Rationalizability, Iterated Dominance, and the Theorems of Radon and Carathéodory
Roy Long
TL;DR
1) The paper investigates the link between rationalizability and iterated strict dominance in finite two‑player games. 2) It proves a tight bound: if a pure strategy in $A$ is strictly dominated by a mixed strategy over a subset $A'\subseteq A$, then it is strictly dominated by a mixed strategy over at most $\min(|A|-1,|B|)$ actions, with a symmetric bound for the opponent. 3) It provides two complementary proofs of this bound: (i) a Radon/Carathéodory convex‑geometry argument yielding the same bound and (ii) a linear‑algebraic perspective based on payoff vectors and conical hulls; both reveal a point–line duality between vector and hyperplane representations. 4) A geometric formulation shows that covering a convex polytope with open half‑spaces reduces to a subcover of at most $d+1$ half‑spaces, and the results extend to infinite action sets; the appendix supplies full proofs of Radon/Carathéodory and a rotation-covering lemma.
Abstract
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 Carathéodory'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.