How Game Theory Affects Decision-Making in Baseball?
The foundation of game theory studies the interactions between players as they make decisions depending upon the actions of the others. Meaning, game theory is a type of strategy a player can use in order to predict the actions of the other participants involved in the game. Players in a game will seek the best response in which they utilize the number one strategy compared to the action of the other player in the game.
At the onset of an at-bat in baseball, the pitcher and batter compete in a one vs. one game as decision-makers. The pitcher tries to instill the best strategy depending on what the batter will do and the batter will do the same. A 3-2 count is one of the most pivotal situations in baseball as one more ball or strike will determine the fate of the at-bat. Both the pitcher and hitter must make a decision in the approach that they are going to take before the next pitch. A pitcher must decide whether they should throw a safe fastball to avoid throwing a ball or an offspeed pitch that could catch the batter off-guard. Likewise, the batter must time and locate their swing based upon whether they perceive that a fastball or an offspeed pitch will be coming in. These four scenarios are detailed in the table below with their respective probabilities and payoffs given.
As detailed in the table, a hitter who accurately predicts a fastball (with the probability “p”), will receive a large payoff due to the benefits in timing and swing location that a hitter will receive from anticipating a fastball. Similarly, a hitter correctly calling an offspeed pitch (with a probability “1-p”), will also see a positive payoff but not as large as if a fastball was thrown and correctly predicted. A pitcher will face a similar situation when they throw a fastball (with a probability “q”), while the hitter anticipates the wrong pitch, and will receive a large payoff by doing so. Similarly, the pitcher will receive a positive payoff when they throw an offspeed pitch (with a probability of “1-q”), and the hitter is wrong in their prediction.
The largest negative tradeoffs come when the pitcher decides to throw a fastball (“q”) as the pitcher will receive the steepest negative payoff when their pitch is accurately predicted, but will have their largest payoff in the contrary. This introduces a mixed strategy equilibrium between the pitcher and batter as there is a best response for each of the players depending on the strategy of the other player against them.
This mixed strategy nash equilibrium is expressed:
Pitcher – Fastball: 4q + – 4(1-q)
8q – 4
Pitcher – Offspeed: -2q + 2(1-q)
-4q + 2
q=1/2
Hitter – Fastball: -4p + 2(1-p)
-6p+2
Hitter – Offspeed: 4p + -2(1-p)
6p – 2
p=⅓
Based on the mixed strategy Nash equilibrium performed above, it is clear that when stepping up to the plate on a 3-2 count, hitters should expect a fastball ½ the time and an off-speed pitch the other 1/2. Moreover, Pitchers should throw a fastball ⅓ of the time on a 3-2 count and throw an offspeed pitch the other ⅔ times to keep the batters on their toes. As shown in the table above, a pitcher is safer throwing an offspeed pitch than a fastball because of the payoffs that result, and on a 3-2 count a pitcher has to be on high alert. In addition, a hitter is safer in the box by anticipating a fastball as the payoff of getting the call wrong is large when predicting an off-speed pitch. This strategy is beneficial for both pitchers and hitters as they remain aware of the tradeoff and payoffs that result in the decisions they make based upon what they believe the other player will do.
Reference: https://www.baseballdatascience.com/game-theory-applications-in-baseball/