Non-probabilistic Correlated Equilibrium as an Expression of Non-bayesian Rationality

We define weak and strong rationality of players in terms of dominance rather than expectation with respect to probabilistic beliefs. We also define certain subsets of strategy profiles as weak or strong non-probabilistic correlated equilibrium, analogously to correlated equilibrium. It is shown that the set of profiles played when there is common knowledge of weak or strong rationality is a weak or strong non-probabilistic correlated equilibrium, correspondingly. The largest weak non-probabilistic correlated equilibrium is the set of profiles that survive iterated elimination of strictly dominated strategies. Rationality can be strengthened for games with perfect information by considering rationality in subgames. A player is substantively rational if she is rational in all subgames, and materially rational if she rational in subgames at vertices that are reached. Aumann (1995, 1998) introduced and studied these two notions for the case of weak rationality. We characterize the weak and the strong versions of these notions of rationality in terms of non-probabilistic correlated equilibria. We show that non-probabilistic correlated equilibria can be also characterized in terms of non-probabilistic belief rather than knowledge.