bargaining with stochastic disagreement payoff

Int J Game Theory (2002) 31:571–591

DOI 10.1007/s001820300140

Bargaining with stochastic disagreement payo¤s*

Taiji Furusawa1, Quan Wen2

1Faculty of Economics, Hitotsubashi University, Kunitachi, Tokyo 186-8601, Japan2Department of Economics, Vanderbilt University, VU Station B #351819,2301 Vanderbilt Place, Nashville, TN 37235-1819, U.S.A. (e-mail: [email protected])

Received April 2002/Final version April 2003

Abstract. We study a bargaining model where (i) players’ interim disagree-ment payo¤s are stochastic and (ii) in any period, the proposer may postponemaking an o¤er without losing the right to propose in the following period.This bargaining model has a generically unique perfect equilibrium payo¤for each player, and the equilibrium outcome is ine‰cient in some cases, fea-turing a stochastically delayed agreement. We show that both the variation ofplayers’ interim disagreement payo¤s and the proposer’s ability to postponemaking an o¤er without losing the right to propose are necessary for the exis-tence of such a unique and ine‰cient perfect equilibrium outcome.

Key words: Negotiation, stochastic bargaining, disagreement payo¤s, delay

1. Introduction

The highly stylized bargaining model of Rubinstein (1982) admits a unique,e‰cient, perfect equilibrium. Despite its strong predictability, however, inef-ficient bargaining outcomes with delayed agreements or even with perpetualdisagreements often occur in reality. It is well known by now that equilibriumoutcomes are quite sensitive to the model structure. It has been shown thatbargaining models of incomplete information often admit ine‰cient equilib-

* We would like to thank Nabil Al-Najjar, Paul Anglin, Hulya Eraslan, Takako Fujiwara-Greve,Hans Haller, Bart Lipman, seminar participants at North American Summer Meeting of theEconometric Society, Midwest Mathematical Economics Conference, Canadian EconomicTheory Conference, Indiana University and York University, an associate editor and two anony-mous referees for their comments and suggestions. Financial support from SSHRC of Canada isgratefully acknowledged.

rium outcomes. Many bargaining models of complete information also pos-sess ine‰cient equilibrium outcomes.1

Similarly to the negotiation model studied by Busch and Wen (1995) andHouba (1997), where players’ interim disagreement payo¤s are strategicallydetermined by a normal form game during the negotiation, we study a bar-gaining model where players’ interim disagreement payo¤s can vary stochas-tically but not strategically. We find that if the proposer can postpone mak-ing an o¤er while retaining the right to propose in the following period, eachplayer will have a generically unique equilibrium payo¤. For certain param-eter values, the unique equilibrium is also ine‰cient, featuring a stochasticallydelayed agreement. Both the variation of disagreement payo¤s and the lessrestricted timing structure in making an o¤er are necessary for the existenceof such a unique and ine‰cient perfect equilibrium outcome in our model.

The model studied in this paper fits situations where bargaining environ-ments stochastically change over time.2 If firms, for example, foresee futurecontingencies and may have complete contracts covering all such contin-gencies, business environments can be made nonstochastic. In reality, how-ever, neither of these conditions holds, and hence environments are inherentlystochastic. When a firm contemplates to purchase another firm, or when twofirms contemplate to merge, they negotiate over the acquisition price, orexchange ratio of their stocks, respectively. During the negotiation, each firmcontinues to earn profits that vary stochastically with the overall performanceof the economy.3 In such changeable environments, the proposers may tem-porarily halt the bargaining process, as Ma and Manove (1993), Perry andReny (1993), Sakovics (1993), Avery and Zemsky (1994b), and Kambe (1999)formulate. When both firms have relatively high profits in the current period,the proposing firm may not want to rush into an agreement since the loss fromtemporary disagreement is relatively small. If this is indeed the case, the pro-posing firm will simply halt the bargaining. The perfect equilibrium in ourmodel has such a feature that two players reach an agreement only when theirinterim disagreement payo¤s are small enough.

Although our model yields a generically unique perfect equilibrium payo¤

1 The possibility of simultaneous o¤ers yields ine‰cient equilibrium outcomes in the models ofChatterjee and Samuelson (1990), Perry and Reny (1993), and Sakovics (1993). The existence of adeadline causes ine‰cient delay as Fershtman and Seidmann (1993), Ma and Manove (1993), andJehiel and Moldovanu (1995) demonstrate. Abreu and Gul (2000) show that the possibility ofplayers’ irrationality yields ine‰cient bargaining equilibrium. Haller and Holden (1990), Fernan-dez and Glazer (1991), Avery and Zemsky (1994b), Busch and Wen (1995), Houba (1997), Busch,Shi and Wen (1998), Kambe (1999), and Manzini (1999) identify endogenous interim disagree-ment payo¤s or bargaining values as possible sources for ine‰cient outcomes. Furusawa and Wen(2001, 2002) show that restricting players’ disagreement actions may not eliminate ine‰cient out-comes. For systematic reviews and discussions, see Osborne and Rubinstein (1990), Muthoo(1999), and Houba and Bolt (2002).2 Within cooperative-game frameworks, Riddell (1981), Chun and Thomson (1990a,b), and Bos-sert, Nosal, and Sadanand (1996) analyze bargaining problems with either stochastic disagreementpayo¤s or stochastic bargaining values. Avery and Zemsky (1994a), Merlo and Wilson (1995,1998), and Eraslan and Merlo (2002) investigate non-cooperative bargaining models with sto-chastic bargaining values.3 In Section 5, we will demonstrate that our model with stochastic disagreement payo¤s is e¤ec-tively equivalent to the case where disagreement payo¤s are fixed but the bargaining value is sto-chastic. Our results can be easily extended to the case where both disagreement payo¤s and bar-gaining value are stochastic.

572 T. Furusawa, Q. Wen

vector, the equilibrium itself varies with respect to the structure of the sto-chastic interim disagreement payo¤s. At one extreme in which players’ higherdisagreement payo¤s are small relative to the bargaining value, the proposeralways makes an acceptable o¤er as in the Rubinstein model. The equilibriumoutcome in this case is e‰cient. At the other extreme in which players’ higherdisagreement payo¤s are so large that their expected disagreement payo¤s arenot strictly dominated by any agreement, players never reach any agreement,which is also an e‰cient outcome. When players’ higher disagreement payo¤slie in a non-degenerate intermediate interval, players will not rush into anyagreement until their lower disagreement payo¤s are realized. This outcomearises even when players’ higher disagreement payo¤s are strictly dominatedby an agreement, which means that any delay is ine‰cient.

