Major League Baseball (MLB) recently limited pitchers to three pickoff attempts, creating a cat-and-mouse game between pitcher and runner. Each failed attempt adds pressure on the pitcher to avoid using another, and the runner can intensify this pressure by extending their leadoff toward the next base. We model this dynamic as a two-player zero-sum sequential game in which the runner first chooses a lead distance, and then the pitcher chooses whether to attempt a pickoff. We establish optimality characterizations for the game and present variants of value iteration and policy iteration to solve the game. Using lead distance data, we estimate generalized linear mixed-effects models for pickoff and stolen base outcome probabilities given lead distance, context, and player skill. We compute the game-theoretic equilibria under the two-player model, as well as the optimal runner policy under a simplified one-player Markov decision process (MDP) model. In the one-player setting, our results establish an actionable rule of thumb: the Two-Foot Rule, which recommends that a runner increase their lead by two feet after each pickoff attempt.

翻译：美国职业棒球大联盟（MLB）近期将投手牵制尝试次数限制为三次，这在投手与跑垒员之间形成了一种"猫鼠游戏"。每次失败的牵制尝试都会增加投手避免再次尝试的压力，而跑垒员则可通过扩大向下一个垒位的离垒距离来加剧这种压力。我们将此动态建模为一个两人零和序贯博弈：跑垒员首先选择离垒距离，随后投手决定是否尝试牵制。我们建立了该博弈的最优性表征，并提出了求解该博弈的价值迭代与策略迭代变体方法。利用离垒距离数据，我们针对给定离垒距离、比赛情境及球员技能条件下的牵制与盗垒结果概率，建立了广义线性混合效应模型进行估计。我们计算了两人模型下的博弈论均衡解，以及简化单人马尔可夫决策过程模型下的最优跑垒员策略。在单人设定中，我们的研究结果确立了一条实用经验法则——"两英尺规则"，该规则建议跑垒员在每次牵制尝试后将离垒距离增加两英尺。