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Bargaining under Incomplete Information
Author(s): Kalyan Chatterjee and William Samuelson
Source: Operations Research, Vol. 31, No. 5, (Sep. - Oct., 1983), pp. 835-851
Published by: INFORMS
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Bargaining under Incomplete Information
KALYAN CHATTERJEE
Pennsylvania State University, University Park, Pennsylvania
WILLIAM SAMUELSON
Boston University, Boston, Massachusetts
(Received July 1981; revised April 1982; accepted April 1983)
This paper presents and analyzes a bargaining model of bilateral monopoly
under uncertainty. Under the bargaining rule proposed, the buyer and the seller
each submit sealed offers that determine whether the good in question is sold
and the transfer price. The Nash equilibrium solution of this bargaining ame
implies an offer strategy of each party that is monotonic in its own reservation
price and depends on its assessment of the opponent's reservation price.
Issues of relative bargaining advantage and efficiency are examined for a
number of special cases. Finally, we discuss the appropriateness of the Nash
solution concept.
THIS PAPER presents a simple model of two-person bargaining
under incomplete information. Applications of the model range from
the settlement of a claim out of court, to union-management negotiations,
to the purchase and sale of a used automobile. The common feature
shared by these examples is that each bargainer, while certain of the
potential value it places on the transaction, has only probabilistic infor-
mation concerning the potential value of the other. For instance, in
haggling over the price of a used car, neither buyer nor seller knows the
other's walk-away price.
Our principal aim is to investigate equilibrium bargaining behavior
under uncertainty and to study how the parties fare, individually and
collectively, under a simple class of bargaining procedures. A main result
is that, given incomplete information, not all mutually beneficial agree-
ments can be attained via bargaining. Even when the buyer values the
good more highly than the seller, a successful sale may be impossible.
Additionally, we present a number of results characterizing the players'
optimal bargaining strategies and give explicit solutions for a number of
examples. These examples provide useful comparative statics results-
indicating the effect on bargaining behavior of changes in the negotiation
Subject classification: 231 bargaining, 232 bargaining.
835
Operations Research 0030-364X/83/3105-0835 $01.25
Vol. 31, No. 5, September-October 1983 ? 1983 Operations Research Society of America

836 Chatterjee and Samuelson
ground rules, the information available to the players, and the degree of
the players' risk aversion.
Complete vs. Incomplete Information. Most game-theoretic literature
on bargaining has assumed that the participants possess complete infor-
mation about the negotiation situation. Important contributions are
provided by Nash [1950], Harsanyi [1956], Schelling [1960], Cross
[1969], and Roth [1979]. Young [1975] provides an excellent summary
and critique of the most important literature on the bargaining problem.
In two-person bargains (to which our discussion will be limited), it is
customary to stipulate that the set of possible actions of the parties and
the player payoffs from any combination of actions are common knowl-
edge. In the model of bilateral monopoly considered here, with the
presumption of complete information, each bargainer, seller and buyer,
knows the other's walk-away price. Then the bargainers negotiate a final
price in the known range of mutually acceptable prices (i.e. between the
seller's minimum and the buyer's maximum acceptable prices). Since any
price in this range can be supported as an equilibrium outcome, bargain-
ing solutions are usually determined either by specifying a concession
mechanism leading to a distinct outcome, or a set of axioms that a
"reasonable" outcome should satisfy.
Despite its many insights, the complete information approach fails to
mirror key features of actual negotiations: 1) the fact that each bargainer
is uncertain about its adversary's payoff, and 2) the occurrence of
"unreasonable" bargaining outcomes-breakdowns in negotiations,
strikes, and work stoppages-when mutually beneficial agreements are
possible. A bargaining model that embraces these features carries a
number of noteworthy implications. First, each party's bargaining strat-
egy will depend directly on the information it possesses about his adver-
sary's possible payoffs. For instance, as shown in later examples, the
more optimistic a player's assessment of his opponent's stake in the
negotiation, 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 in
the breakdown of negotiations altogether-this despite the fact that a
mutually beneficial agreement may exist ex post. Bargaining under
uncertainty will, in general, fail to be Pareto efficient.
The present paper examines these issues by modeling bargaining as a
game of incomplete information following the pioneering approach of
Harsanyi [1967-1968]. The application of these ideas to the bargaining
problem is found in Harsanyi and Selten [1972]. Like Harsanyi and
Selten, we frame the bargaining situation as a noncooperative game and
focus on the resulting set of equilibrium outcomes. In other respects,
however, our methods differ. First, we substitute a single stage bargaining

