We study the emergence of locally suboptimal behavior in finitely repeated games. Locally suboptimal behavior refers to players play suboptimally in some rounds of the repeated game (i.e., not maximizing their payoffs in those rounds) while maximizing their total payoffs in the whole repeated game. The central research question we aim to answer is when locally suboptimal behavior can arise from rational play in finitely repeated games. In this research, we focus on the emergence of locally suboptimal behavior in subgame-perfect equilibria (SPE) of finitely repeated games with complete information. We prove the first sufficient and necessary condition on the stage game G that ensure that, for all T and all subgame-perfect equilibria of the repeated game G(T), the strategy profile at every round of G(T) forms a Nash equilibrium of the stage game G. We prove the sufficient and necessary conditions for three cases: 1) only pure strategies are allowed, 2) the general case where mixed strategies are allowed, and 3) one player can only use pure strategies and the other player can use mixed strategies. Based on these results, we obtain complete characterizations on when allowing players to play mixed strategies will change whether local suboptimality can ever occur in some repeated game. Furthermore, we present an algorithm for the computational problem of, given an arbitrary stage game, deciding if locally suboptimal behavior can arise in the corresponding finitely repeated games. This addresses the practical side of the research question.

翻译：我们研究有限重复博弈中局部次优行为的涌现。局部次优行为指在重复博弈的某些轮次中玩家采取非最优行动（即未在该轮次最大化收益），但能在整个重复博弈中实现总收益最大化。本文旨在回答的核心研究问题是：在有限重复博弈中，局部次优行为何时能从理性决策中涌现。本研究聚焦于完全信息有限重复博弈的子博弈完美均衡（SPE）中局部次优行为的涌现机制。我们证明了关于阶段博弈G的第一个充要条件——该条件确保对所有T及重复博弈G(T)的所有子博弈完美均衡而言，G(T)每一轮的策略组合均构成阶段博弈G的纳什均衡。我们针对三种情形分别给出了充要条件：1）仅允许纯策略；2）允许混合策略的一般情形；3）一方玩家仅能使用纯策略而另一方玩家可使用混合策略的情形。基于这些结果，我们完整刻画了允许玩家使用混合策略时，是否会在某些重复博弈中改变局部次优性的发生条件。此外，针对任意给定的阶段博弈，我们提出了一种判定对应有限重复博弈中能否出现局部次优行为的算法。这解决了该研究问题的实践层面需求。