Ten Little Treasures of Game Theory

and Ten Intuitive Contradictions

By Jacob K. Goeree and Charles A. Holt*

This paper reports laboratory data for games that are played only once. These
games span the standard categories: static and dynamic games with complete and
incomplete information. For each game, the treasure is a treatment in which
behavior conforms nicely to predictions of the Nash equilibrium or relevant refine-
ment. In each case, however, a change in the payoff structure produces a large
inconsistency between theoretical predictions and observed behavior. These con-
tradictions are generally consistent with simple intuition based on the interaction of
payoff asymmetries and noisy introspection about others’ decisions. (JEL C72, C92)

The Nash equilibrium has been the center- sequential Nash equilibrium for dynamic games
piece of game theory since its introduction with incomplete information (e.g., Robert Gib-
about 50 years ago. Along with supply and bons, 1997). The rationality assumptions that
demand, the Nash equilibrium is one of the underlie this analysis are often preceded by
most commonly used theoretical constructs in persuasive adjectives like “perfect,” “intuitive,”
economics, and it is increasingly being applied and “divine.” If any noise in decision-making is
in other social sciences. Indeed, game theory admitted, it is eliminated in the limit in a pro-
has finally gained the central role envisioned by cess of “purification.” It is hard not to notice
John von Neumann and Oscar Morgenstern, and parallels with theology, and the highly mathe-
in some areas of economics (e.g., industrial matical nature of the developments makes this
organization) virtually all recent theoretical de- work about as inaccessible to mainstream econ-
velopments are applications of game theory. omists as medieval treatises on theology would
The impression one gets from recent surveys have been to the general public.
and game theory textbooks is that the field has The discordant note in this view of game
reached a comfortable maturity, with neat clas- theory has come primarily from laboratory ex-
sifications of games and successively stronger periments, but the prevailing opinion among
(more “refined”) versions of the basic approach game theorists seems to be that behavior will
being appropriate for more complex categories eventually converge to Nash predictions under
of games: Nash equilibrium for static games the right conditions.1 This paper presents a
with complete information, Bayesian Nash for much more unsettled perspective of the current
static games with incomplete information, sub- state of game theory. In each of the major types
game perfectness for dynamic games with com- of games, we present one or more examples for
plete information, and some refinement of the which the relevant version of the Nash equilib-
rium predicts remarkably well. These “trea-
sures” are observed in games played only once
* Goeree: Department of Economics, 114 Rouss Hall,
University of Virginia, P.O. Box 400182, Charlottesville, by financially motivated subjects who have had
VA 22904, and University of Amsterdam, Roetersstraat 11, prior experience in other, similar, strategic sit-
1018 WB Amsterdam, The Netherlands (e-mail: uations. In each of these games, however, we
jg2n@virginia.edu); Holt: Department of Economics, 114 show that a change in the payoff structure can
Rouss Hall, University of Virginia, P.O. Box 400182, Char-
lottesville, VA 22904 (e-mail: cah2k@virginia.edu). We
produce a large inconsistency between theoret-
wish to thank Rachel Parkin and Scott Saiers for research
assistance, and Colin Camerer, Monica Capra, Glenn Har-
1
rison, Susan Laury, Melayne McInnes, Theo Offerman, For example, George J. Mailath’s (1998) survey of
Amnon Rapoport, Joel Sobel, and an anonymous referee for evolutionary models cites the failure of backward induction
helpful comments. This research was funded in part by the as the main cause of behavioral deviations from Nash pre-
National Science Foundation (SES–9818683). dictions.
1402
VOL. 91 NO. 5 GOEREE AND HOLT: TREASURES AND CONTRADICTIONS 1403

ical prediction(s) and human behavior. For ex- offs for this game written on a blackboard, and
ample, a payoff change that does not alter the he invented the prisoner’s dilemma story that
unique Nash equilibrium may move the data to was later used in a lecture on game theory that
the opposite side of the range of feasible deci- he gave in the Psychology Department at Stan-
sions. Alternatively, a payoff change may cause ford (Tucker, 1950).
a major shift in the game-theoretic predictions Interestingly, Nash’s response to Dresher and
and have no noticeable effect on actual behav- Flood’s repeated prisoner’s dilemma experi-
ior. The observed contradictions are typically ment is contained in a note to the authors that
somewhat intuitive, even though they are not was published as a footnote to their paper:
explained by standard game theory. In a simul-
taneous effort-choice coordination game, for The flaw in the experiment as a test of
example, an increase in the cost of players’ equilibrium point theory is that the exper-
“effort” decisions is shown to cause a dramatic iment really amounts to having the play-
decrease in effort, despite the fact that any com- ers play one large multi-move game. One
mon effort is a Nash equilibrium for a range of cannot just as well think of the thing as a
effort costs. In some of these games, it seems sequence of independent games as one
can in zero-sum cases. There is just too
like the Nash equilibrium works only by coin- much interaction . . . (Nasar, 1998 p.
cidence, e.g., in symmetric cases where the 119).
costs of errors in each direction are balanced. In
other cases, the Nash equilibrium has consider- In contrast, the experiments that we report in
able drawing power, but economically signifi- this paper involved games that were played only
cant deviations remain to be explained. once, although related results for repeated
The idea that game theory should be tested games with random matching will be cited
with laboratory experiments is as old as the where appropriate. As Nash noted, the advan-
notion of a Nash equilibrium, and indeed, the tage of one-shot games is that they insulate
classic prisoner’s dilemma paradigm was in- behavior from the incentives for cooperation
spired by an experiment conducted at the and reciprocity that are present in repeated
RAND Corporation in 1950. Some of the stra- games. One potential disadvantage of one-shot
tegic analysts at RAND were dissatisfied with games is that, without an opportunity to learn
the received theory of cooperative and zero-sum and adapt, subjects may be especially prone to
games in von Neumann and Morgenstern’s the effects of confusion. The games used in
(1944) Theory of Games and Economic Behav- this paper, however, are simple enough in
ior. In particular, nuclear conflict was not structure to ensure that Nash-like behavior
thought of as a zero-sum game because both can be observed in the “treasure” treatment.
parties may lose. Sylvia Nasar (1998) describes In addition, the study of games played only
the interest at RAND when word spread that a once is of independent interest given the
Princeton graduate student, John Nash, had gen- widespread applications of game theory to
eralized von Neumann’s existence proof for model one-shot interactions in economics and
zero-sum games to the class of all games with other social sciences, e.g., the FCC license
finite numbers of strategies. Two mathemati- auctions, elections, military campaigns, and
cians, Melvin Dresher and Merrill Flood, had legal disputes.
been running some game experiments with their The categories of games to be considered are
colleagues, and they were skeptical that human based on the usual distinctions: static versus
behavior would be consistent with Nash’s no- dynamic and complete versus incomplete infor-
tion of equilibrium. In fact, they designed an mation. Section I describes the experiments
experiment that was run on the same day they based on static games with complete informa-
heard about Nash’s proof. Each player in this tion: social dilemma, matching pennies, and
game had a dominant strategy to defect, but coordination games. Section II contains results
both would earn more if they both used the from dynamic games with complete informa-
cooperative strategy. The game was repeated tion: bargaining games and games with threats
100 times with the same two players, and a fair that are not credible. The games reported in
amount of cooperation was observed. One of Sections III and IV have incomplete infor-
Nash’s professors, Al W. Tucker, saw the pay- mation about other players’ payoffs: in static
1404 THE AMERICAN ECONOMIC REVIEW DECEMBER 2001