Bargaining Under Incomplete Information 837
procedure for the multistage representation of Harsanyi and Selten.
Acting independently and without prior communication, the bargainers
submit price offers. If these offers are compatible (in a sense to be defined
later), a transaction is concluded at a price that depends on the offers; if
they are not, then no transaction takes place. Though abstracting from
the dynamics of the negotiation process, the single stage bargaining
procedure emphasizes the basic strategy trade-off faced by each player.
By making a more aggressive price offer, a player earns a greater profit
in the event of an agreement but, at the same time, increases the risk of
a disagreement (i.e. that his offer is unacceptable to the other).
Second, our approach employs continuous distributions to summarize
the probability beliefs of the bargainers whereas Hansanyi and Selten
focus on discrete distributions. Our aim is to investigate player bargaining
strategies-that is, the mapping between the continuum of possible
values that a player might hold and its price offers. The use of continuous
distributions allows us to characterize a class of equilibria in which these
bargaining strategies are well-behaved (i.e. differentiable). Additionally,
the assumption facilitates the presentation of comparative statics results
for specific examples.
Recently, a number of authors (Rosenthal [1978], Myerson [1979], and
Myerson and Satterthwaite [1981] have examined the performance of
arbitration procedures in the bargaining context. While our model offers
predictions about the frequency of negotiation breakdown, bargaining
efficiency, and relative bargaining advantage, our emphasis is on individ-
ual bargaining strategy and not on arbitration performance.
1. THE BASIC MODEL
Our analysis investigates bargaining behavior in the well-known case
of bilateral monopoly. Suppose that a single seller of an indivisible good
faces a single potential buyer. A successful bargain is concluded if and
only if the good is transferred at a mutually acceptable price. Let us
denote the seller's reservation price-the smallest monetary sum he will
accept in exchange for the good (independent of his level of income).
Similarly, let Ub denote the buyer's reservation price (the greatest sum
he is willing to pay for the good). Since a bargain is struck only when it
is agreeable to both parties, the sale price P must satisfy us < P c Ub if
such an interval exists.
Incomplete information of the bargainers is modeled by the following
assumption: Each party knows his own reservation price, but is uncertain
about his adversary's, assessing a subjective probability distribution over
the range of possible values that his opponent might hold. Specifically,
the buyer regards us as a random variable possessing a cumulative

838 Chatterjee and Samuelson
distribution function Fb(vu) satisfying Fb(vs) = 0 and Fb(08) = 1, and which
is strictly increasing and differentiable on [vs. s]. The seller's knowledge
of Ub is summarized by F8(vb) which satisfies F8(Lb) = 0 and Fs(Ob) = 1,
and which is strictly increasing and differentiable on [NbN Nb]. These
distribution functions are common knowledge in the sense of Aumann
[1976]-that is, each side knows these distributions, knows that they are
known by the other side, knows that the latter knowledge is known, and
so forth.
In this framework, bargaining behavior depends on a player's reser-
vation price vi, his assessment of the opponent's reservation price, Fi(vj),
and the knowledge of the opponent's assessment, Fj(vi), for i = s, b.
We will consider the following simple bargaining procedure:
Bargaining Rule. Seller and buyer submit sealed offers, s and b respec-
tively. If b > s, a bargain is enacted and the good is sold at price, P =
kb + (1 - k)s, where 0 c k c 1. If b < s, there is no sale and no money
trades hands.
Special cases of this rule are of some interest. When k equals 1, the
rule is equivalent to granting the buyer the right to make a first and final
offer that the seller can accept or reject. In this instance, the sale price
is determined solely by the buyer's offer, while the seller's offer serves
only to determine whether there is an agreement or not. The seller's
optimal strategy is to submit offers s = us for all us (i.e. his reservation
price), and an agreement is reached if and only if b > s. Similarly, setting
k = 0 effectively grants the seller the first-offer right. Finally, when k =
1/2, the rule determines a final sale price by splitting the difference
between the player offers. Both offers carry equal weight in determining
the sale price.
Framing the single stage bargain as a noncooperative game, we will
characterize the resulting Nash or Bayesian equilibrium solutions. In the
event of an agreement, each player earns a profit measured by the
difference between the agreed price and his reservation price (P - us for
the seller and Vb- P for the buyer); in the event of no agreement, each
earns a zero profit. Additionally, we assume that each bargainer makes
offers to maximize his expected profit. The notation b = B(vb) and s =
S(vu) indicates that the players' price offers depend on their respective
reservation prices. The functions B( ) and So ) will be referred to as the
player offer strategies. Then, the expected profit of the buyer is given by
rb
Wb(b, Vb) = f (Vb - P)gb(s)ds if b ?> s
= 0 if b < s,
where gb(s) is the density function of seller offers induced by the offer
strategy So ) and the underlying distribution of values Fb(vu). The upper

