178 D.A. Seale et al. J. of Economic Behavior Org. 44 2001 177–200
the reservation value of the other. The party that is most successful in narrowing down the uncertainty about the other’s value is usually best able to press for a favorable transaction
price. The present paper reports on experiments that have uncovered a significant and robust information effect in bilateral bargaining.
In a seminal paper, Chatterjee and Samuelson 1983 constructed Bayesian Nash linear equilibrium strategies LES for the buyer and seller in the sealed-bid k-double auction
mechanism and determined the profit that each person could expect from employing these strategies. In the course of a series of laboratory experiments, we have found general support
for this model. However, with rare exceptions, we have also found that our sellers who were in a weak information position achieved significantly lower gains from trade than they
could have obtained if they bid as LES suggests and our buyers, who were endowed with an information advantage, achieved significantly higher gains.
This paper proceeds as follows. We first describe the sealed-bid k-double auction mecha- nism and then present the LES for two bargaining situations examined in the present study.
Our focus is on asymmetric two-person bargaining with incomplete information Linhart et al., 1992 in which one of the traders has a distinct information advantage. In contrast to
out previous studies Daniel et al., 1998, hereafter DSR; and Rapoport et al., 1998, hereafter RDS, the information structure examined in the present paper favors the seller, not the buyer.
We next present two new experiments that manipulate the amount of information that the seller possesses about the buyer’s reservation values. These experiments are mirror images
of two previous experiments conducted by DSR in which the information advantage was conferred on the buyer. We exploit this feature of the design to measure the increase in profit
that the trader-either buyer or seller-derives from the information advantage. Our results show that in the environment we examine property rights do not matter; rather, it is the infor-
mation advantage that determines the change in profit from equilibrium play. They also show that the learning model proposed by DSR is equally applicable to account for the individual
asks and bids in bargaining situations that favor the seller. The concluding section discusses the applicability of these results to less structured two-person bargaining situations.
2. The sealed-bid kkk-double auction mechanism
Consider a bilateral bargaining situation, where a seller has a single object that he may sell to the buyer if an acceptable price, p, is agreed upon. Assume that V denotes the buyer’s
reservation value, the maximum price she is willing to pay for the item, and C denotes the seller’s reservation value, the minimum price he is willing to accept for the same item.
Both traders are assumed to be risk-neutral, expected utility maximizers, and their utility functions are normalized so that if no trade occurs then the utility of each is zero. If trade
occurs, then the gains from trade for the seller and buyer are p − C and V − p, respectively.
What each trader knows about the reservation value of the other is modeled in the follow- ing fashion. Each trader’s reservation value is a random variable whose value is contained in
some interval. The reservation values C and V for the seller and buyer, respectively, are ran- domly and independently drawn from the distributions F and G. The two distributions F and
G are assumed to be common knowledge, whereas the actual reservation values are private.
D.A. Seale et al. J. of Economic Behavior Org. 44 2001 177–200 179
Under the sealed-bid k-double auction mechanism, the seller submits an ask c and simul- taneously the buyer submits a bid v. If k = 12, as in the present study, trade occurs with
no delay at the price p = v + c2 which is midway between the bid and ask, if v ≥ c. If v c
, then negotiations end and no trade occurs. The best known Bayesian Nash equilibrium solution for the sealed-bid mechanism with
two-sided incomplete information is the LES of Chatterjee and Samuelson 1983. Leininger et al. 1989 have shown that there are many other Bayesian Nash equilibria for this mech-
anism. However, Myerson and Satterthwaite 1983 proved that the LES maximizes the expected ex ante gains from trade achievable by any such mechanism. This provides justi-
fication for its central role in the analysis of bilateral trading.
A strategy for a trader is a real-valued function, called the bid function for the buyer and ask function for the seller, defined on the support of the distribution of this trader’s
reservation value. It specifies an askbid for each reservation value. When F and G are both uniformly distributed over the interval [0, 100], the LES strategies are
ν =
V , if V ≤ 25
25 3
+ 2
3 V ,
if V 25 1
for the buyer, and c =
C,
if C ≥ 75 25 +
2 3
C, if C 75
for the seller. These two equilibrium strategies are portrayed graphically in the top panel of Fig. 1. Both functions are piecewise linear and make only modest departures from the ‘truth
telling’ line, which corresponds to a trader fully revealing his or her reservation value. For the two experimental conditions reported on in this paper, an information asymmetry
has been created which results in more distinct LES functions. In the first condition, called seller’s advantage SA, F is uniformly distributed over the interval [0, 200], and G is
uniformly distributed over the interval [100, 200]. This represents a bargaining situation where the seller knows that the buyer’s best alternative for the object is equally likely to
be any value between 100 and 200 in monetary units. On the other hand, the buyer is less certain as to the seller’s value; it can be anywhere between 0 and 200. The LES strategies
for condition SA are given by
v= 50
3 +
2 3
V , if 100 ≤ V ≤ 200
for the buyer, and c =
C,
if C ≥ 150 50 +
2 3
C, if 50 ≤ C 150
250 3
, if C 50
2 for the seller. These two functions are displayed in the middle panel of Fig. 1.
180 D.A. Seale et al. J. of Economic Behavior Org. 44 2001 177–200
Fig. 1. Linear equilibrium strategies for three experimental conditions.