settings (auctions) and two-stage settings (sig- models with elements of both belief and rein-
naling games). forcement learning (Colin Camerer and Teck-
It is well known that decisions can be af- Hua Ho, 1999). Learning from experience is not
fected by psychological factors such as framing, possible in games that are only played once, and
aspiration levels, social distance, and heuristics beliefs must be formed from introspective
(e.g., Daniel Kahneman et al., 1982; Catherine thought processes, which may be subject to
Eckel and Rick Wilson, 1999). In this paper we considerable noise. Without noise, iterated best
try to hold psychological factors constant and responses will converge to a Nash equilibrium,
focus on payoff changes that are primarily eco- if they converge at all. Some promising ap-
nomic in nature. As noted below, economic proaches to explaining deviations from Nash
theories can and are being devised to explain the predictions are based on models that limit play-
resulting anomalies. For example, the rational- ers’ capacities for introspection, either by lim-
choice assumption underlying the notion of a iting the number of iterations (e.g., Dale O.
Nash equilibrium eliminates all errors, but if the Stahl and Paul W. Wilson, 1995; Rosemarie
costs of “overshooting” an optimal decision are Nagel, 1995) or by injecting increasing amounts
much lower than the costs of “undershooting,” of noise into higher levels of iterated beliefs
one might expect an upward bias in decisions. (Goeree and Holt, 1999; Dorothea Kübler and
In a game, the endogenous effects of such bi- Georg Weizsäcker, 2000). The predictions de-
ases may be reinforcing in a way that creates a rived from these approaches, discussed in Sec-
“snowball” effect that moves decisions well tion V, generally conform to Nash predictions
away from a Nash prediction. Models that in- in the treasure treatments and to the systematic,
troduce (possibly small) amounts of noise into intuitive deviations in the contradiction treat-
the decision-making process can produce pre- ments. Some conclusions are offered in
dictions that are quite far from any Nash equi- Section VI.
librium (Richard D. McKelvey and Thomas R.
Palfrey, 1995, 1998; Goeree and Holt, 1999). I. Static Games with Complete Information
Equilibrium models of noisy behavior have
been used to explain behavior in a variety of In this section we consider a series of two-
contexts, including jury decision-making, bar- player, simultaneous-move games, for which
gaining, public goods games, imperfect price the Nash equilibria show an increasing degree
competition, and coordination (Simon P. of complexity. The first game is a “social di-
Anderson et al., 1998a, b, 2001a; C. Monica lemma” in which the pure-strategy Nash equi-
Capra et al., 1999, 2002; Stanley S. Reynolds, librium coincides with the unique rationalizable
1999; Serena Guarnaschelli et al., 2001). outcome. Next, we consider a matching pennies
A second type of rationality assumption that game with a unique Nash equilibrium in mixed
is built into the Nash equilibrium is that beliefs strategies. Finally, we discuss coordination
are consistent with actual decisions. Beliefs are games that have multiple Nash equilibria, some
not likely to be confirmed out of equilibrium, of which are better for all players.
and learning will presumably occur in such In all of the games reported here and in
cases. There is a large recent literature on in- subsequent sections, we used cohorts of student
corporating learning into models of adjustment subjects recruited from undergraduate econom-
in games that are played repeatedly with differ- ics classes at the University of Virginia. Each
ent partners.2 These models include adaptive cohort consisted of ten students who were paid
learning (e.g., Vincent P. Crawford, 1995; $6 for arriving on time, plus all cash they earned
David J. Cooper et al., 1997), naive Bayesian in the games played. These one-shot games
learning (e.g., Jordi Brandts and Holt, 1996; followed an initial “part A” in which the sub-
Dilip Mookherjee and Barry Sopher, 1997), re- jects played the same two-person game for ten
inforcement or stimulus-response learning (e.g., periods with new pairings made randomly in
Ido Erev and Alvin E. Roth, 1998), and hybrid each period.3 Earnings for the two-hour ses-

2 3
See, for instance, Drew Fudenberg and David K. Le- We only had time to run about six one-shot games in
vine (1998) for a survey. each session, so the data are obtained from a large number
VOL. 91 NO. 5 GOEREE AND HOLT: TREASURES AND CONTRADICTIONS 1405

sions ranged from $15 to $60, with an average 210 ⫺ R respectively. Since R ⬎ 1, the best
of about $35. Each one-shot game began with response is to undercut the other’s decision by 1
the distribution and reading of the instructions (if that decision were known), and therefore, the
for that game.4 These instructions contained upper bound 300 is never a best response to any
assurances that all money earned would be paid possible beliefs that one could have. Conse-
and that the game would be followed by “an- quently, a rational person must assign a proba-
other, quite different, decision-making experi- bility of zero to a choice of 300, and hence 299
ment.” Since the one-shot treatments were cannot be a best response to any possible beliefs
paired, we switched the order of the treasure that rule out choices of 300, etc. Only the lower
and contradiction conditions in each subsequent bound 180 survives this iterated deletion pro-
session. Finally, the paired treatments were al- cess and is thus the unique rationalizable action,
ways separated by other one-shot games of a and hence the unique Nash equilibrium.6 This
different type. game was introduced by Kaushik Basu (1994)
who coined it the “traveler’s dilemma” game.7
A. The One-Shot Traveler’s Dilemma Game Although the Nash equilibrium for this game
can be motivated by successively dropping
The Nash equilibrium concept is based on the those strategies that are never a best response
twin assumptions of perfect error-free decision- (to any beliefs about strategies that have not yet
making and the consistency of actions and be- been eliminated from consideration), this dele-
liefs. The latter requirement may seem tion process may be too lengthy for human
especially strong in the presence of multiple subjects with limited cognitive abilities. When
equilibria when there is no obvious way for the cost of having the higher number is small,
players to coordinate. More compelling argu- i.e., for small values of R, one might expect
ments can be given for the Nash equilibrium more errors in the direction of high claims, well
when it predicts the play of the unique justifi- away from the unique equilibrium at 180, and
able, or rationalizable, action (B. Douglas indeed this is the intuition behind the dilemma.
Bernheim, 1984; David G. Pierce, 1984). Ra- In contrast, with a large penalty for having the
tionalizability is based on the idea that players higher of the two claims, players are likely to
should eliminate those strategies that are never end up with claims that are near the unique
a best response for any possible beliefs, and Nash prediction of 180.
realize that other (rational) players will do the To test these hypotheses we asked 50 subjects
same.5 (25 pairs) to make choices in a treatment with
To illustrate this procedure, consider the R ⫽ 180, and again in a matched treatment
game in which two players independently and with R ⫽ 5. All subjects made decisions in
simultaneously choose integer numbers be- each treatment, and the two games were sepa-
tween (and including) 180 and 300. Both play- rated by a number of other one-shot games. The
ers are paid the lower of the two numbers, and,
in addition, an amount R ⬎ 1 is transferred
from the player with the higher number to the 6
In other games, rationalizability may allow outcomes
player with the lower number. For instance, if that are not Nash equilibria, so it is a weaker concept than
one person chooses 210 and the other chooses that of a Nash equilibrium, allowing a wider range of
possible behavior. It is in this sense that Nash is more
250, they receive payoffs of 210 ⫹ R and persuasive when it corresponds to the unique rationalizable
outcome.
7
The associated story is that two travelers purchase
of sessions where part A involved a wide range of repeated identical antiques while on a tropical vacation. Their lug-
games, including public goods, coordination, price compe- gage is lost on the return trip, and the airline asks them to
tition, and auction games that are reported in other papers. make independent claims for compensation. In anticipation
The one-shot games never followed a repeated game of the of excessive claims, the airline representative announces:
same type. “We know that the bags have identical contents, and we will
4
These instructions can be downloaded from http:// entertain any claim between $180 and $300, but you will
www.people.virginia.edu/⬃cah2k/datapage.html. each be reimbursed at an amount that equals the minimum of
5
A well-known example for which this iterated deletion the two claims submitted. If the two claims differ, we will
process results in a unique outcome is a Cournot duopoly also pay a reward R to the person making the smaller claim
game with linear demand (Fudenberg and Jean Tirole, 1993 and we will deduct a penalty R from the reimbursement to
pp. 47– 48). the person making the larger claim.”
1406 THE AMERICAN ECONOMIC REVIEW DECEMBER 2001

TABLE 1—THREE ONE-SHOT MATCHING PENNIES GAMES

(WITH CHOICE PERCENTAGES)

Left (48) Right (52)

Symmetric Top (48) 80, 40 40, 80
matching Bottom (52) 40, 80 80, 40
pennies
Left (16) Right (84)
Asymmetric Top (96) 320, 40 40, 80
matching Bottom (4) 40, 80 80, 40
pennies
Left (80) Right (20)
Reversed Top (8) 44, 40 40, 80
asymmetry Bottom (92) 40, 80 80, 40

