Bounded Reasoning and Higher-Order Uncertainty

The standard framework for analyzing games with incomplete information models players as if they have an infinite depth of reasoning. This paper generalizes the type spaces of Harsanyi (1967-1968) so that players can have a finite depth of reasoning. The innovation is that players can have a coarse perception of the higher-order beliefs of other players, thus formalizing the small-world idea of Savage (1954) in a type-space context. Unlike in other models of finite-order reasoning, players with a finite depth of reasoning can reason about higher-order events if these events are generated by events of sufficiently low order. In particular, an event F can be common belief if it is entailed by some public event. This is true even if players cannot reason about higher-order statements like "Ann believes that Bob believes that Ann believes...(58 times)...that F"' in isolation. Thus, the usual equivalence between the iterative and the fixed-point account of common belief breaks down when players have a finite depth, and common belief is easier to attain than as suggested by the iterative approach.