Mixed Strategy in Baseball Pitches
https://www.fangraphs.com/community/how-game-theory-is-applied-to-pitch-optimization/
In class we discussed the mixed strategy involved in soccer penalty kicks. A very similar scenario reveals itself when considering the decision making done in baseball by a pitcher and batter. The pitcher is armed with an arsenal of pitches aimed to outplay the batter. On the other hand, the batter is tasked with getting the best possible outcome from a certain pitch. We can simplify this into a model we can reason about. Assume that the pitcher has two strategies: a fastball or a change up. To counter these the batter has a strategy that he knows will give him the best result based on the pitchers pitch. We will call these two options fastball bat and change up bat. The batter wins if he correctly counters a pitch with the analogous batting strategy. As discussed at the end of the above article, developing a pure strategy is not possible in this game for the pitcher as the batter would then develop the pure strategy that mitigates the pure strategy developed by the pitcher. Since there is no equilibrium developed by pure strategy there must be a mixed strategy equilibrium where a mix of strategies are randomly chosen based on certain probability assignments to each strategy.
The nature of these probabilities, however, is based on the actual payoffs of the game which are harder to define. But regardless, a certain percentage of fastballs is developed. As referenced in the article below this percentage was 66.1% back in 2008. But, since this percentage is based on the payoffs of each pair of strategy, if those payoffs change so does the percentage of that strategy. The article below discuses how pitchers have felt less pressured to use fastballs when they are behind in a count. But in reality this is simply game theory in effect. There has been a trend of batters opening up their zone when they are ahead in the count and so the effectiveness of a fastball is affected. Because in these scenarios the payoffs of a pair of strategy is largely determined on probabilities and previous data on the success of either strategy, the effectiveness of a fastball is a variable in determining the payoffs of using a fastball. Thus when the effectiveness of a fastball changes, the pitcher changes the percentage of throws that they throw a fastball on. This effect can actually be seen in the article as pitchers have changed their percentage of fastball throws to 62.5% in response to this trend by batters. This shows how the dynamic nature of certain payoff structures affect the decisions of players in a game. In this case the pitcher needs knowledge on what tendencies batters are developing in order to themselves redevelop their mixed strategy in pitching.
https://sports.vice.com/en_us/article/d7mxkv/the-curious-case-of-baseballs-missing-fastballs