Kaushik Basu (1994) introduced a traveler's dilemma in which two people must independently decide how much to claim for identical objects that have been lost on a flight. The airline, in an attempt to prevent fraud, agrees to pay each the minimum of the two claims, with a penalty for the high claimant and a reward for the low claimant. The Nash equilibrium predicts the lowest possible claim no matter how small the penalty and reward. The intuition that claims may be high is confirmed by laboratory experiments. Average claims are inversely related to the size of the penalty/reward parameter, which is explained by a stochastic generalization of the Nash equilibrium. JEL Classification: C72, C92. Keywords: experiments, decision error, traveler's dilemma, logit equilibrium. The notion of a Nash equilibrium has joined supply and demand as one of the two or three techniques that economists instinctively try to use first in the analysis of economic interactions. Moreover, the Nash equilibrium and closely related game-theoretic concepts are being widely applied in other social sciences and even in biology, where evolutionary stability often selects a subset of the Nash equilibria. Many people are uneasy about the stark predictions of the Nash equilibrium in some contexts where the extreme rationality assumptions seem implausible. Kaushik Basu's (1994) "traveler's dilemma" is a particularly convincing example of a case where the unrelenting logic of game theory is at odds with intuitive notions about human behavior. The story associated with the dilemma is that two travelers purchase identical * Capra: Department of Economics, University of Arkansas, Fayetteville, AR 72701; Goeree and Holt: Department of Economics, University of Virginia, 114 Rouss Hall, Charlottesville, VA 22901; Gomez: Department of Economics, University of Malaga, 29013 Malaga, Spain. This project was funded in part by the National Science Foundation (SBR-9617784). We wish to thank Peter Coughlan, Rob Gilles, Susana Cabrera-Yeto, Irene Comeig, Nadège Marchand, and two anonymous referees for suggestions. 2 antiques while on a tropical vacation. Their luggage is lost on the return trip, and the airline asks them to make independent claims for compensation. In anticipation of excessive claims, the airline representative announces: "We know that the bags have identical contents, and we will entertain any claim between $2 and $100, but you will each be reimbursed at an amount that equals the minimum of the two claims submitted. If the two claims differ, we will also pay a reward of $2 to the person making the smaller claim and we will deduct a penalty of $2 from the reimbursement to the person making the larger claim." Notice that, irrespective of the actual value of the lost luggage, there is a unilateral incentive to "undercut" the other's claim. It follows from this logic that the only Nash equilibrium is for both to make the minimum claim of $2. As Basu (1994) notes, this is also the unique strict equilibrium, and the only rationalizable equilibrium when claims are discrete. When one of us recently described this dilemma to an audience of physicists, someone asked incredulously: "Is this what economists think the equilibrium is? If so, then we should shut down all economics departments." The implausibility of the Nash equilibrium prediction is based on doubts that a small penalty and/or reward can drive claims all the way to an outcome that minimizes the sum of the players' payoffs. Indeed, the Nash equilibrium in a traveler's dilemma is independent of the size of the penalty or reward. Economic intuition suggests that behavior conforms closely to the Nash equilibrium when the penalty or reward is high, but that claims rise to the maximum level as the penalty/reward parameter approaches $0. This paper uses laboratory experiments to evaluate whether average claims are affected by (theoretically irrelevant) changes in the penalty/reward parameter. The laboratory procedures are described in section I. The second and third sections contain analyses of aggregate and individual data. Section IV presents a learning model that is used to obtain maximum likelihood estimates of the learning and decision error parameters. The fifth section considers behavior in the final periods after most learning has occurred, i.e. when average claims stabilize and behavior converges to a type of noisy equilibrium which combines a standard logit probabilistic choice rule with a Nash-like consistency-of-actions-and-beliefs condition. The learning/adjustment and equilibrium models are complementary, and together they are capable of explaining some key features of the data. The final section concludes. 3 The data were collected from groups of 9-12 subjects, with each group participating in a series of traveler's dilemma games during a session that lasted about an hour and a half. This type of experiment had not been done before, and we felt that more would be learned from letting the penalty/reward parameter, R, vary over a wide range of values. Therefore, we used Subjects were recruited from economics classes at the University of Virginia, with the promise that they would be paid a $6 participation fee plus all additional money earned during the experiment. Individual earnings ranged from about $24.00 to $44.00 for a session. We began by reading the instructions for part A (these instructions are available from the authors on request). Although decisions were referred to as "claims," the earnings calculations were explained without reference to the context, i.e. without mentioning luggage, etc. In each period, subjects would record their claim on their decision sheets, which were collected and randomly matched (with draws of numbered ping pong balls) to determine the "other's claim" and "your earnings," and the sheets were then returned. Claims were required to be any number of cents between and including 80 and 200, with decimals being used to indicate fractions of cents. Subjects only saw the claim decision made by the person with whom they were matched in a given period. They were told that part A would be followed by "another decision making experiment" but were not given additional information about part B. The penalty/reward parameter was changed in part B, and random pair-wise matchings were made as before. Part B lasted for 10 periods, except in the first two sessions where it lasted for 5 periods. The part A data are summarized in The two highest lines in Figure 1 plot the average claims for low reward/penalty parameters of 5 and 10 (cents). The first-period averages are close to 180, and they stay high and Session 2 (dashed line) change in behavior. In the two sessions using R = 80 and R = 10, for example, the behavior is approximately reversed, as shown in Figure 2 . 2 There is some evidence of a sequence effect, since the average claims were higher for R = 10 when this treatment came first than when it followed the R = 80 treatment that "locked" onto a Nash equilibrium. In fact, the sequence effect was so strong in one session, with a treatment switch from R = 50 to R = 20, that the data did not rise in part B after converging to the Nash outcome in part A. In all other sessions, the high R treatment resulted in lower average claims, as shown in Table 1 . 1 Of the 720 (=6!) possible ways that the 6 session averages could have been ranked, there are only 6 possible outcomes that are as extreme as the one observed (i.e. with zero or one reversals between adjacent R values). Under the null hypothesis the probability of obtaining a ranking this extreme is: 6/720, so the null hypothesis can be rejected (onetailed test) at the 1 percent level. 2 Obviously, the part B data have not settled down yet after 5 periods, and therefore we decided to extend part B to ten periods in subsequent sessions. Consider again the null hypothesis of no treatment effect, under which higher average claims are equally likely in both treatments. The alternative hypothesis is that average claims are higher for the treatments with a low value of R. This null hypothesis can be rejected at a 3 percent significance level using a standard Wilcoxon (signed-rank) nonparametric test. Thus the treatment effect is significant, even though it does not affect the Nash equilibrium. Basically, the Nash equilibrium provides good predictions for high incentives (R = 80 and R = 50), but behavior is quite different from the Nash prediction under the treatments with low and intermediate values of R. In particular, as shown in Figure 1 , the data for the low-R treatments is concentrated at the opposite end of the range of feasible decisions. The original presentation of the traveler's dilemma as in Basu (1994), involved two travelers with low incentives relative to the range of choices, so in this sense, the intuition behind the dilemma is confirmed. 3 To summarize, the Nash equilibrium prediction of 80 for all treatments fails to account for the most salient feature of the data, the intuitive inverse relationship between average claims and the
For example, Capra et al. REF use what is called a "standard logit model".
Anomalous Behavior in a Traveler ’ s Dilemma ?
Economics