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 o¤er to a valuable partner c, and c can unilaterally deviate to make an o¤er to s. Hence the situation where c and s are not matched cannot be an equilibrium.

7 Hence, the matching function resulting from strategy pro…le x is given by (s) = ms(xs; xC ) for s S and (c) = mc(xc; xS) for c C. 2 2

22 Formally, we specify student s’s payo¤ function us in the current game in such a way that

us(x) > us(x0) if ms(xs; xC )Psms(xs0 ; xC0 ),

us(x) > us(x0) if ms(xs; xC ) = ms(x0 ; x0 ) = xs and x0 = xs. s C s 6 The second line is our key assumption. Other things being equal, a player would like to choose a threshold that is actually realized. The speci…cation of us for the remaining case

( ms(xs; xC ) = ms(x0 ; x0 ) = ms, xs = ms and x0 = ms) is arbitrary. Note that the s C 6 s 6 absence of strategic interaction between students in the outcome (the resulting matching) carries over to the payo¤s: us only depends on xs and xC . With an abuse of notation, we sometimes write us = us(xs; xC ). A college’slexicographic payo¤ uc is similarly de…ned. Let us motivate the assumption of lexicographic preferences. We do not argue that this is a particularly compelling assumption about people’spreferences. 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 payo¤s. 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 payo¤s.

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 su¢ ciently low payo¤ to c (so that c is worse o¤ than in the situation where it …nds 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 speci…ed strategy spaces and payo¤s of our game. We call this a threshold demand game. (Named after the famed simultaneous o¤er bargaining game, the Nash demand game.) Suppose for the moment we do not assume substitutability or responsiveness, and consider the stability as de…ned below. The stability concept here requires no blocking by any subset of players. Formally, a set of students S0 and a set of colleges C0 block a given matching , if (i) S0 and C0 are not matched under ( s S0 (s) C0 = ? and c C0 8 2 \ 8 2 (c) S0 = ?), and (ii) s S0 C0 Chs((s) C0) and c C0 S0 Chc((c) S0). \ 8 2 [ 8 2 [ (S0;C0) is called a blocking coalition. A matching is stable if it is individually rational (same de…nition 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 .) 2 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 de…ned above. In this game all players simultaneously make o¤ers to each other, and players are required to accommodate all accepted o¤ers. Then, a student’spayo¤ only depends on his own o¤ers and colleges’o¤ers, and it is independent of other students’o¤ers. Hence there is no direct strategic interaction between students (and similarly between colleges).

Theorem 10 In a threshold demand game, de…ne the representative student whose strategy consists of students’thresholds xS = (xs)s S XS and whose payo¤ is any function strictly 2 2 increasing in students’ payo¤s: WS((us)s S) and WS is strictly increasing in us for all 2 s S. Similarly de…ne the representative college who has strategy space XC and payo¤ 2 WC ((uc)c C ) which is strictly increasing in uc for all c C. The resulting two-player 2 2 game, the aggregated threshold demand game, has the same set of pure strategy Nash equilibria as the threshold demand game.

Proof. Note that the payo¤ functions of the representative players are given by

S(x) = WS((us(xs; xC )s S)) and 2 C (x) = WC ((uc(xc; xS)c C )). 2

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’sgain from deviation S(xS; x ) S(x ; x ) (a symmetric argument C S C applies to the representative college). Since x is a Nash equilibrium in the threshold demand game, we have us(xs; x ) us(x; x ) for all s. This shows that S(xS; x ) C s C C S(xS ; xC ) 0, because S = WS((us)s S) is increasing in its arguments. Hence x is 2 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 bene…t (us(xs; x ) > us(x; x )), while the payo¤ to any other student s0 = s remains unchanged C s C 6 (us0 (xs ; xC )). By de…nition, representative student gains from deviation to (xs; x s), and 0 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 x0 xS s x0 xs and the S , 8 s representative college’s strategies are ordered by x0 xC c x0 xc. It corresponds C , 8 c to the “top dog”case in Fudenberg and Tirole (1984), where a larger strategy of a player harms the opponent.

