Common belief in approximate rationality

This paper substitutes the standard rationality assumption with approximate rationality in normal form games. We assume that players believe that their opponents might be ε-rational, i.e. willing to settle for a suboptimal choice, and so give up an amount ε of expected utility, in response to the belief they hold. For every player i and every opponents’ degree of rationality ε, we require player i to attach at least probability Fi(ε) to his opponent being ε-rational, where the functions Fi are assumed to be common knowledge amongst the players. We refer to this event as belief in F-rationality. The notion of Common Belief in F-Rationality (CB FR) is then introduced as an approximate rationality counterpart of the established Common Belief in Rationality. Finally, a corresponding recursive procedure is designed that characterizes those beliefs players can hold under CB FR.