It is straightforward that a bargaining model has an ine‰cient equilibriumif there are multiple equilibrium outcomes. Ine‰cient equilibrium outcomesare supported by trigger-type strategies, as in the folk theorem for repeatedgames. The player who deviates from a prescribed ine‰cient outcome wouldbe punished by his worst perfect equilibrium in the continuation game, whichis made possible by the multiplicity of equilibrium outcomes.

Our model, however, predicts a unique and ine‰cient perfect equilibriumoutcome for certain parameter values. The basic intuition behind this resultcan be explained in a simple bargaining model where two players have non-zero disagreement payo¤s and negotiate their shares of a flow value of 1.Consider a period where player 1 makes an o¤er and the players’ interim dis-agreement payo¤s are ðx1; x2Þ. Let d A ð0; 1Þ be the discount factor per period,and for i ¼ 1 and 2, vi be player i ’s continuation payo¤ at the beginning of thenext period (if it is reached). In order to induce player 2 to accept, player 1 hasto o¤er, at least, player 2’s continuation payo¤ after rejection, ð1 � dÞx2 þ dv2.The higher x2 is, the more generous player 1’s o¤er has to be. On the otherhand, player 1’s opportunity cost of an immediate agreement is his continua-tion payo¤, ð1 � dÞx1 þ dv1, which increases with respect to player 1’s dis-agreement payo¤ x1. Player 1 will not make any acceptable o¤er if

ð1 � dÞx1 þ dv1 > 1 � ½ð1 � dÞx2 þ dv2�

x1 þ x2 >1 � dðv1 þ v2Þ

1 � d: ð1Þ

Condition (1) asserts that if the sum of the players’ disagreement payo¤sin the current period or the sum of their continuation payo¤s in the nextperiod is large enough, they will not rush into any agreement in the currentperiod. If ðx1; x2Þ is above the bargaining frontier, delay is possible even whenv1 þ v2 a 1. However, delay is obviously e‰cient in this case. Ine‰cient delayoccurs only if x1 þ x2 < 1 and yet condition (1) holds, which can only arisewhen v1 þ v2 > 1. If there are multiple equilibria, v1 and v2 can be players’payo¤s in di¤erent equilibria and hence v1 þ v2 > 1 is possible. On the otherhand, if the equilibrium is unique, v1 and v2 must be the responder’s and theproposer’s payo¤s in the same equilibrium outcome. Consequently, in eitheralternating-o¤er or randomly-selected-proposer bargaining model, v1 þ v2 a 1when disagreement is always dominated by an agreement. In our model, theproposer can postpone making an o¤er while retaining the right to propose inthe next period. Therefore, if the proposer postpones making an o¤er instead

Bargaining with stochastic disagreement payo¤s 573

of making an unacceptable o¤er, both v1 and v2 correspond to the proposer’spayo¤ in the next period, so v1 þ v2 > 1 even if disagreement is always domi-nated by an agreement.

In a model where the bargaining value is stochastic and the proposer isselected randomly in every period, Merlo and Wilson (1998) show a uniqueequilibrium involving a stochastically delayed agreement. However, delay isalways e‰cient in Merlo and Wilson (1998) since it occurs only when therealized bargaining value is smaller than the expected discounted bargain-ing value in the next period. In this paper, we also obtain a unique equilibriumoutcome with a stochastically delay agreement, but delay is ine‰cient forcertain parameter values. As we will argue, the critical factor that is respon-sible for the di¤erence in these results is whether or not the proposer retainsthe right to propose when he does not make any o¤er in the current period.In their model, the proposer can retain the right only with some probability,whereas in our model he completely retains the right to propose. Inter-preting their formulation within our framework, their specification impliesthat v1 þ v2 a 1. According to condition (1), any delay is associated withx1 þ x2 > 1, and hence it is e‰cient.

Ine‰cient delay occurs also in Avery and Zemsky’s (1994a) model, inwhich the bargaining value is revealed to both players after the proposermakes an o¤er in every period. This asymmetry of information between theproposer and the responder causes the e‰ciency loss.

The rest of this paper is organized as follows. Section 2 presents the formalmodel. In Sections 3 and 4, we characterize the perfect equilibrium and ana-lyze its e‰ciency, respectively. In Section 5, we establish the linkage betweenthe model with stochastic disagreement payo¤s and an alternative model witha stochastic bargaining value, and reinterpret our results within the lattermodel. Section 6 concludes the paper.

2. The model with stochastic disagreement payo¤s

Two players, 1 and 2, bargain over an allocation of a periodic value of 1.Players receive interim disagreement payo¤s in every period until they reachan agreement. As condition (1) suggests, it is the sum of players’ disagreementpayo¤s that matters for the possibility of ine‰cient delay. Without loss ofgenerality, therefore, we assume that the players have a common stochasticdisagreement payo¤ in every period.4 Let xt denote the players’ common dis-agreement payo¤ in period t. We assume that fxtgyt¼1 is an i.i.d. stochasticprocess, such that xt takes 0 with probability p A ½0; 1� and db 0 with proba-bility 1 � p for all t. In the rest of the paper, we often suppress the subscript tand simply let x denote a disagreement payo¤ in a period.

At the beginning of every period before reaching an agreement, bothplayers observe their common disagreement payo¤ in that period. Then, oneof the players, called the proposer, may either make an o¤er as to how toshare the value of 1 from this period onward, or suspend making an o¤er foranother period. If the proposer does not make any o¤er, the players simplycollect their disagreement payo¤s in the current period, and the same processwill repeat in the next period with a newly realized disagreement payo¤. If

4 We will further discuss this simplification in the Concluding Remarks.


the proposer makes an o¤er, on the other hand, then the other player, calledthe responder, can either accept or reject the standing o¤er. If the responderaccepts the o¤er, the game ends, and the players enjoy the constant stream oftheir agreed-upon shares forever. If the responder rejects the o¤er, the playerscollect their disagreement payo¤s in the current period and move on to thenext period, in which a new disagreement payo¤ will be realized and twoplayers will switch their roles in bargaining, i.e., the proposer in the currentperiod becomes the responder in the following period, and vice versa. By con-vention, we assume that player 1 is the proposer in the first period of thegame. Two players are risk-neutral and have a common discount factord A ð0; 1Þ per period. Figure 1 illustrates the game structure of this bargainingmodel.

