Consistent bargaining conjectures in marriage and matching

JOURNAL OF ECONOMIC THEORY 45, 392407 (1988)

Consistent Bargaining Conjectures in Marriage and Matching*

ELAINE BENNETT

University of Kansas, Lawrence, Kansas 66045

Received August 20, 1986; revised July 7, 1987

In this model of bargaining in marriage markets, we assume that each couple bargains in a systematic way over the distribution of the utility of their marriage and that each market participant has accurate conjectures about the utility he or she could obtain through a similar bargaining process with every other potential partner. This paper formulates the bargaining model, characterizes consistent conjectures, and characterizes the matchings which are stable with respect to these conjectures. Journal of Economic Literature Classification Numbers: 022, 026. 0 1988 Academic Press, Inc.

1. INTRODUCTION

This paper presents a model of bargaining in marriage and other bilateral matching markets. In this model each couple bargains in a systematic way over the distribution of utility within their marriage. We assume that each participant has accurate conjectures about the utilities he or she, and each potential partner, can obtain through bargaining with other potential spouses. The bargaining problems of the couples are inter- dependent because one important element in the bargaining is the utility each member could obtain in another marriage. We say that a utility allocation for every potential couple is a consistent conjecture if it is consistent with the bargaining in every couple. Given a consistent conjecture, some marriages will be desirable to form and others will not. We call the resulting set of marriages and utility allocation a bargaining equilibrium. This paper characterizes consistent bargaining conjectures and bargaining equilibria.

The recent literature on matching models has yielded valuable insights into the nature of markets that are thin because the goods in the market

* I am grateful to Jan Aaftink, Sharon Rochford, Alvin Roth, Myrna Wooders, William Zame, and an anonymous referee for their comments on a earlier draft. This research was supported, in part, by the National Science Foundation Grant SES-8500436 and by the University of Kansas General Research Fund Grant 39110038.

392 0022-053 l/88 $3.00 Copyright (0 1988 by Academic Press, Inc. All rights of reproduction m any form reserved.

CONSISTENT BARGAINING CONJECTURES 393

are heterogeneous and indivisible. Matching models are used in [8, 221 to study the matching of men and women (the marriage market), in [S, 9, 16, 173 to study the matching of firms and workers (the job market), in [S] to study the matching of colleges and students (college admissions), and in [ 15, IS] to model the matching of residents and hospitals (the national residents matching program).

In each of these matching models the existence of an equilibrium is demonstrated by providing an algorithm which converges to a stable matching. However, every one of these algorithms converges to an extreme stable matching. This stable matching is both the stable matching most preferred by every member of one side of the market and the one which is least preferred by every member of the other side of the market, e.g., the man-optimal stable matching for marriage markets, the firm-optimal for job markets, and the hospital-optimal for the residency matching program. See [ 183 for further discussion.

These matching algorithms can be though of as describing a certain type of institution for arranging matches in which one side has all the bargaining power; the national residents matching program is an example of such an institution. For other institutions, such as markets, both sides have bargaining power; the stable matchings which occur in such settings are therefore unlikely to be the optimal stable matching for either side of the market. In such situations a bargaining model is more likely to provide relevant insights.

After this introduction, Section 2 presents a bargaining model for marriage markets and shows that there exist consistent conjectures for every marriage market. Section 3 characterizes consistent conjectures. Sec- tion 4 characterizes bargaining equilibria. Section 5 discusses related bargaining models and Section 6 contains proofs of the theorems.

2. BARGAINING IN MARRIAGE MARKETS

We envision the marriage market as taking place through bilateral bargaining in various couples over the division of total utility of their marriages. We assume that bargaining in each couple is regular and predictable. By this we mean that each couple faces a simple bargaining problem of the variety studied by Nash [lo] and by many subsequent writers (see Roth [ 141 for an excellent overview).

In the bargaining problem described by Nash and others, a pair of agents has an attainable utility frontier and a disagreement point. In the marriage market the attainable utility frontier of each couple is the set of all possible allocations of the utility derived from being married to each other and the disagreement utility for each individual is the opportunity

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cost of marrying that particular spouse. A solution concept for the simple bargaining problem, such as the Nash bargaining solution, selects a utility allocation for each disagreement vector. In each couple the bargaining function represents the bargaining process of the couple. While we allow the Nash bargaining solution as one possible bargaining function, we also admit any function that satisfies certain mild restrictions. By using this model of bargaining within each potential couple and by requiring con- sistency across couples, we can predict which couples will marry and the utility allocation within each marriage.

