Strategic uncertainty and the ex-post Nash property in large games - Haomiao Yu; M. Ali Khan; Kali P. Rath; Yeneng Sun (Theoretical Economics, 2015)
This paper elucidates the conceptual role that independent randomization plays in non-cooperative game theory. In the context of large (atomless) games in normal form, we present precise formalizations of the notions of a mixed strategy equilibrium (MSE), and of a randomized strategy equilibrium in distributional form (RSED). We offer a resolution of two long-standing open problems and show: (i) any MSE induces a RSED, and any RSED can be lifted to a MSE, (ii) a mixed strategy profile is a MSE if and only if it has the ex-post Nash property. Our substantive results are a direct consequence of an exact law of large numbers (ELLN) that can be formalized in the analytic framework of a Fubini extension. We discuss how the measurability' problem associated with a MSE of a large game is automatically resolved in such a framework. We also illustrate our ideas by an approximate result pertaining to a sequence of large but finite games.
Keywords: C72; D84; C65; 330; Large game; pure strategy; mixed strategy; randomized strategy in distributional form; Nash equilibrium; ex-post Nash property; saturated probability space; rich Fubini extension; exact law of large numbers (ELLN); asymptotic implementation