[AaronBrown]

Back to the original thread. Here's a start. On fourth down the optimal strategy from the 1 is for the offensive player to run with probability 1, and he wins with probability 1. Although it doesn't affect the optimal solution, the defense might as well play pass, because this could prevent the offensive win if the offensive player makes a mistake. I call this Offense Run 1 - Defense NA - Offense Win 1.

For the rest of the possible yardage points:

2 Offense Run 6/11 - Defense NA - Offense Win 5/11
3 Offense Run 5/9 - Defense NA - Offense Win 10/27
4 Offense Run 5/8 - Defense NA - Offense Win 5/16
5 Offense Run 2/3 - Defense NA - Offense Win 2/9
6 Offense Run 4/5 - Defense NA - Offence Win 2/15
7 Offense NA - Defense Run 0 - Offense Win 0
8 Offense NA - Defense Run 0 - Offense Win 0
9 Offense NA - Defense Run 0 - Offense Win 0
10 Offense NA - Defense Run 0 - Offense Win 0

A couple interesting observations. In each case, only one player's actions matter. If he does the optimal thing, the other player cannot affect the probability of winning. Also, the farther away the offensive player is, the more likely it is that he will run. This is because he can never win from beyond the 1 unless the defense guesses wrong. So he will pick Runs more often than Passes, because these give him less chance of winning, to force the defender to occasionally defend them.

Have you ever heard a football coach say "we like to run our bad plays more often than our good ones, to prevent the defense from gaining any advantage by defending the plays we do well"?