Let N= { 1, . . . . n} denote the set agents. This set contains into two dis- joint subsets: women, W, and men, M. We use C to denote the set of possible marriages: C = ( [i, j] 1 i E W, j E M}. Let vi be the utility to player i from remaining single. For each [i j] in C, let uii be the total utility of the marriage [i j] which can be divided in any way between i and j. We assume that every ui and vii is nonnegative.

In each couple’s bargaining problem the opportunity cost to each individual of marrying one particular spouse is the utility that individual would receive by marrying a different spouse or by remaining single. Therefore in each couple’s bargaining problem we use as the individual’s disagreement value the highest utility that this individual could obtain from another marriage or by remaining single. We use d,fy to denote i’s disagreement value in the couple [i, j] and use d”= (df, dy) to denote the disagreement vector for the couple [i, j]. We say that the disagreement vector d” is a feasible disagreement vector for the couple [i, j] if the sum of the disagreement values does not exceed the value of their coalition, that is d’-‘+ dji < v . .

We ‘assume’& ikach couple decides how they would divide the marital surplus (the utility of the marriage above the partners’ disagreement utility levels) in a regular and predictable way. For each couple [i, j] in C, the bargaining function, f”, captures this systematic division of utility. For each couple [i, j] and for each disagreement vector d”, f”(d”) is the agreed- upon utility allocation in their coalition.

A marriage market M = (C, (ui} iG N, { uii, f”} ci,il E c) specifies the set of possible marriages, the value to each agent of remaining single, the value of each potential marriage, and the bargaining function of each couple.

Let .8= (xy, XI]) denote utility allocation for [i, j]. We say that the utility allocation 9 is a feasible utility allocation for [i, j] if the total allocated utility is exactly equal to the total value of the marriage (i.e., XfJ + Xii = Uq). We use X = (X”) ti.il in c to denote a utility allocation for each potential marriage and refer to x as a conjectured utility allocation (or, more briefly, a conjecture). Since each .x~ specifies two utilities and since there are /Cl couples, each conjecture x contains Q = 2 1CJ components.

An “unprofitable” marriage is one in which both of its members can


obtain higher utilities by marrying other partners. For unprofitable marriages, the disagreement vector for a couple is not feasible. Since unprofitable couples have more attractive opportunities elsewhere, we expect that these couples will not marry and we say that the couple “agrees to disagree.” When a couple agrees to disagree, we use the disagreement vector as their agreement utility allocation. This convention is important because the opportunity cost of some profitable marriage may be based on the utility that one partner can obtain from an unprofitable marriage. As we see below, this convention will allow us to correctly gauge this cost.

Formally, we say that the couple [i, j] agrees to disagree whenever d,? + d,! 2 ui/. Whenever the couple [i, j] agrees to disagree we define their utility allocation, x9, to be equal their disagreement vector d” (i.e., xv= f’( d”) = &J).

Bargaining Functions

Each couple’s bargaining function reflects the bargaining within that couple over the value of their potential marriage. We include as admissible bargaining functions virtually all bargaining solutions that have been proposed for the simple bargaining problem, including the Nash bargaining solution, the Kalai-Smorodinsky bargaining solution, and the egalitarian bargaining solution. (We do extend each of these traditional solution concepts to allow for the possibility that a couple’s disagreement vector is infeasible.) We also consider any function that satisfies certain mild conditions to be an admissible bargaining function.

For any couple [i, j] E C, and for any disagreement vector, d”, we define the marital surplus of [i, j] at d” to be vV -d,!- dy.

We call f”: R* + R* a bargaining function for [i, j], if it satisfies:

(1) No waste: If, at the disagreement vector y, there is a nonnegative marital surplus then f”(y) is a feasible utility allocation for [i, j] (i.e., fy( y) + jjq y) = vii).

(2) Individual rationality: If, at the disagreement vector y, there is a nonnegative marital surplus then fi’( y) 2 y.

(3) Agreeing to disagree: If, at the disagreement vector y, there is a negative marital surplus then f”(y) = y.

(4) Continuity: The functionf’J is a continuous function on R:.