FIGURE 1. CLAIM FREQUENCIES IN A TRAVELER’S DILEMMA

FOR R ⫽ 180 (LIGHT BARS) AND R ⫽ 5 (DARK BARS)
the treatment change does not alter the unique
Nash (and rationalizable) prediction, standard
game theory simply cannot explain the most
ordering of the two treatments was alternated. salient feature of the data, i.e., the effect of the
The instructions asked the participants to devise penalty/reward parameter on average claims.
their own numerical examples to be sure that
they understood the payoff structure. B. A Matching Pennies Game
Figure 1 shows the frequencies for each 10-
cent category centered around the claim label on Consider a symmetric matching pennies
the horizontal axis. The lighter bars pertain to game in which the row player chooses between
the high-R “treasure” treatment, where close to Top and Bottom and the column player simul-
80 percent of all the subjects chose the Nash taneously chooses between Left and Right, as
equilibrium strategy, with an average claim of shown in top part of Table 1. The payoff for the
201. However, roughly the same fraction chose row player is $0.80 when the outcome is (Top,
the highest possible claim in the low-R treat- Left) or (Bottom, Right) and $0.40 otherwise.
ment, for which the average was 280, as shown The motivations for the two players are exactly
by the darker bars. Notice that the data in the opposite: column earns $0.80 when row earns
contradiction treatment are clustered at the op- $0.40, and vice versa. Since the players have
posite end of the set of feasible decisions from opposite interests there is no equilibrium in pure
the unique (rationalizable) Nash equilibrium.8 strategies. Moreover, in order not to be ex-
Moreover, the “anomalous” result for the low-R ploited by the opponent, neither player should
treatment does not disappear or even diminish favor one of their strategies, and the mixed-
over time when subjects play the game repeat- strategy Nash equilibrium involves randomiz-
edly and have the opportunity to learn.9 Since ing over both alternatives with equal
probabilities. As before, we obtained decisions
from 50 subjects in a one-shot version of this
8
This result is statistically significant at all conventional
game (five cohorts of ten subjects, who were
levels, given the strong treatment effect and the relatively randomly matched and assigned row or column
large number of independent observations (two paired ob-
servations for each of the 50 subjects). We will not report
specific nonparametric tests for cases that are so clearly
significant. The individual choice data are provided in the in successive reductions in average claims. With a penalty/
Data Appendix for this paper on: http://www.people. reward parameter of 5, 10, 20, 25, 50, and 80 the average
virginia.edu/⬃cah2k/datapage.html. claims in the final three periods were 195, 186, 119, 138, 85,
9
In Capra et al. (1999), we report results of a repeated and 81 respectively. Even though there is one treatment
traveler’s dilemma game (with random matching). When reversal, the effect of the penalty/reward parameter on av-
subjects chose numbers in the range [80, 200] with R ⫽ 5, erage claims is significant at the 1-percent level. The pat-
the average claim rose from approximately 180 in the first terns of adjustment are well explained by a naive Bayesian
period to 196 in period 5, and the average remained above learning model with decision error, and the claim distribu-
190 in later periods. Different cohorts played this game with tions for the final five periods are close to those predicted by
different values of R, and successive increases in R resulted a logit equilibrium (McKelvey and Palfrey, 1995).
VOL. 91 NO. 5 GOEREE AND HOLT: TREASURES AND CONTRADICTIONS 1407

roles). The choice percentages are shown in TABLE 2—AN EXTENDED COORDINATION GAME
parentheses next to the decision labels in the top
L H S
part of Table 1. Note that the choice percentages L 90, 90 0, 0 x, 40
are essentially “50-50,” or as close as possible H 0, 0 180, 180 0, 40
given that there was an odd number of players
in each role.
Now consider what happens if the row play-
er’s payoff of $0.80 in the (Top, Left) box is in an intuitive manner, with row players’
increased to $3.20, as shown in the asymmetric choices responding to their own payoffs.11 In
matching pennies game in the middle part of this context, the Nash mixed-strategy prediction
Table 1. In a mixed-strategy equilibrium, a seems to work only by coincidence, when the
player’s own decision probabilities should be payoffs are symmetric.
such that the other player is made indifferent
between the two alternatives. Since the column C. A Coordination Game with a Secure
player’s payoffs are unchanged, the mixed- Outside Option
strategy Nash equilibrium predicts that row’s
decision probabilities do not change either. In Games with multiple Nash equilibria pose
other words, the row player should ignore the interesting new problems for predicting behav-
unusually high payoff of $3.20 and still choose ior, especially when some equilibria produce
Top or Bottom with probabilities of one-half. higher payoffs for all players. The problem of
(Since column’s payoffs are either 40 or 80 for coordinating on the high-payoff equilibrium
playing Left and either 80 or 40 for playing may be complicated by the possible gains and
Right, row’s decision probabilities must equal losses associated with payoffs that are not part
one-half to keep column indifferent between of any equilibrium outcome. Consider a coordi-
Left and Right, and hence willing to random- nation game in which players receive $1.80 if
ize.)10 This counterintuitive prediction is dra- they coordinate on the high-payoff equilibrium
matically rejected by the data, with 96 percent (H, H) $0.90 if they coordinate on the low-
of the row players choosing the Top decision payoff equilibrium (L, L), and they receive
that gives a chance of the high $3.20 payoff. nothing if they fail to coordinate (i.e., when one
Interestingly, the column players seemed to player chooses H and the other L). Suppose that,
have anticipated this, and they played Right 84 in addition, the column player has a secure
percent of the time, which is quite close to their option S that yields $0.40 for column and results
equilibrium mixed strategy of 7⁄8. Next, we low- in a zero payoff for the row player. This game is
ered the row player’s (Top, Left) payoff to given in Table 2 when x ⫽ 0. To analyze the
$0.44, which again should leave the row play- Nash equilibria of this game, notice that for the
er’s own choice probabilities unaffected in a column player a 50-50 combination of L and H
mixed-strategy Nash equilibrium. Again the ef- dominates S, and a rational column player
fect is dramatic, with 92 percent of the choices should therefore avoid the secure option. Elim-
being Down, as shown in the bottom part of inating S turns the game into a standard 2 ⫻ 2
Table 1. As before, the column players seemed coordination game that has three Nash equilibria:
to have anticipated this reaction, playing Left 80
percent of the time. To summarize, the unique
Nash prediction is for the bolded row-choice 11
This anomaly is persistent when subjects play the
percentages to be unchanged at 50 percent for game repeatedly. Jack Ochs (1995a, b) investigates a match-
all three treatments. This prediction is violated ing pennies game with an asymmetry similar to that of the
middle game in Table 1, and reports that the row players
continue to select Top considerably more than one-half of
the time, even after as many as 50 rounds. These results are
10
The predicted equilibrium probabilities for the row replicated in McKelvey et al. (2000). Similarly, Goeree et
player are not affected if we relax the assumption of risk al. (2000) report results for ten-period repeated matching
neutrality. There are only two possible payoff levels for pennies games that exactly match those in Table 1. The
column so, without loss of generality, columns’ utilities for results are qualitatively similar but less dramatic than those
payoffs of 40 and 80 can be normalized to 0 and 1. Hence in Table 1, with row’s choice probabilities showing strong
even a risk-averse column player will only be indifferent “own-payoff” effects that are not predicted by the Nash
when row uses choice probabilities of one-half. equilibrium.
1408 THE AMERICAN ECONOMIC REVIEW DECEMBER 2001

both players choosing L, both choosing H, and

a mixed-strategy equilibrium in which both
players choose L with probability 2⁄3.
The Nash equilibria are independent of x,
which is the payoff to the row player when (L,
S) is the outcome, since the argument for elim-
inating S is based solely on column’s payoffs.
However, the magnitude of x may affect the
coordination process: for x ⫽ 0, row is indif-
ferent between L and H when column selects S,
and row is likely to prefer H when column does
not select S (since then L and H have the same
number of zero payoffs for row, but H has a
higher potential payoff). Row is thus more FIGURE 2. EFFORT CHOICE FREQUENCIES FOR A MINIMUM-
likely to choose H, which is then also the opti- EFFORT COORDINATION GAME WITH HIGH EFFORT COST
mal action for the column player. However, (LIGHT BARS) AND LOW EFFORT COST (DARK BARS)
when x is large, say 400, the column player may
anticipate that row will select L in which case
column should avoid H. common starting point will not change the
This intuition is borne out by the experimen- minimum but will reduce one’s payoff by c.
tal data: in the treasure treatment with x ⫽ 0, 96 Similarly, a one-unit decrease in effort will
percent of the row players and 84 percent of the reduce one’s payoff by 1 ⫺ c, i.e., the reduc-
column players chose the high-payoff action H, tion in the minimum is more than the savings
while in the contradiction treatment with x ⫽ in effort costs when c ⬍ 1. Obviously, a
400 only 64 percent of the row players and 76 higher effort cost increases the risk of raising
percent of the column players chose H. The effort and reduces the risk of lowering effort.
percentages of outcomes that were coordinated Thus simple intuition suggests that effort lev-
on the high-payoff equilibrium were 80 for the els will be inversely related to effort costs,
treasure treatment versus 32 for the contradic- despite the fact that any common effort level
tion treatment. In the latter treatment, an addi- is a Nash equilibrium.
tional 16 percent of the outcomes were We ran one treatment with a low effort cost
coordinated on the low-payoff equilibrium, but of 0.1, and the data for 50 randomly matched
more than half of all the outcomes were unco- subjects in this treatment are shown by the dark
ordinated, non-Nash outcomes. bars in Figure 2. Notice that behavior is quite
concentrated at the highest effort level of 170;
D. A Minimum-Effort Coordination Game subjects coordinate on the Pareto-dominant out-
come. The high effort cost treatment (c ⫽ 0.9),
The next game we consider is also a coordi- however, produced a preponderance of efforts
nation game with multiple equilibria, but in this at the lowest possible level, as can be seen by
case the focus is on the effect of payoff asym- the lighter bars in the figure. Clearly, the extent
metries that determine the risks of deviating in of this “coordination failure” is affected by the
the upward and downward directions. The two key economic variable in this model, even
players in this game choose “effort” levels si- though Nash theory is silent.12
multaneously, and the cost of effort determines
the risk of deviation. The joint product is of the 12
fixed-coefficients variety, so that each person’s The standard analysis of equilibrium selection in co-
ordination is based on the John C. Harsanyi and Reinhard
payoff is the minimum of the two efforts, minus Selten’s (1988) notion of risk dominance, which allows a
the product of the player’s own effort and a formal analysis of the trade-off between risk and payoff
constant cost factor, c. In the experiment, we let dominance. Paul G. Straub (1995) reports experimental
efforts be any integer in the range from 110 to evidence for risk dominance as a selection criterium. There
is no agreement on how to generalize risk dominance be-
170. If c ⬍ 1, any common effort in this yond 2 ⫻ 2 games, but see Anderson et al. (2001b) for a
range is a Nash equilibrium, because a uni- proposed generalization based on the “stochastic potential.”
lateral one-unit increase in effort above a Experiments with repeated plays of coordination games
VOL. 91 NO. 5 GOEREE AND HOLT: TREASURES AND CONTRADICTIONS 1409

