A search model of two-sided matching under nontransferable utility

Journal of Economic Theory 113 (2003) 182–198

A search model of two-sided matching undernontransferable utility

Hiroyuki Adachi�

Department of Economics, Toyo University, Hakusan, Tokyo 112-8606, Japan

Received 22 September 1999; final version received 29 August 2002

Abstract

In a decentralized marriage market there are different types of men and women. Agents

sequentially search for mating partners and meet bilaterally in a random fashion. Upon

meeting, the paired agents complete mating if both agree, and separate and continue searching

otherwise. The polarization of interests between men and women appears as in Gale–Shapley

marriage problems; as agents of one sex become more selective about their mates, agents of the

other sex lose. As search costs disappear, the set of equilibrium outcomes in a search model

reduces to the set of stable matchings in a corresponding Gale–Shapley marriage problem.

r 2003 Elsevier Science (USA). All rights reserved.

JEL classification: C71; C72; C78

Keywords: Search; Two-sided matching; Stable matching; Gale–Shapley marriage problem

1. Introduction

There are many markets in which transactions are completed bilaterally. Labormarkets with job-seekers and employers, marriage markets with single men andwomen, and housing markets with tenants and landlords are a few examples. Suchmarkets are called two-sided matching markets and each agent on one side of themarket look for a long-term partner on the other side. As concisely put by Roth andSotomayor [14], ‘‘the term ‘two-sided’ refers to the fact that agents in such marketsbelong, from the outset, to one of the two disjoint sets—for example, firms orworkersy . The term ‘matching’ refers to the bilateral nature of exchange in these

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0022-0531/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved.

doi:10.1016/S0022-0531(03)00085-1

markets—for example, if I work for some firm, then that firm employs me.’’ Intypical two-sided matching markets, goods are indivisible and each agent can tradewith at most one trading partner. The main question in a two-sided matching marketis, Who will trade with whom? And, if utility is transferable, at what prices?

The theory of two-sided matching is pioneered by Gale and Shapley [9] andShapley and Shubik [15]. The former work considers cooperative models of two-sided matching under nontransferable utility and the latter studies those undertransferable utility. The qualifier ‘‘cooperative’’ signifies that their analyses rely onsolution concepts in cooperative game theory such as stability and the core. Manycooperative models of two-sided matching assume (explicitly or implicitly) thepresence of a central matchmaker which recommends a matching to the agents giventhe individuals’ preferences over potential partners.

This paper studies a search model of two-sided matching under nontransferableutility. The term ‘‘search’’ refers to the condition that agents have to incur costs (inthis paper, time) to find a partner in the absence of a central matchmaker. Using amarriage market interpretation, our search model is described as follows. In amarriage market there are different types of men and women. Each agent can getmarried to at most one agent of the opposite sex. The utility to an agent of marryinga partner of the opposite sex depends on both of their types. Agents sequentiallysearch for mating partners and meet bilaterally in a random fashion. Upon meeting,the paired agents complete mating if both agree, and separate and continue searchingotherwise. If a man and a woman complete mating, they leave the marriage marketand ‘‘clones’’ of the two people enter the market, leaving the distribution of singleagents in the market unchanged. To simplify the problem, we assume that each typeof agent uses time-stationary, reservation-utility strategies. This is a behavioralassumption which restricts the set of strategies the agents use. But given that all theother agents use time-stationary, reservation-utility-type strategies, each agent facesa stationary search problem and it is optimal (among all strategies including non-stationary, nonreservation-utility-type strategies) for an agent to employ astationary, reservation-utility-type strategy. Also, this assumption provides a simplecharacterization on when an agent agrees to marry upon meeting an agent of aparticular type. Namely, given other agents’ reservation utilities, each agent willagree to marry a partner he/she meets if his/her gain from marrying the partner ishigher than his/her reservation utility. Therefore, we can say that the higherreservation utility an individual has, the more selective that individual will be towardpotential partners. Since people search for partners, the question of who will matewith whom depends on realizations of random events—random meetings. Givenreservation utilities of the agents in equilibrium, however, we can derive anequilibrium mating correspondence which tells the types of agents each type of agentwill complete mating upon meeting.

The set of equilibria forms a lattice, analogous to the set of stable matchings in aGale–Shapley marriage problem with strict preferences. On one end of the lattice isan equilibrium which each man likes and each woman dislikes as much as any otherequilibrium, whereas on the other end is an equilibrium with the opposite property.In the two-sided matching literature, the lattice property of the equilibria is often

ARTICLE IN PRESSH. Adachi / Journal of Economic Theory 113 (2003) 182–198 183

referred to as the polarization of interests between men and women: men and womenhave conflicting interests while people of the same sex share common interests. Oursearch model gives an intuitive search-theoretic explanation to such a property: asagents of one sex become more selective about their mates, agents of the other sexsuffer more and become less selective, and vice versa.1

There is more than just an analogy between cooperative models and search modelsas search frictions disappear: the set of equilibrium mating correspondences in asearch model with negligible search costs coincides (in a well defined sense) with theset of stable matchings in a corresponding Gale–Shapley marriage problem. Theresult is intuitive in that the equilibria in a marriage market with search frictionsapproach, as search frictions disappear, to the stable matchings in a frictionlessmarriage market. This equivalence result gives a noncooperative interpretation tothe (pairwise) stability condition used in the Gale–Shapley marriage problem.

The literature on cooperative models of two-sided matching is enormous sinceGale and Shapley [9] and Shapley and Shubik [15]. An excellent comprehensivetreatment is given by Roth and Sotomayor [14].

