Statistical Equilibrium of Optimistic Beliefs

Abstract

We study finite normal-form games in which payoffs are subject to random perturbations and players face uncertainty about how these shocks co-move across actions, an ambiguity that naturally arises when only realized (not counterfactual) payoffs are observed. We introduce the Statistical Equilibrium of Optimistic Beliefs (SE-OB), inspired by discrete choice theory. We model players as \textit{optimistic better responders}: they face ambiguity about the dependence structure (copula) of payoff perturbations across actions and resolve this ambiguity by selecting, from a belief set, the joint distribution that maximizes the expected value of the best perturbed payoff. Given this optimistic belief, players choose actions according to the induced random-utility choice rule. We define SE-OB as a fixed point of this two-step response mapping. SE-OB generalizes the Nash equilibrium and the structural quantal response equilibrium. We establish existence under standard regularity conditions on belief sets. For the economically important class of marginal belief sets, that is, the set of all joint distributions with fixed action-wise marginals, optimistic belief selection reduces to an optimal coupling problem, and SE-OB admits a characterization via Nash equilibrium of a smooth regularized game, yielding tractability and enabling computation. We characterize the relationship between SE-OB and existing equilibrium notions and illustrate its empirical relevance in simulations, where it captures systematic violations of independence of irrelevant alternatives that standard logit-based models fail to explain.