D.A. Seale et al. J. of Economic Behavior Org. 44 2001 177–200 181
In our second condition, the seller has a considerably larger advantage SLA. F is now uniformly distributed over the interval [0, 200], whereas G is uniformly distributed over the
considerably narrower interval [180, 200]. In condition SLA, the seller has little doubt as to the value placed by the buyer on the item being traded. The pair of LES is given by
v= 50
3 +
2 3
V , if 180 ≤ V ≤ 200
for the buyer, and
c =
C, if C ≥ 150
50 + 2
3 C,
if 130 ≤ C 150 410
3 ,
if C 130 3
for the seller. These two functions are displayed in the bottom panel of Fig. 1. Contrast the distinct departures from the truth-telling line in the bottom panel with the nearly linear
function in the top panel. The seller never asks less than 136.67 for any item, even one that he values at 0.
2.1. Previous experimental research Following earlier studies by Radner and Schotter 1989, Schotter 1990, and Rapoport
and Fuller 1995, in which the focus was on testing the model of Chatterjee and Samuel- son and on comparisons between the LES and a model postulating truth-telling, DSR and
RDS shifted the focus to information structures that favor the buyer. They did so by in- cluding the support of the seller’s prior distribution F in the support of the buyer’s prior
distribution G. Their intention was to provide an environment in which LES behavior was distinctly different from truth telling. An unexpected finding of this study was that, rela-
tive to the potential payoffs achievable under LES, the seller’s performance was distinctly worse than the buyer’s. Most of the buyers used bidding strategies that generally con-
formed to the LES but with experience bid even more aggressively than predicted by the LES. On the other hand, the sellers tended to bid between the LES and truth telling levels,
thereby achieving profits much lower than those attainable if both parties bid in accordance with LES.
Subsequently, RDS designed and conducted a set of experiments that retained the infor- mation asymmetries of the DRS study but kept the bargaining pairing fixed rather than
random throughout the 50 trials. In contrast to the DSR study, each trader now had the opportunity to establish a reputation against a fixed partner. A tentative explanation of the
DSR results asserts that individual sellers, playing against a different buyer on each trial, could not effectively overcome initial perceptions that they were in a weak position. Buyers,
it was argued, responded to the ‘field’ of sellers and the efforts of individuals were largely futile. However, RDS reported that again the sellers bid cautiously and attained profits that
were at, or even below, the levels of the DSR study.
182 D.A. Seale et al. J. of Economic Behavior Org. 44 2001 177–200
While these studies have answered many questions regarding the strategic behavior of traders, they have also raised an important new one. What are the primary factors governing
the realized gain from trade attained by each party? Radner and Schotter found that sellers more closely approximated the behavior posited by the LES than did the buyers, who tended
not to shave their bids by amounts great enough to satisfy the LES prediction. Consequently, the sellers earned slightly more than they would expect if both traders played either truthfully
or according to the LES. DSR, and subsequently RDS, reported the opposite results and hypothesized that this difference is attributable to the disparity in information between
the seller and buyer that was introduced into their experiments. Radner and Schotter used prior distributions of reservation values with the same support for both traders. As a result,
an explanation of their results must rest on other factors such as the possible perception on the part of both traders that the seller, endowed with the property rights, has intrinsic
power.
To provide a more comprehensive test of the DSR hypothesis, one must also include information conditions favoring the seller. The present study compares the results of the
previous two experiments by DSR to those of two new experimental conditions in which the information disparity favors the seller. Condition SA reverses the advantage provided
to the buyer in Experiment 1 of DSR. The seller’s reservation value is drawn from a wider uniform distribution [0, 200] than is the buyer’s value [100, 200]. Likewise, con-
dition SLA reverses the roles of buyer and seller in Experiment 2 of DSR. It provides an even stronger information advantage to the seller by restricting the buyer’s values to
the interval [180, 200]. Our results show that these changes in the information provided to each trader have a profound effect on the profits they realize. Combined with the re-
sults from the two previous studies, they present a powerful information effect in bilateral bargaining.
What would happen in the experiment of DSR or in the present experiment in which the information disparity is reversed if the traders in the weak position were to play more
aggressively — more in accordance with LES? In a third experiment that we conducted, called condition BAC, the buyer’s reservation values are drawn from a uniform distribution
on [0, 200] and the seller’s from a uniform distribution defined on [0, 100] as in Experiment 1 of DSR in fact, the same sequence of values is used. Significantly, however, and unknown
to the buyers who were only told that they would be randomly matched with a seller on each round, the sellers are programmed to respond with LES asks on every trial of the
experiment. Condition BAC is introduced to determine whether the buyer’s bids will be moderated when the seller is no longer willing to be ‘pushed down’.
Clearly, traders do not solve paired ordinary differential equations to arrive at the LES; rather, their behavior stabilizes through some process of adjustment. The dynamics of play
is a second major focus of the present study. Observing a significant learning effect in the behavior of traders over the course of 50 trials, DSR formulated a learning model to
describe the process by which buyers and sellers change their decisions from round to round. Comparing the predicted and observed decisions of both buyers and sellers, the
reinforcement-based learning model that they proposed accounted for, on the average, al- most 90 percent of the round-to-round variability in the individual bids. A major purpose
of the present study is to determine whether the DSR learning model accounts for the new set of data in which sellers now have the information advantage.
D.A. Seale et al. J. of Economic Behavior Org. 44 2001 177–200 183
3. Method
