Mixed Strategies in Football

One of the most counterintuitive aspects of a mixed strategy Nash equilibrium is that if one of my strategies becomes better, I might end up using it less often. This example is for those who might prefer an analogy to 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. To put some numbers to this, assume that if my opponent 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:

The equilibrium of this game is straightforward. Each of us will "mix" with probability 1/2. Hence, I will choose my plays based on a coin toss each time, and so will the defense.

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:

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?

Well, here's why it works this way. Our intuition suggests that a better running back should imply more runs. But that is ignoring the interactive nature of the game. Our opponents also know that we have a better running back, so they will have more incentive to block him. But if he is blocked more, he isn't as effective. So, to make sure that the other team doesn't block him too much, we have to convince them that we are not going to be using him often. The defense is forced to cover the pass often. Those who are football fans might recognize this logic. You often hear commentators suggest that "the new star running back will really open up their passing game" or "the new running back doesn't do them any good unless they start passing more," suggesting that you have to keep your opponent guessing.

Note that the same is true in any sport in which probabilities are important. A better fastball by a baseball pitcher, a better slice serve in tennis, a better three-point shooter in basketball, might all lead to using these pitches, serves, or players less often.