On the equivalence between iterated application of choice rules and common belief of applying these rules

One central issue tackled in epistemic game theory is whether for a general class of strategic games the solution generated by iterated application of a choice rule gives exactly the strategy profiles that might be realized by players who follow this choice rule and commonly believe they follow this rule. For example, Brandenburger and Dekel (1987) and Tan and Werlang (1988) have established that this coincidence holds for the choice rule of strict undominance in mixtures in the class of finite strategic games, and Mariotti (2003) has established that this coincidence holds for Bernheim's (1984) choice rule of point rationality in the class of strategic games in which the strategy sets are compact Hausdorff and the payoff functions are continuous. In this paper, we aim at studying this coincidence in a general way. We seek to figure out general conditions of the choice rules ensuring it for a general class of strategic games. We state four substantial assumptions on choice rules. If the players' choices rules satisfy - besides the technical assumption of regularity - the properties of reflexivity, monotonicity, Aizerman's property, and the independence of payoff equivalent conditions, then this coincidence applies. This result proves to be strict in the following sense. None of the four substantial properties can be omitted without eliminating the coincidence.