Matching, search, and bargaining

JOURNAL

OF ECONOMIC

THEORY

42. 311-333

Matching,

(1987)

Search,

and Bargaining

ASHER WOLINSKY* Department

of

Economics,

Received

May

The Hebrew

University,

20. 1985; revised

May

Jerusalem,

Israel

5, 1986

First, the paper incorporates search for alternative opportunities into a model of strategic bargaining. Second. this search and bargaining model is embedded in a market matching model. The solutions are presented in a manner that clarifies the role of the search abilities and the details of the bargaining procedure in the determination of the outcomes. The predictions of the model are contrasted with the related literature. The main accomplishment is the attainment of a manageable model which is still rich enough to grasp important differences between environments which are not captured by the standard matching models. Journal o/ Economic Lirerature Classification Number: 022. ’ 19X7 Academic Press. Inc.

1. INTRODUCT10~ In many economic environments in which transactions are concluded in pairwise bargaining the process of bargaining is tied in with search for alternative opportunities. The purposes of this paper are to understand the interrelations between search and bargaining first in the context of an individual bargaining problem and then in the context of a marketmatching model. The first part of the paper (Sects. 2-4) is concerned with the individual search and bargaining problem. The model is based on the strategic bargaining model of Rubinstein [ 131. The added feature is that, during the bargaining process, the parties may also search for alternatives. The search is costly, and its intensity is a decision variable which can be varied over time. Thus, in this bargaining and search game the bargaining positions of the parties are determined endogenously by the parties’ decisions concerning the intensities with which they search and the acceptance criteria that they apply to alternatives. It is shown that, when the parties exchange offers sufficiently rapidly, this bargaining and search game has a unique perfect equilibrium. The equilibrium is characterized and the outcome is * Work on this paper started while the author was a Fellow Studies. 1 would like to thank Charles Wilson and Ariel discussions.

of the Institute for Advanced Rubinstein for very helpful

311 0022-0531/87

$3.00

CopyrIght ,r. 1987 by Academic Press. Inc All rights of reproduction 1” any form reserved

312

ASHER

WOLINSKY

presented in a manner that clarifies how the parties’ search capabilities affect the result. It is observed that the outcome of the bargaining does not depend only on the parties’ relative efficiency in uninterrupted search, but also on how aggressively each party can credibly threaten to search in the event that the agreement in the bargaining is delayed. The second part of the paper presents a market-matching model. The general framework is closely related to the models presented by Mortensen [ 10, 111, Diamond and Maskin [S] and Diamond [7], and is somewhat more distantly related to the work of Rosenthal [12]. The market under consideration is such that transactions are concluded in pairwise meetings between agents of two different types. The determination of the terms of a transaction is thus by nature a bilateral bargaining problem. The resolution of any such individual bargaining situation is affected by the substitution possibilities which depend on the parties’ search capabilities and on the expected behavior of alternative partners. The purpose of that part is to model explicitly the manner in which the market outcome is shaped by the processes of search and bargaining. This is done by embedding the search and bargaining model of Sections 2-4 in a market matching model. The gap that the present model attempts to fill is best understood in reference to the related literature. The works by Mortensen, Diamond, and Maskin cited above treat the basic bargaining component unsatisfactorily. They assume that the bargaining is concluded instantaneously and let the outcome be predicted by Nash’s bargaining solution relative to the disagreement point which is identified with the values attributed by the parties to the prospect of being unmatched. No attempt is made to justify the particular way in which the availability of alternatives affects the outcome of the bargaining. Rubinstein and Wolinsky [14] attempted to remedy this problem by adopting the strategic approach and describing explicitly the manner in which the alternative opportunities enter into the interaction between two parties. But their model did not capture the strategic role of the search activity because it did not treat the search intensity as a decision variable that can be controlled by the agents. This missing element of active search by the agents could play an important role in such markets, and one motivation for developing the present model is thus to incorporate it into the analysis. The market solution obtained for the model of the present paper is different from the corresponding solutions derived in the literature cited above. The qualitative differences owe to the fact that the present model recognizes the possibility that the parties can search during the process of bargaining and can adjust their intensity of search in accordance with their expectations about the eventual agreement. These differences could be deemed unimportant, if one is interested only in the observation that the agreements divide somehow the surplus between the parties. The differences

MATCHING,

SEARCH, AND BARGAINJNG

313

become important, however, if one is also interested in how the nature of the bargaining process and in particular how the manner in which the search activity is incorporated into that process affect the form of the agreement. For example, the previous work does not distinguish between situations in which, upon the formation of the match, the parties withdraw from the market and hence their search capabilities are reduced, and situations in which the search activity need not be interrupted by the formation of a match. These two types of situations presumably lit different markets in which the trade is organized differently, and one may expect that the different institutional arrangements will be reflected in the solution. Indeed, the analysis of the present paper predicts that in the latter scenario the relative search abilities of the parties will affect the market outcomes in a more pronounced way than in the former situation. In fact, it is argued that in the extreme case in which the parties do not search during the bargaining process (see Sect. 7) the market outcomes may be completely independent of the search capabilities, even if these are quite substantial. In a sense the solution obtained in this paper corrects a certain false impression that could have been derived from the Rubinstein-Wolinsky model. The equilibrium in their example coincided with the Mortensen-Diamond-Maskin solution and thus created the impression that the latter solution is supported by the strategic approach as well. However, the analysis of the present paper shows that this is true only for the special example in which the search intensity is not a decision variable and the quality of all matches is the same, which happens to be the example considered by Rubinstein and Wolinsky. Throughout the paper the analysis focuses on steady state situations in which the flow of agents who complete their transactions and leave the market is matched by an equal flow of new agents who join the market. For the most part the flows of arrival are treated as constant and exogenous. In Section 8 the model is extended in the spirit of Gale [9] to include the endogenous determination of flows of entry into the model. This formulation clarifies how the precise nature of the agreement mechanism together with the structure of the arrival flows determine the steady state composition of the market in terms of the stocks of the two types of agents. Given the nature of the agreements, these stocks have to adjust to particular levels SO as to equilibrate the “flow market” in which the entry rates are determined.