The no waste condition requires admissible bargaining functions to assign a Pareto efficient utility allocation whenever there is a marital surplus. Individual rationality requires bargaining functions to allocate each member at least as much as they can obtain elsewhere it it is feasible to do so. When there is no marital surplus, bargaining functions are required to assign the disagreement payoff as the utility allocation of the couple. Con-

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tinuity requires that small changes in the disagreement utilities in any couple result in, at most, small changes in the agreement utility allocations. We consider these to be very mild conditions since, apart from condition 3, we expect bargaining among rational players to enforce these conditions.

Consider a particular class of bargaining functions, those that divide the marital surplus in fixed proportions. Let il= (A,, . . . . A,) be the vector of weights of the players. We call the bargaining functions {f”}ri,i, in o l-bargaining functions if every couple divides its marital surplus in propor- tion to their components of A. We assume that 2 > 0. If there is a marital surplus for the couple [i, j], k’s component of the A-bargaining function (for k = i or k = j) is

ff(d”)=@+(l,/& +,$)(Q-d,!-d’i) J ’

When each couple divides the marital surplus evenly (A = (1, . . . . 1 )), we call the bargaining functions egalitarian bargaining functions.

Disagreement Values

In each couple, each partner uses as his or her disagreement value, the highest utility each could attain by remaining single or through bargaining in another potential marriage. To determine the disagreement utility for player i in the couple [i, j], we calculate the utility i can obtain in each alternative marriage and then take the maximum over these utilities and the utility of remaining single.

Given the utility allocations specified by X, consider the utility i can obtain in [i, k]. When xik is a feasible utility allocation for [i, k], then x? is i’s opportunity cost of not forming [i, k]. Since xik is a feasible utility allocation for [i, k], xfk = vik - xik.

When xV is not a feasible utility allocation (this happens whenever disagreement vector, dik, is infeasible) the opportunity cost of not forming [i, k] is the maximum utility i could actually obtain by marrying k. In order to ensure k’s cooperation, k’s utility must be maintained at a level at least as high as his or her disagreement utility level. What remains, the residual utility, uik - dLk, is available to player i. Thus i’s opportunity cost of not forming [i, k] is vik - dLk. Since the couple [i k] agrees to disagree, xik = dik so i’s opportunity cost based on [i, k] can also be expressed as uik -xF. Thus whether or not xik is a feasible utility allocation for [i, k], the opportunity cost to player i based on [i, k] is uik -xF. Player i’s disagreement value in [i, j] is simply the largest of the opportunity costs based on his or her other- potential marriages and based on remaining single.

Formally i’s disagreement utility in [i, j] based on x is given by

d,?(x) = max(oi, uik - xiklfor [i, k] in C and k #j}.


Consistent Conjectures

The bargaining solution which we define in this paper is based on the notion that all potential partners have accurate conjectures about the utilities each partner could obtain in his or her alternative matches. A conjectured utility allocation for every potential marriage which is consistent with every couple’s bargaining is called a consistent conjecture. In a consistent conjecture, the utility allocation assigned to each couple is precisely the utility the couple would reach through bargaining given their utility possibilities in other potential marriages.

Formally, the conjecture, x, is called a consistent conjecture if for every couple [i, j], x0 = f’J(@(x)).

The conjecture allocation x is called a Lconsistent conjecture (respec- tively egalitarian consistent conjecture) if x is a consistent conjecture for J (egalitarian)-bargaining functions.

The first theorem asserts that there is a consistent conjecture for every marriage market. In particular this means that there is a consistent conjecture for every specification of admissible bargaining functions.

THEOREM 2.1. For every marriage market there exist consistent conjectures.

3. CONSISTENT CONJECTURES

Since a conjecture specifies a utility for each marriage partner in every potential marriage, a conjecture specifies 50 components for a marriage market consisting of only live men and live women. One striking property of the bargaining across potential marriages that leads to consistent conjectures is that it enforces a uniformity of utility allocations for individuals across their potential marriages. At any consistent conjecture, although bargained utility levels may vary widely among men and among women, each individual obtains the same agreement utility level in all of his potential marriages. In some potential marriages this agreement utility level is part of a feasible utility allocation. In others it is the “agreement” utility for couples who agree to disagree. Thus, to characterize a consistent conjecture, we need only specify one component per player-10 components for a market consisting of live men and five women.

