We study an interactive framework that explicitly allows for nonrational behavior. We do not place any restrictions on how players’ behavior deviates from rationality, but rather, on players’ higher-order beliefs about the frequency of such deviations. We assume that there exists a probability p such that all players believe, with at least probability p, that their opponents play rationally. This, together with the assumption of a common prior, leads to what we call the set of p-rational outcomes, which we define and characterize for arbitrary probability p. We then show that this set varies continuously in p and converges to the set of correlated equilibria as p approaches 1, thus establishing robustness of the correlated equilibrium concept to relaxing rationality and common knowledge of rationality. The p-rational outcomes are easy to compute, also for games of incomplete information. Importantly, they can be applied to observed frequencies of play for arbitrary normal-form games to derive a measure of rationality p that bounds from below the probability with which any given player chooses actions consistent with payoff maximization and common knowledge of payoff maximization.

Germano acknowledges financial support from the Spanish Ministry of Economy and Competitiveness (Grant ECO2014-59225-P), as well as from the Barcelona GSE Research Network and the Generalitat de Catalunya. Zuazo-Garin acknowledges financial support from the Spanish Ministry of Science and Technology (Grant ECO2009-11213), from the Spanish Ministry of Economy and Competitiveness (Grant ECO2012-31326) and from the Basque Government (Grants IT568-13 and POS-2015-1-0022).