The literature on search-theoretic models of two-sided matching is rapidlygrowing. Burdett and Wright [8] consider two-sided search models undernontransferable/transferable utility. In their model agents are ex ante identical andgains-to-trade are match-specific random variables. Therefore, the problem ofdetermining which types of agents trade with which types does not arise. Theyobserve that under nontransferable utility equilibrium may not be unique while asimilar model under transferable utility has a unique equilibrium for a givenbargaining power parameter.

When agents are heterogeneous, there are two major problems concerning two-sided matching models with search friction. One is to determine which agents arematched to which agents. The other is to derive the steady state distribution ofunmatched agents in the market. Following McNamara and Collins [10], Bloch andRyder [3], and Morgan [12], we impose the ‘‘clones’’ assumption that keeps thedistribution of agents stationary, and avoid the difficulty caused by the secondproblem. With that simplifying assumption, we concentrate on the first problem andobtain clear results that relate the noncooperative search/matching literature (e.g.[13]) to the cooperative two-sided matching literature (e.g. [9]). Burdett and Coles [5],Smith [18], and Shimer and Smith [16] solved the two problems at the same time.Burdett and Coles [5] endogenize the distributions of agents types in equilibrium,assuming that the utility of an agent from matching depends only on its partner’stype. They show uniqueness of equilibrium, if one exists, and give a sufficientcondition for existence. Smith [18] analyzes a two-sided matching model undernontransferable utility and shows that an equilibrium exists and that log-supermodularity of utility functions implies positive assortative matching. Shimerand Smith [16] consider a similar problem under transferable utility and derivenecessary and sufficient conditions for assortative matching. The previous two

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1Burdett and Wright [8] take notice of the same feature in their two-sided search model where agents are

ex ante identical.

H. Adachi / Journal of Economic Theory 113 (2003) 182–198184

papers thus provide conditions under which the assortative matching result inBecker’s [2] frictionless marriage model extends to the marriage market with searchfrictions.

Most of the above papers emphasize the analyses of steady state equilibria.Exceptions include Burdett and Coles [4] and Smith [17], which focus on transitionaldynamics to equilibrium in search-matching models. While most papers in theliterature assume exogenous types of agents, Burdett and Coles [7] consider a two-sided matching model in which agents invest in quality prior to entering the matchingmarket. For a recent survey on search and matching, refer to Burdett and Coles [6].

The paper is organized as follows. Section 2 sets up the search model. Agents’decision problems are described in Section 3. Section 4 defines market equilibriumand illustrates its properties. In Section 5, we try to investigate a close connectionbetween our search models with negligible search costs and Gale–Shapley marriageproblems. Section 5.1 provides a brief introduction to the Gale–Shapley marriageproblem and a lemma which characterizes the set stable matchings as the fixed pointsof a certain mapping when preferences are strict. Section 5.2 shows that the set ofequilibrium mating correspondences in a search model reduces to the set of stablematchings in a corresponding Gale–Shapley marriage problem as search costsbecome negligible. Some of the limitations of the model and topics for furtherresearch are discussed in Section 6.

2. The search model

Let us consider a marriage market where ex ante heterogeneous single men andwomen are looking for mates. Each man is indexed by his type described by a realnumber, mAMCR: Similarly, each woman is indexed by her type wAWDR:We calla man of type m a man-m and a woman of type w a woman-w: Typically, there aremany men of the same type and many women of the same type. Let M and Wdenote the Borel s-algebras on M and W ; respectively. Also let M#W denote theirproduct s-algebra. The distributions of types of men and women are, respectively,

described by measures #lM and #lW defined on M and W: We assume that thesemeasures are exogenous and stationary over time. The measures of single men and of

single women in the marriage market in each period are thus #lMðMÞ40 and#lW ðWÞ40: It is convenient to define probability measures by lMðGÞ :¼#lMðGÞ=#lMðMÞ for each GAM and lW ðHÞ :¼ #lW ðHÞ=#lW ðWÞ for each HAW:Then ðM;M; lMÞ and ðW ;W; lW Þ each form probability measure spaces.

Each agent can get married with at most one other agent of the opposite sex. Thetype of each agent is private information and agents do not know which agent iswhich type until meeting. Therefore they first need to contact others. Agents meetbilaterally in a random fashion. Time is discrete, and in each period a single man(woman) meets with probability dMA½0; 1�ðdWA½0; 1�Þ a single woman (man), whosetype is a random draw according to lMðlW Þ: To equate the measure of men whomeet women and that of women who meet men, meeting probabilities dM and dW are


assumed to satisfy dM#lMðMÞ ¼ dW

#lW ðWÞ: The paired agents discover the partner’stype upon meeting, and each agent decides whether to agree to mate to the partner ornot. They will mate if both agree. If at least one does not agree to mate, they simplyseparate, forget about the other agent’s identity and look for other partners in thenext period. The utility of staying single for any single period is normalized to zerofor every agent. Mating of a man-m and a woman-w gives a utility of qMðm;wÞ to theman-m and qW ðm;wÞ to the woman-w: A woman-w such that qMðm;wÞo0 is calledunacceptable to a man-m: We impose the nontransferable utility assumption (i.e., noside payments are allowed), and there is no room for negotiations between men andwomen as to the terms of marriage.

