Game Theory in Baseball
Source: http://theincidentaleconomist.com/wordpress/the-game-theory-within-the-game/
This article speaks of game theory in baseball; particularly in the situation in which a pitcher faces off with a runner on first. The author considers this in terms of game theory, while simplifying it by eliminating the effects of all other players besides the runner and pitcher, as well as considering the effectiveness of the runner’s speed and the pitcher’s pickoff move to be equal. In this situation, there are four possibilities: the pitcher tries to pick the runner off and the runner tries to steal, pick off and no steal, no pickoff and the runner tries to steal, and no pickoff and no steal. The author then correctly states that there is no pure Nash Equilibrium in this case. This can be demonstrated in the following chart:
| (Pitcher payoff, Runner payoff) | Steal | No steal |
| Pickoff | 1,-1 | -0.5,0.25 |
| No pickoff | -1,1 | 0,0 |
The (-0.5, 0.25) set as the reward for the case in which the pitcher tries to pick the runner off and the doesn’t steal is an arbitrary number to account for the possibility that the pickoff throw is thrown away (and the runner advances to second), there is a balk called, or to account for the fatigue caused to the pitcher’s arm. As one can see, there is no set of strategies that are best responses for one another. If the pitcher tries to pick the runner off, it is in the runner’s best interest not to steal, and vice-versa. If the pitcher does not try to pick the runner off, it is in the runner’s best interest to steal, and vice-versa. As the author correctly states, mixed strategies come into play. The pitcher must used mixed strategy; likely throwing with some probability based on the runner’s tendency to steal. In this ideal situation, with a lack of a Nash Equilibrium, it makes predicting what the other player will do extremely difficult, leaving perhaps no obvious choice for either player.
Another situation that could be analyzed using game theory in baseball is the approach of a pitcher and batter against one another. Consider this situation: a pitcher is facing a batter, and for simplicity, they are each equally skilled at their respective positions. The pitcher only has two pitches, a fastball and a slider. If the pitcher throws a fastball, he will be conservative, and it will be in the strike zone. If he throws a slider, it will be to try and get a strikeout, and it will be out of the strike zone. If the batter guesses fastball, he swings. If it is a fastball, he gets a single and if it is a slider, he strikes out. If he guesses slider, he does not swing, and strikes out on a fastball, but walks on a slider. The bases are empty, so a walk is just as good as a single. This can be shown as:
| (Pitcher payoff, Batter payoff) | Guess Fastball | Guess Slider |
| Throw Fastball | 0,1 | 1,0 |
| Throw Slider | 1,0 | 0,1 |
This is another situation in which there is no pure Nash Equilibrium. If the pitcher throws one pitch, it would be in the batter’s best interest to guess that pitch. If the batter guesses a pitch, it would be in the pitcher’s best interest to throw the other pitch. Once again, the players must use mixed strategies, with probabilities in which they use each strategy based on many factors, mostly including the pitcher’s history of pitch selection. What both of these situations imply is that baseball is not a game of compromise. There are very few, if any, situations where both teams will “win” in some form. This leads to a lack of Nash Equilibriums throughout the many situations in a baseball game. While the results of baseball games may be largely due to skill, this study of game theory shows that, at its simplest form, baseball may be a game of guessing and chance.