NFL - Written by jeff on Wednesday, July 15, 2009 0:47 - 7 Comments

When you think of Game Theory, does football come to mind? Probably not. For those of you who don’t even know what it is, Game Theory attempts to mathematically capture behavior in strategic situations, in which an individual’s success in making choices depends on the choices of others.

Game Theory is most commonly used in economics. In recent years, Game Theory has become more popular in the football world because football is a Zero-sum game – precisely what Game Theory was created to analyze. But how can Game Theory be used to improve play calling decisions in Football?

At its core, Game Theory tries to find equilibria in games. In an equilibrium, each player of the game has adopted a strategy that they are unlikely to change. The most popular Game Theory concept is the Nash Equilibrium. In a Nash Equilibrium, each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing his or her own strategy unilaterally.

So why is Game Theory a great fit for play calling in the NFL? An article by Advanced NFL Stats sums it up well:

“Coaches tend to find the optimum mix of plays subjectively based on a combination of experience, intuition, and tradition. Game theory, however, can provide the true optimum mix of strategies, assuming the strategy choices are clear and their outcome distributions are known.”

Unpredictably picking between run and pass plays based on play effectiveness data – or in Game Theory what is referred to as a “mixed strategy” – is the optimal solution. The Nash Equilibrium occurs when an offense is choosing its best proportion of play calls, taking into account the defense’s strategy mix, and the defense is choosing its best proportion of plays based on its knowledge of the offense’s strategy mix. This is possible because offensive/defensive play results (yards lost/gained) are measurable and therefore have solvable equilibrium points. So what can we conclude? Coaches and signal-callers may get caught up in the moment while picking plays in a football game. Game Theory shows that picking plays by gut instinct will never be as effective as solving for offensive/defensive equilibrium points and selecting plays based on those points.