An undecidable statement regarding zero-sum games

• 成果类型: Article
• 署名作者: Fey, Mark
• 署名单位: University of Rochester
• 刊物名称: GAMES AND ECONOMIC BEHAVIOR
• ISSN/ISSBN: 0899-8256
• DOI: 10.1016/j.geb.2024.02.004
• 发表日期: 2024
• 页码: 19-26
• 关键词: Zero-sum game; Undecidable; Herodotus paradox; Set theory

摘要

In this paper, we give an example of a statement concerning two-player zero -sum games which is undecidable, meaning that it can neither be proven or disproven by the standard axioms of mathematics. Earlier work has shown that there exist paradoxical two-player zero -sum games with unbounded payoffs, in which a standard calculation of the two players' expected utilities of a mixed strategy profile yield a positive sum. We show that whether or not a modified version of this paradoxical situation, with bounded payoffs and a weaker measurability requirement, exists is an unanswerable question. Our proof relies on a mixture of techniques from set theory and ergodic theory.