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Bargaining under Incomplete Information

Author(s): Kalyan Chatterjee and William SamuelsonSource: Operations Research, Vol. 31, No. 5 (Sep. - Oct., 1983), pp. 835-851Published by: INFORMSStable URL: http://www.jstor.org/stable/170889.

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Bargaining under Incomplete InformationKALYANCHATTERJEE

PennsylvaniaState University,UniversityPark, PennsylvaniaWILLIAMAMUELSON

Boston University,Boston,Massachusetts(ReceivedJuly 1981;revisedApril 1982;acceptedApril 1983)

This paper presents and analyzes a bargainingmodel of bilateralmonopolyunderuncertainty.Under he bargaininguleproposed, he buyerandthe sellereach submitsealed offers that determinewhether he good inquestion s soldand the transferprice.The Nash equilibriumolutionof this bargaining ameimpliesan offer strategy of each party hat is monotonic nits own reservationprice and depends on its assessment of the opponent's reservationprice.Issues of relative bargainingadvantage and efficiency are examined for anumberof special cases. Finally,we discuss the appropriateness f the Nashsolutionconcept.

THIS PAPER presents a simple model of two-person bargainingunderincompleteinformation.Applicationsof the modelrangefromthe settlementof a claimout of court,to union-managementnegotiations,to the purchase and sale of a used automobile. The common featureshared by these examples is that each bargainer,while certain of thepotential value it places on the transaction,has only probabilistic nfor-mation concerning the potential value of the other. For instance, inhagglingover the price of a used car, neither buyernor seller knows theother'swalk-awayprice.Our principal aim is to investigate equilibriumbargainingbehaviorunder uncertainty and to study how the parties fare, individually andcollectively, undera simpleclass of bargainingprocedures.A main resultis that, given incompleteinformation, not all mutuallybeneficial agree-ments can be attained via bargaining.Even when the buyer values thegood more highly than the seller, a successful sale may be impossible.Additionally,we present a number of results characterizing he players'optimalbargainingstrategiesand give explicit solutions for a numberofexamples. These examples provideuseful comparativestatics results-indicatingthe effect on bargainingbehaviorof changesin the negotiationSubject classification: 231 bargaining, 232 bargaining.

835Operations Research 0030-364X/83/3105-0835 $01.25Vol. 31, No. 5, September-October 983 ? 1983OperationsResearchSocietyofAmerica

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836 ChatterjeendSamuelsonground rules, the information available to the players, and the degree ofthe players'risk aversion.

Completevs. IncompleteInformation.Most game-theoretic literatureon bargaininghas assumed that the participantspossess completeinfor-mation about the negotiation situation. Important contributions areprovided by Nash [1950], Harsanyi [1956], Schelling [1960], Cross[1969], and Roth [1979]. Young [1975] provides an excellent summaryand critiqueof the most importantliteratureon the bargainingproblem.In two-person bargains (to which our discussion will be limited), it iscustomaryto stipulate that the set of possible actions of the parties andthe player payoffs fromany combination of actions are common knowl-edge. In the model of bilateral monopoly considered here, with thepresumption of complete information, each bargainer,seller and buyer,knowsthe other'swalk-awayprice.Then the bargainersnegotiatea finalprice in the known rangeof mutually acceptableprices (i.e. between theseller's minimumand the buyer'smaximumacceptableprices).Since anyprice in this range can be supportedas an equilibriumoutcome, bargain-ing solutions are usually determinedeither by specifying a concessionmechanism leading to a distinct outcome, or a set of axioms that a"reasonable" utcome should satisfy.

Despite its many insights, the complete informationapproachfails tomirrorkey featuresof actualnegotiations: 1) the fact that eachbargaineris uncertain about its adversary's payoff, and 2) the occurrence of"unreasonable" bargaining outcomes-breakdowns in negotiations,strikes, and work stoppages-when mutuallybeneficial agreements arepossible. A bargaining model that embraces these features carries anumberof noteworthy implications. First, each party's bargainingstrat-egy will depend directlyon the information t possesses about his adver-sary's possible payoffs. For instance, as shown in later examples, themore optimistic a player's assessment of his opponent's stake in thenegotiation, the more aggressive it can afford to be when bargaining.Second, as has been recently noted by Samuelson [1980] and Crawford[1982], the employment of optimal bargaining strategies may result inthe breakdown of negotiations altogether-this despite the fact that amutually beneficial agreement may exist ex post. Bargaining underuncertainty will, in general,fail to be Pareto efficient.The present paperexamines these issues by modeling bargainingas agame of incomplete information following the pioneering approachofHarsanyi [1967-1968]. The applicationof these ideas to the bargainingproblem is found in Harsanyi and Selten [1972]. Like Harsanyi andSelten, we frame the bargainingsituation as a noncooperativegameandfocus on the resulting set of equilibriumoutcomes. In other respects,however,ourmethods differ.First,we substitutea single stage bargaining



