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, arXiv:2107.10801v4), 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）博弈中纯策略子博弈完美均衡的特征。将博弈的扩展形式规范为五元组结构（Streufert 2023p, arXiv:2107.10801v4），该结构由一组五元组构成，形式化表达了节点、行动、参与者和情境（情境是信息集的推广）之间的抽象关系。由于五元组结构具有集合属性，本文可将博弈形式显式分割为片段形式，每个片段形式始于一个（泽尔腾）子根节点，并包含后续所有非随后的子根节点节点集合。由此，子根节点集合成为价值函数的定义域，片段形式划分则构成了价值递归的框架，该递归是对动态规划中贝尔曼方程的推广。主要结论在上下收敛假设下，建立了价值递归与原博弈子博弈完美均衡之间的关联。最后，推论将子博弈完美性刻画为不存在改进的单片段偏离。