Game Theory in Pitch Selection- Baseball
http://www.fangraphs.com/community/how-game-theory-is-applied-to-pitch-optimization/
The game of baseball is almost entirely based upon the interactions between the pitcher and batter. Both parties are constantly trying to gain an advantage over the other in order for a winning outcome. As the game of baseball has evolved, different strategies have been used in order to exploit a batter’s or pitcher’s weakness. Sabermetrics, or the application of data analytics towards baseball, has become ubiquitous throughout the game. A pitcher can analyze an opposing batter’s swing in order to optimize his pitch selection. Sabermetrics can also explain underlying trends within the game. These can include a pitcher’s true performance becoming masked by a poor defense that allows more hits to the opposing teams, or a batter hitting ground balls as often as another batter, but directly at infielders rather than between them. Hatti James provides a method for pitchers to use game theory in order to win the duel between pitcher and batter.
Most sports are described as zero sum games as there is a direct conflict of interest between opposing teams. For example, an error by a fielder directly benefits the opposing batter. The payoff for the fielder is negative, while that of the batter is positive. Thus, the net payoff sums to 0, hence zero sum game. However, with pitching, one can use sabermetric analysis to predict a batter’s skillset, and therefore exploit their weaknesses. A certain batter may have a tendency to swing at pitches high up in the strike zone. Thus, a pitcher who has analyzed this batter’s tendencies would throw pitches in this area. This is a best response to the batter’s strategy, and establishes a Nash Equilibrium. Neil Paine, an analyst for FiveThirtyEight, used this to create a formula known as the Nash Score, or a way to predict which pitches a pitcher should throw, as well as their frequencies. This score optimizes pitching strategy, ensuring that a pitcher can efficiently rack up strikeouts while limiting the amount of pitches thrown.
In practice, however, the Nash Score would not as effective. Through the principles of Nash Equilibrium, both players in a game have no incentive to change their strategies as these strategies are best responses to one another’s. If a batter believes that pitchers are exploiting his tendency to swing at pitches high in the strike zone, he would adjust, thus letting these pitches go or improving his swing to ensure better contact. Therefore, the equilibrium shifts, and the pitcher is forced to re-adjust after that. The Nash Score assumes that batter will not make adjustments to how pitchers attack them, but in reality, adjustments are key in the game of baseball. Thus, mixed strategies that involve the analysis of an opposing player’s weaknesses as well as the ability to keep the opposing player guessing are the most effective. As James puts it, “a fastball will lose its effectiveness if it’s not balanced against changeup, even if the fastball is a far better pitch on paper.”
In summary, even if a pitcher has attained equilibrium and is de-incentivized to change their strategy, the batter cannot be assumed to continue to give the ideal “best response.” Thus, equilibrium shifts as batters learn to adjust, and pitchers learn to re-adjust. Players continue to choose the best responses for attacking their opponents, and the game of baseball is determined by how their opponent responds back.