# Testing the Minimax Theorem in the Field

## The Interaction between Pitcher and Batter in Baseball

### Abstract

John von Neumann’s Minimax Theorem is a central result in game theory, but its practical applicability is questionable. While laboratory studies have often rejected its conclusions, recent field studies have achieved more favorable results. This thesis adds to the growing body of field studies by turning to the game of baseball. Two models are presented and developed, one based on pitch location and the other based on pitch type. Hypotheses are formed from assumptions on each model and then tested with data from Major League Baseball, yielding evidence in favor of the Minimax Theorem.

### Introduction

The Minimax Theorum

In any two-player zero-sum game of perfect information with finitely many strategies, there is an optimal mixed strategy solution. Furthermore, by performing according to this solution, each player maximizes her minimum possible payoff, and this payoff is unique in the sense that no other outcome involving optimal play can make either player better or worse off. This is what John von Neumann’s Minimax Theorem tells us. The Minimax Theorem is an essential result in game theory. Many of the advances made in game theory over the last century have stemmed from von Neumann’s proof in 1928 . As von Neumann himself said, “there could be no theory of games […] without that theorem” (Casti, 1996). In addition to its role within game theory, the Minimax Theorem has had an impact on the social sciences. Situations involving direct competition are relevant in economics, political science, and psychology. The Minimax Theorem provides us with the tools to analyze these situations.

There are reasons to trust the Minimax Theorem. We are familiar with the importance of unpredictability from our own experiences in competitive situations. For example, Rock-Paper-Scissors is a commonly played game where most people feel the need to vary their actions over time to mimic randomness. Deception is often a natural aspect of competition and playing a mixed strategy seems like a reasonable way to keep an opponent off-balance. Additionally, the suggestion that people maximize their minimum payoff sounds appealing. Most people are risk averse, so it makes sense that they would aim to minimize their potential loss. In many common games, such as Rock-Paper-Scissors, the equilibrium suggested by the Minimax Theorem agrees with our intuition.

However, there are a few reasons we might doubt the practical applicability of the Minimax Theorem. While appealing, mixed strategies may be implausible. In order for a mixed strategy solution to be played, each player must choose a strategy to make the other player indifferent. It is counterintuitive that players choose their strategies based only on their opponents’ payoffs. Furthermore, the performance of a mixed strategy requires genuine random selection of actions. Since we are not random number generators, this is a serious concern. Our attempts at randomization may be influenced by biases. Unsuccessful randomization leads to predictability, which can be exploited in competitive situations. There are suggestions that players engage in “k-level thinking” and try to stay one step ahead of their opponents to gain an advantage. This idea is supported by results in games such as the “two-thirds of the average game.” Lastly, it is important to note that the equilibrium prescribed by the Minimax Theorem is highly unstable. If one player deviates slightly from his equilibrium strategy, the other has an incentive to change his strategy drastically, possibly even ceasing to play a mixture at all.

### Central Question and Structure

In conclusion, it is unclear whether the Minimax Theorem can adequately describe human behavior. This is the central question that will be addressed in this paper. This study uses the interaction between the pitcher and the batter in the game of baseball to test whether experts perform as prescribed by the Minimax Theorem. The remainder of the paper will be structured as follows: we will first present the existing literature on the viability of the Minimax Theorem, differentiating between laboratory and field experiments. We will next present two separate models based on pitch location and pitch type along with justified assumptions about the strategies and payoffs. We will then use the Minimax Theorem to formulate hypotheses for each of the two models. Finally, we will test our hypotheses using empirical data and conclude.

**ABOUT THE AUTHOR**

Christopher Rowe