Bargaining Under Incomplete Information 839
and lower supports of the offer distribution are s and s, respectively.
Similarly, the expected profit of a seller with value vs who makes offer s
and faces a potential buyer using offer strategy B( ) is
rb
wrs(s, v) = f (P - vs)gs(b)db if s <
=0 if s>b.
Conditional on vs. the seller's offer s* is a best response against B( ) if
ws(s*, vs) ? ws(s, vs), for all s. Similarly, a buyer holding Vb makes offer
b* that is a best response against So )if lrb(b*, Vb) - wb(b, Vb), for all b.
Then player i employs a best response strategy if for each vi his offer is
a best response against his opponent's strategy. A pair of best response
offer strategies constitute a Nash (or Bayesian) equilibrium. By defini-
tion, neither player can increase his expected profit by unilaterally
altering his chosen strategy.
2. PROPERTIES OF EQUILIBRIUM STRATEGIES
This section presents a number of results characterizing the equilib-
rium offer strategies of the bargaining game. Offer strategies satisfying
S(V,) > Vb and B(vb) < v, for all v, and Vb constitute a no trade equilibrium.
Obviously, these offer strategies are not very interesting. The results in
this section describe equilibria in which an agreement occurs with a
positive probability.
A fundamental property of equilibrium offer strategies for which trades
occur is that they are increasing in the individual reservation prices. The
higher the value placed on the good by the seller (buyer), the higher the
price he demands (offers). The proof of this result requires no special
assumptions about the offer strategies (e.g. continuity or differentiabil-
ity).
THEOREM 1. Under the sealed offer bargaining rule, the equilibrium
bargaining strategies of the buyer and seller are increasing in the respective
reservation prices.
Proof. We present the proof for only the buyer offer function B( ); the
seller proof is analogous. Consider Vb and Vb' such that Vb =# Vb' and let b
and b' be the buyer's optimal offers for these respective values. Then by
assumption, Wb(b, Vb) ? Wb(b, Vb), and wb(b, Vb') ! Wb(b, Vb'). Combining
these 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 that
7Wb(b, Vb') -lrb(b, 9Vb) = Gb(b)(vb' - Vb) where Gb(b) is the cumulative distri-

