Baseball, An Extensive-Form Game-Theoretic Duel

Abstract

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.