Here’s a broad class of games that captures a typical form of competition.  You and a rival simultaneously choose how much effort to spend and depending on your choices, you earn a score, a continuous variable.  The score is increasing in your effort and decreasing in your rival’s effort.  Your payoff is increasing in your score and decreasing in your effort.  Your rival’s payoff is decreasing in your score and his effort.

In football, this could model an individual play where the score is the number of yards gained.  A model like this gives qualitatively different predictions when the payoff is a smooth function of the score versus when there are jumps in the payoff function.  For example, suppose that it is 3rd down and 5 yards to go. Then the payoff increases gradually in the number of yards you gain but then jumps up discretely if you can gain at least 5 yards giving you a first down. Your rival’s payoff exhibits a jump down at that point.

If it is 3rd down and 20 then that payoff jump requires a much higher score. This is the easy case to analyze because the jump is too remote to play a significant role in strategy.  The solution will be characterized by a local optimality condition.  Your effort is chosen to equate the marginal cost of effort to the marginal increase in score, given your rival’s effort.  Your rival solves an analogous problem.  This yields an equilibrium score strictly less than 20.  (A richer, and more realistic model would have randomness in the score.)  In this equilibrium it is possible for you to increase your score, even possibly to 20, but the cost of doing so in terms of increased effort is too large to be profitable.

Suppose that in the above equilibrium you gain 4 yards. Then when it is 3rd down and 5 this equilibrium will unravel.  The reason is that although the local optimality condition still holds, you now have a profitable global deviation, namely putting in enough effort to gain 5 yards.  That deviation was possible before but unprofitable because 5 yards wasn’t worth much more than 4.  Now it is.

Of course it will not be an equilibrium for you to gain 5 yards because then your opponent can increase effort and reduce the score below 5 again.  If so, then you are wasting the extra effort and you will reduce it back to the old value. But then so will he, etc.  Now equilibrium requires mixing.

Finally, suppose it is 3rd down and inches.  Then we are back to a case where we don’t need mixing.  Because no matter how much effort your opponent uses you cannot be deterred from putting in enough effort to gain those inches.

The pattern of predictions is thus:  randomness in your strategy is non-monotonic in the number of yards needed for a first down.  With a few yards to go strategy is predictable, with a moderate number of yards to go there is maximal randomness, and then with many yards to go, strategy is predictable again. Variance in the number of yards gained in these cases will exhibit a similar non-monotonicity.

This could be tested using football data, with run vs. pass mix being a proxy for randomness in strategy.

While we are on the subject, here is my Super Bowl tweet.