# Game Theory and Professional Mixed-Strategy Models

## Article excerpt

Game theory is a branch of mathematics involving the creation and study of models of situations in which outcomes are interdependent on choices made by two or more actors. A game model requires the following elements:

1. Players. These are assumed to be rational actors weighing costs and benefits as they pursue their own goals. There must be two or more players.

2. Rules of the game. These define the limits of action - what can and cannot be done in the game.

3. Strategies. These are the choices that the players can make within the rules of the game. A strategy is a complete set of choices from beginning to end of the game. For example, if a player can make three different decisions, and for each decision there are two alternatives, he has [2.sup.3] = 8 different strategies for the whole game.

4. Payoffs. These are the outcomes that accrue to players depending on the choice of strategies they and their opponents make. Payoffs may he either ordinal or cardinal.

5. Solutions. A solution is the set of payoffs arising from the strategies that rational players would choose under the rules of the game. Sometimes there are multiple solutions. Indeed, sometimes there are multiple solution concepts, that is, more than one line of reasoning that rational actors might employ.

Game theory emerged as a distinct intellectual enterprise in 1944 with the publication of the Theory of Games and Economic Behavior, by John yon Neumann and Oskar Morgenstern. Its maturity was signalled fifty years later by the award of the 1994 Nobel Prize for economics to three eminent scholars in the field. Game theory is now widely used in economics and political science, and to a lesser extent sociology, psychology, and the other social sciences. Good introductions for nontechnical readers have been published by Dixit and Nalebuff (1991), Davis (1983), and Hamburger (1979).

Curiously, game theory has been little applied to the analysis of athletic events, even though these competitive contests have all the characteristics of games as described above. This paper is intended to show how game theory can contribute to the understanding of sport behavior.(1)

Baseball is well known for the quantity and quality of statistical data associated with the game; and at least one major work (Cook, 1966) has used these data to propound mathematical rules of strategy. However, Cook's approach was not game-theoretical because it did not focus on the interplay of strategic choices made by the offence and defence. Cook looked at offence and defence in isolation, calculating the productivity of various strategies over a long run of games. This amounts to conceiving baseball as a "game against nature," in which players make optimizing choices against random events. But a baseball game is a contest between two teams of opponents, each of which has independent choices to make in the attempt to defeat the other. As such, it is an ideal subject for game-theoretical analysis.

The particular solution concept employed in this paper is known as "solution in mixed strategies." It derives from the minimax theorem, proved by John von Neumann in 1926, which was a turning point in the development of game theory. Von Neumann showed that every two-person, zero-sum game with a finite number of strategies has a solution. When that solution is a pair of pure strategies, it is known as a saddlepoint. Sometimes, however, players do not have a single best strategy, in which case their rational choice is a random mixture of pure strategies in certain calculable proportions (Davis, 1983; Hamburger, 1979; Poundstone, 1992; Rapoport, 1966; Williams, 1982). In this article, the solution of the various models always represents the proportions of at-bats in professional baseball taken by left- and right-handed batters and pitched by left- and right-handed pitchers.

Pitchers vs. Batters

Although baseball is a complex team sport, the focus of attention is the duel between pitcher and batter. …