TABLE 3—TWO VERSIONS OF THE KREPS GAME (WITH CHOICE PERCENTAGES)

Left (26) Middle (8) Non-Nash (68) Right (0)

Basic game Top (68) 200, 50 0, 45 10, 30 20, ⫺250
Bottom (32) 0, ⫺250 10, ⫺100 30, 30 50, 40

Left (24) Middle (12) Non-Nash (64) Right (0)

Positive payoff frame Top (84) 500, 350 300, 345 310, 330 320, 50
Bottom (16) 300, 50 310, 200 330, 330 350, 340

E. The Kreps Game laboratory situation. As before, we randomly

paired 50 subjects and let them make a single
The previous examples demonstrate how the choice. Subjects were told that losses would
cold logic of game theory can be at odds with be subtracted from prior earnings, which were
intuitive notions about human behavior. This quite substantial by that point. As seen from
tension has not gone unnoticed by some game the percentages in parentheses in the top part
theorists. For instance, David M. Kreps (1995) of the table, the Non-Nash decision was se-
discusses a variant of the game in the top part of lected by approximately two-thirds of the col-
Table 3 (where we have scaled back the payoffs umn players. Of course, it is possible that this
to levels that are appropriate for the laboratory). result is simply a consequence of “loss-aver-
The pure-strategy equilibrium outcomes of this sion,” i.e., the disutility of losing some
game are (Top, Left) and (Bottom, Right). In amount of money is greater than the utility
addition, there is a mixed-strategy equilibrium associated with winning the same amount
in which row randomizes between Top and Bot- (Daniel Kahneman et al., 1991). Since all the
tom and column randomizes between Left and other columns contain negative payoffs, loss-
Middle. The only column strategy that is not averse subjects would thus be naturally in-
part of any Nash equilibrium is labeled Non- clined to choose Non-Nash. Therefore, we ran
Nash. Kreps argues that column players will another 50 subjects through the same game,
tend to choose Non-Nash because the other but with 300 cents added to payoffs to avoid
options yield at best a slightly higher payoff losses, as shown in the bottom part of Table
(i.e., 10, 15, or 20 cents higher) but could lead 3. The choice percentages shown in parenthe-
to substantial losses of $1 or $2.50. Notice ses indicate very little change, with close to
that this intuition is based on payoff magni- two-thirds of column players choosing Non-
tudes out of equilibrium, in contrast to Nash Nash as before. Thus “loss aversion” biases
calculations based only on signs of payoff are not apparent in the data, and do not seem
differences. to be the source of the prevalence of Non-
Kreps did try the high-hypothetical-payoff Nash decisions. Finally, we ran 50 new sub-
version of this game on several graduate stu- jects through the original version in the top
dents, but let us consider what happens with part of the table, with the (Bottom, Right)
financially motivated subjects in an anonymous payoffs of (50, 40) being replaced by (350,
400), which (again) does not alter the equi-
librium structure of the game. With this ad-
have shown that behavior may begin near the Pareto- mittedly heavy-handed enhancement of the
dominant equilibrium, but later converge to the equilib- equilibrium in that cell, we observed 96 per-
rium that is worst for all concerned (John B. Van Huyck
et al., 1990). Moreover, the equilibrium that is selected
cent Bottom choices and 84 percent Right
may be affected by the payoff structure for dominated choices, with 16 percent Non-Nash persisting
strategies (Russell Cooper et al., 1992). See Goeree and in this, the “treasure” treatment.
Holt (1998) for results of a repeated coordination game
with random matching. They show that the dynamic II. Dynamic Games with Complete Information
patterns of effort choices are well explained by a simple
evolutionary model of noisy adjustment toward higher
payoffs, and that final-period effort decisions can be As game theory became more widely used in
explained by the maximization of stochastic potential fields like industrial organization, the complex-
function. ity of the applications increased to accommodate
1410 THE AMERICAN ECONOMIC REVIEW DECEMBER 2001

player begins by choosing between a safe deci-

sion, S, and a risky decision, R. If R is chosen,
the second player must choose between a deci-
sion P that punishes both of them and a decision
N that leads to a Nash equilibrium that is also a
joint-payoff maximum. There is, however, a
second Nash equilibrium where the first player
chooses S and the second chooses P. The sec-
ond player has no incentive to deviate from this
equilibrium because the self-inflicted punish-
ment occurs off of the equilibrium path. Sub-
game perfectness rules out this equilibrium by
requiring equilibrium behavior in each sub-
game, i.e., that the second player behave opti-
mally in the event that the second-stage
subgame is reached.
Again, we used 50 randomly paired subjects
who played this game only once. The data for
FIGURE 3. SHOULD YOU TRUST OTHERS TO BE RATIONAL?
this treasure treatment are quite consistent with
the subgame-perfect equilibrium; a preponder-
ance of first players trust the other’s rationality
enough to choose R, and there are no irrational
dynamics and asymmetric information. One of P decisions that follow. The game shown in the
the major developments coming out of these bottom part of Figure 3 is identical, except that
applications was the use of backward induction the second player only forgoes 2 cents by
to eliminate equilibria with threats that are not choosing P. This change does not alter the fact
“credible” (Selten, 1975). Backward induction that there are two Nash equilibria, one of which
was also used to develop solutions to alternating- is ruled out by subgame perfectness. The choice
offer bargaining games (Ariel Rubinstein, percentages for 50 subjects indicate that a ma-
1982), which was the first major advance on this jority of the first players did not trust others
historically perplexing topic since Nash’s to be perfectly rational when the cost of
(1950) axiomatic approach. However, there irrationality is so small. Only about a third of
have been persistent doubts that people are able the outcomes matched the subgame-perfect
to figure out complicated, multistage backward equilibrium in this game.13 We did a third treat-
induction arguments. Robert W. Rosenthal (1981) ment (not shown) in which we multiplied all
quickly proposed a game, later dubbed the “cen- payoffs by a factor of 5, except that the P
tipede game,” in which backward induction over a decision led to (100, 348) instead of (100, 340).
large number of stages (e.g., 100 stages) was This large increase in payoffs produced an even
thought to be particularly problematic (e.g., more dramatic result; only 16 percent of the
McKelvey and Palfrey, 1992). Many of the games outcomes were subgame perfect, and 80 percent
in this section are inspired by Rosenthal’s (1981) of the outcomes were at the Nash equilibrium
doubts and Randolph T. Beard and Beil’s (1994) that is not subgame perfect.
experimental results. Indeed, the anomalies in this
section are better known than those in other sec- B. Should You Believe a Threat
tions, but we focus on very simple games with two That Is Not Credible?
or three stages, using parallel procedures and sub-
jects who have previously made a number of The game just considered is a little unusual in
strategic decisions in different one-shot games. that, in the absence of relative payoff effects,
the second player has no reason to punish, since
A. Should You Trust Others to Be Rational?