A history consists of all past realized disagreement payo¤s, past o¤ers and,of course, rejections. A player’s strategy assigns a feasible action to every pos-sible history. Any strategy profile induces a unique outcome path, whichconsists of all realized disagreement payo¤s before an agreement and theagreement itself (if there is any). An outcome path with an agreement inperiod T is denoted by

pðTÞ ¼ ðxx1; xx2; . . . ; xxT�1; ða1; a2ÞÞ;

where xxt A f0; dg is the realization of xt for 1a t < T , and ai is player i ’s sharein the agreement reached in period T such that a1 þ a2 ¼ 1. By convention, Tis set to be infinity if there is no agreement. The players receive their dis-agreement payo¤s in every period until they reach an agreement, and continueto receive the agreed-upon shares thereafter. Player i ’s (non-expected) averagediscounted payo¤s from pðTÞ is

ð1 � dÞXT�1

t¼1

d t�1xxt þ dT�1ai;

if T b 2, and ai if T ¼ 1.

3. The perfect equilibrium

There are generically three types of equilibrium strategies, depending on thehigher disagreement payo¤ d. If d is small enough, the proposer always makes

Fig. 1. The model where the proposer has the option to wait


an acceptable o¤er in any period. If d is in an intermediate interval, the pro-poser delays making an o¤er till the lower disagreement payo¤ is realized, inwhich case he makes an acceptable o¤er. Finally, if d is large enough, twoplayers disagree forever. We define two threshold values of d that characterizethese three cases as:

d ¼ 1

2ð1 þ dpÞ and d ¼ 1

2dð1 � pÞ : ð2Þ

Note that 0 < d < 1=2 < d for all d A ð0; 1Þ and p A ð0; 1Þ. Proposition 1 com-pletely characterizes the equilibrium of the model.

Proposition 1. Each player’s equilibrium payo¤ in the bargaining model withstochastic disagreement payo¤s is unique if d0 d. There are generically threetypes of equilibrium depending on the value of d.

Case A: When 0a da d, the proposer demands bAðxÞ and the responderrejects the o¤er if and only if his share is less than 1 � bAðxÞ where

bAðxÞ ¼ 1 � ð1 � dÞx� dvA; with vA 11 � ð1 � dÞð1 � pÞd

1 þ d; ð3Þ

whether the realized value of x is 0 or d.

Case B: If da da d, the proposer waits when x ¼ d is realized, and demandsbBð0Þ when x ¼ 0 is realized. The responder rejects the o¤er if and only if hisshare is less than 1 � bBðxÞ where

bBðxÞ ¼ 1 � ð1 � dÞx� dvB; with vB 1pþ ð1 � dÞð1 � pÞd

1 � dþ 2dp; ð4Þ

whether the realized value of x is 0 or d.

Case C: If db d, the proposer either waits or demands more than 1 � bCðxÞ,and the responder rejects the o¤er if and only if his share is less than bCðxÞ where

bCðxÞ ¼ ð1 � dÞxþ dð1 � pÞd; ð5Þ

whether the realized value of x is 0 or d.If d ¼ d, the proposer is indi¤erent when x ¼ d between waiting and making

the acceptable o¤er, so that any mixing between these two actions can be part ofequilibrium in addition to the equilibrium strategies described in Cases A and B.

It is obvious that the equilibrium strategy profile induces the immediateagreement ðbAðxÞ; 1 � bAðxÞÞ in Case A, regardless of the realized value ofx in the first period. In Case B, the players reach an agreement ðbBð0Þ;1 � bBð0ÞÞ in the period when x ¼ 0 is realized for the first time. Finally inCase C, players never reach any agreement.

In the proof of Proposition 1, we first verify that neither player has anyincentive to deviate from the strategy profile described in the proposition, then


show the uniqueness of equilibrium payo¤ profile when d0 d using the tech-nique developed by Shaked and Sutton (1984). The detailed proof is given inthe Appendix. Here we provide some basic intuition behind this result.

Let consider the case where d ¼ 0 as the starting point. The model isthen equivalent to the Rubinstein model except that the proposer may post-pone making an o¤er. Waiting does not change anything but discounts bothplayers’ payo¤s. Therefore, the proposer cannot benefit from waiting. Theequilibrium is the same as that in the Rubinstein model, which continues to bethe case as long as d is small enough.

As d rises, however, the proposer’s temptation to wait increases. Let vdenote the proposer’s continuation payo¤ at the beginning of the next period.Then the proposer must o¤er, at least, ð1 � dÞxþ dv (as the responder’s share)to the responder in order to induce acceptance. As d increases, so do the pro-poser’s continuation payo¤ v in the next period and the acceptable o¤erð1 � dÞxþ dv to the responder in the current period. On the other hand, theproposer’s payo¤ from waiting, which also equals ð1 � dÞxþ dv, increaseswith respect to d. Therefore, the proposer is more willing to wait as d rises.When d becomes higher than d, the proposer’s payo¤ from waiting exceeds hispayo¤ from making an acceptable o¤er upon the realization of x ¼ d. Con-sequently, the proposer chooses to wait when x ¼ d is realized. However, theproposer makes an acceptable o¤er when x ¼ 0 is realized, since in this casethe responder is willing to accept a lower o¤er and the proposer is less willingto wait as the current disagreement payo¤ is low.

As d exceeds d, even the proposer’s payo¤ from waiting when x ¼ 0 isrealized, dv, becomes higher than his payo¤ from making the acceptable o¤er,1 � dv. Notice that in this case, ð1 � dÞd þ dv > 1 � ð1 � dÞd � dv so that theproposer also prefers waiting to making an acceptable o¤er when x ¼ d isrealized. The proposer will not make any acceptable o¤er in any period.In this case, the players’ expected disagreement point ðð1 � pÞd; ð1 � pÞdÞ isstrictly above the bargaining frontier, so that the expected disagreement pointeven when x ¼ 0 is realized (where the expectation is taken for the payo¤sfrom the next period) must be above the bargaining frontier, i.e., dð1 � pÞdb1=2, which is equivalent to db d.

Proposition 1 also shows that when d < d, the model has a unique equi-librium except when d ¼ d. In the knife-edge case of d ¼ d, the proposerhas the same payo¤ when x ¼ d is realized from either making an acceptableo¤er or waiting. Consequently, the proposer may choose any mixed strategybetween making the acceptable o¤er and waiting when x ¼ d is realized.Although the proposer has the same payo¤ in any mixed strategy equilibrium,the responder’s payo¤ is monotonically decreasing in the probability that theproposer chooses to wait. When db d, every player has a unique equilibriumpayo¤, but there are multiple equilibrium strategy profiles. Throughout theperpetual disagreement, the proposer may either make an unacceptable o¤eror make no o¤er at all.

