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The Game Theory of Baseball

The duel between a pitcher and batter is arguably the most important aspect of baseball and as such it is very important to understand the dynamics at play between the pitcher and batter. A pitcher must decide which type of pitch to throw (fastball, changeup, slider, etc.) and where to aim each pitch (high, low, inside, outside, etc.). A batter must decide what type of pitch and pitch location to expect and based on the expectations and split second read of the pitch if it is beneficial to swing or take the pitch.

 

The duel between a pitcher and batter can be seen as a two-player simultaneous game in which the strategies of the pitcher are the different pitch types and locations and the strategies of the batter are to swing or take the pitch based on the batters expectations of pitch type and location and split second reads. For simplicity purposes, we will ignore the split second reads in the time that the pitch is traveling towards the plate and assume that the batter already decided to swing or not to swing prior to the pitch. In this situation, neither player has a dominant strategy as players will know what to expect and act accordingly based on historical patterns. Thus, the Nash equilibrium can only be achieved by a mixed strategy of both players.

 

In the paper by Gelblum, Laucys, Saperstein, and Sodha, the authors dove into the specific situations that occurred during the 8th inning of the Dodgers v Giants game on April 18th, 2010. The most interesting finding of the paper talks about the mixed strategy equilibrium of pitch selection and expectation, not including pitch location for simplicity reasons. Not surprisingly, the expected outcome of RAA (Runs Above Average) for the batter is only positive when he correctly guesses the pitch that is being thrown and is only positive for the pitcher when the batter incorrectly guesses the pitch coming. Based on the batter’s negative RAA payoffs for incorrectly guessing pitches, the batter is least negatively impacted by not guessing a fastball than not guessing either a changeup or slider. Thus, the mixed strategy Nash equilibrium occurs with the batter, Manny Ramirez guessing Fastball 6.25%, slider at 50.02% and changeup at 43.73% frequencies while the pitcher pitches fastballs 43.47%, slider 34.90% and the changeup at 21.63% frequencies. The batter over expects the pitches that have worse payoffs for the batter.

 

Source: The Game Theory of Baseball

http://faculty.haas.berkeley.edu/rjmorgan/mba211/2010%20Final%20Projects/RPS%20Game%20Theory%20Final%20-%20Baseball.pdf

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