The power of backward induction is illus- 13

See Beard and Beil (1994) for similar results in a
trated in the top game in Figure 3. The first two-stage game played only once.
VOL. 91 NO. 5 GOEREE AND HOLT: TREASURES AND CONTRADICTIONS 1411

ments. Alternatively, the occurrence of the high

fraction of P decisions in the bottom game of
Figure 4 may be due to negative emotions that
follow the first player’s R decision, which re-
duces the second player’s earnings (Matthew
Rabin, 1993). Notice that this earnings reduc-
tion does not occur when the first player
chooses R for the game in the bottom part of
Figure 3, which could explain the lower rate of
punishments in that game.
The anomalous results of the contradiction
treatments may not come as any surprise to
Selten, the originator of the notion of subgame
perfectness. His attitude toward game theory
has been that there is a sharp contrast between
standard theory and behavior. For a long time
he essentially wore different hats when he did
theory and ran experiments, although his 1994
FIGURE 4. SHOULD YOU BELIEVE A THREAT THAT
IS NOT CREDIBLE?
Nobel prize was clearly for his contributions in
theory. This schizophrenic stance may seem
inconsistent, but it may prevent unnecessary
anxiety, and some of Selten’s recent theoretical
the first player’s R decision also benefits the work is based on models of boundedly rational
second player. This is not the case for the game (directional) learning (Selten and Joachim
in Figure 4, where an R decision by the first Buchta, 1998). In contrast, John Nash was re-
player will lower the second player’s payoff. As portedly discouraged by the predictive failures
before, there are two Nash equilibria, with the of game theory and gave up on both experimen-
(R, P) equilibrium ruled out by subgame per- tation and game theory (Nasar, 1998 p. 150).
fectness. In addition to not being credible, the
threat to play P is a relatively costly punishment C. Two-Stage Bargaining Games
for the second player to administer (40 cents).
The threat to play P in the top part of Figure Bargaining has long been considered a cen-
4 is evidently not believed, and 88 percent of tral part of economic analysis, and at the same
the first players choose the R strategy, with time, one of the most difficult problems for
impunity. The threat is cheap (2 cents) for the economic theory. One promising approach is to
game in the bottom part of the figure, and out- model unstructured bargaining situations “as if”
comes for 25 subject pairs are evenly divided the parties take turns making offers, with the
between the subgame-imperfect outcome, the costs of delayed agreement reflected in a shrink-
incredible threat outcome, and the subgame- ing size of the pie to be divided. This problem is
perfect outcome. Cheap threats often are (and particularly easy to analyze when the number of
apparently should be) believed. Again we see alternating offers is fixed and small.
that payoff magnitudes and off-the-equilibrium- Consider a bargaining game in which each
path risks matter. player gets to make a single proposal for how to
Since the P decisions in the bottom games of split a pie, but the amount of money to be
Figures 3 and 4 only reduce the second player’s divided falls from $5 in the first stage to $2 in
payoff by 2 cents, behavior may be affected by the second. The first player proposes a split of
small variations in payoff preferences or emo- $5 that is either accepted (and implemented) or
tions, e.g., spite or rivalry. As suggested by rejected, in which case the second player pro-
Ernst Fehr and Klaus Schmidt (1999) and Gary poses a split of $2 that is either accepted or
E Bolton and Axel Ockenfels (2000), players rejected by the first player. This final rejection
may be willing to sacrifice own earnings in results in payoffs of zero for both players, so the
order to reduce payoff inequities which would second player can (in theory) successfully de-
explain the P choices in the contradiction treat- mand $1.99 in the second stage if the first player
1412 THE AMERICAN ECONOMIC REVIEW DECEMBER 2001

prefers a penny to nothing. Knowing this, the behavior in bargaining games hinge on models
first player should demand $3 and offer $2 to of preferences in which a person’s utility de-
the other in the first stage. In a subgame-perfect pends on the payoffs of both players, i.e., dis-
equilibrium, the first player receives the amount tribution matters (Bolton, 1998; Fehr and
by which the pie shrinks, so a larger cost of Schmidt, 1999; Bolton and Ockenfels, 2000;
delay confers a greater advantage to the player Miguel Costa-Gomes and Klaus G. Zauner,
making the initial demand, which seems reason- 2001). The role of fairness is illustrated dramat-
able. For example, a similar argument shows ically in the experiment reported in Goeree and
that if the pie shrinks by $4.50, from $5 to Holt (2000a), who obtained even larger devia-
$0.50, then the first player should make an tions from subgame-perfect Nash predictions
initial demand of $4.50. than those reported here by giving subjects
We used 60 subjects (six cohorts of ten sub- asymmetric money endowments that were paid
jects each), who were randomly paired for each independently of the bargaining outcome. These
of the two treatments described above (alternat- endowments were selected to accentuate the
ing in order and separated by other one-shot payoff inequities that result in the subgame-
games). The average demand for the first player perfect Nash equilibrium, and hence their effect
was $2.83 for the $5/$2 treatment, with a stan- was to exaggerate fairness issues without alter-
dard deviation of $0.29. This is quite close to ing the equilibrium prediction. The result (for
the predicted $3.00 demand, and 14 of the 30 seven different one-shot bargaining games) was
initial demands were exactly equal to $3.00 in for demands to be inversely related to the sub-
this treasure treatment. But the average demand game-perfect Nash predictions.
only increased to $3.38 for the other treatment
with a $4.50 prediction, and 28 of the 30 de- III. Static Games with Incomplete Information
mands were below the prediction of $4.50. Re-
jections were quite common in this contradiction William Vickrey’s (1961) models of auctions
treatment with higher demands and correspond- with incomplete information constitute one of the
ingly lower offers to the second player, which is most widely used applications of game theory. If
not surprising given the smaller costs of rejecting private values are drawn from a uniform distribu-
“stingy” offers. tion, the Bayesian Nash equilibrium predicts that
These results fit into a larger pattern surveyed bids will be proportional to value, which is gen-
in Douglas D. Davis and Holt (1993 Chapter 5) erally consistent with laboratory evidence. The
and Roth (1995); initial demands in two-stage main deviation from theoretical predictions is the
bargaining games tend to be “too low” relative tendency of human subjects to “overbid” (relative
to theoretical predictions when the equilibrium to Nash), which is commonly rationalized in terms
demand is high, say more than 80 percent of the of risk aversion, an explanation that has lead to
pie as in our $5.00/$0.50 treatment, and initial some controversy. Glenn W. Harrison (1989), for
demands tend to be close to predictions when instance, argues that deviations from the Nash
the equilibrium demand is 50 –75 percent of the equilibrium may well be caused by a lack of
pie (as in our $5.00/$2.00 treatment). Interest- monetary incentives since the costs of such devi-
ingly, initial demands are “too high” when the ations are rather small: the “flat maximum cri-
equilibrium demand is less than half of the pie. tique.” Our approach here is to specify two
Here is an example of why theoretical explana- auction games with identical Nash equilibria, but
tions of behavior should not be based on exper- with differing incentives not to overbid.
iments in only one part of the parameter space, First, consider a game in which each of two
and why theorists should have more than just a bidders receives a private value for a prize to be
casual, secondhand knowledge of the experi- auctioned in a first-price, sealed-bid auction. In
mental economics literature.14 Many of the di- other words, the prize goes to the highest bidder
verse theoretical explanations for anomalous for a price equal to that bidder’s own bid. Each
bidder’s value for the prize is equally likely to
14
be $0, $2, or $5. Bids are constrained to be
Another example is the development of theories of
generalized expected utility to explain “fanning out” pref- integer dollar amounts, with ties decided by the
erences in Allais paradox situations, when later experiments flip of a coin.
in other parts of the probability triangle found “fanning in.” The relevant Nash equilibrium in this game
VOL. 91 NO. 5 GOEREE AND HOLT: TREASURES AND CONTRADICTIONS 1413

TABLE 4—EQUILIBRIUM EXPECTED PAYOFFS FOR THE (0,2,5) TREATMENT (EQUILIBRIUM BIDS MARKED WITH AN ASTERISK*)

Bid ⫽ 0 Bid ⫽ 1 Bid ⫽ 2 Bid ⫽ 3 Bid ⫽ 4 Bid ⫽ 5

Value ⫽ $0 0* ⫺0.5 ⫺1.66 ⫺3 ⫺4 ⫺5
Value ⫽ $2 0.33 0.5* 0 ⫺1 ⫺2 ⫺3
Value ⫽ $5 0.83 2 2.5* 2 1 0

TABLE 5—EQUILIBRIUM EXPECTED PAYOFFS FOR THE (0,3,6) TREATMENT (EQUILIBRIUM BIDS MARKED WITH AN ASTERISK*)

