**Abstract:**

Inherent in standard game-theoretic models is that players have infinite
hierarchies of beliefs, that is, beliefs about the state of nature,
beliefs about others' beliefs about nature, and so on. Can the standard
framework nevertheless be used to model situations in which players
potentially have a finite depth of reasoning? This paper extends the
standard Harsanyi framework to allow for higher-order uncertainty about
players' depth of reasoning. The basic principle is that players with a
finite depth of reasoning cannot distinguish states that differ only in
players' beliefs at high orders. I apply the new framework to the
electronic mail game of Rubinstein (1989). Coordination on the
Pareto-efficient action is possible when there is higher-order
uncertainty about players' depth of reasoning, unlike in the standard
case, provided that one player thinks it is sufficiently likely that the
other player has a finite (though potentially very high) depth of
reasoning. Finally, I construct a type space that allows for bounded
reasoning that contains the universal type space (which generates all
infinite belief hierarchies) as a subspace, showing that the present
framework fully generalizes the Harsanyi formalism.