In this paper we are only concerned with stationary equilibrium in a marriage marketwhere the distributions of single men and women are exogenously given and stationaryover time. To keep the stationarity we impose the replacement assumption that if aman-m and a woman-w get married, they both exit the marriage market and clones of aman-m and a woman-w will come into the pool of singles. Also we assume that if someagents would not mate with any other agents with probability one they simply stay inthe marriage market. This assumption is plausible particularly for a marriage marketwhere entering the market costs practically nothing because agents are indifferentbetween staying single forever in the market and staying out of the market. But it is arestrictive assumption for other markets where participation costs are positive.

All the single men and women are assumed to be risk neutral, have a commondiscount factor rAð0; 1Þ; and maximize their expected utilities. All of the aboveinformation is common knowledge.

We assume the following throughout the paper.

Assumption 2.1. (i) M and W are closed and bounded in R: (ii) qMðm;wÞ andqW ðm;wÞ are M#W-measurable, and qMðm;wÞ; qW ðm;wÞpboN for some b:

3. Agents’ decision problems

Let us consider agents’ decision problems. Let vMðmÞ denote the expected utility ofa man-m at the end of the period who remains single and searching for matingpartners. Similarly, vW ðwÞ denotes the expected utility of a woman-w at the end ofthe period who is seeking for partners. For the moment let us suppose these expectedutilities are given and consider the agents’ decision problem on when they shouldagree to mate to a particular partner upon meeting. If vMðmÞ and vMðwÞ are theexpected utilities to a man-m and a woman-w of staying single, it is optimal for aman-m to agree to marry a woman-w upon meeting her if and only ifqMðm;wÞXvMðmÞ; and for a woman-w to agree to mate with a man-m uponmeeting if and only if qW ðm;wÞXvW ðwÞ:2 Therefore, we can regard vMðmÞ andvW ðwÞ as a man-m’s and a woman-w’s reservation utilities.

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2Here we assume that if agents are indifferent between getting married this period and remaining single

they choose to get married.


Given the agents’ expected utilities vMðmÞ and vW ðwÞ of remaining single, define aset Yðm; vW Þ of women who will agree to mate with a man-m upon meeting by

Y ðm; vW Þ ¼ fwAW j qW ðm;wÞ vW ðwÞX0g: ð3:1ÞSimilarly, define a set Xðw; vMÞ of men who agree to mate with a woman-w by

X ðw; vMÞ ¼ fmAM j qMðm;wÞ vMðmÞX0g: ð3:2ÞLet wY ðm;w; vW Þ and wX ðm;w; vMÞ; respectively, denote the indicator functions forsets Yðm; vW Þ and Xðw; vMÞ: Note the following monotonicity property of Yðm; vW Þand Xðw; vMÞ:

Y ðm; v0W ÞDY ðm; v00W Þ if v0W ðwÞXv00W ðwÞ for every w; ð3:3Þ

X ðw; v0MÞDXðw; v00MÞ if v0MðmÞXv00MðmÞ for every m: ð3:4ÞThis property shows that the reservation utilities vMðmÞ and vW ðwÞ of a man-m and awoman-w represent their selectiveness toward their mates. The higher reservationutility an agent has, the more selective about whom to mate he/she is.

For given reservation utilities v � ðvM ; vW Þ; we want to predict the types of womenwith which a particular man-m will complete mating, and the set of men a particularwoman-w will mate with upon meeting. To such an end, we can formally define amating correspondence m in the following way. Given v � ðvM ; vW Þ; define for each m

mðmÞ :¼ fwAW j qW ðm;wÞXvW ðwÞ and qMðm;wÞXvMðmÞg;if the set is nonempty, and mðmÞ :¼ fmg otherwise. Similarly, define for each w

mðwÞ :¼ fmAM j qW ðm;wÞXvW ðwÞ and qMðm;wÞXvMðmÞgif the set is nonempty, and mðwÞ :¼ fwg otherwise. That is, mðmÞ is the set of woman’stypes a man-m will and can mate upon meeting if mðmÞDW ; and mðmÞ ¼ fmgindicates that a man-m will stay single forever. Note that a mating correspondencehas the property that, for any mAM and wAW ; wAmðmÞ if and only if mAmðwÞ:

So far we have treated the agents’ reservation utilities as given and looked at theirdecision problems about with whom they agree to mate. Now we will determine eachagent’s reservation utility, or expected utility of remaining single, given all otheragents’ reservation utilities. Consider a man-m’s search problem. The expected utilityvMðmÞ of a man-m at the end of the period who is searching optimally for partners isdefined by the following Bellman’s equation:

vMðmÞ ¼ r dM

ZwY ðm;w; vW ÞmaxðqMðm;wÞ; vMðmÞÞ dlW ðwÞ

��

þ vMðmÞlW ðYCðm; vW ÞÞ�þ ð1 dMÞvMðmÞ

�; ð3:5Þ

where YCðm; vW Þ denotes the complement of Yðm; vW Þ with respect to W : Theequation says that the expected utility to a man-m from staying single this periodmust be equal to the right-hand side of the equation. With probability 1 dM a man-m will not meet any woman next period, which keeps him single and gives anexpected utility of vMðmÞ: With the remaining probability dM he meets a woman,


who is either one of those women who would like to mate with him, Yðm; vW Þ; orone of those who would not, YCðm; vW Þ: Recall that wY ðm;w; vW Þ is the indicatorfunction for Yðm; vW Þ: When he meets a woman who belongs to the former group,he will and can marry her if he wishes, i.e., if qMðm;wÞXvMðmÞ; or he will stay singleand get vMðmÞ if he does not want to marry her, i.e., if qMðm;wÞovMðmÞ: When hemeets a woman who would not want to marry him (which happens with probability

lW ðYCðm; vW ÞÞÞ; he has to stay single and get an expected utility of vMðmÞ whetherhe likes her or not. r is the agents’ common discount factor.

