The theory of strong dominance in games is well understood, but that of admissibility (weak dominance) much less so. Yet admissibility gives much sharper predictions in many games of applied interest. This paper gives epistemic foundations for admissibility concepts.

Specifically, we develop an epistemic structure for games, in which rational players choose admissible (not weakly dominated) strategies. With this definition of rationality, we formulate the condition of rationality and mth-order assumption of rationality (RmAR). Rationality and common assumption of rationality (RCAR) then means RmAR for all m.

We prove three results on admissibility in games:

(i) RCAR characterizes a new solution concept, which we call a self-admissible set.
(ii) RmAR, if formulated in a complete structure, characterizes m+1 rounds of iterated admissibility.
(iii) Under a nontriviality condition, RCAR is impossible in a complete structure.

The impossibility result appears to indicate a limit on players' ability to reason about all possibilities in a game. Admissibility asks a player to take all states into consideration. RCAR asks players to assume "RmAR for the other players" for all m. Completeness asks players to consider all types. Evidently, not all of this is possible.