Logit Equilibrium Models of Anomalous Behavior: What to Do When the Nash Equilibrium Says One Thing and the Data Say Something Else

Every experimentalist will sooner or later come across a situation in which results from initial "baseline" treatments conform nicely to the Nash equilibrium, but subsequent changes in parameters push the data in ways not predicted by Nash. This may happen when one begins by giving theory its "best shot," reserving stress tests for later. Such tests often involve changing a parameter that, on the basis of intuition, is likely to alter behavior, but which has no effect on the Nash equilibrium. For example, behavior in a symmetric matching-pennies game conforms to the Nash prediction of mixing with equal probabilities. However, changing a player's own payoff parameters will typically change that player's choice probabilities (Ochs, 1995; Goeree, Holt, and Palfrey, 2002b), in spite of the fact that in a Nash equilibrium a player's mixed strategy only depends on the other players' payoffs. In "Ten Little Treasures of Game Theory and Ten Intuitive Contradictions," Goeree and Holt (2001) report a variety of games in which behavior conforms nicely to Nash predictions in one treatment (the "treasures") but deviates sharply from Nash predictions in the other treatment (the "contradictions"). A particularly striking example of the contrast between economic intuition and the cold logic of game theory is the "traveler's dilemma" described by Basu (1994). The dilemma is based on a situation in which two vacationers have purchased identical objects, which are then lost on the flight home. The airline tells them to fill out claim forms independently, with the promise that both claims will be paid if they match. Otherwise, both travelers are only reimbursed at the lower of the claims, with a small penalty for the high claimant and an equally small reward for the low claimant. Even with a very low penalty and reward, each person has an incentive to "undercut" any anticipated common claim amount, and so the only Nash equilibrium (in pure or mixed strategies) is at the lowest possible claim. The implausibility of this prediction becomes apparent when one considers very low values of the penalty and reward parameter. Capra, Goeree, Gomez, and Holt (1999) report a traveler's dilemma experiment in which claims are required to be between 80 and 200 cents, so the Nash equilibrium involves claims of 80 cents. With the penalty for the high claimant and the reward for the low claimant set at 50 cents, claims converged to near-Nash levels, as indicated by the blue bars in Figure 1. But with a relatively low penalty-reward level of 10 cents, the frequency distribution of claims (represented by the green bars) is much higher. Note that this distribution is concentrated at the opposite side of the feasible set from the Nash