Common belief in approximate rationality

This paper substitutes the standard rationality assumption with approximate rationality in normal form games. Players are assumed to be \(\varepsilon\)-rational, i.e. willing to settle for a suboptimal choice, and so give up an amount \(\varepsilon\) of expected utility, in response to the belief they hold about their opponents' choices. For every player \(i\) and every opponents' degree of rationality \(\varepsilon\), we require player \(i\) to attach at least probability \(F_i(\varepsilon)\) to his opponent being \(\varepsilon\)-rational, where the functions \(F_i\) are assumed to be common knowledge. We refer to this event as belief in \(F\)-rationality. The notion of Common Belief in \(F\)-Rationality (CBFR) 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 CBFR.