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BargainingUnder ncomplete nformation 837procedure for the multistage representation of Harsanyi and Selten.Acting independentlyand without prior communication, he bargainerssubmit price offers.If these offers arecompatible (in a sense to be definedlater), a transactionis concludedat a pricethat dependson the offers; fthey are not, then no transaction takes place. Though abstractingfromthe dynamics of the negotiation process, the single stage bargainingprocedureemphasizesthe basic strategy trade-off faced by each player.By makinga more aggressiveprice offer, a player earns a greater profitin the event of an agreement but, at the same time, increasesthe risk ofa disagreement (i.e. that his offer is unacceptable o the other).Second, our approachemployscontinuous distributions to summarizethe probabilitybeliefs of the bargainerswhereas Hansanyi and Seltenfocuson discrete distributions.Ouraim is to investigateplayerbargainingstrategies-that is, the mapping between the continuum of possiblevalues that a player mighthold and its priceoffers. The use of continuousdistributionsallows us to characterizea class of equilibria n whichthesebargainingstrategies are well-behaved(i.e. differentiable).Additionally,the assumptionfacilitates the presentationof comparative tatics resultsfor specific examples.Recently,a numberof authors(Rosenthal [1978], Myerson [1979], andMyerson and Satterthwaite [1981] have examined the performance ofarbitrationprocedures n the bargainingcontext. While our model offerspredictions about the frequency of negotiation breakdown,bargainingefficiency, and relativebargainingadvantage,ouremphasis is on individ-ual bargainingstrategyand not on arbitrationperformance.

1. THE BASIC MODELOuranalysis investigates bargainingbehaviorin the well-knowncaseof bilateralmonopoly. Supposethat a single seller of an indivisiblegoodfaces a single potential buyer. A successful bargainis concludedif andonly if the good is transferredat a mutually acceptable price. Let usdenote the seller's reservationprice-the smallest monetarysum he willaccept in exchange for the good (independentof his level of income).Similarly, let Ub denote the buyer's reservationprice (the greatest sumhe is willing to pay for the good). Since a bargainis struckonly when itis agreeableto both parties, the sale price P must satisfy us< P c Ub ifsuch an intervalexists.Incomplete informationof the bargainers s modeledby the followingassumption:Eachpartyknows his ownreservationprice,but is uncertainabout his adversary's,assessing a subjectiveprobabilitydistributionoverthe range of possible values that his opponent might hold. Specifically,the buyer regards us as a random variable possessing a cumulative



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838 ChatterjeendSamuelsondistributionfunctionFb(vu) satisfyingFb(vs) = 0 andFb(08) = 1, andwhichis strictly increasingand differentiableon [vs. s].The seller'sknowledgeof Ub is summarizedby F8(vb) which satisfies F8(Lb) = 0 and Fs(Ob) = 1,and which is strictly increasing and differentiable on [NbNNb]. Thesedistributionfunctions are common knowledgein the sense of Aumann[1976]-that is, each side knowsthese distributions,knowsthat they areknownby the other side, knowsthat the latter knowledge s known,andso forth.In this framework,bargainingbehaviordepends on a player's reser-vation price vi, his assessment of the opponent'sreservationprice,Fi(vj),and the knowledgeof the opponent's assessment,Fj(vi),for i = s, b.

We will consider the following simplebargainingprocedure:BargainingRule. Seller andbuyersubmit sealedoffers,s and b respec-tively. If b > s, a bargain is enacted and the good is sold at price, P =kb + (1 - k)s, where0 c k c 1. If b < s, there is no sale and no moneytrades hands.Special cases of this rule are of some interest. When k equals 1, theruleis equivalentto grantingthe buyerthe rightto make a first and finaloffer that the seller can accept or reject. In this instance, the sale priceis determinedsolely by the buyer's offer, while the seller's offer servesonly to determine whether there is an agreement or not. The seller'soptimal strategy is to submit offers s = usfor all us(i.e. his reservationprice),and an agreement s reached f and only if b > s. Similarly,settingk = 0 effectively grants the seller the first-offerright.Finally,when k =1/2, the rule determines a final sale price by splitting the differencebetween the playeroffers. Both offerscarry equal weight in determiningthe sale price.Framingthe single stage bargain as a noncooperative game, we willcharacterize he resultingNash orBayesianequilibrium olutions. In theevent of an agreement, each player earns a profit measured by thedifferencebetweenthe agreedpriceand his reservationprice (P - us forthe seller and Vb- P for the buyer);in the event of no agreement,eachearns a zero profit. Additionally,we assume that each bargainermakesoffers to maximize his expected profit. The notation b = B(vb) and s =S(vu) indicates that the players' price offers dependon their respectivereservationprices. The functions B( ) and So) will be referred o as theplayeroffer strategies. Then, the expectedprofit of the buyer is givenby

