Rationalizability, Iterated Dominance, and the Theorems of Radon and Carath\'eodory

The game theoretic concepts of rationalizability and iterated dominance are closely related and provide characterizations of each other. Indeed, the equivalence between them implies that in a two player finite game, the remaining set of actions available to players after iterated elimination of strictly dominated strategies coincides with the rationalizable actions. I prove a dimensionality result following from these ideas. I show that for two player games, the number of actions available to the opposing player provides a (tight) upper bound on how a player's pure strategies may be strictly dominated by mixed strategies. I provide two different frameworks and interpretations of dominance to prove this result, and in doing so relate it to Radon's Theorem and Carath\'eodory's Theorem from convex geometry. These approaches may be seen as following from point-line duality. A new proof of the classical equivalence between these solution concepts is also given.