2. THE TWO-PERSON

BARGAINING

AND SEARCH GAME

The game is played by two parties 1 and 2 who bargain over the partition of a sum of size m. The bargaining process takes place over time; the

314

ASHER

WOLINSKY

discrete time periods are of length A and will be denoted by t, t = 0, 1, 2, 3 ,.... In each period one of the parties is selected randomly (with probability + and independently of previous selections) to propose’ a partition of the sum, and the other party responds immediately by accepting the offer or rejecting it. If the offer is accepted, it is implemented and the game ends. The Outside Opportunities The parties can also search for outside opportunities. Upon encountering an outside opportunity a party finds out about its value and has to decide immediately whether or not to adopt it. Adoption of such an opportunity ends the process in the sense that the party withdraws from the bargaining and does not search for further opportunities (say, a party can engage only in one project). The details of the search process are as follows. With probability s:A party i will encounter an outside opportunity during period t, where s: is interpreted as the search intensity employed by party i in that period. Let c,(s)A be the cost to party i of employing intensity s in a given period and assume that c,( ) is increasing, twice differentiable, with c,(O) = 0, c:(O) = 0, c;‘(s) > 0 and bounded away from zero for all s > 0. The (monetary) values of the outside opportunities for party i are realizations of i.i.d. random variables with distribution function G;(. ) which is continuously differentiable and has its support on [0, M,], where M, 3 M. The Order qf Events Suppose that the process has lasted up to period t (i.e., by t the parties have not yet reached an agreement or adopted outside opportunities). At the beginning of period t the parties enter the bargaining stage of that period: a chance move determines the identity of the proposer (the probability for any party to be selected is 4) who then makes an offer to which the other party responds with acceptance or rejection. Acceptance terminates the bargaining. Upon rejection the parties proceed into the search stage of period t. In this stage the parties choose first their search intensities, s;, and incur the costs c,(sf)A, and then they may encounter their outside opportunities and consider their adoption. If both parties do not adopt their outside opportunities they will proceed into the bargaining stage of period t + 1. I The particular specification of the procedure employed here is not essential for the analysis. In fact, a large variety of alternative procedures which are stationary and essentially symmetric will lead to the same results. It is convenient to use the present specification since its complete symmetry simplifies the analysis.

MATCHING,

SEARCH,

AND

BARGAINING

315

Outcomes and Preferences The participation of a party in the process ends at some period t after the party has obtained a sum .X (either by reaching an agreement or by adopting an outside opportunity) and has incurred a stream of search costs. The parties are assumed maximizers of expected utility. Party ?s utility from a stream (x,, X, ,..., .X,,...) is given by C,Z?~ 6:s,, where 6, are discount factors. For later reference we shall write 6, = e-‘I’, where r, is party is instantaneous rate of time preference. hformation,

Strategies, and Equilibrium

The distribution function G;, the cost functions ci and the preferences as captured by 6, are common knowledge. The parties, however, do not observe the actual search history of their rivals (i.e., they do not know the search intensities chosen by the rival and the timing and values of the outside opportunities he has encountered). At each time period t there are four consecutive instances at which a party might have to make a decision. The different decisions are: (1) What proposal to make (if the bargaining was resumed and the party was selected to propose); (2) Whether or not to accept the rival’s proposal (if the rival was selected to propose): (3) which search intensity, s;. to employ (if no agreement was reached); (4) whether or not to adopt an outside opportunity (if such has just been encountered). A strategy for party i is a sequence of decision rules which describe the party’s behavior at each time period and each of the above decision points, conditional on the party’s history up to that point. The equilibrium concept adopted here is Perfect Equilibrium. This is a pair of strategies-one for party 1 and one for party 2-such that the strategy of a party is the best response to the other party’s strategy after any possible history of the game.

3.

THE

EQUILIBRIUM

ANALYSIS

This section presents and analyzes the equilibrium of the bargaining game studied here. First, let us ignore the bargaining process and consider the search component of the model in isolation. Suppose that agent i looks only for outside opportunities employing a search policy characterized by a constant search intensity s per time period and a constant reservation value x (i.e., the agent adopts the first opportunity whose value exceeds x). Let Vi(s, x) denote the present discounted value of this search policy, and let Qi( v) = 1 - G,( 4’) (note that dGj( y) = - dQi( y)).

316

ASHERWOLINSKY

Vi(S, x) = Si[ 1 -s dQ,(.u))]

-sA s

V,(S, X)

M’ y dQi( y) - c;(s) A. .x

(1)

That is,

V;(s,x) = -sA JM,Y dQi(~)-ci(s) A 1 -S,[l

--s de,(x)]

’

(2)

Let F,, and S;, and 2, be such that V, = V,(S,, 2;) = Max P’;(s, x). .A.Y

(3)

Note that by a standard result in search theory Pi is the overall optimum value of this search problem and not only the optimum over the constant reservation value policies considered here. Thus, in the above bargaining model Vi is the value for party i of remaining without a partner. Next, let us turn to the full problem which involves both bargaining and search. The main result concerning the bargaining and search game is stated by PROPOSITION 1. (i) If V, + V2 > m the unique equilibrium is such that the parties do not attempt to reach an agreement and just search optimally among their respective outside opportunities.

(ii) If P, + V2 < m there exists u perfect equilibrium in which the parties reach an agreement. The equilibrium strategies are characterized by silt numbers,~,,,cz,s,,.~2,.~,,.~2suchthat P, , i responds,he agrees only to offers +ilhichgive him at leust m - N’~, k # i; between consecutive bargaining sessionsparty i searches brith intensity si and adopts an outside opportunity I# its value exceeds ._ . a party who remains without a partner searches optimally with si, .Y;.

(iii) If P, + Pz < m and tf the length of single period, A, is sufficiently small, then the perfect equilibrium described above is unique. Proof: The skeleton of the proof is brought below, The more technical points are proved in the Appendix.

