棒球：一个扩展式博弈对决
Baseball, An Extensive-Form Game-Theoretic Duel
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中文总结 AI 辅助
该研究将棒球打席建模为带随机行动与不完全信息的有限两人零和扩展式博弈，利用MLB Statcast数据实现序列形式表示，通过线性规划与反向归纳计算极小极大均衡，得到相关策略与价值度量。
中文摘要 AI 辅助
我们将棒球的打席表述为一个有限两人零和、带有随机行动与不完全信息的扩展式博弈。连续的投球周期通过不断变化的球数连接，击球手在未观察到投手当前投球选择的情况下选择行动。终端结果通过跑垒期望效用估值，该效用包含打席期间的得分以及半局的延续价值。我们实现了该博弈的序列形式表示，其线性规划公式随博弈树线性增长，可对远超标准型实际范围的博弈树进行精确的极小极大均衡计算。自然的转移概率通过美国职业棒球大联盟（MLB）Statcast数据估计。我们还建立了两种互补的动态规划解释：对于具有完美回忆的一般有限两人零和博弈，我们证明序列形式最佳响应程序的对偶变量可分解为可达权重与条件延续价值；在棒球模型额外的状态马尔可夫假设下，我们证明可通过对12种非终端球数进行反向归纳计算完整的极小极大均衡。所得计算结果产生均衡策略、延续价值以及非最优行动的战略成本的条件与可达加权度量。
英文摘要
We formulate a baseball plate appearance as a finite two-person, zero-sum, extensive-form game with chance moves and imperfect information. Successive pitch cycles are connected through the evolving count, while the Batter chooses an action without observing the Pitcher's current pitch selection. Terminal outcomes are valued through a run-expectancy utility that incorporates both runs scored during the plate appearance and the continuation value of the half-inning. We implement the sequence-form representation of the game, whose linear-programming formulation grows linearly with the game tree and permits exact minimax equilibrium computations for trees far beyond the practical range of the normal form. Nature's transition probabilities are estimated from MLB Statcast data. We also establish two complementary dynamic-programming interpretations. For general finite two-person zero-sum games with perfect recall, we show that the dual variables of the sequence-form best-response programs decompose into reach weights and conditional continuation values. Under the additional state-Markov assumptions of the baseball model, we prove that the full minimax equilibrium can be computed by backward induction over the twelve non-terminal counts. The resulting computations produce equilibrium strategies, continuation values, and conditional and reach-weighted measures of the strategic cost of non-optimal actions.