4. The e‰ciency of the perfect equilibrium

In contrast to many existing bargaining models where ine‰ciency resultsfrom the multiplicity of equilibrium, the equilibrium in our model is generi-cally unique when it is ine‰cient. The ine‰ciency results from stochastically


delayed agreement as the proposer chooses to wait whenever d A ½d; 1=2Þ isrealized. Delay does not always imply ine‰ciency, however. As we havedemonstrated in the previous section, the perpetual disagreement outcomewhen db d is an e‰cient outcome since the players’ payo¤ profile alwayslie above the bargaining frontier. In what follows, we investigate the e‰ciencyof the equilibrium outcome in details by comparing the equilibrium outcomewith the e‰cient outcome of the model. We argue that ine‰ciency arises asthe proposer chooses to wait whenever x ¼ d A ½d; 1=2Þ is realized. On theother hand, the equilibrium always yields an e‰cient outcome when x ¼ 0 isrealized.

Let VðdjxÞ, for x ¼ 0 and d, denote the players’ ex post joint equilibriumpayo¤ immediately after the realization of the players’ disagreement payo¤ x.Then, the players’ ex ante joint equilibrium payo¤ at the beginning of anyperiod can be written as

VðdÞ1 p � Vðdj0Þ þ ð1 � pÞ � VðdjdÞ:

Proposition 1 implies that VðdjdÞ and VðdÞ take set values, while all othervalues of Vðdj0Þ, VðdjdÞ, and VðdÞ are singleton. Although VðdjdÞ and VðdÞare correspondences of d, we abuse notations slightly to simplify the exposi-tion by writing them as if they were functions (or a single-valued correspon-dence) whenever their values are singleton.

Proposition 1 outlines three di¤erent types of equilibrium outcomes,depending on the value of d. Treating the case of d ¼ d separately, we derivethe values of VðdjxÞ for the following four cases:

Case 1: If 0a d < d, the players reach an agreement immediately regardlessof the realization of x. Therefore, Vðdj0Þ ¼ VðdjdÞ ¼ 1.

Case 2: If d < da d, the players reach an agreement when x ¼ 0 is realized,so Vðdj0Þ ¼ 1. When x ¼ d is realized, the proposer waits so that VðdjdÞ isdetermined by

VðdjdÞ ¼ 2ð1 � dÞd þ d½ pþ ð1 � pÞVðdjdÞ�;

which gives us

VðdjdÞ ¼ dpþ 2ð1 � pÞd1 � dð1 � pÞ : ð6Þ

Case 3: If db d, the players never reach any agreement so they simply collecttheir disagreement payo¤s in every period.

VðdjxÞ ¼ 2½ð1 � dÞxþ dð1 � pÞd�:

Case 4: If d ¼ d, there are multiple equilibrium outcomes. Recall that theproposer makes an acceptable o¤er when x ¼ 0 is realized, and randomizesbetween making an acceptable o¤er and waiting when x ¼ d is realized.Therefore, it is straightforward that Vðdj0Þ ¼ 1. When x ¼ d is realized, onthe other hand, VðdjdÞ depends on the proposer’s mixed proposing strategy.At one extreme, the proposer always makes an acceptable o¤er, so the maxi-


mum of VðdjdÞ is equal to 1. At the other extreme, the proposer alwayschooses to wait. Thus, it follows from equation (6) that

VðdjdÞ ¼ dpþ 2ð1 � pÞd1 � dð1 � pÞ ¼ 1 � dþ dpð1 þ dpÞ

ð1 þ dpÞ½1 � dð1 � pÞ� :

In summary, the players’ ex post joint equilibrium payo¤s are

Vðdj0Þ ¼ 1 if 0a da d

2dð1 � pÞd if da d;

�ð7Þ

VðdjdÞ ¼

1 if 0a d < ddpþ2ð1�dÞd

1�dð1�pÞ ; 1h i

if d ¼ d

dpþ2ð1�dÞd1�dð1�pÞ if d < da d

2ð1 � dpÞd if da d:

8>>>>><>>>>>:

ð8Þ

It is easy to see that both Vðdj0Þ and VðdjdÞ are upper hemi-continuous cor-respondences as Figure 2 depicts.

Next, we derive e‰cient outcomes that maximize the players’ joint payo¤s.Similarly to the ex post joint equilibrium payo¤s, we define WðdjxÞ as theplayers’ joint payo¤ in the e‰cient outcome after x is realized. Intuitively, thee‰ciency calls for (i) an immediate agreement in any period if d is small, (ii)an immediate agreement only when x ¼ 0 if d is medium, and (iii) perpetualdisagreement if d is large. It is apparent that for the first type of outcome, animmediate agreement whether x ¼ 0 or d yields Wðdj0Þ ¼ WðdjdÞ ¼ 1. Forthe second type of outcome, Wðdj0Þ ¼ 1 while WðdjdÞ satisfies WðdjdÞ ¼2ð1 � dÞd þ d½ pþ ð1 � pÞWðdjdÞ�, which yields

WðdjdÞ ¼ dpþ 2ð1 � dÞd1 � dð1 � pÞ :

Fig. 2. Player’s joint equilibrium payo¤


Finally, for the third type of outcomes, WðdjxÞ ¼ 2½ð1 � dÞxþ dð1 � pÞd�,which leads to

Wðdj0Þ ¼ 2dð1 � pÞd and WðdjdÞ ¼ 2ð1 � dpÞd:

Consequently, the joint e‰cient payo¤s are derived as

Wðdj0Þ ¼ maxf1; 2dð1 � pÞdg

¼ 1 if 0a da d

2dð1 � pÞd if da d;

�ð9Þ

WðdjdÞ ¼ max 1;dpþ 2ð1 � dÞd

1 � dð1 � pÞ ; 2ð1 � dpÞd� �

¼1 if 0a da 1

2dpþ2ð1�dÞd

1�dð1�pÞ if 12 a da d

2ð1 � dpÞd if da d:

8>><>>:

ð10Þ

Comparing (7) with (9), we immediately recognize that the equilibriumoutcome is always e‰cient when x ¼ 0 is realized. In the situation where theproposer chooses to wait when x ¼ 0 is realized, the proposer also chooses towait when x ¼ d is realized. However, perpetual disagreement in this case ise‰cient since the players’ payo¤ profile from perpetual disagreement is abovethe bargaining frontier as Figure 2(a) shows.