MATCHING,

SEARCH,

AND

317

BARGAINING

(i) Obviously, party i will never accept an agreement that gives him less than 8,, which is the expected payoff of disagreeing and searching optimally for an outside opportunity. When F, + Vr > m there is no agreement that gives both parties more than P, and therefore the parties just search optimally for outside opportunities. (ii) The key to the result lies in the following system of equations in the six variables \v,, br2, s,, s2, x,, s2. 111- )\‘A = [I-.s,p,(.~,)d][l-.s~~2(.1.~)n]~s,~(l~~,+t??-l~~,)

.tt,

1 +S,s,Q,(.u,) A[l-.s,Q,(s,,A] r'j,,i#k= I,2. +

[

-s,

j tr

s c/Q,(x) - (.,(s,) A

(4)

-P;(.Y;)lI I - .sxQJsr) A] s, 4 (H’,+ 111 - )(‘A) - I (” s dQ,(x) - c:(.s,) \I -S,Q,(.u,) s;=[l

sk Qk(sr

) A r, = 0,

i#k=

1, 2.

-s/,Q~(.~/.)A]6,f(~l~,+ttt-,1~,)+6,.s,Q,(.~,)A~,,

(5) i#k=

I, 2. (6)

The two equations (4) are the basic equations of the bargaining and will be made clear shortly. Equations (5) and (6) are the first order conditions for the maximization of the r.h.s. of (4) with respect to s, and s,. It can be verified that if H’,, N’,, s,, s,, s,, .Y? is a solution for system (4)-(6) satisfying V, d m - cr2 6 )r, d m - Vz, then the strategies described in the proposition constitute a perfect equilibrium. To see this note that since the strategy of party k is stationary, the best response of party i is stationary’ as well and hence can be only one of the following three: (a) always demand MI,, accept no - )I’~ and search optimally; (b) always demand II’,, reject HZ- N’~ and search optimally: (cl always demand more than \I‘,, reject m - u‘L and optimally.

search

(Any other possible stationary rule is obviously inferior to one of the above three.) A routine calculation shows that the expected values of rules (a) and (b) for party i are the same and equal to f( IIT, + m - ~7~). Also, the expected value of (c) is v, which is smaller than 4 (1~; + tit - NT/,).Therefore, strategy (a) is a best response after any history, and the choice of strategies ’ When the strategy of party X is stationary. problem. It is well known that for such a problem Derman 161.

party i faces a stationary there exists a stationary

Markov decision optimal policy. see

318

ASHER

WOLINSKY

type (a) by both parties is a perfect equilibrium. Finally, to see that s, and .Y; are indeed the optimum search components associated with these equilibrium strategies, observe that the r.h.s. of (4) captures the expected payoff to party i in the subgame that starts immediately after a proposal was rejected. Equations (5) and (6) then establish that, given )v,, LC~, So and .Y~, the choice of s, and s, maximizes this expected payoff. Finally, by direct application of Brouwer’s fixed point theorem we have that if 8, + vz < M, then the system (4))(6) has a solution such that v, < 1)~- M’~ < bri d trr- Vx. This observation together with the preceding paragraph imply that if V, + V2
The rest of the proof deals with uniqueness.

Cluim. There exists 2 > 0, such that for any d > d, the equilibrium is unique. (The proof is in the Appendix.) Since by the preceding discussion there always exists an equilibrium type described in the proposition, it follows from the claim that, for this equilibrium is the unique perfect equilibrium.

perfect of the J < d’, Q.E.D.

Thus, when 8, + Vz < ~1 and d is sufficiently small (d < J), the unique perfect equilibrium is such that the bargaining is concluded immediately with the agreement (hi,, , nz - \v, ) or (m - \I’?, IVY) according to whether party 1 or 2 is the first to propose. Note that the requirement that A is sufficiently small is not needed for proving the existence of equilibrium. It is used to prove the uniqueness, but we do not know whether it is necessary for that either. To see briefly the role of this assumption recall that in bargaining games of the alternating offers variety the perfect equilibrium is usually generically unique. In the present model, however, the possibility of search may spoil the uniqueness. The choice of the equilibrium intensity s, is affected both by the expected agreement and directly by the intensity sk that the other party is expected to adopt. If the latter effect is sufficiently strong there can conceivably exist multiple equilibria such that in one equilibrium both s, and s2 are higher than in another equilibrium. However, when d is made sufficiently small the direct effect of s, on So becomes sufficiently weak and this removes the possibility of multiple equilibria.

4. THE SOLUTION The equilibrium agreements M’, depend of course on A, wyi = M,~(A). In many settings it is reasonable to assume that the length of a single bargaining period, A, is small. We shall therefore let A recede to zero and view the limiting equilibrium outcome as the solution for the considered bargaining

MATCHING,

SEARCH,

AND

319

BARGAINING

situation. Since for small values of d the equilibrium is unique, the sequence of outcomes is well defined. Upon solving system (4))(6), taking the limits and letting u’, denote lim d -0 w,(d) we get r2 +sIQI(wl)+.~2Q2(wJ IV, = ‘ne-1(‘2=mr,

+r,

+2s,Q,(w,)+2s2Q,(~v2)

- Cr2+~~Q~(w)l V,b,, tt:2)-s,Q1(~~,)8,] ' rI +r2 +2s,Q,(w,)+2.s2Q2(~',)

+ where s, is given by

-

(8)

(s - w,) de;(x) = c.‘(s,)

and V,(s, ~1) is the limiting value of (2), i.e., V,(s, ~7)= lim Vi(s, 111) (9)

Despite its complicated appearance formula (7) is quite intuitive. To see this let us rewrite (7) as ~3; = d, + cx,(m - d, - d?),

i=

1, 2.

where rA +s,Q,(~‘,)+.s~Q~(,l,~)

That is, uli is the asymmetric Nash bargaining solution with bargaining powers CI, and CI~applied to the problem of dividing the sum r~z,given the disagreement point (d,, d,). Note that the bargaining power, x,, is larger the more patient is party i relative to the other party (i.e., the smaller is r, relative to rk). Note further that dj is a weighted sum of the present value assigned by i to search during the bargaining, V,(s,, MI,), and the present value to i of remaining without a partner, P,. The comparative statics implications of the above are straightforward: a party’s bargaining position and hence his share are greater the more patient and the more efficient searcher the party is and the better are the alternative opportunities he faces. (For further discussion of this point see Sect. 7; for a general discussion of the relations between the Nash Solution and the strategic solution see Binmore [2], Binmore, Rubinstein and Wolinsky [4]).