Since, in any consistent conjecture, each agent receives the same utility level in all of his potential marriages, we can think of this utility level as a reservation price in terms of utility for that player’s participation in a potential marriage.

Formally, we say that agent i has the reservation utility price, pi, at the

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conjecture x, if for every marriage [i, j] in C, pi = X$ We say that the consistent conjecture x is generated by the reservation price vector p in RN,, if each agent i has the reservation utility price pi.

THEOREM 3.1. For every marriage market, every consistent conjecture is generated by reservation utility price vector.

The next theorem asserts that any vector that generates a consistent conjecture is both realizable and maximal. Realizability guarantees that, at any consistent conjecture, no player will agree to disagree in every potential marriage unless he can obtain his reservation utility price by remaining single. Maximality guarantees that individuals would want to belong to any couple in which their reservation utility prices are realizable because, given the reservation prices of the other players, no individual can obtain a higher utility level.

We say the vector p E RN is realizable for the marriage market M if for every agent iE N either pi = vi or there is a je N with [i, j] E C for which pi + pj = uii. We say that p E RN is maximal for the marriage market M if for every agent i E N, pi 2 vi and for every [i, j] E C, pi + pj B ui,.

THEOREM 3.2. Every reservation price vector that generates a consistent conjecture is realizable and maximal.

One should note that realizability does not guarantee that a price vector will be a feasible utility allocation for any partition of agents.

The next result characterizes ,I-consistent conjectures. For each couple [i, j] E C, we define the excess of i in [i, j] at p by

e!(p) = max{vj - pi, IP - p,-p,Ifor[i,k]inCandk#j}.

THEOREM 3.3. The vector p generates a A-consistent conjecture f and only if p is realizable, maximal, and there exists a A > 0 such that for each [i, j] e C, Ajey(p) = Il,ey(p).

4. BARGAINING EQUILIBRIA

After bargaining over utility allocations, some individuals marry and others remain single. Since we assume that individuals either remain single or monogamously marry, the resulting set of coalitions partitions the set of players. A matching is a partition of the set of players into singles and couples and a feasible allocation of utility for each of these coalitions. We use the vector p E RN to denote a utility allocation and for each [i, j] we use p” to denote the pair (pi, pi).


Formally, we call (p, T) a matching if for each S in T, either ISI = 1 and pi = ui or else S = [i, j] for some [i, j] in C, and pi’ is a feasible utility allocation for [i, j].

We say that the matching (p, T) is based on the conjecture x if each couple in T receives its conjectured utility allocation. Formally, (p, T) is a matching based on x if for every [i, j] ET, pi’ = x? Notice that, since matchings are required to have feasible utility allocations, no couple can agree to disagree in a matching based on x and every single player obtains his or her utility of being single.

The matching (p, T), based on a consistent conjecture, is a bargaining equilibrium if no players would rather be married to each other than to their assigned mates in T, given their conjectures. Formally, the matching (p, T) is a bargaining equilibrium if (p, T) is based on a consistent conjecture x and there exist no i andj such that pp <xy and pT<x,? with one of these inequalities holding strictly.

THEOREM 4.1. For every marriage market a bargaining equilibrium exists.

We say that a bargaining equilibrium is an egalitarian bargaining equilibrium if it based on an egalitarian consistent conjecture or a l-bargaining equilibrium if it is based on a A-consistent conjecture.

We use the letter p both for the utility allocation of a matching and also for reservation utility price vectors. The next theorem shows that these uses are consistent for bargaining equilibria because every bargaining equilibrium utility allocation is a reservation utility price vector which generates the associated consistent conjecture.

THEOREM 4.2. Zf the bargaining equilibrium (p, T) is based on the consistent conjecture x then the vector p generates x.

The definition of a bargaining equilibrium guarantees that given the conjectured utilities, no alternative couple would prefer each other. It can be shown that bargaining equilibria actually possess the stronger stability property that no man or woman would conjecture that he or she would be happier either single or in another marriage, given his or her conjectured utility in these states. This stability property is stronger than the core stability which we next consider.