840 Chatterjee and Samuelson
bution function of s. In turn, it follows that 7rb(b , Vb)- 7b(b, Vb) =
Gb(b')(vb' -Vb). Substituting these expressions into (2) yields [Gb(b') -
Gb(b)][Vb' - Vb] = 0. For Vb' > vb, then Gb(b') > Gb(b). Since Gb() is a
distribution function, b' > b when the strict inequality holds. If Gb(b') =
Gb(b), the buyer's profit maximization behavior ensures that b' > b. If b'
were smaller than b, a buyer holding Ub could lower his offer to b' and
increase his profit in the event of a bargain without affecting the
probability of a bargain. This would imply that b' and not b was optimal
for Vb-a contradiction. This illustrates the basic principle that a player
should never place an offer in an interval where the other player makes
no offer.
Our principal objective is to characterize the class of equilibria for
which player offer strategies are well-behaved. We shall consider the
family A of equilibria for which the following assumption is met: Each
offer strategy is bounded above and below and is strictly increasing and
differentiable except possibly at these offer bounds. Theorems 2 and 3
that follow characterize class A equilibria.
THEOREM 2. In a class A equilibrium, over intervals of reservation prices
for which the offer strategies are strictly increasing, So ) and B( ) must
satisfy 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 equilibrium strategy. His expected profit
is
rb
71b(b, Vb) = 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)), where S1 is the inverse offer function, i.e.
Vs = S1(b). Equivalently, Gb(b) = Fb(y), where y is a dummy variable
defined so that b = S(y). After noting that gb(b) = fb(y)/S'(y) and Vb =
B`1(Sy)), we can rewrite the first order condition as
[B-1(S(y)) - S(y)]fb(b) - kS'(y)Fb(y) = 0,
which, after some rearrangement, is identical to (3a).
Similarly, the seller's first order condition is
arslas = (vs - s)gs(s) + (1 - k)(1 - Gs(s)) = 0.
Then by employing the dummy variable x = B '(s) and making the
appropriate substitutions, we arrive at (3b).

Bargaining Under Incomplete Information 841
Equations 3a and 3b are "linked" differential equations indicating that
the player strategies are interdependent. The equations are a precise
expression of the fact that a player's optimal price offer depends not only
on vi and Fi(vj), but also indirectly on Fj(vi) since the latter influences
the opponent's strategy.
We can now characterize the equilibrium strategies at the offer bounds.
For reference purposes, we denote the solutions to (3a) and (3b) over an
unrestricted range of reservation prices by S( ) and B( ) respectively. We
denote the corresponding inverse functions similarly. Since these func-
tions are strictly increasing, the maximum and minimum price offers are
S(v&) and S(L) for the seller and B(vb) and B(vb) for the buyer.
To illustrate the nature of the boundary solutions consider the case
when B(Mb) > S(v,) and B(vb) < S(vs). In this case, B( ) describes the
buyer's optimal strategy only for the interval of reservation prices where
the probability of a successful bargain is positive but smaller than unity.
When the buyer employs B), a bargain first becomes a certainty when
B(Vb) = S(&s)-when the buyer's offer just matches the greatest possible
seller offer. Equivalently, this occurs at value Vb = BA-(S(Vs)). Clearly,
the buyer improves upon B( ) by employing the constant offer strategy
B(vb) = S(0s), for Vb > B-(S(1)), since he increases his profit in instances
when he knows a successful bargain is certain. In this case, S(vs) estab-
lishes the upper bound for both the buyer and seller offers.
It is straightforward to check that the seller's best response strategy is
unchanged with this modification in the buyer's optimal strategy. Against
the buyer's modified strategy, the seller's expected profit is
rs(s, Vs) = k f bgs(b)db + k f S(vs)gs(b)db
rb
+ [(1-k)s - vs] gs(b)db.
Differentiating this expression with respect to s yields precisely the first
order condition of Theorem 2.
Similarly, in the case that B(vb) < S(s), a bargain is impossible for
sufficiently low buyer reservation prices. For lower and lower buyer
values, the probability of a bargain first goes to zero at Vb = S(8s). (At a
reservation price greater than this, it is suboptimal for the buyer to
match the lowest seller offer since by bidding instead in the interval
(S(V8), Vb), he can earn a positive profit. Thus, the buyer makes the
truthful offer B(vb) = Vb only at Vb =9S(L).) For Vb < S(v8), the buyer's
precise offer strategy is largely irrelevant since in equilibrium no bargain
will ever be concluded. A buyer with Vb in this range faces only one
restriction-that his offer be smaller than B(vb) in order to preserve the

