Game Theory in America’s Pastime: Baseball
In class, we have delved into the idea of game theory and how it is a driving factor behind making choices in a competitive environment where your choice directly affects (and affected by) someone else’s choice. In this article, the author looks at game theory application in, what I would consider to be the greatest game on earth, baseball. This article goes through only a few examples of the countless amount of games that are played within baseball. A few examples include a pitcher versus a hitter in a 3-0 count, a pitcher versus a hitter in a 3-2 count, and two teams battling to sign a top free agent. The game I’m choosing to focus on is the pitcher versus a hitter in a 3-0 count game, as I was in this same scenario for Big Red’s baseball team not too long ago.
For those unfamiliar with baseball, this game is looking at the count during an at bat of a single hitter. The count starts at 0-0 and progresses as each pitch is thrown by the pitcher. If the first pitch is a strike, the count would go to 0-1 but if the first pitch is a ball the count will instead go to 1-0. A hitter is automatically walked if the count reaches 4 balls before either the ball is put in play or the “3 strikes and you’re out” saying is put into effect. A 3-0 count is what’s known as the most optimal count for a hitter because it puts pressure on the pitcher to not throw a ball, meaning the pitch will most likely be a strike and therefore a good a pitch for the hitter to swing at and hit.
The game in the article shows the payoff combinations for the hitter and pitcher when the pitcher either throws a pitch down the middle (a strike) or on the corner (a potential ball) and when the hitter swings or doesn’t swing. A higher payoff here for the hitter corresponds to a better chance of getting a hit while the higher payoffs for the pitcher corresponds to the pitcher getting a strike. If the pitcher throws it down the middle and the hitter swings, his high payoff of 5 means he most likely got a hit. If the pitcher throws it down the middle but the hitter doesn’t swing, the hitter has a payoff of -1 meaning he most likely missed the ball. This game clearly does not exhibit a dominant strategy as both the hitter and pitcher are competing against each other for opposite outcomes. The Nash equilibrium to this game is a mixed strategy meaning both the hitter and pitcher should randomize what they do for their best possible outcomes. The NE shows the hitter should swing 14% of the time due to the fact that this count is an optimal count for the hitter. This means that, even if he doesn’t hit this pitch, he will have more opportunities with succeeding pitches. On the other hand, the pitcher should throw a pitch down the middle 13% of the time. This number is so low because it is incredibly difficult for a pitcher to battle his way out of a 3-0 count and get the hitter out. Without getting into too much detail about the ins and outs of baseball, sometimes it is in your best interest to throw 4 balls to a hitter, and walk them, then it is to give them a pitch to hit.
This game shows that game theory in baseball is very different than most sports due to the fact that the best mixed strategies are often not 50/50 splits but rather very obscure numbers showing how complicated baseball truly is. A soccer player kicking a free kick generally has a 50/50 chance of scoring depending on if he kicks left or right and whether the goalie moves left or right. In baseball, every pitch and every play has intricate outcomes that ultimately will determine who wins and who loses.