rbWb(b,Vb) = f (Vb - P)gb(s)ds if b ?>s= 0 if b < s,

wheregb(s) is the density function of seller offers inducedby the offerstrategySo) and the underlyingdistribution of values Fb(vu). The upper



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BargainingUnder ncomplete nformation 839and lower supports of the offer distribution are s and s, respectively.Similarly,the expectedprofit of a seller with value vswho makesoffer sand faces a potential buyer using offer strategyB( ) is

rbwrs(s,) = (P - vs)gs(b)db if s b.

Conditional on vs. the seller's offer s* is a best responseagainst B( ) ifws(s*, vs) ? ws(s, vs), for all s. Similarly,a buyer holding Vbmakes offerb* that is a best response against So)if lrb(b*, Vb) - wb(b,Vb), for all b.Then player i employs a best responsestrategyif for each vi his offer isa best response against his opponent'sstrategy.A pair of best responseoffer strategies constitute a Nash (or Bayesian) equilibrium.By defini-tion, neither player can increase his expected profit by unilaterallyalteringhis chosen strategy.

2. PROPERTIESOF EQUILIBRIUMTRATEGIESThis section presents a number of results characterizing he equilib-rium offer strategies of the bargaining game. Offerstrategies satisfying

S(V,) > Vb andB(vb) < v, for all v, and Vbconstitutea no tradeequilibrium.Obviously,these offer strategies are not very interesting.The results inthis section describe equilibria in which an agreement occurs with apositive probability.A fundamentalpropertyof equilibriumofferstrategies for whichtradesoccuris that they areincreasingin the individualreservationprices.Thehigherthe value placedon the goodby the seller (buyer),the higher theprice he demands (offers). The proof of this result requiresno specialassumptions about the offer strategies (e.g. continuity or differentiabil-ity).THEOREM1. Under the sealed offer bargaining rule, the equilibriumbargaining strategies of the buyer and seller are increasing in the respectivereservation prices.

Proof.We present the prooffor only the buyerofferfunctionB( ); thesellerproof is analogous. ConsiderVband Vb'such that Vb=# Vb' and let band b' be the buyer's optimal offers for these respectivevalues. Then byassumption, Wb(b, Vb) ? Wb(b, Vb), and wb(b, Vb') Wb(b,Vb'). Combiningthese inequalities,we find

Wb(b, Vb) - ?7b(b', Vb) + b(b', Vb') - 1b(b, Vb') ? 0. (2)The expression for the buyer's expected profit in (1) implies that7Wb(b, b') -lrb(b, 9Vb) = Gb(b)(vb' - Vb) where Gb(b) is the cumulative distri-



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840 Chatterjee nd Samuelsonbution function of s. In turn, it follows that 7rb(b , Vb)- 7b(b, Vb) =Gb(b')(vb' -Vb). Substitutingthese expressionsinto (2) yields [Gb(b') -Gb(b)][Vb' - Vb] = 0. For Vb' > vb, then Gb(b') > Gb(b). Since Gb() is adistributionfunction,b' > b when the strict inequalityholds. If Gb(b') =Gb(b),the buyer'sprofit maximizationbehavior ensures that b' > b. If b'were smaller than b, a buyer holding Ub could lower his offer to b' andincrease his profit in the event of a bargain without affecting theprobabilityof a bargain.This wouldimplythat b' and not b was optimalfor Vb-a contradiction.This illustrates the basic principlethat a playershould neverplace an offer in an intervalwhere the other playermakesno offer.Our principal objective is to characterizethe class of equilibriaforwhich player offer strategies are well-behaved.We shall consider thefamily A of equilibriafor which the followingassumptionis met: Eachoffer strategy is bounded above and below and is strictly increasing anddifferentiableexcept possibly at these offer bounds. Theorems 2 and 3that follow characterizeclass A equilibria.THEOREM . In a class A equilibrium, over intervals of reservation pricesfor which the offer strategies are strictly increasing, So) and B( ) mustsatisfy the linked differential equations

kFb(y)S'(y) + fb(y)S(y) = B1(S(y))fb(y) (3a)(1 - k)(1 - F.(x))B'(x) - fs(x)B(x) = -S-1(B(x))f,(x). (3b)