842 Chatterjee and Samuelson
zero profit equilibrium. If this restriction failed to hold, the buyer would
earn negative expected profits against the seller's best response.
In sum, the pair S( ) and B( ) of Theorem 2 describe class A equilibria
except at boundary conditions that occur as follows:
THEOREM 3. In a class A equilibrium, boundary conditions are:
1) If B(vb) > S(&,), then 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) <S(gs), then B(Vb) C B(O) for Vb < S(vs).
Thus, the range of "serious" offers is bounded from below by max[S(vL),
B(Lb)1 and from above by min[S(os), IB(b)]
3. ANALYSIS OF SPECIAL CASES
While Theorems 2 and 3 characterize equilibrium strategies that are
well-behaved, this by no means exhausts the set of all equilibrium
solutions. For instance, it is possible to construct other equilibria involv-
ing discontinuous offer strategies that place probability masses on spe-
cific offer values. Furthermore, even the linked differential equations of
Theorem 2 resist analytic solutions for all but the most elementary
distribution functions. The examples that follow are limited to distribu-
tions belonging to the uniform family. We begin with an examination of
"identical" bargains-cases in which Fs( ) = Fb( ).
Example 1. Suppose the parties bargain under the sealed offer rule
with Fs(v) = Fb(v) = v/v. We have the following results:
a) The equilibrium offer strategies, So ) and B( ), are given by:
S(v8) = vs/(2 - k) + ((1 - k)/2)v for 0 c v. < ((2 - k)/2)0
S(v,) 2 v,/(2 - k) + ((1 - k)/2)vt for ((2 - k)/2) v < vs < 0
B(vb) < vb/(1 + k) + (k(l - k)/2(1 + k))0 for 0 c Vb < ((1- k)/2)v
B(vb) = vb/(l + k) + (k(l - k)/2(1 + k))o for ((1 - k)/2) V C Vb C 0.
b) The probability that a bargain is reached equals (-k2 + k + 2)/8
and achieves it maximum value, 9/32, at k = 1/2.
c) i) The seller's expected profit is 7rs(k) = (0/48)(2 - k)2(1 + k) which
is strictly decreasing in k.
ii) The buyer's expected profit is 7rb(k) = (0/48)(1 + k)2(2 - k) which
is strictly increasing in k.
iii) The sum of the parties' profits is 7rs + 7rb = (v/16)(1 + k)(2 -k)
which has a maximum of (9/64)0, at k = 1/2.
By simple inspection one can check that linear offer strategies above
constitute solutions of Equations 3a and 3b when both distribution

Bargaining Under Incomplete Information 843
functions are linear. Once the equilibrium strategies have been estab-
lished, the results in parts b and c follow from straightforward compu-
tations. Note that part c describes each player's ex ante profit prior to
the "draw" of either reservation price.
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 general properties. The
buyer "shades" his offer below his true reservation price except when the
probability of a bargain just becomes zero. (In Figure 1, the buyer strategy
intersects the 450 line precisely at the seller's lowest offer; point B is
level with point A.) Similarly, the optimal seller strategy calls for a "mark
up." 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 V
Figure 1. An identical bargain (k - Y/2).

