The Game Theory of Bunting
Last week, the movie “Moneyball” debuted in theaters. Based on Michael Lewis’s book of the same name, it follows the story of how Oakland A’s General Manager Brad Pitt Billy Beane built a competitive team for the 2002 season with a payroll that was just a fraction of the league’s other top teams. Beane was the first GM to adopt the principles of sabremetrics when making player personnel decisions; however, the new wave of statistical analysis had begun many years before with the work of Bill James and the Society for American Baseball Research. The movie emphasizes the aspect of sabermetrics that deals primarily with player valuation, but the application of economics principles, particularly game theory, has the potential to impact other areas of the game.
Bunting has been an established tactic in baseball for over a century. In particular, bunting has two distinct uses: bunting for a base hit and bunting to advance runners already on base (known as a sacrifice bunt, since the player that bunts will usually be out). Through study of win expectancies, it has been demonstrated that bunting to advance runners is almost always a poor strategy (due to the fact that giving the opposition a free out has a significant negative impact on win expectancy), so we’ll consider the case of bunting for a base hit. The decision to bunt or not can be seen as a simple game between the batter and the opposing manager.
The batter’s possible strategies are to bunt or swing away (not bunt), and the opposing manager’s strategies are to position the defense to guard against the bunt (by moving the infield in), or play the infield at normal depth. If the defense is in and the player bunts, he is more likely to be out, but if he swings away he is more likely to get a hit, compared to the defense playing at normal depth. This setup is very similar to the game theory problem involving two generals that we studied earlier in the semester. There is no Nash equilibrium, since the best response to the defense being in would be to not bunt, while the best response to not bunting would be to play the defense at normal depth. This indicates that both sides should employ a randomized mixed strategy, as in the general game.
However, one important factor has been neglected in this analysis: the defense must make its move before. That means that the batter has full information regarding the move of the defense. Therefore, the batter should always choose the strategy that maximizes payoff based on the defense’s choice. What does this mean for the manager of the defense? Since the batter has superior information, he can always counter whatever strategy the defense chooses. The defensive manager’s best play would then be to hedge his bets – minimize the batter’s payoff by playing the defense at the exact depth that equalizes the payoff for bunting and swinging away. This depth would obviously be dependent on the skills of the batter (speed, power, contact skills, etc.), but if all managers were to position their defense at the optimal depth, they would probably be able to save a few runs over the course of a season, helping their team to win more games.
Article: http://collegian.csufresno.edu/2011/10/03/%E2%80%98moneyball%E2%80%99-yet-to-affect-managers/