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Lead distance under a pickoff limit in Major League Baseball: A sequential game model

Scott Powers, Sivaramakrishnan Ramani, Jacob Hahn, Andrew J. Schaefer

TL;DR

This paper analyzes MLB’s pickoff-limit rule by modeling the pitcher–runner interaction as a two-player zero-sum sequential game, where the runner selects a lead distance and the pitcher chooses to attempt a pickoff or throw a pitch. It couples a data-driven transition-probability framework—built from generalized linear mixed-effects models for runner outcomes and state transitions—with undiscounted value/policy iteration to characterize game-theoretic equilibria and optimal runner policies. A one-player MDP variant further yields an actionable heuristic, the Two-Foot Rule, recommending a two-foot increase in lead after each pickoff attempt. The study estimates that adopting these strategies could yield substantial run gains (up to about 12 extra runs per season) while highlighting the difference between observed conservative behavior and game-theoretic optima; the work provides a practical, data-backed guideline for teams under the new rule changes.

Abstract

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.

Lead distance under a pickoff limit in Major League Baseball: A sequential game model

TL;DR

This paper analyzes MLB’s pickoff-limit rule by modeling the pitcher–runner interaction as a two-player zero-sum sequential game, where the runner selects a lead distance and the pitcher chooses to attempt a pickoff or throw a pitch. It couples a data-driven transition-probability framework—built from generalized linear mixed-effects models for runner outcomes and state transitions—with undiscounted value/policy iteration to characterize game-theoretic equilibria and optimal runner policies. A one-player MDP variant further yields an actionable heuristic, the Two-Foot Rule, recommending a two-foot increase in lead after each pickoff attempt. The study estimates that adopting these strategies could yield substantial run gains (up to about 12 extra runs per season) while highlighting the difference between observed conservative behavior and game-theoretic optima; the work provides a practical, data-backed guideline for teams under the new rule changes.

Abstract

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.
Paper Structure (16 sections, 5 theorems, 39 equations, 4 figures, 6 tables)

This paper contains 16 sections, 5 theorems, 39 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. The two-player sequential game model
  4. Transition probability model
  5. Probability model for runner outcomes
  6. Pickoff attempt probability
  7. Pickoff success probability
  8. Stolen base attempt probability
  9. Stolen base success probability
  10. Probability model for state transitions
  11. Numerical study
  12. Runner outcome probability model
  13. Two-player sequential game
  14. One-player MDP and the Two-Foot Rule
  15. Discussion and strategic insights
  16. ...and 1 more sections

Key Result

Lemma 1

Suppose Assumption assum:game_halts holds. Then $V^{\pi_\text{R},\pi_\text{P}}(s)$ is well-defined for all $\pi_\text{R} \in \overline\Pi_\text{R}, \pi_\text{P} \in \overline\Pi_\text{P}$, and $s \in {\cal S}$. Also, $V^{\pi_\text{R},\pi_\text{P}}$ is uniformly bounded over $\overline\Pi_\text{R} \t

Figures (4)

  • Figure 1: The league-wide distribution of lead distance from first base when second base is unoccupied, comparing 2022 with 2023. The 2023 data are split by prior disengagements because, beginning in 2023, pitchers were limited to three disengagements, influencing pitcher and runner behavior. In 2022, there was no limit on the number of disengagements.
  • Figure 2: Pickoff attempt probability (left) and success probability (right) modeled as functions of lead distance. The left figure corresponds to the model detailed from Section \ref{['sec:prob-po-attempt']}, assuming zero balls, zero strikes, zero outs, and a median pitcher random effect. The right figure corresponds to the model from Section \ref{['sec:prob-po-success']}, which includes only an intercept, a fixed effect for lead distance and a random effect for pitcher identity.
  • Figure 3: Stolen base success probability modeled as function of lead distance, split by season and runner effect (combining the fixed effect of runner sprint speed and the random effect of runner identity). This figure shows the probability estimated by the model detailed in Section \ref{['sec:prob-sb-success']}, assuming zero balls, zero strikes, zero outs, and a median value for the combined effect of pitcher identity (random effect), catcher arm strength (fixed effect), and catcher identity (random effect).
  • Figure 4: (Left) The relationship between arm strength and catcher combined effect on stolen base success probability (relative to average, on the log-odds scale). The combined effect is the sum of the fixed effect due to arm strength and the catcher random effect. (Right) The relationship between sprint speed and runner combined effect on stolen base success probability (relative to average, on the log-odds scale). The combined effect is the sum of the fixed effect due to sprint speed and the runner random effect.

Theorems & Definitions (11)

  • Lemma 1
  • proof
  • Definition 1
  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 1 more