Barring a rule change outlawing them, it seems defensive shifts are here to stay. In fact, teams are using them increasingly and becoming more efficient at them in the process. This is not a good sign for predictable hitters such as David Ortiz, Brian McCann and the plethora of other pull-happy hitters. Smart baseball minds such as Tom Tango and Bill James have written about ways that hitters can essentially fight back by bunting or simply becoming a more well-rounded hitter. In this post I will attempt to build on some of these findings. More specifically I will be using game theory to help prove that hitters have a way out of their proverbial pickle.

First, let’s discuss game theory.

Simply put, game theory is the study of strategic decision making. Baseball is a game of strategy and there are hundreds of decisions made by players, managers, and umpires each and every game. It makes sense that the best response to some of these baseball decisions could be derived by using game theory. It is mainly practiced in business and government settings, and was informally introduced to popular culture through the movie A Beautiful Mind which captured the life of mathematician and renowned game theorist John Nash.

As I mentioned earlier, game theory suggests we can create structured games that will allow us to make better decisions. The games can be set up to resemble real life, meaning they can be sequential or non-sequential and contain as little or as much information as necessary. Games consist of players, the possible actions of each player, and the payoffs for each outcome. There are many ways to go about solving these games and I will demonstrate a couple of these methods later on. There are games in which the answer is very clear, and others that require more in-depth reasoning. A typical game is laid out in matrix form and it is actually one of the structures I have chosen to include in my baseball example.

Now let’s talk about David Ortiz and probability.

Back in 2012, Tom Tango wrote a little article on his website about David Ortiz using the bunt to beat the shift. In typical Tango fashion, he used a bit of secondary research along with his ninja math skills to determine that Ortiz should be bunting more often when encountering the defensive shift. Tango came up with the conclusion that a great hitter, such as Ortiz, would need to be able to lay a bunt down successfully 45-50% of the time in order for it to make sense to bunt in a shift situation. This means that if Ortiz was able to lay down a bunt at an efficiency greater than 50% he should continue to bunt, because he would not be able to reach a greater outcome by swinging away. To my surprise, I learned that Ortiz has a bunt success rate of 50%. This is, however, only based on a sample of 12 attempts over his career, but as Tango states any hitter that can lay a bunt down 50% of the time against the shift should continue to bunt. So why is Ortiz (and others) not bunting more often?

Russel Carleton of Baseball Prospectus suggests that it could be the testosterone factor. In the player’s mind he is getting paid to hit home runs, not bunt singles. In truth, players are starting to bunt more in shift situations, but are they doing it enough to make a difference? If bunting becomes a more frequently used approach to beating the shift, will teams alter their defensive tactics? Game theory can answer some of these questions, so let’s get back to that.

I talked about using a matrix as a way to set up a game. Let’s try to use this approach to examine a basic example of how game theory could be used to analyze baseball. Take a gander at the 2X2 matrix below.

This game was set up to resemble the actions and payoffs that a batter and the defense would face in a situation that might call for a shift. Imagine that you are David Ortiz. If the defense shifts and you swing away the defense wins (1) because you essentially give them an out (-1). The same goes if the defense does not shift and you bunt. However, if the defense shifts and you bunt down the line you win and gain a base runner (1) while the defense loses that out (-1). For the sake of this simplistic example, we also assume that if the defense does not shift and you swing away, you gain a base runner and the defense loses an out. Obviously the last part is not realistic, but it works for trying to explain the theory.

The game looks simple enough, but if we look close it is actually a little more difficult than what it might seem. In this case there are no pure strategies. For example, if the team shifts you would lay down a bunt. However, knowing this fact, the team would not shift, which in turn means you would swing away. It seems like this game would end in a draw. Enter John Nash and his beautiful mind. Nash came up with a neat little way to solve these types of games. His theorem suggests that all finite games must have an equilibrium, even ones that do not have a pure strategy (ones that are obvious). If there are no pure strategies then a mixed strategy can be used. A mixed strategy is a probability distribution over two or more pure strategies. Don’t get confused here. Bottom line, if we can solve for a mixed strategy Nash equilibrium, then we can determine the best responses or actions that can be taken for this game.

Before we solve for the mixed strategy Nash equilibrium, let’s alter the payoffs (the numbers in the matrix) so that they look more like the payoffs in a real baseball game.

As you can see, the only box I altered was the bottom left box (or the No Shift/Swing away cell. I changed this to -2, 2 instead of -1, 1 because there is a chance that swinging away with no shift on will lead to an extra base hit. I assumed that in the other situations the best that a hitter could do would be a single. Obviously this not accurate since there is a smaller chance the player will actually get an extra base hit, but for this first example it will have to work. I intend to expand on this in the next post with a better way to use game theory to solve the shift.

The new matrix still has no pure strategies so we must use the mixed strategy approach. This will require us to use the mixed strategy algorithm which is not complicated, but boring, so I will spare you the gory details and just list the equation below for reference:

Solving for the defense:

EU_SwingAway = σ_Shift(-1) + (1-σ_Shift)(2)

EU_Bunt = σ_Shift(1) + (1-σ_Shift)(-1)

σ_Shift(-1) + (1-σ_Shift)(2) = σ_Shift(1) + (1-σ_Shift)(-1)

σ_Shift = 3/5

This means that if the defense applies the shift 3/5ths of the time and does not apply the shift 2/5ths of the time then the batter is indifferent about whether to bunt or hit away. Regardless of his choice he still winds up with the same expected outcome.

Solving for the batter:

EU_Shift = σ_SwingAway(1) + (1-σ_SwingAway)(-1)

EU_NoShift = σ_SwingAway(-2) + (1-σ_SwingAway)(1)

σ_SwingAway(1) + (1-σ_SwingAway)(-1) = σ_SwingAway(-2) + (1-σ_SwingAway)(1)

σ_SwingAway = 2/5

In the case of the hitter, this solution means that if he swings away 2/5ths of the time and bunts 3/5ths of the time, then the defense is indifferent about whether to shift or not shift. Regardless of their choice the defense still winds up with the same expected outcome.

So in the case of David Ortiz, defenses are shifting at a rate not consistent with the mixed strategy Nash equilibrium. Ortiz is seeing the shift nearly every plate appearance, but the game theory solution says they should be mixing it up. If Ortiz were to actually attempt a few bunts and maintain his bunt success rate then he could exploit the defenses’ strategy causing them to either alter their shift – which is happening with other players – or stop shifting as much. The former is a more likely outcome. However, Ortiz must consider his payoffs and realize that if he bunts every time he too would be playing the game outside of the equilibrium and defenses would compensate and ultimately take back the advantage.

There is one major flaw with analyzing the shift using this form of game theory, and it’s that we are assuming the game is played with simultaneous moves where all information is available to both sides. In reality there is a sequential order to these moves: player two (the hitter) reacts to the initial move from player one (the defense). The defense shows their cards first and the hitter makes a decision to bunt or swing away based on the information given to him at that moment.

In a later post, I will take a look at an alternative game theory structure that might be better suited for solving this type of game. This example was, however, a good way for those who are not familiar with game theory to get a basic understanding of how it could be applied.

For those of you looking for other ways to use game theory, Matt Swartz, a contributor for several baseball outlets gave a presentation on game theory at the 2013 SABR Analytics Conference.