Proof. Start with the buyer's equilibriumstrategy.His expectedprofitis

rb71b(b, b) = f [vb - kb - (1 - k)s] gb(s)ds.The first order condition for a maximum is

Owb/ib = (Vb - b)gb(b) - kGb(b)= 0.Note that Gb(b)= Fb(S1(b)), whereS1 is the inverse offer function, i.e.Vs= S1(b). Equivalently, Gb(b) = Fb(y), where y is a dummy variabledefined so that b = S(y). After noting that gb(b) = fb(y)/S'(y) and Vb=B`1(Sy)), we can rewritethe first orderconditionas

[B-1(S(y)) - S(y)]fb(b) - kS'(y)Fb(y) = 0,which, after some rearrangement,s identical to (3a).Similarly,the seller's first ordercondition is

arslas = (vs - s)gs(s) + (1 - k)(1 - Gs(s)) = 0.Then by employing the dummy variable x = B '(s) and making theappropriate ubstitutions,we arriveat (3b).



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BargainingUnder ncomplete nformation 841Equations 3a and 3b are "linked"differentialequations indicatingthatthe player strategies are interdependent.The equations are a preciseexpression of the fact that a player's optimal priceofferdependsnot onlyon vi and Fi(vj), but also indirectly on Fj(vi) since the latter influencesthe opponent's strategy.We can nowcharacterize he equilibrium trategiesat the offerbounds.For referencepurposes,we denote the solutions to (3a) and (3b) over anunrestrictedrangeof reservationprices by S( ) andB( ) respectively.Wedenote the corresponding nverse functions similarly.Since these func-tions are strictly increasing,the maximumand minimumpriceoffersareS(v&) nd S(L) for the seller and B(vb) and B(vb) for the buyer.To illustrate the nature of the boundary solutions consider the casewhen B(Mb)> S(v,) and B(vb) < S(vs). In this case, B( ) describes thebuyer's optimal strategy only for the interval of reservationprices wherethe probabilityof a successfulbargainis positive but smaller than unity.When the buyer employsB), a bargainfirst becomesa certaintywhenB(Vb)= S(&s)-when the buyer'sofferjust matches the greatest possibleseller offer. Equivalently,this occurs at value Vb = BA-(S(Vs)).Clearly,the buyer improves upon B( ) by employingthe constant offer strategy

B(vb) = S(0s),for Vb B-(S(1)), since he increaseshisprofitin instanceswhen he knows a successfulbargainis certain. In this case, S(vs) estab-lishes the upperbound for both the buyerand seller offers.It is straightforwardo check that the seller's best response strategy isunchangedwith this modification n the buyer'soptimal strategy.Againstthe buyer'smodifiedstrategy,the seller'sexpected profit isrs(s,Vs) k bgs(b)db k S(vs)gs(b)db

rb+ [(1-k)s - vs] gs(b)db.Differentiating this expression with respectto s yields precisely the firstordercondition of Theorem 2.Similarly, in the case that B(vb) < S(s), a bargain is impossible forsufficiently low buyer reservation prices. For lower and lower buyervalues, the probabilityof a bargainfirst goes to zero at Vb = S(8s). (At areservation price greater than this, it is suboptimal for the buyer tomatch the lowest seller offer since by bidding instead in the interval(S(V8), Vb), he can earn a positive profit. Thus, the buyer makes thetruthful offer B(vb) = Vb only at Vb =9S(L).) For Vb< S(v8), the buyer'spreciseofferstrategyis largely rrelevantsince in equilibriumno bargainwill ever be concluded. A buyer with Vb in this range faces only onerestriction-that his offer be smaller than B(vb) in order to preserve the



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842 Chatterjee nd Samuelsonzero profit equilibrium. f this restrictionfailed to hold, the buyerwouldearnnegative expectedprofits against the seller'sbest response.In sum, the pair S( ) andB( ) of Theorem2 describeclassA equilibriaexcept at boundaryconditionsthat occur as follows:THEOREM 3. In a classA equilibrium,boundary onditionsare:1) If B(vb)> S(&,), hen B(vb) = S(&,)for Vb> Bl(S(Os))-2) If B(Ob)< S(s), then S(vs) - S(vs) for Vs> B(Ob).3) If B(Cb) > S(U), then S(v8)= B(Vb) for vs < S (B(Qb))-4) If B(vb)