A woman-w’s search problem is treated in a similar way. The woman-w’sreservation utility, vW ðwÞ; solves the following Bellman’s equation:

vW ðwÞ ¼ r dW

ZwX ðm;w; vMÞmaxðqW ðm;wÞ; vW ðwÞÞ dlMðmÞ

��

þ vW ðwÞlMðXCðw; vMÞÞ�þ ð1 dW ÞvW ðwÞ

�; ð3:6Þ

where XCðw; vMÞ denotes the complement of X ðw; vMÞ with respect to M:

4. Market equilibrium

Before giving a definition of market equilibrium, let us fix some notation. LetLðMÞ denote the space of equivalence classes of measurable functions defined on

ðM;M; lMÞ with L1-norm jjxjjM ¼R

x dlM : Similarly, let LðWÞ denote the space ofequivalence classes of measurable functions defined on ðW ;W; lW Þ with L1-norm

jjyjjW ¼R

y dlW : Although an element in L is not a function but an equivalence class

of functions, we speak, as in the convention, interchangeably of some measurablefunction and the equivalence class of measurable functions containing it. Forx; x0ALðMÞ and y; y0ALðWÞ; we denote xXx0 if xðmÞXx0ðmÞ for a.e. m; and yXy0 ifyðwÞXy0ðwÞ for a.e. w: Let VM :¼ fxALðMÞ: 0pxðmÞpbg and VW :¼fyALðWÞ: 0pyðwÞpbg; where 0 and b are constant functions (or, the classes offunctions equivalent to them). Next consider a product space LðMÞ � LðWÞ with aproduct norm defined by jjðx; yÞjj ¼ maxðjjxjjM ; jjyjjW Þ: Let us consider a subset

V :¼ VM � VW of LðMÞ � LðWÞ:We define a partial ordering^M on V by v^Mv0 ifvMXv0M and vWpv0W ; where v � ðvM ; vW Þ; v0 � ðv0M ; v0W ÞAV :

We require in equilibrium that each agent searches for partners optimally giventhe reservation utilities of all the other types of men and woman. Precisely put,

Definition 4.1. A profile of reservation utilities for men and women, v� ¼ðv�MðmÞ; v�W ðwÞÞAV ; constitutes a market equilibrium if it satisfies (3.5) and (3.6)

along with (3.1) and (3.2).

Once equilibrium v� ¼ ðv�MðmÞ; v�W ðwÞÞ is computed we can get the set Yðm; v�W Þ ofwomen who agree to marry to a man-m in equilibrium from (3.1) and the setXðw; v�MÞ of men who would agree to marry a woman-w in equilibrium from (3.2).

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Moreover, for an equilibrium v; we can derive an associated mating correspondencem: We call such a m an equilibrium mating correspondence. Note that an equilibriummating correspondence m only tells which set of types of partners each type of agentswill and can marry upon meeting. Exactly which type of people an agent marriesdepends on the realization of stochastic events—random meetings. However, it ispossible to calculate the probabilities an agent of one sex will get married withparticular types of agents of the other sex, and the expected waiting time until gettingmarried for each type of agent.

The next proposition shows that the set of equilibria is nonempty and forms acomplete lattice, as an immediate implication of Tarski’s fixed point theorem [19]:Let /L;XS be a complete lattice, T be an increasing3 function from L to itself, andL� be the set of all fixed points of T : Then the set L� is nonempty and /L�;XS is acomplete lattice.

Proposition 4.2. Under Assumption 2.1, the set V � of equilibria is nonempty and

/V�;^MS forms a complete lattice.

Proof. We can think of a market equilibrium as a fixed point of a mapping T �ðT1;T2Þ; where T1 : V-VM and T2 : V-VW are defined by the right-hand sides of(3.5) and (3.6), respectively:

T1ðvM ; vW ÞðmÞ :¼ r dM

ZwY ðm;w; vW ÞmaxðqMðm;wÞ; vMðmÞÞ dlW ðwÞ

��

þ vMðmÞlW ðYCðm; vW ÞÞ�þ ð1 dMÞvMðmÞ

�:

T2ðvM ; vW ÞðwÞ :¼ r dW

ZwX ðm;w; vMÞmaxðqW ðm;wÞ; vW ðwÞÞ dlMðmÞ

��

þ vW ðwÞlMðXCðw; vMÞÞ�þ ð1 dW ÞvW ðwÞ

�:

We want to prove the proposition by applying Tarski’s fixed point theorem. First, weshow that T is increasing in v � ðvM ; vW Þ with respect to^M : Consider v0 ¼ ðv0M ; v0W Þand v00 ¼ ðv00M ; v00W ÞAV such that v0%Mv00 (i.e., v0Mpv00M and v0WXv00W ). Then we can see

by noting (3.3) that

T1ðv0M ; v0W ÞðmÞ ¼ r dM

ZwY ðm;w; v0W ÞmaxðqMðm;wÞ; v0MðmÞÞ dlW ðwÞ

��

þ v0MðmÞlW ðYCðm; v0W ÞÞ�þ ð1 dMÞv0MðmÞ

�;

p r dM

ZwY ðm;w; v00W ÞmaxðqMðm;wÞ; v00MðmÞÞ dlW ðwÞ

��

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3The term increasing is used in a weak sense throughout the paper.