Comparing (8) with (10), we find that when x ¼ d is realized, the perfectequilibrium is not e‰cient, i.e., VðdjdÞ < WðdjdÞ, for d A ðd; 1=2Þ. Althoughe‰ciency calls for an immediate agreement both when x ¼ 0 and d if d is inthis range, the proposer chooses to wait when x ¼ d is realized based on hisindividual rationale in equilibrium. To assess how serious this ine‰ciency is,we first note that d is decreasing in p. Therefore, as p increases, the rangeof d that is associated with ine‰cient equilibrium, i.e., ðd; 1=2Þ, expands.5However, the maximum expected e‰ciency loss, defined by L ¼ maxd Ex �½WðdjxÞ � VðdjxÞ�, changes non-monotonically with respect to p. It followsfrom (7)–(10) that

L ¼ 1 � minVðdÞ ¼ 1 � pð1 þ dpÞ þ ð1 � dÞð1 � pÞð1 þ dpÞ½1 � dð1 � pÞ� :

It is easy to see that L > 0 for all d A ð0; 1Þ and p A ð0; 1Þ, and L ¼ 0 for p ¼ 0and p ¼ 1. For any d A ð0; 1Þ, the maximum e‰ciency loss L has a uniquemaximum with respect to p as Figure 3 illustrates. Either a high or low valueof p implies a low variance of the disagreement payo¤s, pushing our modelcloser to the bargaining model with a fixed disagreement point, in which theequilibrium outcome is e‰cient.

5 An increase in p raises the chance of agreement in the next period, and hence raises the expectedpayo¤ from waiting. Thus, the proposer may become willing to wait when x ¼ d even for asmaller d.


The players’ discount factor d has similar e¤ects on the e‰ciency loss. Anincrease in d also widens the range of d that corresponds to the ine‰cient equi-librium outcome. The maximum expected e‰ciency loss L also increases ini-tially from 0 as d increases from 0, and then decreases to 0 as d approaches 1.

In the rest of this section, we show that the two distinct features in ourmodel, the variation of disagreement payo¤s and the proposer’s option to waitwhile retaining the right to propose, are necessary for the existence of theunique ine‰cient equilibrium. First, we examine the necessity of the variationof players’ disagreement payo¤s. Suppose now that the proposer has theoption to wait but the disagreement point is fixed and lies strictly below thebargaining frontier. Without loss of generality, we assume that the disagree-ment point is ð0; 0Þ. This situation is a special case of our model when d ¼ 0or equivalently p ¼ 1. Case A of Proposition 1 applies so that there is noine‰cient equilibrium. Waiting does not change the bargaining environmentexcept that it discounts players’ payo¤s. Thus the proposer does not benefitfrom waiting. The Rubinstein solution is the unique equilibrium in this case.

Proposition 2. Suppose that the proposer has the option to postpone making ano¤er while retaining the right to propose, and that the players’ disagreementpayo¤s are fixed over time. Then there is a unique, e‰cient, perfect equilibrium.

Next, we demonstrate that the variation of disagreement payo¤s alonedoes not cause any ine‰cient equilibrium outcome. Suppose that the proposerdoes not have the option to wait. The proposer in any period will be the re-sponder in the following period. Whether the proposer makes an acceptableor an unacceptable o¤er, there must be one equilibrium outcome from whichboth players’ continuation payo¤s are derived. The continuation payo¤ vectorcorresponds to a feasible payo¤ vector of the game, and hence any equilib-rium outcome must be e‰cient.

Proposition 3. Suppose that the proposer does not have the option to postponemaking an o¤er without losing the right to propose in the following period.Then, the perfect equilibrium is unique in terms of payo¤s, and it is e‰cient.

Fig. 3. The maximum loss of e‰ciency when d ¼ 0:6


The equilibrium strategy profile is the same as that of Proposition 1, exceptthat in this case the lower threshold of d equals 1=2 and the proposer makesan unacceptable o¤er when he does not want to reach an agreement in thecurrent period. As far as the higher disagreement point is below the bar-gaining frontier, i.e., d < 1=2, the players reach an agreement immediately.When 1=2a da d, the proposer makes an acceptable o¤er if x ¼ 0, butmakes an unacceptable o¤er if x ¼ d. The resulting delay when x ¼ d, how-ever, is e‰cient since db 1=2. When db d so that even the present value ofthe players’ joint expected payo¤ when x ¼ 0 exceeds 1, perpetual disagree-ment occurs in equilibrium, which is also e‰cient. The proof of Proposition 3is relegated to the Appendix.

5. An alternative model with a stochastic bargaining value

The bargaining model with stochastic disagreement payo¤s and a fixed bar-gaining value, which we have studied so far, is equivalent to the model withfixed disagreement payo¤s and a stochastic bargaining value, which Averyand Zemsky (1994a), Merlo and Wilson (1995, 1998), and Eraslan and Merlo(2002) study. After all, players bargain over how to share the surplus, thedi¤erence in values from agreement and disagreement. In order to make pos-sible the comparison between the implications of our model and those of theexisting literature, we demonstrate that all of our results are carried over to abargaining model with stochastic bargaining value. Then Propositions 2 and 3imply that the two distinct features of our model lead to the unique, ine‰cientequilibrium outcome, which is new to the bargaining literature.

In the primary model, the surplus to be divided in period t is 1 � 2xt,where xt equals 0 with probability p, and d with probability 1 � p. In order totransform the primary model into the one with a stochastic value, we assumethat the bargaining value follows an i.i.d. stochastic process fytgyt¼1 whereyt ¼ 1 � 2xt for all t. Accordingly, yt (or y for simplicity) takes two possiblevalues: 1 with probability p and w1 1 � 2d ða1Þ with probability 1 � p. Thevalue of w can be negative since the disagreement point in the primary modelcan be above the bargaining frontier. The bargaining protocol in the modelwith a stochastic value is the same as that in the primary model.

We interpret this model as a variant of the primary model such thatplayers explicitly make o¤ers in term of the share of the periodic surplus y.Case A in the primary model, characterized by da d, is now represented bywb w, where

w ¼ 1 � 2d ¼ dp

1 þ dp:

Direct application of Proposition 1 indicates that the proposer always makesan acceptable o¤er in this case. Case B ðda da dÞ is now characterized bywawa w, where

w ¼ 1 � 2d ¼ � 1 � dð1 � pÞ1 � p

ð< 0Þ:

The proposer makes an acceptable o¤er when y ¼ 1 is realized, and waitswhen y ¼ w is realized. Finally in Case C ðdb dÞ, which is now characterizedby waw, two players will never reach any agreement in equilibrium.