844 Chatterjee and Samuelson
gaining rule imply that the buyer's (seller's) conditional expected profit
increases (decreases) as the square of his reservation price.
The employment of shading and mark up strategies in equilibrium
precludes sales in many circumstances when mutually beneficial bargains
exist. A bargain is possible provided vs < Vb but will only be concluded
when 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 another way, the probability
of a successful bargain would be 1/2 if both parties made truthful offers
(since F8 = Fb). The maximum attainable probability when optimal offer
strategies are used is 9/32. Similarly, expected group profit under truthful
offers is 1/6; under the equilibrium strategies it is at most /64.
One conclusion of Example I is that one's intuitive notion of who
gains from different kinds of compromises can be mistaken. Suppose
that the bargaining rule is k 1/2. A suggestion is made to change the
rule to k = ,/4 so that the sale price is determined according to P =
(3/4)b + (1/4)s. In whose interest, buyer or seller, is such a change? It
would appear at first blush that this change benefits the seller and harms
the buyer. Since a sale is made only when there is some "bargaining
space" between the two offers (i.e., when b - s ! 0), the move would
seem to signify a surrender of this space by the buyer. Should not the
switch signify an increase in profit for the seller and a reduction for the
buyer?
The answer is no. The fallacy of this line of reasoning is that it
implicitly assumes that the bargaining strategies are unaffected by the
rule change. It is true that the seller would gain and the buyer would lose
if the strategies employed for k = 1/2 were continued to be used for k =
3/4. This is not the case, however, as the formulas of Example 1 demon-
strate. When k increases, both the buyer and the seller shade their offers,
the seller moving closer to his reservation price, the buyer toward a
greater understatement. The net result is to confer an advantage on the
buyer.
In Example 1, the bargain is identical (F, = Fb) and, furthermore, the
probability distribution is symmetric. In this instance, the bargaining
rule that calls for splitting the difference (k = 1/2) has a natural appeal.
It is efficient (i.e., it maximizes expected group profit), and it treats the
parties symmetrically. In fact, Myerson and Satterthwaite show that the
split-the-difference bargaining rule is an optimal mechanism-that is, it
attains the greatest group expected profit of all possible bargaining
procedures. For these reasons splitting the difference between offers is
attractive for bargains that are identical and symmetric.
The next example characterizes the equilibrium strategies for noni-
dentical bargains when the buyer and seller distributions are uniform.
Example 2. Suppose that vs is uniformly distributed on the interval [0,

Bargaining Under Incomplete Information 845
0s] and that Vb is uniformly distributed on [tb, Ub], where 0 c Vb c Os : Ob.
Then we have the following results:
a) The equilibrium offer strategy of the seller is specified by:
S(Vs) = vb/(l + k) + (k(l - k)/2(1 + k))Vb
for 0 vs < ((2-k)/(l + k))Vb
-((1 - k)(2 - k)/2(1 + k))Vb
S(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)bb
S(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)Ob
B(vb) = Vb/( + k) + (k(l - k)/2(1 + k))Ob
for ((1 - k)/2)bb < Vb < ((1 + k)/(2 - k))vs + ((1 - k)/2)Ob
B(Vb) = Vsl/(1 - k) + ((1 -k)/2)Vb
for ((1 + k)/(2 - k))Os + ((1 - k)/2)Ob < Vjb < Vb
The offer strategies are sensitive to changes in the underlying distri-
butions in the natural way. Other things equal, increases in Vb, Os, and Ub
imply larger (or no smaller) buyer and seller offers-that is, a shift to
uniformly higher possible buyer or seller reservation prices (in the sense
of stochastic dominance) results in higher offers. As a second example
consider a player's offer response to a mean preserving spread in the
distribution of his opponent's reservation price. For instance, if the seller
becomes more uncertain about the buyer's value (such that Avb =
-_AVb > 0), the range of his offers increases in equilibrium (i.e. he makes
higher (lower) offers than before when he holds sufficiently large (small)
reservation prices). Furthermore, a simple calculation shows that, with
the increase in uncertainty, the seller on average is worse off than before.
An analogous result holds for an uncertain buyer. Figures 2 and 3
illustrate a pair of typical equilibria.
A number of points concerning the efficiency of the sealed offer
bargaining rule can be summarized. First, in the nonidentical setting the
rule of thumb of splitting the difference between offers will no longer
maximize expected group profit (except coincidentally). When the dis-

846 Chatterjee and Samuelson
tributions FS( ) and Fb( ) differ, the seller and buyer obviously play
asymmetric roles. Clearly then, the expected group profit as a function
of the bargaining rule k will vary with vs, vs. Lb, and Vb. To take an
extreme case, suppose there is no uncertainty surrounding the buyer's
reservation price. The bargaining rule that calls for the seller to make
the first and final offer (k = 0) extracts 100% of the potential group
profit (since the seller offers s = Vb when vs , Vb). But a buyer who makes
a 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. In
short, 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 Vb
V SVb b
Figure 2. Nonidentical bargain, k 1/2.