320

ASHER WoLINSKY 5. A MATCHING

AND BARGAINING

MODEL

The above is a partial equilibrium analysis in the sense that the values of the alternative opportunities faced by the parties are exogeneous. However, in the cases for which this model might be relevant an alternative opportunity is often an alternative match whose value to the party is also determined in bargaining. It is therefore desirable to extend the model in a way that recognizes this fact. Our next task is then to embed the above described bargaining and search a model in a wider market model. The model is based on the familiar matching models considered by Mortensen, Diamond and Maskin (see, e.g., Mortensen [ 10, 111, Diamond and Maskin [8] and Diamond [7]). There are large numbers of agents of two different types denoted by 1 and 2. Associated with a match of two agents of opposite types there is surplus to be divided between them. Ex ante all unmatched pairs have identical expected surplus. However, when two agents meet they observe immediately a statistic m E [0, M] which is interpreted as the “quality” or “value” of the match. The magnitude of m is viewed as a random draw from a distribution F(. ), assumed continuously differentiable. Unmatched agents search for an appropriate match. Matched agents may either consummate their match and divide the associated surplus, or reject the match and continue to search. When a match is consummated the parties leave the market. The search technology is as described in Section 2 except that now the probability that an agent has of meeting a partner does not depend only on his search intensity, but also on the search activity of other agents and on their numbers. The meeting probability that an agent of type i has, at a given period, will be described by (s, + sj)) A, where si is the search intensity employed by the agent at that period, and ,Y: captures the effect of the search activity of the agents of the opposite type at that period. As before, the intensity s, involves a cost c;(s,). A particular interpretation of the above is that si A is the probability that an agent of type i will initiate a contact with an agent of the opposite type, while .$’ A is the probability that the agent will be contacted. This implies a particular relation between .$’ and the search activity of agents of the opposite type. If there are Ni agents of type i= 1, 2 and if all agents of type k #i search with intensity sk, then the total number of meetings initiated by them is N,s, A and hence

Nk sp=-s~.. N,

(13)

In the related literature the assumptions concerning the search together with (13) is referred to as linear meeting technology. Consider the numbers of agents of each type who are present at the

MATCHING,

SEARCH.

AND

321

BARGAINING

market. The evolution of these numbers depends on the endogenous rate at which matches are consummated and agents depart, on the flows of arrival of new agents and on the initial numbers. In order to concentrate on the issues that interest us here it is convenient to assume that the flows of arrival are exogeneous, constant over time and equal for the two types: at each period there are b new agents of each type. Furthermore, the analysis will focus on steady state situations in which the numbers of agents are constant over time. That is, the steady state numbers are such that the endogenous rate at which matches form is equal to the exogeneous rate of arrival. The MDM

Solution

The solution suggested by Mortensen, Diamond, and Maskin in the above cited articles (the MDM solution) consists of an agreement schedule (w,(Hz), M’~(Hz)) that specifies the division of the surplus of a match as a function of its quality m and the other parameters. This solution’s requirements are: (i) each agent’s search strategy is optimal, given M.,( .f and the search strategies of the other agents; (ii) each agreement (w,(m), Wan) is a solution to the bilateral bargaining problem faced by the matched agents; (iii) the system is at a steady state. The idea is that (w,(m), u,,(m)) is determined in bargaining between the matched parties. The main version of the solution treats the division of the surplus between matched parties as a static bargaining problem and requires ( M’,(I)z), M’?(~z)) to be Nash’s solution relative to the disagreement outcome of continued search. This solution is derived as follows. Given an increasing schedule w,(m) and a constant sr, let V,(.F,, s; sY) denote the expected value for i of a search policy characterized by constant search intensity s, and constant reservation value .Y (i.e., the agent continues to search until he finds a match whose quality is s or more). Let Q(y) 5 1 - F( JJ) and observe that VJs,, x; $7) = -(s,

+ ~7, A jM wi(y) I;

+ Si[ 1 - Q(x)(s;

and that at the limit,

de(y)

- c;(s,) A

+ s:‘) A] Vi(s,, .u; sf),

(14)

as A approaches zero, we have

ri V,(s;, x; ~7) = -(s, -Y,(si,

+ $) j”” [wi( y) Y x: sp)] dQ(y)-c;(s,).

(15)

322

ASHER WOLINSKY

Given the rates of arrival b and the initial numbers of agents NY and NY the solution is characterized by a schedule (I,, am), intensities S, and S, and numbers N, and N2 that satisfy P, = V,(S,, fi; sp) = max Vi(si, ~73;sp), ,,.m w,(m)= P, +; (m- P, - P2), .$ = SkNk/N,,

i#k=

i= 1, 2,

j=l ) I) 3

1, 2,

(16) (17) (18)

N,(S, + sp) Q(g) = h,

(19)

N, -N,

(20)

=N’:-N’;,

The meaning of a solution is that the system is in a steady state in which all unmatched agents of type i search with the constant intensity S, and all matches of quality m >ti = V, + P, are concluded with an agreement according to (w,(m), M’?(W)). Conditions (19) and (20) refer to the steady state numbers of agents: (19) states that the flow of departures as captured by the 1.h.s. is equal to the exogenous flow of arrival; (20) requires that the steady state numbers are consistent with the initial conditions. Conditions (16) and (17) refer to the bargaining and search: (16) requires that each agent’s search strategy as characterized by Si and 8, is optimal, given w,(. ) and s?; (17) establishes (w,(m), wl(m)) as the Axiomatic Nash solution relative to the disagreement point (V,, P,) which captures the parties’ alternative opportunities of continued search. Finally, (18) requires that .sp is consistent with the search intensity of agents of the opposite type. Mortensen considered the symmetric case in which NY = fl and showed the existence of a unique solution (M’,(. ), H’~(. )), S, , SZfor the above system (see, Mortensen [ 111).

