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Game Theory in Football

Justin Wolfers from the New York Times has done some game-theoretical analyses of the Seattle Seahawks’ final play in the Super Bowl, played on February 1st. They discuss that both could have used a bit more explicitness on a big issue in game theory: how optimal strategies evolve over the course of a game, especially near the end. The way this topic relates to class is by using the concept of game theory and Nash equilibrium and relating them to real life situations.

It brings attention to the counterintuitive aspects of a mixed strategy Nash Equilibrium in a football game which isthat if one of my strategies becomes better, I might end up using it less often.

In football suppose that my team, which is on offense, has an equally effective running and passing game. Similarly, my opponent can defend against the run and the pass equally well. Lets say the other team, calls my play, meaning that the opposing team defends against the run when I plan on running, or defends against the pass when I plan on passing, I expect to lose five yards. On the other hand, if the opponent calls the wrong play (expects me to run when I pass, or expects me to pass when I run), I will definitely get 5 yards on the play. The game looks like this:

Run Pass
Offense Run -5 , 5 5 , -5
Pass 5 , -5 -5 , 5

The equilibrium of this game is simple. Each of us will “interact” with probability 1/2.

Now, imagine that my team recruits a very fast running back. The only difference now is that if the defense expects me to pass when my team selects to run, we will gain 10 yards instead of 5. The new game looks like this:

Run Pass
Offense Run -5 , 5 10 , -10
Pass 5 , -5 -5 , 5

The only difference between this game and the previous one is that the run has become a better strategy than it was before. Let us see what the new equilibrium is. Suppose that I, on offense, will run with probability q and pass with probability (1-q). We know that the equilibrium value of q may be found by making the opponent indifferent between his two strategies. So:

if Defense defends against the run: Expected gain = 5q – 5(1-q) = 10q – 5
if Defense defends against the pass: Expected gain = -10q + 5(1-q) = 5 – 15q

At equilibrium, we must have these expected gains equal to each other, so:

10q-5 = 5 – 15q   –>   25q = 10   –>   q=2/5

Before my new star running back, my team ran half of the time. Now, we are only running 2 out of every 5 plays, despite the fact that our running game is better.

Pretty counterintuitive, isn’t it?

At the end of the article Justin relates this back to the game by saying if the player had said he thought the play would fail but that he was sending a signal, hoping his opponents would look out for a pass also on third down, “Monday-morning quarterbacks might not be calling him a strategic ignoramus”.



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