Strategic uncertainty and the ex post Nash property in large games

• 成果类型: Article
• 署名作者: Khan, M. Ali; Rath, Kali P.; Sun, Yeneng; Yu, Haomiao
• 署名单位: Johns Hopkins University; University of Notre Dame; National University of Singapore; Toronto Metropolitan University
• 刊物名称: THEORETICAL ECONOMICS
• ISSN/ISSBN: 1933-6837
• DOI: 10.3982/TE1492
• 发表日期: 2015-01-01
• 页码: 103-129
• 关键词: 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

摘要

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 longstanding open problems and show that (i) any MSE induces a RSED and any RSED can be lifted to a MSE, and (ii) a mixed strategy profile is aMSE 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 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 present an approximate result pertaining to a sequence of large but finite games.