6. A SOLUTION BASED ON STRATEGIC BARGAINING

The bargaining component of the above model is treated somewhat mechanically: there is no sound justification for the manner in which the alternative opportunities and the search activities should affect the outcome. The purpose of this section is to present an alternative solution in which the bilateral bargaining component and the manner in which the search affects the bargaining are modelled explicitly using the strategic approach presented above. Specifically, condition (17) in the above solution is replaced by a condition that expresses (w,(m), wl(m)) as the limiting (as A -+ 0) perfect equilibrium outcome in a bargaining and search game between matched parties the quality of whose match is m. Consider an agreement schedule (u’,(m), \c*?(m)) such that w';(m) is

MATCHING,

SEARCH,

AND

323

BARGAINING

increasing, and suppose that in all matches that are concluded with an agreement the surplus is divided according to this schedule. Suppose now that in a particular match of quality m the parties did not settle for the agreement prescribed by the schedule and instead started a process of alternating offers, responses and search for alternative matches as described in Section 2. Assume that both parties expect that their payoffs in alternative matches are described by the schedule (w,(m), M’,(M)), and that upon meeting an alternative partner a party has to decide immediately whether to withdraw from the process and enter the new match or to reject it and continue with the ongoing bargaining process (i.e., an agent cannot negotiate with two or more partners at the same time). Note that this bargaining game is the one described in Section 2 with the special features: (i) the outside alternatives are identified here with the opportunities of meeting alternative partners and their distributions (denoted in Sect. 2 by G,) are derived here from the schedules w;(~I) and the distribution function F; (ii) the probability of encountering an alternative includes here an autonomous component $. Thus, it follows from Proposition 1 that when the length of a single period, A, is sufficiently small, this game has a unique perfect equilibrium. This equilibrium is characterized by w,* = we, s,* = s*(m), x* = s,*(m), i= 13 2 >where M,,* is the sum demanded by party i whenever it is his turn to make an offer; .Y,* .1s the intensity with which party i plans to search between consecutive bargaining sessions; s,* is party is reservation quality for acceptance of alternative matches, i.e., if party i encounters a match of quality greater than s,* = -u,?(m) he will withdraw from the bargaining and enter this new match. The expected equilibrium payoff to party i is $ (w,? + WI- M.: ). The values of M’*, s,*, x,* are the unique solution for the following system which is the version of system (4)-(6) that corresponds to the present case.

xQ(x:)d[l

-(s:+.~~)Q(x,f+)A]

P,,

i#k=

1,2, (21)

-Q(,~~,[l-(s~+~~)Q(~~)d]~~~(~~',*-trn--~,*) -i’:‘u’;(.x) dQ(.x)-c;(s:)-h,Q(x,?)(s; yi = 0, i#k= 1, 2.

+s;)

Q(xf)

AP, (22)

324

ASHER

WOLINSKY

UJ;(.X*)=[~-((~;+~;)Q(x*)A]s;+*+rn-w*) +S;(sk*+s~)AB;,

i#k=

1,2.

(23)

The values of F, and sp, i = 1, 2, are given by (15), (16), and ( 18) and of course depend on the postulated schedule (w,(m), am). Upon solving system (21)-(23) and letting A recede to zero we get lim, + o w,+= lim(nz - bt’t), and this common value is expressed by the version of formula (7) which suits the case at hand. Consider now a schedule (w,(m), wJm)) such that, for each m, w;(m) is the limiting (as A -+ 0) perfect equilibrium payoff in the sense described above. If such a schedule exists, then the counterpart of formula (7) must hold for all m. That is. w;(m) = m

~2 + (s, +s’f+s2 +s:) Q(m) rI + rI + 2(s7 + s, + s2 + s:) Q(m)

+ [r, + (~1 + .$) Q(m)1 V,(s,, m; ~7)+ (s2 + si) Q(m) V, rI + r2 + 2($‘+

s, + s2 + s$) Q(m)

Crz+ ($2 + .$) Q(m)1 V,(s,, m; s!j) + (s, +sy) Q(m) Vr ’ r1 +r, +2(sy+s, +s, +sy)Q(m)

(24)

where s, is given by - j” [wi(x) n,

- wj(m)]

dQ(x) = c,!(s,)

(25)

and where V,, Vi and S,, sp are given by ( 15), ( 16) and ( 18) respectively. PROPOSITION 2. There exists a schedule (w,(x), wz(x)) satisfying (24), (16), and (18))(20). (The proposition is prooed in the Appendix using a contraction map theorem presented by BlackweN[S].)

Thus, the solution suggested here replaces the agreement schedule (17) by the agreement schedule (24) which is based on the limiting perfect equilibrium in the strategic bargaining game. The behaviour described by this solution is as before: an unmatched agent of type i searches with intensity Si that maximizes the agent’s expected benefit, given the agreement schedule (24) and the search activity of his potential partners as captured by .$‘; matches of quality m 3 ti = P, +. p> are consummated and the surplus is divided according to the agreement schedule (24); matches of lesser quality are not consummated and the parties continue to search. Note that the proposition does not establish the uniqueness of the

MATCHING,

SEARCH,

AND

325

BARGAINING

market solution. My conjecture is that the solution is unique, but I have been able to prove it only for the symmetric case in which, NY = N: (see Wolinsky [ 163). 7.