We say that the matching (p, T) is a core matching if for every i E N, pi > vi and for every [i, j] EC, pi + pj 2 vii. (Recall that the utility allocation in any matching is required to be feasible for every single and couple in T). We say that the vector p is in the core of the marriage market if there exists a partition T such that (p, T) is a core matching. The core stability notion requires that a matching be stable not only with respect to

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the particular conjectured utilities but also with respect to any reallocation of utility within any potential alternative marriage.

The next theorem shows that every bargaining equilibrium utility allocation is in the core. The following theorem shows that every core matching is a bargaining equilibrium for some selection of bargaining functions for each couple.

THEOREM 4.3. Every bargaining equilibrium is a core matching.

THEOREM 4.4. Every core matching is a bargaining equilibrium for some choice of bargaining functions for each couple.

One can also show if p is in the interior of the core, then the core matching (p, T) is a A-bargaining equilibrium for some choice of 1.

5. DISCUSSION

In [ 133, Rochford considers the bargaining over utility allocations which might take place within the marriages of a fixed assignment T of men to women. Each couple in T bargains over the marital surplus using an egalitarian bargaining function. Relative to a given matching (p, T), the disagreement value for each player in a given couple is the most he or she can obtain by remaining single or in another marriage, subject to the constraint that his or her potential partner’s must receive no less in that marriage than at p.

Rochford’s model fixes an assignment without explaining within the model how this particular assignment comes about. The assignments she uses, however, are precisely those which are stable with respect to egalitarian consistent conjectures. Since one can also show that the disagreement functions in this paper reduce to Rochford’s disagreement functions for each couple of the given assignment, it is not too surprising that the utility allocations of egalitarian bargaining equilibria coincide with Rochford’s solution. This model in this paper can be viewed as a extension and completion of Rochford’s model. (For another extension of Rochford’s model in a different direction see [19]).

In [6], Crawford and Rochford present a distinctly different model of bargaining in marriage markets. Each couple uses the egalitarian bargaining function and each member uses, as his or her disagreement value, the utility he or she would obtain through bargaining in the marriage he or she would make in a certain reduced marriage market. The reduced marriage market for each couple is obtained by setting the value of the given couple to zero and keeping all other couple’s values fixed.


This model has a significant drawback. In the reduced marriage market, the utility a player can obtain from his or her original marriage cannot affect his or her utility allocation in an alternative marriage because, in the reduced market, the value of the original marriage is arbitrarily set to zero. This seems rather unrealistic. For example, when a faculty member wishes to obtain an offer from another university in order to improve his bargaining position at his current institution, he brings to the negotiations with the alternative university his current salary. Crawford and Rochford’s model does not allow this type of aspect of bargaining to be considered in the reduced game negotiations.

Since their reduced marriage market does not reflect the complete set of bargaining possibilities of the original marriage market, the Craw- ford-Rochford solution can lead to less intuitive predictions of the matchings that take place. Consider their example of a four-player marriage market. There are two women (numbered 1, 3) and two men (numbered 2, 4). The single state has ui =O. For the possible marriages, t‘ ,2 =6, u,~ =2, v32 = 8, and v3., = 5. The unique Crawford-Rochford solution assigns woman 1 to man 4 and woman 3 to man 2, with a utility allocation of (19/16, 17/4, 15/4, 13/16). However, this matching is not stable (players 1 and 2 can each obtain a higher utility by marrying each other rather than by marrying their assigned spouses, as can players 3 and 4), it does not maximize total utility, and it is not Pareto optimal.

By contrast, every bargaining equilibrium for this marriage market sup- ports the same set of marriages: { [ 1,2], [3,4] }. Every bargaining equilibrium is in the core, is Pareto optimal, and maximizes aggregate utility. When all couples use the egalitarian bargaining function, the unique bargaining equilibrium utility allocation is (l/2, 1 l/2, 3,2).

Multilaterai Bargaining and Aspirations

Certain elements in this paper correspond to aspirations and aspiration solution concepts. Theorem 3.2 shows that every consistent conjecture is generated by a vector (of reservation utility prices) which is both realizable and maximal. These vectors are aspirations. It is not difficult to show that a vector is an egalitarian consistent conjecture if and only if it is generated by an equal gains aspiration. It can also be shown (although not so easily) that for every marriage market the set of all utility allocations of A-bargaining equilibria (as A varies over its range) coincides with the intersection of the core and the set of partnered aspirations for the associated sidepayment game. For aspirations and aspiration solution concepts for sidepayment games see [ 1,2] and for nonsidepayment games see [43.