H. Adachi / Journal of Economic Theory 113 (2003) 182–198 189

þ v00MðmÞlW ðYCðm; v00W ÞÞ�þ ð1 dMÞv00MðmÞ

�;

¼T1ðv00M ; v00W ÞðmÞ:

And similarly,

T2ðv0M ; v0W ÞðwÞXT2ðv00M ; v00W ÞðwÞ:

Therefore Tv0%MTv00:Under Assumption 2.1, T maps LðMÞ � LðWÞ into itself. We can also see that T

maps from V into itself by noticing that T1ð0; bÞðmÞX0;T1ðb; 0ÞðmÞpb;T2ð0; bÞðwÞpb and T2ðb; 0ÞðwÞX0: Note that /V ;^MS is a complete lattice. &

As an immediate corollary, we have

Corollary 4.3. There exists a greatest (with respect to ^M ) equilibrium %v � ð%vM ;%vW Þ

and a smallest (with respect to ^M ) equilibrium%v � ð

%vM ; %vW Þ: Each man likes and each

woman dislikes %v as much as any other equilibrium and the opposite is true for%v:

We can call such equilibria, %v and%v;M- and W -optimal equilibria, respectively, as

in Gale–Shapley marriage problems. In general, it is difficult to identify all equilibria.However, when sets M and W are both finite, we can easily locate the M- andW -optimal equilibria, %v and

%v: Suppose for the moment that sets M and W are both

finite.4 Then mapping T is continuous and the following iterative procedure willwork to locate the M- and W -optimal equilibria. To find the M-optimal equilibrium

%v; construct a sequence %vn � ð%vnM ;

%vn

W ) in V defined by %vn :¼ T %vn1 for nX1 and

%v0 :¼ ðb; 0Þ where b and 0 are constant functions. The limit of the sequence %v :¼lim %vn; is the M-optimal equilibrium. To obtain the W -optimal equilibrium, define a

similar sequence%vn � ð

%vn

M ; %vnW Þ starting with

%v0 :¼ ð0; bÞ: The limit of the sequence,

%v :¼ lim

%vn; is then the W -optimal equilibrium.

Generally speaking, M- and W -optimal equilibria, %v and%v; are different from each

other, or there are multiple equilibria. Because, %v^Mv0^M%v holds for any equilibrium

v0AV�; we can tell immediately that, if we have %v ¼M%v; equilibrium is unique. But it

is in general hard to see in advance if a certain parametric model has a uniqueequilibrium.

We can carry out comparative statics analysis about the effect of changes inmeeting probability d � ðdM ; dW Þ on v � ðvMðmÞ; vW ðwÞÞ: First, we fix a partialordering on the set of parameters. Let D � ½0; 1� � ½0; 1�: Define a partial ordering

XD on D by d00 � ðd00M ; d00W ÞXDd0 � ðd0M ; d0W Þ if d00MXd0M and d00Wpd0W : The next

proposition says that if it gets easier for men to meet women and more difficult forwomen to meet men, each man becomes more selective and each woman lessselective.

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4This is the assumption presumed in Section 5.2.


Proposition 4.4. Take two pairs of parameters d00 ¼ ðd00M ; d00W Þ and d0 ¼ ðd0M ; d0W ÞAD

such that d00M #l00MðMÞ ¼ d00W #l00W ðWÞ; d0M #l0MðMÞ ¼ d0W #l0W ðWÞ; l00MðGÞ ¼ l0MðGÞ for

each GAM; and l00W ðHÞ ¼ l0W ðHÞ for each HAW:5 Let%v00 ¼ ð

%v00M ; %v00W Þ and %v00 ¼

ð%v00M ;%v00W Þ; respectively, be the W- and M-optimal equilibria corresponding to d00: Define

%v0 ¼ ð

%v0M ; %v0W Þ and %v0 ¼ ð%v0M ;

%v0W Þ similarly. If d00XDd

0; then%v00^M

%v0 and %v00^M %v0:

Proof. The proposition follows from the fact that mapping T is increasing in d withrespect to XD: See [11, Theorem 1]. &

The proposition does not answer what would happen to vM and vW if meetingprobabilities for both men and women, dM and dW ; increase. The increases in bothdM and dW do not necessarily raise vM and vW for the following reason. Suppose anincrease in dM raises vMðmÞ for any m: Then men becomes more selective and agreeto mate with a smaller set of women. It will decrease the possibility of women beingaccepted by men and may decrease women’s expected utility vW even when dW

increases. By the same argument, we can see that even if it becomes easier for agentsto meet others, it does not necessarily increase the probability of agents actuallycompleting mating with someone and thus does not necessarily decrease the expectedperiods of staying single. This problem seems particularly relevant to labor-jobmatching models.

5. Negligible search costs and the connection to Gale–Shapley marriage problems

In this section we want to unveil a close, formal connection between cooperativegame theoretic marriage problems (CMPs) studied by Gale and Shapley and search-theoretic marriage problems (SMPs) we have considered: the set of equilibriummating correspondences in a given SMP with negligible search costs coincides withthe set of stable matchings in a corresponding CMP.

5.1. Digression on Gale–Shapley marriage problems

In this subsection, we give a brief introduction to Gale–Shapley marriageproblems following the exposition by Roth and Sotomayor [14], and a fixed-pointcharacterization of stable matchings for a Gale–Shapley marriage problem withstrict preferences, which will help us recognize the relationship between CMPs andSMPs.