DISCUSSION

This section discusses the strategic solution presented in the previous section. We attempt to clarify the distinction between the strategic solution and the MDM-solution by looking more closely at the forces that determine the outcome of the bilateral bargaining. We also consider the consequences of changing one of the model’s basic assumptions concerning the relations between the bargaining and matching processes. The Relations with the MDA4 Solution

To clarify the differences between our solution and the MDM solution, observe that the agreement schedule (24) can also be described in terms of Nash’s bargaining solution. For Y, = r2 = r we can rewrite (24) as Wi(rn) = d,(m) + ; [m - d;(m) -d,(m)],

if/k=

1, 2,

(26)

That is, (w,(m), M’~(Hz)) is Nash’s bargaining solution for the problem of dividing the sum m, given the disagreement point (d,, dz). (This observation and the following discussion are related to the points made by Binmore [3], Binmore, Rubinstein and Wolinsky [4] concerning the relations between the axiomatic and strategic approaches to bargaining.) Note that, in contrast to the MDM solution, the disagreement poing d; = d;(m) does not coincide with the expected value of remaining without a partner, Pi. Rather, di is a weighted sum of V, and V;(s,, ty1:sj’). Since for all m> 7, + Vz we have Yj(sj, m; $)< Y,, it follows that di < Vi (with equality holding only for ,n = ti = V, + Pz). To understand why the disagreement point is not identified with Pi, note that, in the underlying dynamic process, the value of being in a state of perpetual disagreement is not the same as the value of being without a partner. This is because matched agents who have not yet reached an agreement do not follow the same search policy as do unmatched agents: they search with intensity si -CSi and accept an alternative match only if its quality exceeds n? > nl. The reason is that they still expect to reach the agreement (w,(m), wz(m)) with their current partner. Therefore, the value for agent i to be matched and in a state of disagreement is composed of the value of searching while in state of disagreement, Vi(s,, nz; $), and the value of

326

ASHER

WOLINSKY

remaining without a partner, V,, weighted by the probability. (sk + $) Q(m), that the partner will indeed find a better match. Note that the term dj captures exactly this value of being in a state of disagreement. Now, when two agents are matched we expect that the outcome of their bilateral bargaining will depend on what their credible threats are. If P, + Vr
on the Relations between the Bargaining

and the Matching

So far it was assumed that the parties to a match could search for alternative partners while in the process of bargaining. Let us now consider a situation in which matched parties do not participate in the matching process while they bargain. That is, upon meeting and realizing the quality of their match the parties remove themselves form the market and start bargaining. While bargaining the parties do not meet alternative partners, but each party can always decide to end this bargaining process in which case both parties return to the matching process and look for new matches.

MATCHING,

SEARCH,

AND

327

BARGAINING

One reason for considering this variation is that it clarifies how the solution depends on the manner in which the bargaining and mathcing processes are interlaced. The analysis of this variation further differentiates the predictions of the present model and those of the MDM solution, which in not equipped to distinguish between the scenario considered throughout and the one considered here. Let (IV,(m), W,(nz)) be the agreement schedule in this case and let Vi be defined by (16) with respect to this schedule (i.e., V, is the expected utility to party i from being unmatched). As before, matches of quality m < v, + v2 are not consummated and the parties to such a match separate immediately and continue their search. For m 2 V, + 8, we have

W,(m)

= 117 -

WJtn)

=

.; n1 -

r2

vz

1 -r, +r, 111

if

V2> - rl

if

V,
r, +r2 t-3

r1 +r,

112

(28)

112,v, < - 1’1 n1. r, +r,

That is, in this case the solution is given by (16) (28), and (18)-(20). The explanation for this solution uses the “outside option” argument pointed out by Binmore [3]. Note that IV,(m) = (r?/(r, + r,))m is the limiting (as A -+ 0) perfect equilibrium outcome for the bargaining game between two agents who have to divide the sum m and who possess no “outside options” such as the possibility of meeting alternative partners. In the present case the parties have the “outside option” of withdrawing from the bargaining and returning to the matching process. However, if the value of this “outside option,” Vi, is smaller than (rk/(r, + r?))m for both i= 1 and i= 2, then the availability of the outside option does not affect the equilibrium outcome of the particular bargaining game. This is because the “threat” by any one party to withdraw and realize his outside option is not credible, since the outside option is inferior to the expected equilibrium agreement. The availability of the outside option becomes important when some party prefers it to the outcome that would obtain in its absence, i.e., for some i, Pi > (rk/(r, + r,))m. In such a case the perfect equilibrium outcome is a “corner solution” which gives that party the sum 8,. Finaily, note that in this case as well the strategic solution as captured by (28) does not coincide with the MDM-solution. The difference between the model of this subsection and the RubinsteinWolinsky model that supports the MDM-solution is that in the Rubinstein-Wolinsky model matched agents who are in a state of disagreement continue to meet alternative partners just the same as unmatched agents.

328

ASHER WOLINSKY 8. ENDOGENEOUS

ENTRY DECISIONS

Throughout we modelled the arrivals of new agents as taking place at an exogeneous and constant rate. Of course, in many cases for which this type of analysis is relevant the entry is a matter of a decision and is affected by the market conditions. In the above we chose to neglect this part of the modeling in order to obtain a relatively simple model that will enable us to focus on the combination of search and bargaining. Obviously, however, this model can be embedded in a wider model in which the rates of entry are determined endogenously. Such an extension of a pairwise meetings model was recently presented by Gale [9]. In what follows we shall explain how the model of the present paper can be extended in that spirit. Suppose that in each period new agents of types 1 and 2 arrive on the scene. Upon arrival these agents have to decide once and for all whether to enter the above described pairwise meetings market or to pursue an alternative activity. The assumption is that these two choices are mutually exclusive and an agent cannot pursue both. An agent who enters the pairwise meetings market will search around and eventually transact in the manner described throughout the paper. The agents differ with regard to the value they attribute to the alternative activity.3 Let L,(V) denote the number of new agents for whom the value of pursuing the alternative activity is less than or equal to V. It is assumed that Li( V) is constant over time, continuous and strictly increasing with L,(O) = 0. Clearly, a new agent of type i chooses to enter the market if and only if the expected value of so doing, V,, is greater than the value he attributes to the other activity. Thus, the rates of entry into the pairwise meetings model are L,( V,), i = 1, 2. The steady state solution of the extended model consists of a schedule (W,(M), use), intensities S,, reservation levels Vi, numbers N, and arrival rate h that satisfy (16)-( 19) and in addition L,(P,)=Lz(V,)=b.