4.2 General o¤er demand game We required the lexicographic payo¤ and threshold strategies, but now allow any o¤ers and assume another version of lexicographic preferences. Speci…cally, we may assume that (i) players can make o¤ers to any subset of partners (ii) if player i has two strategies (sets

of o¤ers) xi and xi0 that yields the same match for him and one is larger than the other

(xi xi0 ), then i prefers the larger strategy. Let us call this general o¤er demand game. Now consider the following conjecture:

Conjecture 1 The set of outcomes achieved by the pure strategy Nash equilibria of general o¤er 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 o¤ than being unmatched. (Note that there are complementarities among the potential partners. The preferences here violate substitutability.) Consider the following strategy pro…le in general

o¤er demand game: xs = c , xs = c0 , xc = s0 , xc = s0 . These “criss-cross” o¤ers f g 0 f g f g f g result in no matching. If s changes his o¤er to either c; c0 or c0 , he is matched only f g f g to c0, and hence he is worse o¤. 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 o¤ 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 o¤er demand game is stable.

Proof. Let be the matching achieved by a pure strategy Nash equilibrium of general o¤er demand game. It is individually rational, because otherwise a player is better o¤ by making o¤ers to a smaller subset. Suppose is not stable. Then, in the one-to-many case,

the outcome must be blocked by one college c and the set of students S0 (in the one-to-many

case, any blocking coalition contains such a subset). By de…nition of blocking, c and S0 are not matched in the Nash equilibrium. By the lexicographic payo¤ assumption, there is no

s S0 such that no o¤er is being made between c or s. (If such a case exists, c can make 2 an o¤er to s. This does not change c’s match, and c is strictly better o¤ by making a

larger set of o¤ers.) Let us suppose that c is making o¤ers to T S0 and T are making no o¤er to c. Then, any student s T is better o¤ by deviating to a new set of o¤ers 2 Chs((s) c). This is a contradiction, so such a set T does not exist. Hence it must be [ the case that all students in S0 are making o¤ers to c and c is not making any o¤er to S0 in the equilibrium. This is also a contradiction, because c can pro…tably deviates to a new

o¤ers Chc((c) S0). [ 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 o¤er demand game coincides with the pairwise stable matchings.

Proof. Step 1 (Nash stable): Let be the matching achieved by a pure strategy ) Nash equilibrium of general o¤er demand game. It is individually rational, because otherwise a player is better o¤ by making o¤ers to a smaller subset. Suppose is not pairwise stable. Then, the outcome must be blocked by a pair (c; s). By de…nition of blocking, c and s are not matched in the Nash equilibrium. By the lexicographic payo¤ assumption, it must not be the case that no o¤er is being made between c or s. (In such a case, c could make an o¤er to s. This would not change c’smatch, and c could be strictly better o¤ by making a larger set of o¤ers.) Let us suppose that c is making an o¤er to s but s is not

making an o¤er to c. Then, s is better o¤ by deviating to a new set of o¤ers Chs((s) c), [ a contradiction. The remaining case, where c is not making an o¤er to s but s is making an o¤er to c, also leads to a contradiction. Hence any Nash outcome is pairwise stable. Step 2 (Stable Nash): Let be a stable matching and de…ne a strategy pro…le in ) 0 0 general o¤er demand game x by xi = (i) for all i. This may not be a Nash equilibrium,

26 because a player can gain by making an additional o¤er that is rejected (by the lexicographic payo¤ assumption, a player is better of making the larger set of o¤ers.) We modify x0 to obtain a Nash equilibrium x that also achieves . Let x = c C c Chs((s) c) . s f 2 j 2 [ g By the individual rationality of ((s) = Chs((s))), xs contains (s). Let xc be the union of (c) and all o¤ers that are rejected under x : x = (c) s S c = x . By S c [ f 2 j 2 sg construction x achieves outcome . Also by construction, there is no pair of players s and c between whom no o¤er is made under x. This implies that, if any player i deviates from xiand makes any additional o¤er, it is accepted. It is easy to see that any player i is strictly worse o¤ by o¤ering a strict subset of xi (it either reduces the set of o¤ers without changing i’smatching partners, or i would lose a valuable partner in (i):) We now show that player i is worse o¤ by deviating to xi " xi. First, consider the case i S. Then, after the 2 deviation, i is matched to the union of xi (i) and T xi c C c = Chi((i) c) . \ \ f 2 j 2 [ g If i prefers this to (i), Chi((i) T ) must contain at least one element in T . Otherwise, [ Chi((i) T ) = Chi((i)) = (i) (the last equality follows from the individual rationality [ of ), contradicting our premise that [xi (i)] T (a subset of (i) T ) is better than \ [ [ (i) for i. However, by responsiveness Chi((i) T ) should contain no elements in T , [ because c T satis…es c = Chi((i) c). Hence we have established that a student cannot 2 2 [ gain by a unilateral deviation at x. It remains to show that any college i C cannot 2 pro…tably deviate to xi " xi. After the deviation, i is matched to the union of xi (i) \ and K xi s S i = (s) and i Chs((s) i) . Note that i is matched to K because \f 2 j 2 2 [ g under x, any student s is making an o¤er to any college c that satisfy c Chs((s) c). 2 [ By the stability of , any student s in K satis…es s = Chi((i) s) (otherwise, s and i 2 [ could block ). Then, we can employ the same argument as in the previous case (i S) 2 to show that any deviation to xi " xi is unpro…table for i C. 2