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BargainingUnder ncomplete nformation 843functions are linear. Once the equilibrium strategies have been estab-lished, the results in parts b and c follow from straightforward ompu-tations. Note that part c describes each player'sex ante profit prior tothe "draw" f either reservationprice.For easy reference, Figure 1 plots the equilibrium strategies for the

split-the-difference rule, k = 1/2.Regardless of the specific rule, however,the equilibria of Example 1 satisfy a number of generalproperties. Thebuyer"shades"his offerbelow his true reservationpriceexcept whentheprobabilityof a bargain ust becomeszero.(In Figure1, the buyerstrategyintersects the 450 line precisely at the seller's lowest offer; point B islevel with pointA.) Similarly,the optimal sellerstrategycalls for a "markup." The combination of linear distributions,offer strategies, and bar-

12

10 - ~ ~ ~ ~ ~ ~9

I I I L I I | I | I I _ _ _

8 1 891 11

VsVbs VFigure 1. An identicalbargain (k - Y/2).



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844 Chatterjee nd Samuelsongaining rule imply that the buyer's (seller's)conditionalexpected profitincreases (decreases)as the squareof his reservationprice.The employment of shading and mark up strategies in equilibriumprecludessales in manycircumstanceswhenmutuallybeneficialbargainsexist. A bargain is possible provided vs < Vbbut will only be concludedwhen s c b, or when vs, ((2 - k)/(1 + k))vb - ((1 - k)(2 - k)/2(1 +k))o, using the results in part a. To put this anotherway,the probabilityof a successfulbargainwould be 1/2 if both parties made truthful offers(since F8= Fb). The maximumattainableprobabilitywhen optimalofferstrategiesare used is 9/32. Similarly, expectedgroupprofit undertruthfuloffers is 1/6;under the equilibriumstrategiesit is at most /64.One conclusion of Example I is that one's intuitive notion of whogains from different kinds of compromisescan be mistaken. Supposethat the bargainingrule is k 1/2.A suggestion is made to change therule to k = ,/4 so that the sale price is determinedaccordingto P =(3/4)b+ (1/4)s.In whose interest, buyer or seller, is such a change? Itwould appearat first blush that this changebenefits the sellerand harmsthe buyer. Since a sale is made only when there is some "bargainingspace"between the two offers (i.e., when b - s 0), the move wouldseem to signify a surrenderof this space by the buyer. Should not theswitch signify an increasein profit for the seller and a reductionfor thebuyer?The answer is no. The fallacy of this line of reasoning is that itimplicitly assumes that the bargainingstrategies are unaffected by therule change. It is true that the seller wouldgain andthe buyer would loseif the strategies employedfor k = 1/2were continued to be used for k =3/4. This is not the case, however, as the formulasof Example 1 demon-strate.When k increases,both the buyerand the sellershade theiroffers,the seller moving closer to his reservation price, the buyer toward agreater understatement.The net result is to conferan advantageon thebuyer.In Example 1, the bargainis identical (F, = Fb)and, furthermore, heprobability distribution is symmetric. In this instance, the bargainingrule that calls for splitting the difference(k = 1/2) has a naturalappeal.It is efficient (i.e., it maximizes expectedgroupprofit), and it treats theparties symmetrically.In fact, Myerson and Satterthwaite show that thesplit-the-differencebargainingrule is an optimal mechanism-that is, itattains the greatest group expected profit of all possible bargainingprocedures.For these reasons splitting the difference between offers isattractiveforbargainsthat are identicaland symmetric.The next example characterizesthe equilibrium strategies for noni-denticalbargainswhen the buyerand seller distributionsare uniform.Example2. Suppose that vsis uniformly distributedon the interval [0,



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BargainingUnder ncomplete nformation 8450s]and that Vbs uniformlydistributedon [tb, Ub],where 0 c Vbc Os : Ob.Then we have the followingresults:a) The equilibriumoffer strategyof the seller is specified by:

S(Vs) = vb/(l + k) + (k(l - k)/2(1 + k))Vbfor 0 vs < ((2-k)/(l + k))Vb

-((1 - k)(2 - k)/2(1 + k))VbS(v) =v -(2 -k) + ((1 - k)/2)Ob

for ((2 - k)/(l + k))Vb- ((1 - k)(2 - k)/2(1+ k))bb < s ((2 - k)/2)bbS(vs) > vs(2 - k) + ((1 - k)/2)Ob for ((2 - k)/2) Ub < Vs < Os

b) The equilibrium offer strategy of the buyer is specified by:B(vb) < Vb/(l + k) + (k(l - k)/2(1 +k))Vb

for 0 c Vb < ((1 - k)/2)ObB(vb) = Vb/( + k) + (k(l - k)/2(1 + k))Ob

for ((1 - k)/2)bb < Vb < ((1 + k)/(2 - k))vs + ((1 - k)/2)ObB(Vb) = Vsl/(1 - k) + ((1 -k)/2)Vb

for ((1 + k)/(2 - k))Os+ ((1 - k)/2)Ob < Vjb < VbThe offer strategies are sensitive to changes in the underlying distri-butions in the naturalway. Otherthings equal, increases in Vb,Os, nd Ubimply larger (or no smaller) buyer and seller offers-that is, a shift touniformlyhigherpossible buyeror seller reservationprices (in the senseof stochastic dominance) results in higher offers. As a second exampleconsider a player's offer response to a mean preserving spread in thedistributionof his opponent'sreservationprice.Forinstance, if the sellerbecomes more uncertain about the buyer's value (such that Avb =

-_AVb > 0), the rangeof his offers increasesin equilibrium i.e. he makeshigher (lower)offersthan before when he holds sufficiently large (small)reservationprices). Furthermore,a simple calculation shows that, withthe increase in uncertainty,the seller on average s worse off than before.An analogous result holds for an uncertain buyer. Figures 2 and 3illustratea pair of typical equilibria.A number of points concerning the efficiency of the sealed offerbargainingrulecan be summarized.First, in the nonidenticalsetting therule of thumb of splitting the differencebetween offers will no longermaximize expected group profit (except coincidentally).When the dis-



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846 ChatterjeendSamuelsontributions FS() and Fb( ) differ, the seller and buyer obviously playasymmetricroles. Clearly then, the expected group profit as a functionof the bargainingrule k will vary with vs, vs. Lb, and Vb. To take anextreme case, suppose there is no uncertainty surrounding he buyer'sreservation price. The bargainingrule that calls for the seller to makethe first and final offer (k = 0) extracts 100%of the potential group

profit (since the seller offers s = Vbwhen vs Vb).But a buyer who makesa first and final offer (k = 1) will shade his offer below his true value;consequently, a number of mutually beneficial bargains will be missed.Here, a seller's first offer is superior to any other bargaining rule. Inshort, the bargaining rule that maximizes group profit must be deter-mined on a case-by-case basis.Second, a tedious but straightforward calculation can confirm that the

Vb

Seller / *

O~~~vb v~~"s VbVSVb b

Figure 2. Nonidenticalbargain,k 1/2.



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BargainingUnder ncomplete nformation 847expected profit of the seller (buyer) is decreasing (increasing) n k. Thisresult, which held in the identicalbargainexample, continuesto hold fornonidentical uniformly distributedbargains.Player Risk Aversion. Additional insight into the nature of the bar-gaining equilibriumcan be gained by relaxing the assumption that thebargainers are risk neutral. Suppose that the seller and the buyer,

respectively, earn von Neumann-Morgenstern utilities u,(P - vs) andUb(Vb - P) in the event of an agreement and zero utilities otherwise.Then it is easy to check that the new first order conditions are

-bUb(Vb- b) -k Ub(Vb- kb - (I1-k) s)gb(s)s = O.

Vb

s, b e l r Buyer

0 Vb VS VbVS,Vb I

Figure 3. Nonidenticalbargain,k = 1/2.



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848 ChatterjeendSamuelsonand

uS(s - vs) -(1-k) us(kb + (1 - k)s - vs)gs(b)db= 0.With these expressions,we can present equilibria n the case that: 1) theplayers' utility functions display constant relative risk aversion, u(y) =ya, where0 < a < 1, and 2) the underlyingdistributionsareuniform. Asa simple illustration, suppose the players have the same degreeof riskaversion,the underlyingdistributionsareboth uniformon [0, 1], and thebargaining rule is k = 1/2.Then the equilibrium strategies are linear:

