Backward induction and beliefs about oneself

According to decision theory, the rational initial action in a sequential decision-problem may be found by ‘backward induction’ or “folding back’. But the reasoning which underwrites this claim appeals to the agent's beliefs about what she will later believe, about what she will later believe she will still later believe, and so forth. There are limits to the ‘depth’ of people's beliefs. Do these limits pose a threat to the standard theory of rational sequential choice? It is argued, first, that the traditional solutions of certain games depend on knowledge which exceeds depth limits, and that these solutions therefore cannot be shown rational in the usual sense. Then, for that related reason even ‘folding back’ solutions of one-person problems cannot be! A revision of our notion of rational choice is proposed, analogous to the reliabilist account of knowledge of Goldman and others, by which this paradox is resolved.