(29)

Condition (29) describes what Gale calls the “flow market” equilibrium. Formally, condition (29) replaces condition (20). This means that the difference N, -N, is not arbitrary anymore. Rather, the stocks N, and N2 have to adjust so that the corresponding P, and B, will equilibrate the “flow market” as required by (29). If, for example, the shapes of L,(V) and

3 The heterogeneity of agents in the present case paper. Here the agents are alike within the considered to their outside opportunities. In Gale’s model agents values after they have entered the market as well. For the purposes of Gale’s analysis) the present approach

is modelled differently than in Gale’s market and they differ only with respect differ with respect to their reservation our purposes (and it seems that even for facilitates a much simpler analysis.

MATCHING,

SEARCH,

AND

329

BARGAINING

L,(V) dictate that for (29) to be satisfied V, has to be relatively low, then the difference N, -N, will have to be sufficiently high to depress VI to the required level. APPENDIX

Proof sf the Claim

The main task is to prove that the system (4)-(6) has a unique solution. This is accomplished by the following Assertions l-3. Once this is proven it is possible to use quite directly an idea that appears in Shaked and Sutton [ 151 to prove the uniqueness of the perfect equilibrium. 1. There exists A0 > 0 such that for all A < A’, the system of

ASSERTION

,four equations (5)-(6) has a unique solution (sI, s2, s,, x2) for any choice of $ (w, + m - w’Z) in the interval [V, 1m - B?]. Proof: The Jacobian J of system (5), (6) with respect to (sI, .F~,x,, .Y?) at a point where (5))(6) hold is

J=

d&s,)

-Q,QzS,f’,A

-Q, Q,bPzA

c”(.sz)

0

Qzh,f’,A 0

Q,&P,A

0 -s, Q;Q2d2P2A

1

-szQ, Q;h, P, A 0 .s2Q;6, P,A

’

1

s,Q;62P2A

(A.1 1 where P, = 4 (w; + nz - ~1~)- Vi, Q, = Q,(x,) and Q,! = LIQ~(s,)/cLY~. Note that all the nonzero terms off the diagonal of J are products of A and some bounded terms. Note further that the elements on the diagonal are not multiplied by A and by our assumptions c,!‘(s,) is bounded away from zero. Therefore, there exists a sufficiently small A0 > 0 such that, for all A < A0 and for any value i (w; + nz - We) E [vi, m - Pklj, the determinant of every principal submatrix of J is positive. By a theorem concerning the uniqueness of a solution for a system of equations (see, e.g., Arrow and Hahn [ 1, p. 2361, when A < A0 system (5), (6) has a unique solution s, , s2, x , , .Y? for any value of $ (MI, + m - wk) E [V,,nz-

i;i,].

1

For A < A0 define the functions R,: [Pi, m - PA] -+ [ 8,,

nz -

a,]

by

R,(z) = [l - s,Qi(.\-,, A] x {[l -.sxQ/:(xA, +

\i

A] 6,r+6,sxQx(xn)

.Y dQ,(.u, - c,(s,) A , I

Av,/

(A.2)

330

ASHER WOLINSKY

where (sr, s2, x , , x2) is the unique solution for the four equations (5) (6) with f (M:, + m - wk) = Z. That is Ri(z) is the r.h.s. of equation (4) evaluated at$(M~i+m--~k)==andat (s,,s,,x,,x,)asdescribedabove. ASSERTION 2. There exists 2 > 0 such that, for any A < 1, the Junction R,(z) is increasing in all z E [V,, m - Pk].

Proof

Substitute $ (IV, + m - wk) = z into system (5) (6) and note that functions of Z. By implicit $1, s2, -Yl,. Y2 which solve that system are implicit differentiation we get ds,/dz as a ratio of two determinants: the denominator is det .I; the numerator is another determinant in which the terms above the diagonal are of the order of A, while the terms on the diagonal are bounded away from zero and their product is negative. By following the same reasoning of assertion 1 it can be shown that there exists 2 E (0, A”) such that for all A < d” the determinant in the numerator is negative. Hence, for any A < 2, ds,/dz < 0. Since Eq. (5) can be expressed as

M, -.I”,, (X -

.v;) dQi(x) = C:(si)

and since c;(s,) is convex it follows that sgn[dx;/dz] Hence, for all A < d, dx,/dz > 0. Note from (6) and (A.2) that I$(,-) can be written as

= -sgn[ds,/dz].

XdQi(X)-Cf(si)]A. Yil

R;(ii=[l-~iQi(xi)A].~;+[~~I

(A.3)

Therefore, y

= [ 1 - siQi(xi)

y,

A] 2

(x-x,)

dQi(x) - c;(si)

A 2.

By (5) the coefficient of ds;/dz is zero. Hence, for A < d”, (A.4) ASSERTION 3. For any A cd, unique solution.

the system of six equations (4)-(6) has a

MATCHING,

ProoJ

SEARCH,

AND

331

BARGAINING

Note from (A.4) and the inequalities

dsk/dz > 0, dxk/dz < 0 that

for A
= [l -s,Qj(.u,) +

1

-skQk(xk)

A][1

A] Ahi[P,-z]

.skQ~(xC)~+Q,(xk)~

< [I -s;Qi(x;)

By the definition (4)-(6) if

A][1

-1

-skQ,(xk)

of R,(z) the pair VU,, ~1~is part of a solution m-w2=R,[$(w,

(A.5)

A] < 1.

for system

+m-w2)],

m - U’, = R2t-4 (u’z + m - M’] ,].