The theory of multilateral bargaining for a more general class of bargaining problems is presented in [S].

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6. PROOFS OF THE THEOREMS

Proof of Theorem 2.1. Let A4 be a marriage market. Each f” is a continuous function defined on R: and each dV is a continuous function defined on R$ (recall that Q = 2 ICI). Delinef by f = (f”)ci.j, E c and d by d= Wp,,,,t. Then f and d are functions from Ry to RP,. Continuity off and d follow from the continuity of each fi’ and d”. Let z be the highest utility any individual can obtain either by remaining single or by marrying; i.e., z=max{u;, uiilig N, [i, j] EC}. Let Z be the Q-fold product of [0, z]: Z = [0, ~1~. It is easy to see that the compositionf 0 d restricted to Z maps into Z. The set Z is compact and convex, and f 0 d is continuous. Hence, by Brouwer’s fixed point theorem, f od has a fixed point. Clearly, every fixed point of this mapping is a consistent conjecture. [

Proof of Theorem 3.1. Let x E R$ be a consistent conjecture. We will show that xP = of, for every i E W and every j, k E M-the proof for i E M and j, k E W is similar. First observe that the proof is trivial if 1 M( = 1. For [MI > 1, fix any ie W. Select Jo A4 so that x7> xjk, for every k E M.

Case 1. x~~.u’J=v~. Then for any keM, d,?3vV--x~=x~. Since bargaining functions are individually rational, xik 2 dp. Thus, -vik > xy. Since xji is maximal, X? > xik. Hence, xjk = .Y!.

Case 2. xy+ xY> vii and xy > v,. Since xV is not a feasible utility allocation for [i, j], i and j must agree to disagree and hence x0 = d”. Let S be the coalition from which d” is obtained. S cannot by [i] since if this were true then xf = dy = ui, which contradicts one of the assumptions for this case. So, S = [i, t] for some t E M\{ j}. If xi’ + xf’ = ui, then dy = xi’. Hence x7=x?. We then apply case 1 with j = t to show that i receives the same utility allocation in each of her potential marriages. If xr + x: > ai, then xi’ > u ,r --XI’ = d;j=+, whi h c contradicts the maximality of xy.

Case 3. x7 + .xP > vii and XI = v,. Since i can always remain single, for any potential spouse, k, djk >, ui. Since utility allocations are individually rational, ?c;~ >/ d,!k. Thus, vi = xy > xi” > d$a vi. Hence, xi” = ui = xf and i receives the same utility allocation with any potential spouse.

These cases exhaust the possibilities for player i. In each case player i receives the same agreement value. Let this value be pi. Clearly p = (p,, . . . . p,) generates x and every consistent conjecture is price generated. a

Proof of Theorem 3.2. Suppose p generates the consistent conjecture x. We first prove maximality. Fix any agent iE N. Without loss of

generality let ie W. We wish to show that p, > vi and pi + pj > vV for every j. For any Jo M, p, = x7 > d,?(x) 2 u,. Fix any Jo M. In order to be an


admissible bargaining function, xu+ x7> uii. This means that pi + pj = xy + x; 2 uo, which completes the proof of maximality.

We next show realizability. Fix any agent i E N. Without loss, assume ie W. We wish to show that either pi = vi or else there exists a potential spouse Jo M such that pi + pj = uii. Fix a particular potential spouse j. If pi + p, = v;,, then i can realize her price in [i, j]. Otherwise (by maximality) pi + pj > uii. Since, in this case, i and j agree to disagree, pii = nii. Let S be the coalition from which di/ obtains its value. If S= [i],

pi = df = u,, so i can realize her price by remaining single. If S = [i, k] for some k, then d:J = uik - pk. Since pi = dy, we have that uik = pk + pi, which means that player i can realize her price by marrying player k. Since the choice of player i was arbitrary, p is realizable. 1

Proof of Theorem 3.3. First suppose that p generates a A-consistent conjecture, call it x. By Theorem 3.2, p is realizable and maximal. Fix any [i, j] e C.

