This paper uses value functions to characterize the pure-strategy subgame-perfect equilibria of an arbitrary, possibly infinite-horizon game. It specifies the game's extensive form as a pentaform (Streufert 2023p, coming revision of arXiv:2107.10801), which is a set of quintuples formalizing the abstract relationships between nodes, actions, players, and situations (situations generalize information sets). Because a pentaform is a set, this paper can explicitly partition the game form into piece forms, each of which starts at a (Selten) subroot and contains all subsequent nodes except those that follow a subsequent subroot. Then the set of subroots becomes the domain of a value function, and the piece-form partition becomes the framework for a value recursion which generalizes the Bellman equation from dynamic programming. The main results connect the value recursion with the subgame-perfect equilibria of the original game, under the assumptions of upper- and lower-convergence. Finally, a corollary characterizes subgame perfection as the absence of an improving one-piece deviation.

翻译：本文利用价值函数刻画任意（可能无限期界）博弈中的纯策略子博弈完美均衡。将博弈的扩展式表述为五元组形式（Streufert 2023p，arXiv:2107.10801的修订版），即一组对节点、行动、参与者和情境（情境是对信息集的泛化）间抽象关系进行形式化的五元组集合。由于五元组形式是基于集合的定义，本文可将博弈形式显式划分为若干片段形式——每个片段形式始于一个（泽尔腾意义上的）子根节点，并包含其后全部节点（后续子根节点之后的节点除外）。由此，子根节点集合成为价值函数的定义域，而片段形式划分则构成价值递归的框架，该递归是对动态规划中贝尔曼方程的一般化。在上下收敛性假设下，主要成果建立了价值递归与原始博弈子博弈完美均衡之间的关联。最终推论将子博弈完美性刻画为不存在改进性单片段偏离。