A Gale–Shapley marriage problem is described as follows. In a marriage market,there are two finite and disjoint sets M and W : M ¼ fm1;m2;y;mng is the set ofmen, and W ¼ fw1;w2;y;wkg is the set of women. Each man has preferences over

ARTICLE IN PRESS

5Here l00M ; l0M ; l00W and l0W are probability measures constructed from measures #l00M ; #l0M ; #l00W and #l0W ;

respectively, as in the first paragraph in Section 2. Note that equilibrium conditions (3.5) and (3.6) depend

on probability measures lM and lW ; not directly on #lM and #lW :

H. Adachi / Journal of Economic Theory 113 (2003) 182–198 191

the women and each woman has preferences over the men. An agent may preferto remain single rather than get married to some agent of the opposite sex. Soa typical man m’s preference ordering 4m is represented by an ordered list overthe set W,fmg: We assume preferences are rational. We denote w4mw0 tomean m prefers w to w0; and wXmw0 to mean m likes w at least as well as w0: We alsowrite w ¼m w0 to mean man m is indifferent between mating w and w0; and writew ¼ w0 to mean w is the same person as w0: Woman w is acceptable to man m if helikes her at least as well as staying single, i.e., if wXmm: An individual is said to havestrict preferences if he or she is not indifferent between any two acceptablealternatives. We assume that every individual has strict preferences. This assumptionbrings a great simplification to the problem; we can use the indifference statementw ¼m w0 and the identity statement w ¼ w0 interchangeably (if w and w0 areacceptable to man m).

An outcome of the marriage market is described by a rule that matches an agent toan agent:

Definition 5.1. A matching m is a one-to-one correspondence from the set M,W

onto itself of order two (that is, m2ðxÞ ¼ x) such that if mðmÞam then mðmÞAW andif mðwÞaw then mðwÞAM:6

Definition 5.2. A matching m is stable if it satisfies(IR) mðmÞXmm; 8mAM and mðwÞXww; 8wAW ; i.e., a matching m is individually

rational.(S) )ðm;wÞ such that w4mmðmÞ and m4wmðwÞ; i.e., a matching m is not blocked by

any pair of a man and a woman.

Example 5.3 (Roth and Sotomayor [14, Example 2.4]). M ¼ fm1;m2;m3g and W ¼fw1;w2;w3g have the following preferences:

Pðm1Þ: w2;w�1;w3;m1 Pðw1Þ: m�

1;m3;m2;w1

Pðm2Þ: w1;w�3;w2;m2 Pðw2Þ: m�

3;m1;m2;w2

Pðm3Þ: w1;w�2;w3;m3 Pðw3Þ: m1;m3;m�

2;w3

Everyone has strict preferences and prefers marrying any one of the opposite sexrather than being single. Man m1; for instance, prefers w2 most, then w1; and soforth. Matching m0 indicated by the asterisks � is one of the two stable matchings.Observe that, say, the mate of m1 under m0; m0ðm1Þ ¼ w1; is the greatest element (withrespect to m1’s preferences) among the potential partners who like m1 at least as wellas their mates under m0; fwAW j m1Xwm0ðwÞg,fm1g: This property holds for everyagent. The observation is instrumental in proving that when preferences are strict theset of stable matchings can be identified with the set of fixed points of a mappingdefined by the right-hand side of (5.1) below.

ARTICLE IN PRESS

6We use the same Greek letter m for a matching in a Gale–Shapley marriage problem and an equilibrium

mating correspondence in a search theoretic marriage problem, as this will cause no confusion.


The rest of the subsection summarizes the fixed-point characterization of stablematchings for a Gale–Shapley marriage problem with strict preferences, which willbe useful later. See Adachi [1] for more details.

Definition 5.4. A pair of functions v � ðvM ; vW Þ is called a pre-matching if vM :M-W,M and vW : W-M,W such that if vMðmÞam then vMðmÞAW and ifvW ðwÞaw then vW ðwÞAM:

Let VM and VW denote the set of all such functions vM and vW ; respectively. LetV :¼ VM � VW denote the set of all pre-matchings v: We can think of VM as the setof vectors in �mAMðW,fmgÞ and of VW as that in �wAW ðM,fwgÞ: We can definethe following relationships between pre-matchings and matchings.

Definition 5.5. (i) For a given matching m; a function v � ðvM ; vW Þ defined byvMðmÞ :¼ mðmÞ and vW ðwÞ :¼ mðwÞ for all mAM and wAW is called a pre-matching v

defined by a matching m:(ii) We say a pre-matching v induces a matching m if a function m defined by

mðmÞ :¼ vMðmÞ and mðwÞ :¼ vW ðwÞ is a matching.(iii) A matching m and a pre-matching v are said to be equivalent if m defines v and v

induces m:Note that every matching defines an equivalent pre-matching while a pre-matching

may fail to induce a matching.Now consider the following set of equations:

vMðmÞ ¼ max4m

fwAW j mXwvW ðwÞg,fmg; 8mAM; ð5:1aÞ

vW ðwÞ ¼ max4w

fmAM j wXmvMðmÞg,fwg; 8wAW : ð5:1bÞ

In (5.1a) maximization is taken with respect to each man m’s preference ordering4m

over the set W,fmg under the constraint mXwvW ðwÞ: Since there are a finitenumber of agents and we have assumed strict preferences, the right-hand side of (5.1)is well defined and singleton for each m and each w:

Next lemma from Adachi [1] shows that in a Gale–Shapley marriage problem withstrict preferences the set of stable matchings is identified with the set of solutions to (5.1):

Lemma 5.6. Suppose every agent has strict preferences. Then

(i) If a matching m is stable, then the pre-matching v defined by m solves (5.1).(ii) If a pre-matching v solves (5.1), then v induces a matching m; which is stable.