Bid ⫽ 0 Bid ⫽ 1 Bid ⫽ 2 Bid ⫽ 3 Bid ⫽ 4 Bid ⫽ 5

Value ⫽ $0 0* ⫺0.5 ⫺1.66 ⫺3 ⫺4 ⫺5
Value ⫽ $3 0.5 1* 0.83 0 ⫺1 ⫺2
Value ⫽ $6 1 2.5 3.33* 3 2 1

with incomplete information about others’ pref- as indicated by the location of optimal bids for
erences is the Bayesian Nash equilibrium, each value. Even though the equilibria are the
which specifies an equilibrium bid for each pos- same, we expected more of an upward bias in
sible realization of a bidder’s value. It is bids in the second (0, 3, 6) treatment. The
straightforward but tedious to verify that the intuition can be seen by looking at payoff losses
Nash equilibrium bids are $0, $1, and $2 for a associated with deviations from the Nash equi-
value of $0, $2, and $5 respectively, as can be librium. Consider, for instance, the middle-
seen from the equilibrium expected payoffs in value bidder with expected payoffs shown in the
Table 4. For example, consider a bidder with a second rows of Tables 4 and 5. In the (0, 3, 6)
private value of $5 (in the bottom row) who treatment, the cost of bidding $1 above the
faces a rival that bids according to the proposed equilibrium bid is $1 ⫺ $0.83 ⫽ $0.17, which is
Nash solution. A bid of 0 has a one-half chance less than the cost of bidding $1 below the equi-
of winning (decided by a coin flip) if the rival’s librium bid: $1 ⫺ $0.50 ⫽ $0.50. In the (0, 2, 5)
value, and hence the rival’s bid, is zero, which treatment, the cost of an upward deviation from
happens with probability one-third. Therefore, the equilibrium bid is greater than the cost of a
the expected payoff of a zero bid with a value of downward deviation; see the middle row of
$5 equals 1⁄2 ⴱ 1⁄3 ⴱ ($5 ⫺ $0) ⫽ $5/6 ⫽ 0.83. Table 4. A similar argument applies to the high-
If the bid is raised to $1, the probability of win- value bidders, while deviation costs are the
ning becomes 1⁄2 (1⁄3 when the rival’s value is $0 same in both treatments for the low-value bid-
plus 1⁄6 when the rival’s value is $2). Hence, the der. Hence we expected more overbidding for
expected payoff of a $1 bid is 1⁄2 ⴱ ($5 ⫺ the (0, 3, 6) treatment.
$1) ⫽ $2. The other numbers in Table 4 are This intuition is borne out by bid data for the
derived in a similar way. The maximum expected 50 subjects who participated in a single auction
payoff in each row coincides with the equilibrium under each condition (again alternating the or-
bid, as indicated by an asterisk (ⴱ). Note that the der of the two treatments and separating the two
equilibrium involves bidding about one-half of auctions with other one-shot games). Eighty
the value.15 percent of the bids in the (0, 2, 5) treatment
Table 5 shows the analogous calculations for matched the equilibrium: the average bids for
the second treatment, with equally likely private low-, medium-, and high-value bidders were $0,
values of $0, $3, or $6. Interestingly, this in- $1.06, and $2.64, respectively. In contrast, the
crease in values does not alter the equilibrium average bids for the (0, 3, 6) treatment were $0,
bids in the unique Bayesian Nash equilibrium, $1.82, and $3.40 for the three value levels, and
only 50 percent of all bids were Nash bids. The
bid frequencies for each value are shown in
15
The bids would be exactly one-half of the value if the
Table 6. As in previous games, deviations from
highest value were $4 instead of $5, but we had to raise the Nash behavior in these private-value auctions
highest value to eliminate multiple Nash equilibria. seem to be sensitive to the costs of deviation. Of
1414 THE AMERICAN ECONOMIC REVIEW DECEMBER 2001

TABLE 6—BID FREQUENCIES whether the sender was of type A or B. Every-

(EQUILIBRIUM BIDS MARKED WITH AN ASTERISK*) body knew that the ex ante probability of a type
A sender was one-half. The sender, knowing
(0, 2, 5) Treatment (0, 3, 6) Treatment
his/her own type would choose a signal, Left or
Bid Frequency Bid Frequency Right. This signal determined whether the pay-
Value ⫽ 0 0* 20 Value ⫽ 0 0* 17 offs on the right or left side of Table 7 would be
used. (The instructions used letters to identify
Value ⫽ 2 1* 15 Value ⫽ 3 1* 5 the signals, but we will use words here to facil-
2 1 2 11
3 0 3 2 itate the explanations.) This signal would be
communicated to the responder that was
Value ⫽ 5 1 1 Value ⫽ 6 1 0 matched with that sender. The responder would
2* 5 2* 3 see the sender’s signal, Left or Right, but not the
3 6 3 4
4 2 4 6
sender’s type, and then choose a response, C, D,
5 0 5 1 or E. The payoffs were determined by Table
6 0 6 1 7, with the sender’s payoff to the left in each
cell.
First, consider the problem facing a type A
course, this does not rule out the possibility that sender, for whom the possible payoffs from
risk aversion or some other factor may also have sending a Left signal (300, 0, 500) seem, in
some role in explaining the overbidding ob- some loose sense, less attractive than those
served here, especially the slight overbidding for sending a Right signal (450, 150, 1,000).
for the high value in the (0, 2, 5) treatment.16 For example, if each response is thought to be
equally likely (the “principle of insufficient
IV. Dynamic Games with Incomplete reason”), then the Right signal has a higher
Information: Signaling expected payoff. Consequently, type A’s pay-
offs have been made bold for the Right row in
Signaling games are complex and interesting the top right part of Table 7. Applying the
because the two-stage structure allows an op- principle of insufficient reason again, a type B
portunity for players to make inferences and sender looking at the payoffs in the bottom
change others’ inferences about private infor- row of the table might be more attracted by
mation. This complexity often generates multi- the Left signal, with payoffs of (500, 300,
ple equilibria that, in turn, have stimulated a 300) as compared with (450, 0, 0).17 There-
sequence of increasingly complex refinements fore, sender B’s payoffs are in bold for the
of the Nash equilibrium condition. Although it Left signal. In fact, all of the type B subjects
is unlikely that introspective thinking about the did send the Left signal, and seven of the ten
game will produce equilibrium behavior in a type A subjects sent the Right signal. All
single play of a game this complex (except by responses in this game were C, so all but three
coincidence), the one-shot play reveals useful of the outcomes were in one of the two cells
information about subjects’ cognitive processes. marked by an asterisk. Notice that this is an
In the experiment, half of the subjects were equilibrium, since neither type of sender
designated as “senders” and half as “respond- would benefit from sending the other signal,
ers.” After reading the instructions, we began by and the respondent cannot do any better than
throwing a die for each sender to determine the maximum payoff received in the marked
cells. This is a separating Nash equilibrium;
the signal reveals the sender’s type.
16 The payoff structure for this game becomes a
Goeree et al. (2001) report a first-price auction exper-
iment with six possible values, under repeated random little clearer if you think of the responses as one
matching for ten periods. A two-parameter econometric of three answers to a request: Concede, Deny, or
model that includes both decision error and risk aversion
provides a good fit of 67 value/bid frequencies and shows
17
that both the error parameter and risk-aversion parameter These are not dominance arguments, since the responder
are significantly different from zero. David Lucking-Reiley can respond differently to each signal, and the lowest payoff
(1999) mentions risk aversion as a possible explanation for from sending one signal is not higher than the highest payoff
overbidding in a variety of auction experiments. that can be obtained from sending the other signal.
VOL. 91 NO. 5 GOEREE AND HOLT: TREASURES AND CONTRADICTIONS 1415

TABLE 7—SIGNALING WITH A SEPARATING EQUILIBRIUM (MARKED BY ASTERISKS) (SENDER’S PAYOFF, RESPONDER’S PAYOFF)

Response to Left signal Response to Right signal

C D E C D E
Type A sends Left 300, 300 0, 0 500, 300 Type A sends Right 450, 900 150, 150 1,000, 300
(*)
Type B sends Left 500, 500 300, 450 300, 0 Type B sends Right 450, 0 0, 300 0, 150
(*)

TABLE 8—SIGNALING WITHOUT A SEPARATING EQUILIBRIUM (SENDER’S PAYOFF, RESPONDER’S PAYOFF)

Response to Left signal Response to Right signal

C D E C D E
Type A 300, 300 0, 0 500, 300 Type A 450, 900 150, 150 1,000, 300
sends Left sends Right
Type B 300, 300 300, 450 300, 0 Type B 450, 0 0, 300 0, 150
sends Left sends Right