S = -( 1/2)/(2f2 _ 1/2) + [1 -(1/2f)]vsb = (1 - 1/2f3)/(4f2 - 1) + [1 - (1/2f)]vbwhereA = 21/a -1/2. In this symmetric example,the offerstrategieshavethe same slope, and the buyer's intercept is smaller than the seller's.Considerthe effect on the bargaining strategies as the players becomeincreasinglyrisk averse:As a decreases, : increasescausingthe slope ofeach offer strategy to increase (and the intercept to fall). The result isgenerallylower seller offers and higher buyer offers. The intuition un-derlyingthis result is that, with risk aversion,the marginal ncrement inprofit associatedwith a slighty moreagressiveoffer (a higherseller offeror a lowerbuyer offer) is weighted less heavily than the possible loss, ifas a result of the change, an agreementis precluded (b < s). This factleads risk-averse bargainersto make offers closer to their true valuesthan their risk-neutralcounterparts.In the limit as a approaches0 (i.e.the players become infinitely risk averse), the equilibrium strategiesapproach he strategiess = vsand b = Vb for all vsand Vb. Consequently,b 2 s if and only if Vb UU,and so all mutually beneficial agreementsareattained. Short of this extreme case, however, potentially beneficialbargainswill be lost.In the case of differing degrees of risk aversion, one can show that,other things equal, an increase in the risk aversionof the seller (buyer)implies uniformly lower (higher) offers by both parties in equilibrium.Not surprisingly, he opponent'sbest response to more truthful bids bythe player who has become more risk averse is to make more aggressiveoffershimself.

4. CONCLUDING EMARKSIn what respects do the equilibria described in Theorems 2 and 3provide an accurate representation of bargaining under uncertainty?Since the conclusions of any model dependon its premises, it is well toexamine the extent to whichthe model's assumptions captureor approx-



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BargainingUnder ncomplete nformation 849imate the actual bargainingconditions. First, uncertaintyconcerning heopponent's walk away price is frequently encountered in bargainingsituations. It is the task of each bargainer o assess the likely reservationprice of his opponent. Indeed, the better the bargainer's informationabout his opponent, the better he can expect to fare in the negotiations.For instance, the man on the street is better equipped o bargainwith acar dealer if he possesses the "book"that lists the prices the dealerhimself has paid for various models and if he has assessed the currentstate of demandfor automobiles.Probabilityassessment becomesmorecomplicated n an environmentwith stochastic dependence between the player values. For instance,suppose that each bargainer'svalue for the good depends on future(unknown)market conditions as well as on his personalcharacteristics.Additionally, suppose that each bargainer possesses differential infor-mation bearing on the good's potential value. Then, based on thisinformation,eachplayermust estimate his opponent'svalue and his ownvalue in the event there is an agreementor not. In particular,the factthat the playerswould concludean agreement (or not) is informative ofthe good'sultimate value.While the single-stage bargainingrule fails to capturethe pattern ofreciprocalconcessions observedin everydaypractice,it is a useful ideal-ization and a starting point for other investigations. A more generalmodelwouldallow the bargainersmultipleroundsin which to exchangeoffers (potentially incurrng"transaction"costs in the process). To theextent that the exchange of offers conveys informationabout the reser-vation prices, one would expect agreementsto become more frequent.For example, when a bargain is unsuccessful at the initial stage, eachparty could revise his probabilityassessment of the other's reservationprice and could make a concession in his next offer. These kinds ofmultistage bargainscan be properly analyzedwithin our frameworkanddeserve further attention. (For an example, see Sobel and Takahashi[1980].)In our view, there are strong argumentsfor the monotonic equilibriaof Theorems 2 and 3 even when other equilibriamay exist. An individualis likely to accept the following propositionas reasonable:"The higherthe seller's value the more he demands for the good; the higher thebuyer'svalue the more he is willing to offer."Acceptingthis principleand believing firmlythat any bargainingopponent will accept it as well,the individualthen seeks a best monotonic offerstrategy.Monotonic offer strategies may also provide a focal point for thebargainingprocess.As a simple example, supposethe buyer'sreservationprice is uniformly distributed on the interval [0.5, 1] and the seller'sprice is similarly distributed on the interval [0, 0.5]. In this case, theconstant offers S(v,) = B(vb) = 0.5 constitute an equilibriumand guar-