Case 1. Suppose xy + XT = vi/. Then d! + d,? d uy. Since d” is feasible, xy=k?(diJ) = dji+ [A,/(l; +A,)]. [v. -djl-df] and xjj=fl(d”) = d,? + [A,/(& + ;1,)] . [v,~ -d,Y - dji]. From this we obtain Ai[xf-- dy] = l,[xT-d/l. Substituting for each x and d, after a little algebra we obtain

~,max{~j-p,,~jk-pj-pkICi,kl~C,k#j} =i,max{u,-p,,vk, -Pk-pjl[j,klEC,k#i}.

However, this is just E.,ey(p) = lie:(p).

Case 2. Suppose xi” + .x;> viJ. Then dji + dy> vii which means that .$ = d,? and .$? = d,V. Since xy - d;! = 0 = xy- d/: lj[pi - d,V] = nj[pj - df]. Hence AjeF( p) = Aiey( p). Since these cases are exhaustive, this completes the proof of sufficiency.

To show necessity suppose there is such a price vector p. Define x by x4 = (p,, p,) for each [i, j] in C. Fix [i, j] in C.

Case 1. Suppose xy + x7 > viJ. Since 3L/-ey = Lie:,

lj max{u; - Pi, uik - Pi - pk 1 [i, kl EC, k # j}

= ;lj max{ u, - pj, ok, - pk - p, 1 [j, k] E c, k # i}.

Simple substitutions and a little algebra yield

lJpi - di/] = l,[p, - dJ/] (1)

Equivalently, [pi - d,!!]/Aj = [p, - dT]/il,. Let this common value be denoted by c. By the assumption for Case 1, pi + pi = uii. Using this fact

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and (1) we solve for c and obtain c = [l/(2, + Aj)] [oli - d;j - d,!!]. Solving for Pk for k=i orj:

pk = dk” + [&/(A; + A,)][u, - d: - d;].

Since the right hand side is the A-bargaining function, f$(d”), we have that xV = f”(d”(x)), as required.

Case 2. Suppose xy + ~7 > vi/. Since p is realizable, i and j obtain pi and p, as their agreement utilities in other potential marriages. Hence do = pg. Since the disagreement utility vector is not a feasible utility allocation for [i, j], the couple under any bargaining function must agree to disagree, and, as required dii= xv. Hence xv= f”(d”). Since these cases are exhaustive, we have shown that for every [i, j] in C, xi’=f”(d”(x)), hence p generates a A-consistent conjecture. 1

Proof of Theorem 4.1. Let M be a marriage market. Shapley and Shubik [22] showed that the core of every marriage market

is nonempty, convex, and compact. Let p be any core vector for M and let T* be a partition for which (p, T*) is a core matching; T* consists of couples S, , S2, . . . . S, and single individuals S,, i, . . . . S,. Define the set A by A = (q E RNlq is the core of M and (q, T*) is a matching}. Notice that A is nonempty, compact, and convex.

For each t = 1, . . . . K, write S, = [i, j], and define 4’ = (&, . . . . &), a function which maps A to RN, by

&(q) = f;,‘(d”(q))

d;(q) =&W”(q))

d;(q) = qk for k#iorj.

We observe that 4’ alters only the i and j components of q. We assert that each 4’ actually maps A into itself. To see this, fix q E A.

We first show that (d’(q), T*) is a matching. Since the components of 4’(q) are the same as the components of q with the possible exception of #i(q) and d;(q), we need only verify that &(q) + d;,!(q) = vii. We first show that dv< q”. Suppose to the contrary that, say, djJ > qi. Let S be the coalition containing i from which this disagreement value is obtained. If S = i, then q cannot be in the core because q, < vi. If S= [i, k] for some k then q cannot be in the core because dy = vik - qk > qi which means that qi + qk -c u,k. Since, by assumption, q is in the core, we conclude that d” 6 qv. Since the disagreement vector is feasible for [i, j], f”(d”(q)) is a feasible utility allocation for [i, j]. Hence 4: + 4; = uij.

To complete the proof that 4’(q) is in the core, we show that d’(q) is not “blocked” by an individual or couple. Since every bargaining function is


required to be individually rational, f”(d”(q)) 2 d”. Hence, t&(q) 2 max{vi, oik - qJfor [i, k] E C and kf j}. This implies that #j(q) 2 0,; moreover, c&(q) = qk, for k # i orj, so it is also true that &(q) 2 uik -4;(q) for every k #j. Hence #i(q) + d;(q)> ujk, for all k #j. Since &(q) + b;(q) = uii, d*(q) is in the core.