5 Discussion

5.1 Monotone methods for stable matching The current paper is the …rst 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 …nds that the space of “pre-matchings”allows for an increasing function, and that the set of …xed 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 (Hat…eld

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 …rst

consider the solutions to the system of two equations, G1 : X2 X1 and G2 : X1 X2: ! ! x = G (x ) 1 1 2 ; ( x2 = G2(x1)

or, the …xed 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 …nd out the …xed point.

Lemma 5 (Composition Lemma): De…ne a composition mapping associated with G by

Hi(xi) Gi(Gj(xi)), i = j. The …xed points of Hi coincide with the …xed points of G, for 6 i = 1; 2. More precisely,

(x; x) = G(x; x) x = Hi(x) (and x = Gj(x)). 1 2 1 2 , i i j i Proof. Consider the case i = 1. Obviously,

x = G (x ) x = G (G (x )) 1 1 2 1 1 2 1 : ( x2 = G2(x1) , ( x2 = G2(x1)

Now we present the mappings employed by the existing literature, whose …xed 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 o¤er is a (possibly empty) subset of colleges, xs C, and let xS = (xs)s S be the pro…le of students o¤ers. Similar de…nitions 2 apply for colleges’o¤ers.

1. Hat…eld-Milgrom mapping HM:

HMs(xC ) C rs(xC ) for all s S; n 2 HMc(xS) S rc(xS) for all c C; ( n 2

where rs(xC ) is the set of rejected o¤ers by student s. More precisely, let ACs(xC ) be the set of colleges which would accept s (i.e., which are making an o¤er to s) at

college o¤er pro…le xC . Recall that s( ) denotes the rejected set function of student s R ( s(Y ) = Y Chs(Y ), where Chs is student s’schosen set function). Then, rs(xC ) R 8 See also Fleiner (2003).

28 s(ACs(xC )). (Similar de…nition applies to rc.) Let HMS(xC ) = (HMs(xC ))s S and 2 R 9 HMC (xS) = (HMc(xS))c C and HM = (HMS;HMC ). 2 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; f 2 j 2 [ g 2 Tc(xS) s S s Chc(ACc(xS) s) for all c C; ( f 2 j 2 [ g 2

where ACi is de…ned in item 1 above. TS(xC ) = (Ts(xC ))s S and TC (xS) = (Tc(xS))c C 2 2 and T = (TS;TC ).

Hat…eld and Milgrom (2005) and Echenique and Oviedo (2006) showed that the set of all …xed points of those mappings coincide with the set of all stable matchings. More precisely, if (xS ; xC ) is a …xed point of HM or T , then, a stable matching is obtained by matching any student and any college who are making an o¤er to each other at (xS ; xC ). Conversely, any stable matching can be obtained by a …xed 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 …rst consider our demand games, where students and colleges simultaneously make o¤ers.

Theorem 11 Hat…eld-Milgrom mapping HM is the best reply mapping of the general o¤er 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) = 2 tc(brc(x)) for every c C where bri is the best reply function (in the target partners) of 2 agent i in the threshold demand game.

9 To be precise, their model is slightly di¤erent in two ways from ours: …rst, 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 di¤erences and refer to the Hat…eld-Milgrom model as the adaptation of their model to many-

to-many matching without contracts. also, Hat…eld 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 …xed points as HM, as the composition lemma shows. 10 Ostrovsky (2007) considers a model in which there is a …nite partially ordered set of agents and contracts are incorporated, and his function reduces to the mapping T here when we focus on many-to-many matching without contracts. 11 A comment on a …ner point: The …xed points of HM and T are generally di¤erent (the …xed points of the former have more o¤ers by de…nition), but the matchings obtained by the …xed point o¤er pro…les are the same for HM and T .