5.2. Negligible search costs lead to stable matchings

In this subsection we want to prove that as search frictions disappear the set ofequilibrium mating correspondences in a given SMP reduces to the set of stablematchings in a corresponding CMP. To make SMPs comparable to CMPs, werestrict the set of agents’ types to be finite; M ¼ fm1;m2;y;mng and W ¼


fw1;w2;y;wkg: Let lMðmÞ and lW ðwÞ denote the probability measures over M andW ; respectively. Assume lMðmÞ40 and lW ðwÞ40 for every m and w:We say a CMPand a SMP correspond to each other if sets M and W in a CMP are identical to thosein a SMP, and if utility functions qMðm;wÞ and qW ðm;wÞ in a SMP preserve theordinal preferences over acceptable partners in a CMP. As before, the utility to eachagent of remaining single for a period is normalized to zero in a SMP. Note thatthere are many SMPs corresponding to a particular CMP while there is only oneCMP that corresponds to an SMP. In this section we confine our attention to CMPsand SMPs with strict preferences.

CMPs and SMPs differ in problems they deal with and approaches they take. InSMPs, we talk about anonymous agents with certain characteristics (characteristicsmatter, but not identities). Each agent chooses his/her actions given the rules of thegame, and we want to predict which types of partners a particular type of agent willmate upon meeting (a mating correspondence m assigns a set of types of partners toeach type of agent). In CMPs, we deal with agents with identities, and cooperativegame-theoretic concepts are used, and we wish to predict who mates with whom (amatching m assigns an individual to each agent).

Despite those interpretational differences between the two approaches, if we makea formal correspondence between them as in the above paragraph, we can show anintimate connection between the set of equilibrium mating correspondences in aSMP with negligible search costs and the set of stable matchings in a correspondingCMP; they are equivalent. This implies that when search costs are small enough anequilibrium mating correspondence m in a SMP reduces to a matching, a functionassigning exactly one type of agent to each type of agent. For an intuitiveexplanation, think of a SMP in which each agent is very patient. Because search costsdo not count much, a man do not have to compromise on whom to marry and canwait until his best partner appears. But he can only marry a woman if she also agrees,so the best he can do is to wait for a woman of the type he most prefers who acceptshim, and marry her. However, this is how one characterizes the set of stablematchings in a CMP with strict preferences using (5.1) in Lemma 5.6.

Proposition 5.7. Consider a SMP with strict preferences and a corresponding CMP.

As search costs become negligible ðr-1Þ; the set of equilibrium mating correspon-

dences m in a SMP coincides with the set of stable matchings in a corresponding CMP.

Proof. Consider a SMP described above. Let r40 be the rate of time preference

corresponding to discount factor r; r ¼ ð1þ rÞ1: One of the equilibrium conditions,(3.5), reduces to

vMðmÞ ¼ 1

1þ rdM

XwAY ðm;vW Þ

maxðqMðm;wÞ; vMðmÞÞlW ðwÞ

8<:

24

þ lW ðYCðm; vW ÞÞvMðmÞ

9=;þ ð1 dMÞvMðmÞ

35;


3 ðr þ dMÞvMðmÞ ¼ dM

XwAYðm;vW Þ

maxðqMðm;wÞ; vMðmÞÞlW ðwÞ

8<:

þ lW ðYCðm; vW ÞÞvMðmÞ

9=;;

3 rvMðmÞ ¼ dM

XwAY ðm;vW Þ

maxðqMðm;wÞ vMðmÞ; 0ÞlW ðwÞ: ð5:2Þ

The other equilibrium condition (3.6) works out similarly and is omitted. Then asr-0 (i.e., r-1Þ; the left-hand side of the last line of (5.2) approaches zero, and sodoes the right-hand side. Therefore it must be that

max maxwAYðm;vW Þ

ðqMðm;wÞ vMðmÞÞ; 0� �

-0;

3vMðmÞ max maxwAYðm;vW Þ

qMðm;wÞ; 0� �

-0:

That is, for any e40; there is a %rAð0; 1Þ such that, for all rAð %r; 1Þ; jvMðmÞ maxðmaxwAY ðm;vW Þ qMðm;wÞ; 0Þjoe; 8mAM: Then choose an e40 such that

eomin minmAM;w;w0AW ;waw0

jqMðm;wÞ qMðm;w0Þj;�

minm;m0AM;mam0;wAW

jqW ðm;wÞ qW ðm0;wÞj�:

Since sets M and W are finite and agents have strict preferences, such an e exists.Then

wAW vMðmÞ max maxwAY ðm;vW Þ

qMðm;wÞ; 0� ��

��oe��

� �

¼ wAW vMðmÞ ¼ max maxwAYðm;vW Þ

qMðm;wÞ; 0� ��

� �:

The equation in the last parentheses is nothing but a cardinal expression of (5.1a),provided that we represent agents preferences with utility functions qM and qW andwe normalize the utility of remaining single to zero. Recall that Lemma 5.6 tells thatthe set of stable matchings in a CMP can be identified to the solutions to (5.1). &

This equivalence result gives a noncooperative interpretation to the stabilitycondition used in the Gale–Shapley marriage problem. While we did not impose thestability condition per se in defining equilibrium in the search model, the equilibria inthe search model converge, as search frictions diminish, to the stable matchings inthe frictionless Gale–Shapley marriage problem.

