For example, consider a "matching pennies" game in which a soccer player can either kick to the East side or the West side, and the goalie can either dive East or West. Suppose that a "match" (the goalie dives to meet the kick) gives payoffs of 1 to the goalie and -1 to the kicker, whereas a mismatch gives -1 to the goalie and +1 to the kicker. Each player wants to be unpredictable, and the only Nash equilibrium for this symmetric game is for each player to choose each direction with probability 1/2 (otherwise at least one player would want to change their behavior). This prediction is confirmed in laboratory matching pennies games. Now consider what happens if the fans of the goalie's team, who are seated on the East side of the field, announce publicly that they will donate ten thousand dollars to the team scholarship fund if the goalie blocks a kick on the East side. This only increases the goalie's payoff for the (dive-East, kick-East) outcome, but it cannot increase the Nash equilibrium probability that the goalie dives East, since the kicker must remain indifferent over the two kick directions in a Nash equilibrium with randomized decisions. This unintuitive prediction, that a change in a player's own payoff for one of the outcomes will not change that player's probability of choosing that outcome, is dramatically rejected in the experiments based on asymmetric matching pennies games. Thus the Nash equilibrium works well in the symmetric game and it predicts poorly in the asymmetric matching pennies game.
If you look at the mathematical expressions in the traveler's dilemma paper, you will probably be skeptical about whether such abstract formulas can actually predict the behavior of human subjects who typically rely on intuition instead of pencil-and-paper calculations. People in the experiments adapt and learn, if one sees high claims in the first rounds of the experiment, it is reasonable to expect high claims in subsequent rounds. We use models of "belief" learning with noisy decisions to explain the pattern of adjustments to equilibrium
Our research introduces randomness into decision making, which causes decisions to be only imperfectly related to measured economic incentives. Although the extent of this randomness is a matter of empirical estimation, the incorporation of such noise is a common element of the techniques that we use to describe the behavior of our laboratory subjects. This narrative has described three complementary modifications of classical game theory. The models of introspection, learning/evolution, and equilibrium contain the common stochastic elements that represent errors or unobserved preference shocks. These three approaches are like the "three friends" of classical Chinese gardening (pine, cherry, and bamboo), they fit together nicely, each with a different purpose. Models of iterated noisy introspection are used to explain beliefs and choices in games played only once, where surprises are to be expected, and beliefs are not likely to be consistent with choices. With repetition, beliefs and decisions can be revised via learning or evolution. Choice distributions will tend to stabilize when there are no more surprises in the aggregate, and the resulting steady state constitutes a noisy equilibrium. This general approach can usefully incorporate a broad range of human motivations, including altruism, fairness, and risk aversion, factors which improve the predictive power of equilibrium models in some of the applications we have considered. To summarize, the major scientific insights are:
1) We have developed of a new model of noisy introspection to explain anomalous behavior in games played once. This introspective solution exists and reduces to the logit quantal response equilibrium in the special case where uncertainty does not increase with successive iterations of introspective thinking.
2) We have used the logit equilibrium to explain anomalous data patterns in a range of games, especially those with a continuum of strategies, where we derive existence, uniqueness, and comparative statics proofs. The advantage of this approach is that the same formal model tracks behavior that closely conforms to Nash equilibrium predictions in one treatment and deviates sharply in another.
3) We have developed ways to incorporate broader motivations into the stochastic equilibrium models, e.g. altruism, inequity aversion, and risk aversion, with applications to public goods, bargaining, and auctions.
4) We have used learning models to explain the time patterns of directional adjustment in specific laboratory experiments. We have characterized a steady-state learning equilibrium and its relationship to the (static) logit quantal response equilibrium.
5) For potential games, we use an entropy measure of dispersion to construct a "stochastic potential function" that is maximized by a noisy evolutionary process. All maxima of this stochastic potential are logit equilibria, which generalize the popular notion of "risk dominance" for selecting among multiple equilibria in 2x2 games.