Bargaining Under Incomplete Information 847
expected profit of the seller (buyer) is decreasing (increasing) in k. This
result, which held in the identical bargain example, continues to hold for
nonidentical uniformly distributed bargains.
Player Risk Aversion. Additional insight into the nature of the bar-
gaining equilibrium can be gained by relaxing the assumption that the
bargainers are risk neutral. Suppose that the seller and the buyer,
respectively, earn von Neumann-Morgenstern utilities u,(P - vs) and
Ub(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
-b
Ub(Vb- b) -k Ub(Vb- kb - (I1-k) s)gb(s) ds = O.
Vb
s, b l Sel ler X
Buyer
0 Vb VS Vb
VS,Vb I
Figure 3. Nonidentical bargain, k = 1/2.

848 Chatterjee and Samuelson
and
uS(s - vs) -(1-k) f us(kb + (1 - k)s - vs)gs(b)db = 0.
With these expressions, we can present equilibria in the case that: 1) the
players' utility functions display constant relative risk aversion, u(y) =
ya, where 0 < a < 1, and 2) the underlying distributions are uniform. As
a simple illustration, suppose the players have the same degree of risk
aversion, the underlying distributions are both uniform on [0, 1], and the
bargaining rule is k = 1/2. Then the equilibrium strategies are linear:
S = -( 1/2)/(2f2 _ 1/2) + [1 -(1/2f)]vs
b = (1 - 1/2f3)/(4f2 - 1) + [1 - (1/2f)]vb
where A = 21/a -1/2. In this symmetric example, the offer strategies have
the same slope, and the buyer's intercept is smaller than the seller's.
Consider the effect on the bargaining strategies as the players become
increasingly risk averse: As a decreases, : increases causing the slope of
each offer strategy to increase (and the intercept to fall). The result is
generally lower seller offers and higher buyer offers. The intuition un-
derlying this result is that, with risk aversion, the marginal increment in
profit associated with a slighty more agressive offer (a higher seller offer
or a lower buyer offer) is weighted less heavily than the possible loss, if
as a result of the change, an agreement is precluded (b < s). This fact
leads risk-averse bargainers to make offers closer to their true values
than their risk-neutral counterparts. In the limit as a approaches 0 (i.e.
the players become infinitely risk averse), the equilibrium strategies
approach the strategies s = vs and b = Vb for all vs and Vb. Consequently,
b 2 s if and only if Vb U vU, and so all mutually beneficial agreements are
attained. Short of this extreme case, however, potentially beneficial
bargains will be lost.
In the case of differing degrees of risk aversion, one can show that,
other things equal, an increase in the risk aversion of the seller (buyer)
implies uniformly lower (higher) offers by both parties in equilibrium.
Not surprisingly, the opponent's best response to more truthful bids by
the player who has become more risk averse is to make more aggressive
offers himself.
4. CONCLUDING REMARKS
In what respects do the equilibria described in Theorems 2 and 3
provide an accurate representation of bargaining under uncertainty?
Since the conclusions of any model depend on its premises, it is well to
examine the extent to which the model's assumptions capture or approx-