Because of this equivalence, all the results regarding e‰ciency can alsobe carried over to this alternative bargaining model with stochastic value.Proposition 1 implies that if 0 < wa w, ine‰cient delay occurs when y ¼ wis realized. Proposition 3 implies if the proposer does not have the option towait in this alternative model, similarly to the model of Merlo and Wilson(1998), the unique perfect equilibrium even with a delayed agreement is al-ways e‰cient.

6. Concluding remarks

We have considered a bargaining model with discounting in which (i) players’disagreement payo¤s are stochastically determined in every period at eitherð0; 0Þ or ðd; dÞ, and (ii) the proposer in any period can postpone making ano¤er without losing the right to propose in the following period. Except whend ¼ d, this model admits a unique equilibrium payo¤ for every player. Ford A ðd; dÞ where d < 1=2 < d, the proposer postpones making an o¤er when-ever the realized disagreement payo¤ is d. Therefore, if d A ðd; 1=2Þ, the uniqueequilibrium is ine‰cient since the outcome involves a stochastic delayedagreement even though the surplus from an agreement is strictly positive.

The model studied in this paper is highly stylized, but our results are quiterobust in many aspects of the model specification. For example, the assump-tion that players receive a common disagreement payo¤ is innocuous. As wemay infer from (1), it is players’ joint disagreement payo¤s that a¤ects the pro-poser’s strategy between waiting and making an o¤er. Indeed, we can easilyextend our analysis to the case where players’ disagreement payo¤s ðx1; x2Þare arbitrary random variables. Under certain conditions, such as the varia-tion of x1 þ x2 is large enough, stochastic delay in reaching an agreement willarise in the perfect equilibrium. In such an equilibrium, the proposer makesan acceptable o¤er when the realized joint disagreement payo¤ is less thana threshold, and he waits otherwise. The resulting outcome is ine‰cient ifthe joint disagreement payo¤ is greater than the threshold but less than 1. Wehave also argued that the model with stochastic disagreement payo¤s is equiv-alent to the one with a stochastic bargaining value. It is straightforward togeneralize our analysis to more realistic models where both disagreementpayo¤s and a bargaining value are stochastic.

Appendix

Proof of Proposition 1: First, we verify that neither player has any incentive todeviate from the strategy profile described in Proposition 1 in each of the threecases. Then with Shaked and Sutton’s (1984) technique, we demonstrate thatProposition 1 completely characterizes the perfect equilibrium outcome of themodel. Due to the symmetry in strategies between players 1 and 2, we onlyneed to analyze the proposer’s and the responder’s strategies in the three cases.

Case A: According to the strategy profile when da d, the proposer’s expectedpayo¤ in any period before the realization of x is

pbAð0Þ þ ð1 � pÞbAðdÞ ¼ 1 � ð1 � dÞð1 � pÞd1 þ d

; ð11Þ


which equals vA by (3). The responder’s expected payo¤ before x is realized isthen 1 � vA.

In any period where an o¤er is made, the current responder will be theproposer in the next period after the responder rejects the o¤er. The responderwill then reject any o¤er if and only if his share in the o¤er is less than hiscontinuation payo¤ from rejecting the o¤er, which is

ð1 � dÞxþ dvA ¼ 1 � bAðxÞ:

In other words, the responder will reject any o¤er if and only if the proposerdemands more than bAðxÞ in any period whether x ¼ 0 or d is realized.

The proposer’s payo¤ from the strategy profile is bAðxÞ for x ¼ 0 and d. Ifthe proposer makes an unacceptable o¤er, he will be the responder in the nextperiod with the expected payo¤ of 1 � vA. Thus, the proposer’s payo¤ frommaking an unacceptable o¤er is ð1 � dÞxþ dð1 � vAÞ, which is less than bAðxÞdue to the fact that xa d < 1=2 for x ¼ 0 and d. If the proposer chooses towait, he will still be the proposer in the next period with the expected payo¤of vA. The proposer’s payo¤ from waiting is then ð1 � dÞxþ dvA, which is lessthan or equal to bAðxÞ due to xa da d. Consequently, the proposer neitherwaits nor makes any unacceptable o¤er in any period when 0a da d.

Case B: According to the strategy profile, the proposer’s expected payo¤ vB inany period before the realization of x satisfies

vB ¼ pð1 � dvBÞ þ ð1 � pÞ½ð1 � dÞd þ dvB�;

which yields the value of vB as given in (4). The responder’s payo¤ fromrejecting an o¤er is then ð1 � dÞxþ dvB ¼ 1 � bBðxÞ by (4). Thus, the re-sponder rejects any o¤er if his share is less than 1 � bBðxÞ whether x ¼ 0 or d.

When x ¼ d is realized, the proposer’s payo¤ from waiting is ð1 � dÞd þdvB. The proposer’s payo¤ from making an unacceptable o¤er is ð1 � dÞd þdr, where r represents the responder’s expected payo¤ before the realization ofx. Since r satisfies r ¼ pdvB þ ð1 � pÞ½ð1 � dÞd þ dr�, we have

r ¼ pdvB þ ð1 � pÞð1 � dÞd1 � ð1 � pÞd :

Notice that da d implies ra vB, which in turn implies that the proposer’spayo¤ from making an unacceptable o¤er is smaller than that from waiting. Ifthe proposer makes an acceptable o¤er, his payo¤ will be at most bBðdÞ,which is less than or equal to ð1 � dÞd þ dvB due to db d. Consequently, theproposer prefers waiting in any period where x ¼ d is realized.

When x ¼ 0 is realized, the proposer’s payo¤s from waiting, making anyunacceptable o¤er and making the acceptable o¤er are dvB, dr, and bBð0Þ,respectively. Since r < vB and dra dvB a bBð0Þ due to da d, the proposerprefers making the acceptable o¤er bBð0Þ when x ¼ 0 is realized.