Evade. With some uncertainty about the send- tive criterion rules out these beliefs, and selects
er’s type, Evade is sufficiently unattractive to the separating equilibrium observed in the trea-
respondents that it is never selected. Consider sure treatment.19
the other two responses and note that a sender The game in Table 8 is a minor variation on
always prefers that the responder choose Con- the previous game, with the only change be-
cede instead of Deny. In the separating equilib- ing that the (500, 500) in the bottom left part
rium, the signals reveal the senders’ types, the of Table 7 is replaced by a (300, 300) pay-
responder always Concedes, and all players are off.20 As before, consider the sender’s ex-
satisfied. There is, however, a second equilib- pected payoffs when each response is
rium for the game in Table 7 in which the presumed to be equally likely, which leads
responder Concedes to Left and Denies Right, one to expect that type A senders will choose
and therefore both sender types send Left to Right and that type B senders will choose
avoid being Denied.18 Backward induction ra- Left, as indicated by the bold payoff numbers.
tionality (of the sequential Nash equilibrium) In the experiment, 10 of the 13 type A senders
does not rule out these beliefs, since a deviation did choose Right, and 9 of the 11 type B
does not occur in equilibrium, and the respon- senders did choose Left. But the separation
dent is making a best response to the beliefs. observed in this contradiction treatment is not
What is unintuitive about these beliefs (that a a Nash equilibrium.21 All equilibria for this
deviant Right signal comes from a type B) is
that the type B sender is earning 500 in this
(Left, Concede) equilibrium outcome, and no
deviation could conceivably increase this pay-
off. In contrast, the type A sender is earning 300 19
Brandts and Holt (1992, 1993) report experimental
in the Left side pooling equilibrium, and this data that contradict the predictions of the intuitive criterion,
type could possibly earn more (450 or even i.e., the decision converged to an equilibrium ruled out by
that criterion.
1,000), depending on the response to a devia- 20
Unlike the paired treatments considered previously,
tion. The In-Koo Cho and Kreps (1987) intui- the payoff change for these signaling games does alter the
set of Nash equilibria.
21
The respondents would prefer to Concede to a Right
18
To check that the responder has no incentive to devi- signal and Deny a Left signal. Type B senders would there-
ate, note that Concede is a best response to a Left regardless fore have an incentive to deviate from the proposed sepa-
of the sender’s type, and that Deny is a best response to a rating equilibrium and send a Right signal. In the
deviant Right signal if the responder believes that it was sent experiment, half of the Left signals were Denied, whereas
by a type B. only 2 of the 12 Right signals were Denied.
1416 THE AMERICAN ECONOMIC REVIEW DECEMBER 2001

contradiction treatment involve “pooling,” often, is sharply contradicted by the data in

with both types sending the same signal.22 the middle part of Table 1.23,24
Another possibility is that behavior in one-
V. Explaining Anomalous Behavior shot games conforms to a simple heuristic. In-
in One-Shot Games deed, some experimental economists have
suggested that subjects in the initial period of a
Although the results for the contradiction repeated game choose the decision that maxi-
treatments seem to preclude a game-theoretic mizes their security level, i.e., the “maximim”
explanation, many of the anomalous data pat- decision. For example, in the Kreps game of
terns are related to the nature of the incentives. Table 3, the frequently observed Non-Nash de-
This suggests that it may be possible to develop cision maximizes column’s security. The strong
formal models that explain both treasures and treatment effects in the matching pennies games
contradictions. Below we discuss several recent cannot be explained in this way, however, since
approaches that relax the common assumptions in all three treatments each player’s minimum
of perfect selfishness, perfect decision-making payoff is the same for both decisions. A similar
(no error), and perfect foresight (no surprises). argument applies to the coordination game in
As noted in Section II, the anomalies ob- Table 2. Moreover, the security-maximizing
served for the dynamic games in Figures 3 and choices in the traveler’s dilemma and the
4 are consistent with models of inequity aver- minimum-effort coordination game are the low-
sion (Fehr and Schmidt, 1999; Bolton and Oc- est possible decision, which is contradicted by
kenfels, 2000), which assumes that people like the high claim and effort choices in the contra-
higher payoffs for themselves and dislike earn- diction treatments. Subjects may be risk averse
ing less than the other person (“envy”) or earn- in unfamiliar situations, but the extreme risk
ing more (“guilt”). Inequity aversion also seems aversion implied by maximum security is gen-
to play a role when players bargain over the erally not observed. Furthermore, heuristics
division of a fixed amount of money (Goeree based on reciprocity or a status quo bias do not
and Holt, 2000a). However, it cannot explain apply to single-stage, one-shot games where
observed behavior in the contradiction match- there is neither a precedent nor an opportunity
ing pennies treatments. Consider, for exam- to reciprocate. Nor can loss aversion be the
ple, the “320” version of the matching primary cause, since losses are impossible in
pennies game in Table 1. Since the column most of the games reported here, and the pos-
player is averse to the (320, 40) outcome, the sibility of a loss had no effect in the Kreps
column player would only be willing to ran- game.
domize between Left and Right if the attrac- As an alternative to simple heuristics, one
tiveness of Right is increased by having the could try to model players’ introspective
row player play Bottom more often than the thought processes. Previous models have typi-
0.5 probability that would make a purely self- cally specified some process of belief forma-
ish column player indifferent. This prediction, tion, assuming that players best respond to the
that the row player should play Bottom more

22 23
For example, it is an equilibrium for both types to send Goeree et al. (2000) report formal econometric tests
Right if a Left signal triggers a C or a D response. The D that reject the predictions of inequity aversion models in the
response to Left is appropriate if the respondent thinks the context of a group of repeated asymmetric matching pennies
deviant signal comes from a type B sender, and the C games.
24
response is appropriate if the deviant is thought to be of type Payoff inequity aversion also has no effect in the
A. Beliefs that the deviant is of type A are intuitive, since minimum-effort coordination game; any common effort
type A earns 450 in equilibrium and could possibly earn level is still a Nash equilibrium. To see this, note that a
more (500) by switching to Left (if an E response follows). unilateral effort increase from a common level reduces
A second pooling equilibrium involves both types sending a one’s own payoff and creates an disadvantageous inequity.
Left signal to which the respondents Concede. A deviant Similarly, a unilateral decrease from a common effort level
Right signal is Denied, which is appropriate if the respon- reduces one’s payoff and creates an inequity where one
dent thinks the deviant signal comes from a type B sender. earns more than the other, since their costly extra effort is
Again these beliefs are intuitive since the type B sender wasted. Thus inequity aversion cannot explain the strong
could possibly gain by deviating. effect of an increase in effort costs.
VOL. 91 NO. 5 GOEREE AND HOLT: TREASURES AND CONTRADICTIONS 1417

resulting beliefs.25 The experiments reported reported by Nagel (1995).27 It is easy to verify
above indicate that magnitudes (not just signs) that level one rationality also provides good
of payoff differences matter, and it is thus nat- predictions for both treasure and contradiction
ural to consider a decision rule for which choice treatments in the traveler’s dilemma, the
probabilities are positively but imperfectly re- minimum-effort coordination game, and the
lated to payoffs. The logit rule, for example, Kreps game. There is evidence, however, that at
specifies that choice probabilities, pi , for op- least some subjects form more precise beliefs
tions i ⫽ 1, ... ,m, are proportional to exponen- about others’ actions, possibly through higher
tial functions of the associated expected levels of introspection.28 In the matching pen-
payoffs, ␲ ei : nies games in Table 1, for example, a flat prior
makes column indifferent between Left and
exp共 ␲ ei / ␮ 兲 Right, and yet most column players seem to
(1) p i ⫽ , i ⫽ 1, ... , m, anticipate that row will choose Top in the 320
¥ exp共 ␲ e / ␮ 兲 version and Bottom in the 44 version of this
j
j ⫽ 1,...,m
game.
Of course, what the other player does de-
where the sum in the denominator ensures that pends on what they think you will do, so the
the probabilities sum to one, and the “error next logical step is to assume that others make
parameter,” ␮, determines how sensitive choice responses to a flat prior, and then you respond to
probabilities are to payoff differences.26 that anticipated response (Selten, 1991). This is
In order to use the “logit best response” in Stahl and Wilson’s (1995) “level two” rational-
(1), we need to model the process of belief ity. There is, however, no obvious reason to
formation, since belief probabilities are used to truncate the levels of iterated thinking. The no-
calculate the expected payoffs on the right side tion of rationalizability discussed above, for ex-
of (1). By the principle of insufficient reason ample, involves infinitely many levels of
one might postulate that each of the others’ iterated thinking, with “never-best” responses
actions are equally likely. This corresponds to eliminated in succession. But rationalizability
the Stahl and Wilson (1995) notion of “level seems to imply too much rationality, since it
one” rationality, which captures many of the predicts that all claims in the traveler’s dilemma
first-period decisions in the “guessing game” will be equal to the minimum claim, indepen-
dent of the penalty/reward parameter. One way
25
to limit the precision of the thought process,
Perhaps the best-known model of introspection is Har-
sanyi and Selten’s (1988) “tracing procedure.” This proce- without making an arbitrary assumption about
dure involves an axiomatic determination of players’ the number of iterations, is to inject increasing
common priors (the “preliminary theory”) and the construc- amounts of noise into higher levels of iterated
tion of a modified game with payoffs for each decision that thinking (Goeree and Holt, 1999; Kübler and
are weighted averages of those in the original game and of
the expected payoffs determined by the prior distribution.
Weizsäcker, 2000). Let ␾␮ denote the logit best-
By varying the weight on the original game, a sequence of response map (for error rate ␮) on the right side
best responses for the modified game are generated. This of (1). Just as a single logit response to beliefs,
process is used to select one of the Nash equilibria of the p0 , can be represented as p ⫽ ␾␮(p0), a series of
original game. Gonzalo Olcina and Amparo Urbano (1994) such responses can be represented as:29
also use an axiomatic approach to select a prior distribution,
which is then revised by a simulated learning process that is
essentially a partial adjustment from current beliefs to best
27
responses to current beliefs. Since neither the Harsanyi/ In our own work, we have used a noisy response to a
Selten model nor the Olcina/Urbano model incorporates any flat prior as a way of starting computer simulations of
noise, they predict that behavior will converge to the Nash simulations of behavior in repeated games (Brandts and
equilibrium in games with a unique equilibrium, which is an Holt, 1996; Capra et al., 1999, 2002; Goeree and Holt,
undesirable feature in light of the contradictions data re- 1999).
28
ported above. Costa-Gomes et al. (2001), for example, infer some
26
As ␮ goes to zero, payoff differences are “blown up,” heterogeneity in the amount of introspection by observing
and the probability of the optimal decision converges to 1. the types of information that subjects acquire before making
In the other extreme, as ␮ goes to ⬁, the choice probabilities a decision.
29
converge to 1/m independently of expected payoffs. See R. Goeree and Holt (2000b) use continuity arguments to
Duncan Luce (1959) for an axiomatic derivation of the logit show that the limit in (2) exists even if the (increasing) error
choice rule in (1). parameters are person specific.
1418 THE AMERICAN ECONOMIC REVIEW DECEMBER 2001