(A.6)

(A.7)

Since for A 4. Therefore, for A < d” the locus defined by (A.6) intersects the locus defined by (A.7) only once. The above together with Assertion 1 and the fact that a solution exists imply that, for A cd, system (4))(6) has a unique solution. 1 It is now possible to present the main idea of the proof. Consider the sets B;, i= 1,2. B, = { 2 ) There exists a perfect equilibrium in a subgame starting with i’s proposal such that I’ is the expected utility to party 1 at this equilibrium 1. Let D, = Sup B, and di = Inf Bi and note that V, 6 d, 6 Di
332

ASHER

WOLINSKY

For the specific details of applying reader is referred to Wolinsky [ 161. Proof of Proposition

this method to the present case the Q.E.D.

2

Given constant levels of $3, i= 1, 2, the r.h.s. of (24) together with (25) (15) and (16) define a map T that transforms an increasing function wi defined on [0, M] and bounded by M into another such function. Checking for Blackwell’s sufficient conditions (see Blackwell [S]) for the operator T to be a contraction map, it is seen that T is obviously monotone. Also for function w, and a constant B>O, T(wi(m)

+ B) d T(w;(m))

2(s, + sy + s2 + s;, Q(m)

+

rI +r, +2(s, +sy+s,

+si) Q(m)

B.

(A.8)

To verify (A.8) note first that (25) implies that sj, sz are the same for w, and wi + B. This observation and (15) imply that the value of the term [ri + (si + sp) Q(m)] Vl(si, m; $7) calculated for u’~ + B (for ~3,-B) is greater (smaller) by (s, + sp) B than the value of this term when calculated for ~1~. It can also be verified that the difference between the values of V, calculated for w, + B and wi respectively is less than B. The above observations together with expression (24) yield inequality (A.8). Thus, there exists a b E (0, 1) such that T(wi + B) 6 T( wi) + /?B, for any increasing function u’; defined on [0, M] and any B> 0 in the relevant range, so that T is a contraction map. This proves that, given sy and s:, there exists a unique schedule ~1; satisfying (25) and (24) and hence unique values P, and S,, i = 1, 2. The remaining step is to show that there exist .sy and $ such that together with the corresponding P, and Si, i= 1, 2, satisfy conditions (18))(20) as well. Let the number S be such that cl(S) > M*, i = 1,2, and note that S, is always smaller than S. For any (s,, .s2) in [0, S] x [0, S] define sp = .sp(s,, s2) by b-si(NP-nS1)Qb)

s.

’

h-s,(x-NY)

o

Q(x)'

1 '

64.9)

where x is the unique solution for

IY*~(r-x)dQ(y)=c;(s,)+c;(s~) if such a solution exists and is 0 otherwise. Recall from above that given sp = sp(s,, s2) there exists a unique schedule w,(m) and intensities Si = S,(s], s2) which correspond to that schedule. Define a function H from [0, S] x [0, S] into itself by H[(s,, s?)] = (S,(s,, s2), S,(s,, sz)). The

MATCHING,

SEARCH,

AND

BARGAINING

333

function H is continuous and hence, by Brouwer’s theorem it has a fixed point (S1, S,). Clearly (S,, S2) # (0,O). Now, it can be verified that a fixed point S,, S2 together with sp(SI, S,) as defined by (A.9), Ni = [b/Q(m) S,(P, - NP)]/(F, + S,), and the corresponding schedule w,(m) constitute a solution satisfying the full system (16), (25), (lS)-(20). Q.E.D.

REFERENCES 1. K. J. ARROW AND F. K. HAHN, “General Competitive Analysis,” Holden-Day, San Francisco, 197 I. 2. K. G. BINMORE, “Perfect Equilibria in Bargaining Models,” ICERD, London School of Economics, D.P. 82/58, 1982. 3. K. G. BINMORE, Bargaining and coalitions, in “Game Theoretic Models of Bargaining” (A. Roth, Ed.), Cambridge Univ. Press, London, 1985. 4. K. G. BINMORE. A. RUBINSTEIN, AND A. WOLINSKY, The Nash bargaining solution in economic modelling, Rand. J. Econ. (1986). in press. 5. D. BLACKWELL. Discounted dynamic programming, Ann. Moth. Statist. 36 (1965), 226-233. 6. C. DERMAN. “Finite State Markovian Decision Processes,” Academic Press, New York, 1970. 7. P. A. DIAMOND. Wage determination and efficiency in search equilibrium, Ret]. of Econ. Stud. 49 ( 1982). 217.-227. 8. P. A. DIAMOND AND E. MASKIN. An equilibrium analysis of search and breach of contract, 1. Steady states, Bell J. Econ. 10 (1979). 9. D. GALE, “Limit Theorems for Markets with Sequential Bargaining,” CARESS Working Paper No. 85-15. 1985. 10. D. T. MORTENSEN, Specific capital and labor turnover, Bell J. Gon. 9 (1978), 572-586. Il. D. T. MORTENSEN, The matching process as a noncooperative bargaining game, in “The Economics of Information and Uncertainty” (J. J. McCall, Ed.), Univ. of Chicago Press, Chicago, 1982. 12. R. W. ROSENTHAL. Sequences of games with varying opponents, Economerrica 47 (1979). 1353-l 366. 13. A. RUBINSTEIK. Perfect equilibrium in a bargaining model, Ecorwnw/rir~u 50 ( 1982). 97-l 10. 14. A. RUBINSTEIN. ANII A. WOLINSKY. Equilibrium in a market with sequential bargaining. Econornetrica 53 (1985). 1133-l 151. 15. A. SHAKEI) AND J. SUTTON. Involuntary unemployment as a perfect equilibrium in a bargaining model, Econometrica 52 ( 1984), 135 l-1 364. 16. A. WOLINSKY. “Matching, Search and Bargaining,” Research Report No. 151, The Hebrew University, 1985.

MZi42!2-I0