29 Proof. Consider the general o¤er demand game and let x = (xS; xC ) be a given strategy pro…le. Then, by the lexicographic preferences as imposed in the de…nition of the game, a best response of an arbitrary student s S is 2

Chs(ACs(xC )) [C ACs(xC )] = C rs(xC ): [ n n Similarly, the best response of an arbitrary college c C is 2

Chc(ACc(xS)) [S ACc(xS)] = S rc(xS): [ n n Therefore the best reply mapping of the general o¤er game is identical to mapping HM.

Next, consider the threshold demand game and let x = (xS; xC ) be a given strategy pro…le. Then, by the lexicographic preferences as imposed in the de…nition of the game, a

best response of an arbitrary student s S is brs(x) = Chs(ACs(xc)). Therefore 2

ts(brs(x)) = c C c Chs(Chs(ACs(xC )) c) : f 2 j 2 [ g

Since preferences of s is substitutable by assumption, Chs(Chs(ACs(xC )) c) = Chs(ACs(xC ) [ [ c) for any c C. Therefore we obtain 2

ts(brs(x)) = c C c Chs(ACs(xC ) c) = Ts(xC ): f 2 j 2 [ g 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 …nal o¤ers game, where students make o¤ers …rst and then colleges respond.

Theorem 12 The largest best reply functions of students’…nal o¤ers game is the composition mapping associated with the mixture of Hat…eld-Milgrom and T mappings (6). More precisely: Let brS(xS) = (brs(x s)) be the largest best reply function in the students’…nal o¤ers game. Then,

brS(xS) = HMS(TC (xS)).

By Theorem 12, together with Proposition 1 before, we obtain the following corollary.

30 Corollary 3 The set of matchings obtained by the …xed points of the following mapping coincides with the set of stable matchings12:

xS0 = HMS(xC ) (C rs(xC ))s S 2 : (6) ( xC0 = TC (xS) ( s S s Chc(ACc(xS) s )c C f 2 j 2 [ g 2 Remark 2 By the composition lemma, the …xed points of the largest best reply mapping of students’ …nal o¤ers game corresponds to the set of …xed points of the mixed mapping (6). Intuitively, in this game, student s’s largest best reply to other students’ o¤ers are calculated in two steps. The …rst step is to determine which college accepts s, given other students’ o¤ers. A college would accept s if and only if s is a welcome addition to the given o¤ers 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 s(As(x s)), where s is the R R rejected set function of student s and As(x s) is the set of colleges which would accept s, given other students’o¤ers x s in the students’…nal o¤ers game. Since HMs(TC (xS)) = C rs(TC (xS)) = C s(ACs(TC (xS))), we need to show R

As(x s) = ACs(TC (xS)): (7)

By the de…nition of As(x s), college c is in As(x s) if and only if

s Chc(ACc(?; x s) s); (8) 2 [ where (?; x s) denote a pro…le where student s is making an empty set o¤er and others are making o¤ers x s. Note that ACc(?; x s) is simply the set of students who are making an o¤er to c under x s. In contrast, the right hand side of the desired equality (7) is the set of colleges which are making o¤ers under TC (xS). By de…nition, college c is making an o¤er to s under

TC (xS) if and only if

s Chc(ACc(xS) s): (9) 2 [ 12 The same claim also applies to mapping

x = T (x ) S0 S C : ( xC0 = HMC (xS)

31 Hence, we are done if we show (8) and (9) are equivalent. When s ACc(xS), the right 2 hand sides of (8) and (9) are both equal to Chc(ACc(xS)). When s = ACc(xS), the right 2 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’…nal o¤ers game. This corresponds to mapping T , as in the threshold demand game.

Theorem 13 The threshold best reply functions of students’…nal o¤ers 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’…nal o¤ers game. Then,

brS(xS) = TS(TC (xS)).

The proof is similar to the previous one and therefore we only provide a sketch. The …rst 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’o¤ers is given by mapping TC . Then

students apply threshold best reply TS, and the end result is the threshold best reply in the students’…nal o¤ers game.

Remark 3 By the discussion of this section, we have shown that the mathematical struc- tures of the threshold demand game and general o¤er demand game share critical features with existing monotonicity-based approaches to stable matching. The …nal o¤er 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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