Game Theory and Baseball

Game Theory and Baseball

Ever since 1977, there have been links between mathematics and baseball. The chief reason for this is because of a man named Bill James, who coined the term “sabermetrics”, which is the application of statistical analysis to baseball records used to compare and evaluate players’ and teams’ success. This boiled down to Bill James having the ability to find value in cheap players that were written off by the MLB, and change the way organizations scout and draft players (he is now a chief advisor for the Red Sox). I’m sure most of you have seen or heard of the movie, Moneyball (excellent flick and pretty much Jonah Hill’s audition for serious movies later on).  Since the moneyball revolution in the early 2000s, all teams have been looking for more ways to get a leg up on the competition, and it seems that game theory is next in line. In the last five years, game theory has been applied to baseball to try to improve future performance in a number of ways, such as the following: Pitch Selection, Batting Discipline, and Player Drafting.

Nash Equilibrium and Drafting

The first application of game theory that we will run through will utilize our good friend John Nash, and how the Nash equilibrium can apply to drafting prospects out of high school and college. In 2012, the MLB instituted a new rule regarding bonuses for drafted players. This rule essentially created slots for groups of players based on their pick number that determined the maximum bonus they could receive, for example let’s say all players between picks 15 and 25 are allowed a max of $1,000,000. Each team has a total amount that they can spend on draftees, so if a team goes big in the early rounds, then they will have to be more frugal on later in the draft. The experts broke this down to find a Nash equilibrium. So lets say two teams, the LA Dodgers and San Francisco Giants are debating on picks early in the draft between picking a proven college player, and a top high school prospect. Each team must decide in advance if they need to negotiate a contract with a high school player, so they don’t know what the other team will do. The figure below shows us how the teams will act:

Dodgers\Giants	College draftee	High-school draftee
College draftee	5,5	1,7
High-school draftee	7,1	3,3

[1]

As we can see with our trained Nash eyes, both teams will end up choosing high school draftees although they could guarantee a higher payoff elsewhere. The reason behind this numbers is that high school players are a bigger risk/reward payoff, meaning if both teams picked a college star, it is cheaper and he will contribute right away rather than spending years in the minor leagues. However, it is optimal for both teams to pick the high school player because if the Giants choose a college star, and there is a future superstar from high school then the Dodgers must choose a high school player. Both teams will end up drafting high school players because although it is a riskier option, it is the “smart” move (basically due to the Mike Trout effect, an MLB MVP at the age of 23).

Pitching

Next we will look at the applications of game theory to pitch selection and how some pitchers can improve performance. This application works by taking one player and analyzing each pitch and measuring its effectiveness against all others thrown by that pitcher. With each pitcher in the graph below, his first pitch (most thrown) and second pitch were analyzed, with scores ranging from -3.2 to +5. A positive score means the pitch is more effective so should be thrown more, and the opposite for a negative pitch. The number below is a graph of the best and worst pitchers measured:

[2]

The closer each pitch number is to 0.0, the closer the pitcher is to throwing the optimal number of fastballs, sliders, curveballs etc. As we can see from the graph Tanner Roark is a smart pitcher, and Ross Detwiler not so much.

Batter Up

Game Theory has also been applied to help batters out in key situations, such as at the end of games with a 3-2 count (on the next pitch the batter either walks, gets a hit, or is out). The prisoner’s dilemma box applies to this situation, but we won’t find a Nash equilibrium in this situation. Let’s say for the sake of the game that if a pitcher throws a strike and the batter swings, he will get a hit and the pitcher loses. If the pitcher throws a ball and the batter swings, then the batter is out and the pitcher wins. If a batter decides not to swing, he is gambling that the pitcher will throw a ball and not strike him out. The box below shows us the predicament the players are in:  

Pitcher\Batter	Swing	Take
Strike	-1,1	1,-1
Ball	1,-1	-1,1

[3]

Because there is no zone where both players are satisfied with their decision, a mixed strategy must be played in order to find equilibrium, which means deciding a move on probability. Long story short, (I will explain the math in class) we find an equilibrium in the long run if both the pitcher and batter choose each strategy 50% of the time. Both players will win half the time, and because the other player is choosing each option 50% at random, neither can improve utility by altering strategy.

As we can see, there are many applications of math to this nation’s pastime and this game theory connection is relatively new for baseball so I’m sure there is more to come in the future! So that is a bit of game theory and baseball, again if you liked reading this post at all, go watch Moneyball right now, the trailer is linked here. See ya Tuesday folks.

-HGC

Sources:

http://www.hardballtimes.com/game-theory-and-baseball-part-1-concepts/

http://www.hardballtimes.com/game-theory-is-the-next-moneyball/

http://www.hardballtimes.com/game-theory-and-baseball-part-2-introduction-to-pitch-selection/

http://fivethirtyeight.com/features/game-theory-says-r-a-dickey-should-throw-more-knuckleballs/

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[1] http://www.hardballtimes.com/game-theory-and-baseball-part-1-concepts/

[2] http://fivethirtyeight.com/features/game-theory-says-r-a-dickey-should-throw-more-knuckleballs/

[3] http://www.hardballtimes.com/game-theory-and-baseball-part-1-concepts/