Case C: The proposer’s expected equilibrium payo¤ at the beginning of anyperiod vC satisfies

vC ¼ ð1 � dÞð1 � pÞd þ dvC ;


which yields vC ¼ ð1 � pÞd. Thus, the responder’s continuation payo¤ afterrejecting an o¤er is

ð1 � dÞxþ dð1 � pÞd;

which is the same as bCðxÞ under the prescribed strategy after the realizationof x. The responder will reject any o¤er if his share is less than bCðxÞ. So theproposer’s payo¤ from making an acceptable o¤er is at most 1 � bCðxÞ. If theproposer chooses to wait or make an unacceptable o¤er, his payo¤ will bebCðxÞ. It follows from db d that

1 � bCðxÞ ¼ 1 � ð1 � dÞx� dð1 � pÞda ð1 � dÞxþ dð1 � pÞd ¼ bCðxÞ;

for x ¼ 0 and d. The above inequality asserts that the proposer never makesany acceptable o¤er in this case.

From the first half of this proof and the setup of the model, we know thatthe set of equilibrium payo¤s in the model is non-empty and bounded. Fol-lowing Shaked and Sutton’s (1984) technique, we now derive the range ofthe players’ equilibrium payo¤s. Let MðxÞ and mðxÞ be the supremum andinfimum of the proposer’s equilibrium payo¤s, respectively, in any period withdisagreement payo¤ x ¼ 0 or d. Let M and m denote the expected values ofMðxÞ and mðxÞ, respectively, i.e.,

M ¼ pMð0Þ þ ð1 � pÞMðdÞ; m ¼ pmð0Þ þ ð1 � pÞmðdÞ: ð12Þ

By definition, we have MðxÞbmðxÞ for x ¼ 0 and d and hence Mbm.Now consider the proposer’s strategy in a period with disagreement payo¤

x. If the proposer chooses to wait, he collects x in the current period and willstill be the proposer in the next period with a continuation payo¤ between ofm and M. Therefore, the proposer receives neither less than ð1 � dÞxþ dm,nor more than ð1 � dÞxþ dM.

Alternatively, if the proposer chooses to make an o¤er, the responder inthe current period will be the proposer in the next period when the o¤er isrejected. The responder certainly rejects any o¤er which gives him less thanhis lowest possible payo¤ after rejection, ð1 � dÞxþ dm. Consequently, theproposer cannot receive more than 1 � ð1 � dÞx� dm from making an o¤er.On the other hand, the responder certainly accepts any o¤er which gives himmore than his highest possible payo¤ after rejection, ð1 � dÞxþ dM. Thus, theproposer would not receive less than 1 � ð1 � dÞx� dM from making an o¤er.

Since the proposer chooses between waiting and making an o¤er, MðxÞand mðxÞ must satisfy the following inequalities for x ¼ 0 and d:

MðxÞamaxfð1 � dÞxþ dM; 1 � ð1 � dÞx� dmg;

mðxÞbmaxfð1 � dÞxþ dm; 1 � ð1 � dÞx� dMg:ð13Þ

Note that for x ¼ 0 and d, we have

ð1 � dÞxþ dMaðbÞ 1 � ð1 � dÞx� dm


if and only if

ð1 � dÞxþ dmaðbÞ 1 � ð1 � dÞx� dM:

In solving MðxÞ and mðxÞ from (13), therefore, we only have to check whichof ð1 � dÞxþ dM and 1 � ð1 � dÞx� dm is greater in order to identify all pos-sible cases.

Given that x takes two possible values, 0 and d, it appears that there arefour di¤erent cases to consider in (13). In fact, however, we only have threecases to consider since ð1 � dÞd þ dMa 1 � ð1 � dÞd � dm implies dMa

1 � dm. The three cases, called Cases I, II and III, correspond to Cases A, Band C that we have already analyzed, respectively.

Case I ð1 � dÞd þ dMa 1 � ð1 � dÞd � dm:Inequality system (13) becomes

Mð0Þa 1 � dm;

mð0Þb 1 � dM;

MðdÞa 1 � ð1 � dÞd � dm;

mðdÞb 1 � ð1 � dÞd � dM:

ð14Þ

Then (12) implies that expected values of MðxÞ and mðxÞ satisfy

Ma 1 � ð1 � dÞð1 � pÞd � dm;

mb 1 � ð1 � dÞð1 � pÞd � dM:

Solving these inequalities for M and m, we have

1 � ð1 � dÞð1 � pÞd1 þ d

amaMa1 � ð1 � dÞð1 � pÞd

1 þ d;

which yields that M ¼ m. Now, substituting M ¼ m into (14) yields

1 � dmamð0ÞaMð0Þa 1 � dm;

1 � ð1 � dÞd � dmamðdÞaMðdÞa 1 � ð1 � dÞd � dm;

which imply that each player has a unique equilibrium payo¤ after x is real-ized. Substituting the value of M ¼ m into the inequality that defines Case I,we obtain

2ð1 � dÞda 1 � 2d½1 � ð1 � dÞð1 � pÞd�1 þ d

;

which holds if and only if da d. Therefore, Case I is equivalent to Case A.


In this Case I, observe the fact that M ¼ m ¼ vA and MðxÞ ¼ mðxÞ ¼bAðxÞ for x ¼ 0 and d. As we have argued in Case A, the proposer prefersmaking the acceptable o¤er that demands bAðxÞ to waiting, whether x ¼ 0 ord. It is straightforward that if d0 d, the equilibrium described in the propo-sition is unique, yielding the unique payo¤ vector. If d ¼ d, the proposer isindi¤erent between making an acceptable o¤er and waiting when x ¼ d isrealized. So any mixing between waiting and making the acceptable o¤er canbe part of the perfect equilibrium.

Case II dMa 1 � dm and ð1 � dÞd þ dMb 1 � ð1 � dÞd � dm:Conditions (13) become

Mð0Þa 1 � dm;

mð0Þb 1 � dM;

MðdÞa ð1 � dÞd þ dM;

mðdÞb ð1 � dÞd þ dm:

ð15Þ

Again, (12) implies that the expected values of MðxÞ and mðxÞ satisfy

Ma pð1 � dmÞ þ ð1 � pÞ½ð1 � dÞd þ dM�;

mb pð1 � dMÞ þ ð1 � pÞ½ð1 � dÞd þ dm�;

which yield

pþ ð1 � dÞð1 � pÞd1 � dþ 2dp

amaMapþ ð1 � dÞð1 � pÞd

1 � dþ 2dp:

Therefore, we have M ¼ m. Substituting the value of m and M in this caseinto (15) gives us the unique equilibrium payo¤s for x ¼ 0 and d, respectively.We further substitute the value of M ¼ m into the two inequalities that defineCase II to obtain

2d½ pþ ð1 � dÞð1 � pÞd�1 � dþ 2dp

a 1;

2d½ pþ ð1 � dÞð1 � pÞd�1 � dþ 2dp

b 1 � 2ð1 � dÞd:

The last two inequalities are equivalent to da d and db d, respectively, sothat Case II corresponds to Case B.