Each 4’ is a continuous function (sincef” and dti are continuous) from A to itself. Define @ to be the composition of the K functions: @ = bKo ... 0 4’. Since each 4’ is continuous, @ is a continuous mapping from the A to itself. Since A is nonempty, convex, and compact, @ has a fixed point, call it q*. By construction (q*, T*) is a matching and q* is in the core.

To show that q* generates a consistent conjecture we define the conjecture x E RQ by xy = q* for every i E N and every [i, j] E C. We show x is a consistent conjecture by showing that for every [i, j] in C, xV is their agreement given X. For [i, j] ET*, .x~ is, by construction of q*, the agreement utility allocation for [i, j]. Consider a couple [i, j] not in T*. Since i can obtain his full price in some coalition in T*, his disagreement value in [i, j] is q:. Similarly J’S disagreement value is q,?. Since q* is in the core of this marriage market, q: + q,* > uij. Since there is no marital surplus for the couple [i, j], the conditions on admissible bargaining functions require that f”(d”(x)) = do = q*” = xi’. Hence xV is the agreement utility allocation for [i, j]. Since xv is the agreement utility allocation for every [i, j] in C, x is a consistent conjecture.

To show that the matching (q*, T*) is a bargaining equilibrium we observe that (q*, T*) is based on the consistent conjecture x, and that for every [i, j] in C, qii 3 zcV so q is stable with respect to these conjectured utility allocations. Hence (q*, T*) is a bargaining equilibrium for M. This completes the proof of Theorem 4.1. U

Proof of Theorem 4.2. For each player i, either one of two cases holds.

Case 1. Suppose that for some j, [i, j] E T. By definition of a bargaining equilibrium, for every couple [i, j] E T, pi = xy. Since x is a consistent conjecture xjk = xy for all k for which [i, k] E T. Hence pi is i’s reservation utility price.

Case 2. Suppose [i] ET. Since (p, T) is a matching pi = ui. For every j for which [i, j] E C, xy 2 dji > oi. Thus xy > pi. We will show that if xf > pi, then (p, T) cannot be a bargaining equilibrium. Now either [j] ET or there is a k for which [j, k] ET. In the first instance a similar argument for j shows that xj 2 pj. Since xv 3 p”, and xy > pi, (p, T) is not a bargaining equilibrium. In the second instance [j, k J E T. Then by Case 1, x/ > p,, and again (p, T) is not a bargaining equilibrium. Since this contradicts our initial assumption, xy = p, and pi is I’s price.

406 ELAINE BENNETT

Since each player has a reservation utility price, the price vector generates X. a

Proof of Theorem 4.3. Let (p, T) be a bargaining equilibrium for the marriage market A4. From Theorem 4.2 we know that p generates a consistent conjecture, call it X. Since p generates a consistent conjecture we know that every couple [i, j] in C, p” = .I? and that p is maximal. Since p is maximal, pi b ui for every i in N. Since xv is an outcome of a bargaining function, ,# + xi’ > uii for every [i, j] E C. Hence, p, + pi b v,-, and (p, T) is a core matching.

Proof of Theorem 4.4. Assume that (p, T) is a core matching. We con- struct bargaining functions for each coalition such that (p, T) is a bargaining equilibrium.

Define the conjecture x based on p by xri = pV for every [i, i] E C. For each [i, j], let a” denote the disagreement utility vector for the couple at the conjecture I and let sii= vti - uf - # be the marital surplus at x. Since x is fixed each aii is also fixed. We define bargaining functions by different methods according to whether there is a positive marital surplus in [i, j].

For [i, j] with sii > 0, definef” byf”(z) = z if zi + zj > uii and by f{(i) = zk + [(xf - u{)/sii](uj, - zi - zj) otherwise for k = i or j.

For [i, j] with sV d 0, definef” byf”(z) = z if zi + zi > uii and by ft(z) = ik + (l/Z)(u, - z, -z,) otherwise for k = i or j.

Clearly these functions satisfy the conditions on bargaining functions and .I!= f”(d”(x)). Hence .Y is a consistent conjecture for these bargaining functions. Since p generates a consistent conjecture, maximality guarantees that (p, T) is a bargaining equilibrium.

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Consistent bargaining conjectures in marriage and matching