Example 5.8. Consider a SMP that corresponds to Example 5.3. M ¼ fm1;m2;m3gand W ¼ fw1;w2;w3g: We represent utility functions qMðm;wÞ and qwðm;wÞ;


respectively, by the following 3� 3 matrices:

qM ¼2 3 1

3 1 2

3 2 1

0B@

1CA and qW ¼

3 1 2

2 1 3

3 1 2

0B@

1CA;

where in matrix qM the nth row represents man mn’s utilities over women and inmatrix qW the nth row represents woman wn’s utilities over men. The utility ofstaying single for a period is normalized to zero for each type of agent. Set meetingprobabilities to be dM ¼ dW ¼ 1: We consider two SMPs associated with two valuesof the discount factor, r ¼ 0:85 and 0.86. The two SMPs each correspond to theCMP in Example 5.3.

With r ¼ 0:85: The discount factor is small enough that the M- and W -optimalequilibria coincide with each other and thus equilibrium is unique. We represent theequilibrium v � ðvMðmÞ; vW ðwÞÞ by the following vectors; vM ¼ ð1:98; 1:31; 1:98Þ andvW ¼ ð1:98; 1:98; 0:65Þ (hereafter values are rounded). Let m be the associated matingcorrespondence. m can be represented by a 3� 3 matching configuration matrix a withvalues zeros and ones, defined as follows: for each ðm;wÞ; set aðm;wÞ :¼ 1 if wAmðmÞ(or, equivalently mAmðwÞ), and aðm;wÞ :¼ 0 otherwise. That is, m w element ofmatrix a is 1 if a man-m and a woman-w complete mating upon meeting, and it is 0otherwise. Then a looks like

a ¼1 1 0

0 0 1

1 1 0

0B@

1CA:

The first row, for instance, tells that mðm1Þ ¼ fw1;w2g: So the equilibrium matingcorrespondence is not single valued.

With r ¼ 0:86: For a discount factor as large as r ¼ 0:86; equilibrium is no longerunique. In the M-optimal equilibrium %v � ð%vMðmÞ;

%vW ðwÞÞ; %vM ¼ ð2:02; 1:34; 2:02Þ

and%vW ¼ ð1:34; 1:34; 0:67Þ: The associated mating correspondence %m is represented

by the following matching configuration matrix %a:

%a ¼0 1 0

0 0 1

1 0 0

0B@

1CA:

In the W -optimal equilibrium%v � ð

%vMðmÞ; %vW ðwÞÞ;

%vM ¼ ð1:34; 1:33; 1:34Þ and vW ¼

ð2:02; 2:02; 0:67Þ: The associated mating correspondence%m is expressed by the

matching configuration matrix%a;

%a ¼

1 0 0

0 0 1

0 1 0

0B@

1CA:

Note that %m and%m are indeed matchings in the sense of Gale–Shapley, and they are,

respectively, identical to the M- and W -optimal stable matchings in Example 5.3.


6. Discussion

This paper presents a simple search model of the two-sided matching problemunder the nontransferable utility assumption. The search model contrasts with theGale–Shapley marriage model in that agents search for partners by themselves in theabsence of a central matchmaker. The set of equilibria in a search model has a latticeproperty analogous to the set of stable matchings in a Gale–Shapley marriageproblem, exhibiting the polarization of interests inherent in two-sided matchingmarkets. Moreover, when search costs are negligible there is a certain equivalencebetween the set of equilibria in a search model and the set of stable matchings in acorresponding Gale–Shapley marriage problem. In this sense, our model relates thenoncooperative search/matching literature (e.g. [13]) to the cooperative two-sidedmatching literature (e.g. [9]).

Some of the limitations of this model are the following. First, by imposing thereplacement assumption that if agents get married they are replaced by their clones,we essentially assume exogenously given stationary distributions of agents’ types.Therefore, we cannot explain how we get such distributions in the first place. Second,we ignore the participation constraints of agents and assume that every agent stays inthe market even though he or she surely has no prospect of getting married. Thismay not be a big limitation in the context of marriage, but it is certainly troublesomein labor markets where participation decisions of agents are crucial. Third, agents’types are exogenously given and there is no room for agents to choose their owntypes. In labor markets, however, potential workers try hard to improve their skillsor choose their specialities given their preferences over jobs and the distribution ofexisting jobs. Fourth, we can give no prediction as to which of the equilibria is morelikely to emerge. Our model shows the possibility that one society can have differentmarriage market equilibria, but it cannot explain which of the equilibria is realizeddepending on the history of the society. See [4,17], which study transitional dynamicsto equilibrium in search-matching models.

Also, we have assumed that agents contact one another in a totally random wayand agents often end up wasting time by meeting undesirable partners. In a realmarriage market there often exist commercial services that make it easier for singlepeople to meet others. With such services agents could meet a desirable set of agentsexclusively or frequently and it might eliminate part of the inefficiency that arisesfrom the random meeting assumption. Bloch and Ryder [3] and Morgan [12] giveanalyses on such issues in their models and argue that an institution which segmentsthe market improves social welfare in their models. Further analysis along this line inour model would be interesting.

Acknowledgments

I thank two anonymous referees and an associate editor for helpful comments andsuggestions. This paper is based on chapter 3 of my dissertation submitted to


SUNY-Buffalo. I thank Professors Mitchell Harwitz, Peter Morgan, and Alex Anasfor their guidance. Of course, all remaining errors are mine.

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A search model of two-sided matching under nontransferable utility