In addition, notice the fact that M ¼ m ¼ vB and MðxÞ ¼ mðxÞ ¼ bBðxÞfor x ¼ 0 and d. Our early analysis of Case B implies that the proposer pre-fers making the acceptable o¤er bBð0Þ when x ¼ 0, but prefers waiting whenx ¼ d. Given the proposer’s strategies, it is straightforward to verify thatthe equilibrium described in the proposition is unique, and yields the uniqueequilibrium payo¤ when d0 d.


Case III dMb 1 � dm:Conditions (13) become

Mð0Þa dM;

mð0Þb dm;

MðdÞa ð1 � dÞd þ dM;

mðdÞb ð1 � dÞd þ dm:

ð16Þ

Then, the expected values of MðxÞ and mðxÞ satisfy

ð1 � pÞdamaMa ð1 � pÞd;

which implies that M ¼ m ¼ ð1 � pÞd. Substituting M ¼ m ¼ ð1 � pÞd into(16) yields the unique equilibrium payo¤. Condition dMb 1 � dm is equiva-lent to db d, which implies that Case III corresponds to Case C.

Notice that M ¼ m ¼ vC and MðxÞ ¼ mðxÞ ¼ bCðxÞ for x ¼ 0 and d.Perpetual disagreement is the only way to achieve these values as equilibriumpayo¤s. In such outcomes, the proposer either makes unacceptable o¤ers ormakes no o¤er at all. Equilibrium strategies, therefore, are not unique in thiscase. Q.E.D.

Proof of Proposition 3: It is straightforward that neither player has any incen-tive to deviate from the strategy profile described in Proposition 3. Here, weshow the uniqueness of the players’ equilibrium payo¤. As in the proof ofProposition 1, the following argument also suggests that players have noincentive to deviate from the prescribed strategies.

Let MðxÞ and mðxÞ be the supremum and infimum of the proposer’sequilibrium payo¤s, respectively, in any period with disagreement payo¤ x.Also let M and m denote the expected values of MðxÞ and mðxÞ, respectively,as (12) depicts. Similarly, we define the responder’s counterparts as RðxÞ, rðxÞ,R, and r, respectively.

Since the responder’s equilibrium payo¤s cannot be higher than the su-premum of continuation payo¤s when he rejects the standing o¤er, we have

RðxÞa ð1 � dÞxþ dM:

On the other hand, the infimum of the responder’s continuation payo¤s mustsatisfy

rðxÞb ð1 � dÞxþ dm:

Consequently, we have

Ra ð1 � dÞð1 � pÞd þ dM and rb ð1 � dÞð1 � pÞd þ dm:

In contrast to our primary model where the proposer may postpone mak-ing o¤ers, the proposer now has only two options in any period: either mak-ing an acceptable o¤er or making an unacceptable o¤er. When he makes an


unacceptable o¤er in a period with disagreement payo¤ x, the proposer’spayo¤ cannot be lower than

ð1 � dÞxþ drb ð1 � dÞxþ dð1 � dÞð1 � pÞd þ d2m;

and cannot be higher than

ð1 � dÞxþ dRa ð1 � dÞxþ dð1 � dÞð1 � pÞd þ d2M:

If the proposer makes an acceptable o¤er, the responder will certainly acceptany o¤er less than ð1 � dÞxþ dM and reject any o¤er less than ð1 � dÞxþ dm.Therefore, the proposer’s payo¤ when he makes an acceptable o¤er is boundedbetween

1 � ð1 � dÞx� dM and 1 � ð1 � dÞx� dm:

Since the proposer chooses between making an acceptable and unaccep-table o¤ers, MðxÞ and mðxÞ must satisfy the following inequalities for x ¼ 0and d:

MðxÞamaxfð1� dÞxþ dð1� dÞð1� pÞd þ d2M; 1�ð1� dÞx� dmg; ð17Þ

mðxÞbmaxfð1� dÞxþ dð1� dÞð1� pÞd þ d2m; 1�ð1� dÞx� dMg: ð18Þ

Notice that

ð1 � dÞxþ dð1 � dÞð1 � pÞd þ d2Mb ðaÞ1 � ð1 � dÞx� dm

if and only if

ð1 � dÞxþ dð1 � dÞð1 � pÞd þ d2mb ðaÞ1 � ð1 � dÞx� dM:

Similarly to the proof of Proposition 1, we need to consider three cases asclassified below in order to solve (17) and (18) for MðxÞ and mðxÞ. Case A iswhen d is so small that

ð1 � dÞd þ dð1 � dÞð1 � pÞd þ d2Ma 1 � ð1 � dÞd � dm;

which implies that dð1 � dÞð1 � pÞd þ d2Ma 1 � dm (corresponding to thecase where x ¼ 0). Case B is when d is medium so that

dð1 � dÞð1 � pÞd þ d2Ma 1 � dm,

ð1 � dÞd þ dð1 � dÞð1 � pÞd þ d2Mb 1 � ð1 � dÞd � dm:

Case C is when d is so large that

dð1 � dÞð1 � pÞd þ d2Mb 1 � dm;

which also implies that ð1 � dÞd þ dð1 � dÞð1 � pÞd þ d2Mb 1 � ð1 � dÞd �dm.


From a similar analysis to the second half of the proof of Proposition 1, wefind that MðxÞ ¼ mðxÞ for x ¼ 0 and d, which implies that the model alwayshas a unique equilibrium in terms of payo¤s. The equilibrium payo¤ varieswith the value of d in the three cases.

Case A: M ¼ m ¼ 1 � ð1 � dÞð1 � pÞd1 þ d

if 0a da1

2;

Case B: M ¼ m ¼ pþ ð1� dÞð1� pÞ½1þ dð1� pÞ�dð1þ dÞ½1� dð1� pÞ� if

1

2a da

1

2dð1� pÞ ;

Case C: M ¼ m ¼ ð1 � pÞd if1

2dð1 � pÞ a d:

We have shown that for any value of d, the perfect equilibrium yields aunique payo¤ for each player. Equilibrium strategies, however, are not uniquein Cases B and C since there are many possible unacceptable o¤ers. Sincedelay occurs only when the realized value of x exceeds 1=2, the perfect equi-librium is always e‰cient. Q.E.D.

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