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850 ChatterjeendSamuelsonantee that the players always concludeda bargain.Each party choosesnot to push further its demand,nor to retreat,expectinghis opponenttofeel the same way. One must not overlook,however,the existence of asecond equilibriumin monotonic strategies. In this case, both partiesmake offers that are smaller than 0.5 for sufficiently low reservationprices and offers greater than 0.5 for high prices. Thus, it is prudentfora buyer who values the good highly to make a generousoffer, therebyincreasing the likelihoodof an agreementwhen facinga seller who maynot cooperatewith an offer of S(v,) = 0.5.Equilibriumtheory can furnish no definitive conclusion indicatingwhich outcome represents the "more logical" bargaining focal-point.Observe, however, that the monotonic equilibrium is responsive tochanges in the underlyingdistributionsFS and Fb, while the constantstrategy equilibrium s not. If, for instance, the buyer'svalue is drawnuniformly from the interval [0.5, 5] instead of [0.5, 11, the constantstrategy equilibriumS(v,) = B(vb) = 0.5 becomes much less compelling.Is this a logical resting point-a position from which neither playerexpects the other to retreat? It would seem that the buyer could beexpected to concede (settling for a smallerbut still substantialprofit) ifthe seller insisted. The monotonic equilibrium s free from these criti-cisms since it depends explicitly on the underlyingdistributions.Withthe shift in possible buyer values, the offers of the buyerbecome moregenerous and those of the seller more demanding. This response isconsistent with the expectation that such a shift should benefit bothparties (not just the buyer).Finally, when Us> Qb, there is no guaranteethat a mutuallybeneficialagreementis always available.In this case of a "close"bargain,there isno constant strategy equilibrium. ndeed,the case of an identical bargainF, = Fb points out the main difficulty with a constant offer strategy. Theemployment of an aggressive offer will likely preclude an agreement,while an offer making too great a concession will be unprofitable. Inshort, players will adopt monotonic strategies because they are moreefficient.We would argue that close bargains occur most frequentlyin actualpractice. In most bargainingsituations, the opportunityfor the seller orthe buyerto cease negotiationsandto considera third-party ransactionis available, at least implicitly.It is unusualfor the buyer to be the onlypotential customer for the good or for the seller to have a monopolyposition. In this instance, each party's reservationprice will reflect, atleast partially, the price (net of search transaction costs) that he couldexpect to obtain elsewhere. To the extent to which beliefs about theprices available from third parties transactionsdiffer, or access to theseparties differs,the reservationprices of the buyer and seller will diverge.Nevertheless,one wouldexpect such an opportunity o minimizethe gap



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BargainingUnder ncomplete nformation 851betweenthe possible reservationpricesof the parties.This suggeststhatclose bargains may be of the greatestpracticalimportance.

ACKNOWLEDGMENTWe would like to thank Jerry Green, Takao Kobayashi, and JohnPratt for helpful commentsand especiallyHoward Raiffafor suggestingthe problem.

REFERENCESAUMANN, R. J. 1976.Agreeing o Disagree.Ann. Statist. 4, 1236-1239.CRAWFORD,. P. 1982.A Theory of Disagreement n Bargaining.Econometrica50, 602-637.CROSS,. G. 1969. The Economicsof Bargaining.Basic Books, New York.HARSANYI,. C. 1956.Approaches o the BargainingProblem Before and Afterthe Theory of Games.Econometrica 4, 144-157.HARSANYI,. C. 1967-1968. Games with Incomplete Information Played byBayesian Players. Mgmt. Sci. 14, 159-182, 321-334, 486-502.HARSANYI,. C., ANDR. SELTEN. 972.A GeneralizedNash Solution for Two-Person BargainingGameswith IncompleteInformation.Mgmt.Sci. 18, P80-P106.MYERSON, . B. 1979. Incentive Compatibilityand the BargainingProblem.Econometrica 7, 61-73.MYERSON,R. B., AND M. SATTERTHWAITE.981. Efficient Mechanisms forBilateral Trading,"DiscussionPaper 469S, NorthwesternUniverstiy.NASH, . C. 1950.The BargainingProblem.Econometrica18, 151-162.ROSENTHAL,. W. 1978.Arbitrationof Two-Party DisputesunderUncertainty.Rev.Econ. Studies 45, 595-604.ROTH,A. E. 1979.AxiomaticModelsof Bargaining.Springer-Verlag.SAMUELSON, W. 1980.First-OfferBargains.Mgmt. Sci. 26, 155-164.SCHELLING,. 1960. The Strategyof Conflict.HarvardUniversity Press, Cam-

bridge,Mass.SOBEL,J., AND I. TAKAHASHI.980. A Multi-Stage Model of Bargaining, Discus-sion Paper 80-25, Universityof California,San Diego.YOUNG,. R. (ED.)1975.Bargaining:FormalTheoriesof Negotiations.Universityof Illinois Press, Champaign-Urbana,ll.

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