Bargaining Under Incomplete Information 849
imate the actual bargaining conditions. First, uncertainty concerning the
opponent's walk away price is frequently encountered in bargaining
situations. It is the task of each bargainer to assess the likely reservation
price of his opponent. Indeed, the better the bargainer's information
about his opponent, the better he can expect to fare in the negotiations.
For instance, the man on the street is better equipped to bargain with a
car dealer if he possesses the "book" that lists the prices the dealer
himself has paid for various models and if he has assessed the current
state of demand for automobiles.
Probability assessment becomes more complicated in an environment
with stochastic dependence between the player values. For instance,
suppose that each bargainer's value for the good depends on future
(unknown) market conditions as well as on his personal characteristics.
Additionally, suppose that each bargainer possesses differential infor-
mation bearing on the good's potential value. Then, based on this
information, each player must estimate his opponent's value and his own
value in the event there is an agreement or not. In particular, the fact
that the players would conclude an agreement (or not) is informative of
the good's ultimate value.
While the single-stage bargaining rule fails to capture the pattern of
reciprocal concessions observed in everyday practice, it is a useful ideal-
ization and a starting point for other investigations. A more general
model would allow the bargainers multiple rounds in which to exchange
offers (potentially incurrng "transaction" costs in the process). To the
extent that the exchange of offers conveys information about the reser-
vation prices, one would expect agreements to become more frequent.
For example, when a bargain is unsuccessful at the initial stage, each
party could revise his probability assessment of the other's reservation
price and could make a concession in his next offer. These kinds of
multistage bargains can be properly analyzed within our framework and
deserve further attention. (For an example, see Sobel and Takahashi
[1980].)
In our view, there are strong arguments for the monotonic equilibria
of Theorems 2 and 3 even when other equilibria may exist. An individual
is likely to accept the following proposition as reasonable: "The higher
the seller's value the more he demands for the good; the higher the
buyer's value the more he is willing to offer." Accepting this principle
and believing firmly that any bargaining opponent will accept it as well,
the individual then seeks a best monotonic offer strategy.
Monotonic offer strategies may also provide a focal point for the
bargaining process. As a simple example, suppose the buyer's reservation
price is uniformly distributed on the interval [0.5, 1] and the seller's
price is similarly distributed on the interval [0, 0.5]. In this case, the
constant offers S(v,) = B(vb) = 0.5 constitute an equilibrium and guar-

850 Chatterjee and Samuelson
antee that the players always concluded a bargain. Each party chooses
not to push further its demand, nor to retreat, expecting his opponent to
feel the same way. One must not overlook, however, the existence of a
second equilibrium in monotonic strategies. In this case, both parties
make offers that are smaller than 0.5 for sufficiently low reservation
prices and offers greater than 0.5 for high prices. Thus, it is prudent for
a buyer who values the good highly to make a generous offer, thereby
increasing the likelihood of an agreement when facing a seller who may
not cooperate with an offer of S(v,) = 0.5.
Equilibrium theory can furnish no definitive conclusion indicating
which outcome represents the "more logical" bargaining focal -point.
Observe, however, that the monotonic equilibrium is responsive to
changes in the underlying distributions FS and Fb, while the constant
strategy equilibrium is not. If, for instance, the buyer's value is drawn
uniformly from the interval [0.5, 5] instead of [0.5, 11, the constant
strategy equilibrium S(v,) = B(vb) = 0.5 becomes much less compelling.
Is this a logical resting point-a position from which neither player
expects the other to retreat? It would seem that the buyer could be
expected to concede (settling for a smaller but still substantial profit) if
the seller insisted. The monotonic equilibrium is free from these criti-
cisms since it depends explicitly on the underlying distributions. With
the shift in possible buyer values, the offers of the buyer become more
generous and those of the seller more demanding. This response is
consistent with the expectation that such a shift should benefit both
parties (not just the buyer).
Finally, when Us> Q b, there is no guarantee that a mutually beneficial
agreement is always available. In this case of a "close" bargain, there is
no constant strategy equilibrium. Indeed, the case of an identical bargain
F, = Fb points out the main difficulty with a constant offer strategy. The
employment of an aggressive offer will likely preclude an agreement,
while an offer making too great a concession will be unprofitable. In
short, players will adopt monotonic strategies because they are more
efficient.
We would argue that close bargains occur most frequently in actual
practice. In most bargaining situations, the opportunity for the seller or
the buyer to cease negotiations and to consider a third-party transaction
is available, at least implicitly. It is unusual for the buyer to be the only
potential customer for the good or for the seller to have a monopoly
position. In this instance, each party's reservation price will reflect, at
least partially, the price (net of search transaction costs) that he could
expect to obtain elsewhere. To the extent to which beliefs about the
prices available from third parties transactions differ, or access to these
parties differs, the reservation prices of the buyer and seller will diverge.
Nevertheless, one would expect such an opportunity to minimize the gap

Bargaining Under Incomplete Information 851
between the possible reservation prices of the parties. This suggests that
close bargains may be of the greatest practical importance.
ACKNOWLEDGMENT
We would like to thank Jerry Green, Takao Kobayashi, and John
Pratt for helpful comments and especially Howard Raiffa for suggesting
the problem.
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