(2) p ⫽ lim ␾ ␮ 1 共 ␾ ␮ 2 共 ... ␾ ␮ n 共p0 兲兲兲, example. Several of the models discussed above
n3⬁ do a fairly good job of organizing the qualitative
patterns of conformity and deviation from the
where ␮1 ⱕ ␮2 ⱕ ... , with ␮⬁ converging predictions of standard theory, but there are
to infinity.30 This assumption captures the idea obvious discrepancies. We hope that this paper
that it becomes increasingly complex to do will stimulate further theoretical work on mod-
more and more iterations.31 Since ␾␮ for ␮ ⫽ ⬁ els of behavior in one-shot games. One poten-
maps the whole probability simplex to a single tially useful approach may be to elicit beliefs
point, the right side of (2) is independent of the directly as the games are played (Theo Offer-
initial belief vector p0. Moreover, the introspec- man, 1997; Andrew Schotter and Yaw Narkov,
tion process in (2) yields a unique outcome even 1998).
in games with multiple Nash equilibria. Note
that the choice probabilities on the left side of VI. Conclusion
(2) generally do not match the beliefs at any
stage of the iterative process on the right. In One-shot game experiments are interesting
other words, the introspective process allows because many games are in fact only played
for surprises, which are likely to occur in one- once; single play is especially relevant in appli-
shot games. cations of game theory in other fields, e.g.,
For games with very different levels of com- international conflicts, election campaigns, and
plexity such as the ones reported here, the error legal disputes. The decision makers in these
parameters that provide the best fit are likely to contexts, like the subjects in our experiments,
be different. In this case, the estimates indicate typically have experience in similar games with
the degree of complexity, i.e., they serve as a other people. One-shot games are also appeal-
measurement device. For games of similar com- ing because they allow us to abstract away from
plexity, the model in (2) could be applied to issues of learning and attempts to manipulate
predict behavior across games. We have used it others’ beliefs, behavior, or preferences (e.g.,
to explain data patterns in a series of 37 simple reciprocity, cooperativeness). This paper re-
matrix games, assuming a simple two-parame- ports the results of ten pairs of games that are
ter model for which ␮ n ⫽ ␮ t n , where t deter- played only once by subjects who have experi-
mines the rate at which noise increases with ence with other one-shot and repeated games.
higher iterations (Goeree and Holt, 2000b). The The Nash equilibrium (or relevant refinement)
estimated value (t ⫽ 4.1) implies that there is provides accurate predictions for standard ver-
more noise for higher levels of introspection, a sions of these games. In each case, however,
result that is roughly consistent with estimates there is a matched game for which the Nash
obtained by Kübler and Weizsäcker (2000) for prediction clearly fails, although it fails in a way
data from information-cascade experiments. that is consistent with simple (non-game-
The analysis of introspection is a relatively theoretic) intuition. The results for these expe-
understudied topic in game theory, as compared rienced subjects show:
with equilibrium refinements and learning, for
(1) Behavior may diverge sharply from the
unique rationalizable (Nash) equilibrium in
30
The case of a constant parameter (␮1 ⫽ ␮2 ⫽ ... ⫽ ␮)
a social (traveler’s) dilemma. In these
is of special interest. In this case, the process may not games, the Nash equilibrium is located on
converge for some games (e.g., matching pennies), but one side of the range of feasible decisions,
when it does, the limit probabilities, p*, must be invariant and data for the contradiction treatment
under the logit map: ␾ ␮ ( p*) ⫽ p*. A fixed point of this
type constitutes a “logit equilibrium,” which is a special
have a mode on the opposite side of this
case of the quantal-response equilibrium defined in range. The most salient feature of the data
McKelvey and Palfrey (1995). It is in this sense that the is the extreme sensitivity to a parameter
logit equilibrium arises as a limit case of the noisy intro- that has no effect on the Nash outcome.
spective process defined in (2). (2) Students suffering through game theory
31
For an interesting alternative approach, see Capra
(1998). In her model, beliefs are represented by degenerate classes may have good reasons when they
distributions that put all probability mass at a single point. have trouble understanding why a change
The location of the belief points is, ex ante, stochastic. in one player’s payoffs only affects the
VOL. 91 NO. 5 GOEREE AND HOLT: TREASURES AND CONTRADICTIONS 1419

other player’s decision probabilities in a compare actual and optimal decision-making.

mixed-strategy Nash equilibrium. The data This normative-based defense is not convincing
from matching pennies experiments show for games, however, since the best way for one
strong “own-payoff” effects that are not to play a game depends on how others actually
predicted by the unique (mixed-strategy) play, not on how some theory dictates that ra-
Nash equilibrium. The Nash analysis seems tional people should play. John Nash, one of
to work only by coincidence, when the pay- the other Nobel recipients, saw no way
off structure is symmetric and deviation around this dilemma, and when his experi-
risks are balanced. ments were not providing support to theory,
(3) Effort choices are strongly influenced by he lost whatever confidence he had in the
the cost of effort in coordination games, an relevance of game theory and focused on
intuitive result that is not explained by stan- more purely mathematical topics in his later
dard theory, since any common effort is a research (Nasar, 1998).
Nash equilibrium in such games. Moreover, Nash seems to have undersold the importance
as Kreps conjectured, it is possible to de- of his insight, and we will be the first to admit
sign coordination games where the majority that we begin the analysis of a new strategic
of one player’s decisions correspond to the problem by considering the equilibria derived
only action that is not part of any Nash from standard game theory, before considering
equilibrium. the effects of payoff and risk asymmetries on
(4) Subjects often do not trust others to be incentives to deviate. But in an interactive, stra-
rational when irrationality is relatively cost- tegic context, biases can have reinforcing ef-
less. Moreover, “threats” that are not cred- fects that drive behavior well away from Nash
ible in a technical sense may nevertheless predictions, and economists are starting to ex-
alter behavior in simple two-stage games plain such deviations using computer simula-
when carrying out these threats is not tions and theoretical analyses of learning and
costly. decision error processes. There has been rela-
(5) Deviations from Nash predictions in alter- tively little theoretical analysis of one-shot
nating-offer bargaining games and in pri- games where learning is impossible. The mod-
vate-value auctions are inversely related to els of iterated introspection discussed here offer
the costs of such deviations. The effects of some promise in explaining the qualitative fea-
these biases can be quite large in the games tures of deviations from Nash predictions enu-
considered. merated above. Taken together, these new
(6) It is possible to set up a simple signaling approaches to a stochastic game theory enhance
game in which the decisions reveal the sig- the behavioral relevance of standard game the-
naler’s type (separation), even though the ory. And looking at laboratory data is a lot less
equilibrium involves pooling. stressful than before.

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