Two players start at forty paces apart. Each has a gun loaded with one bullet. For simplicity we will assume that at each distance the player at the left has the option of shooting first. If s/he does not shoot then the player at the right has the opportunity to shoot. If neither player shoots, then the two players each take a step toward the other and the game continues.

If one player shoots and hits, the game is over. That player wins. If the player shoots and misses, then the other player gets to advance to some predesignated distance where he or she can shoot and the chance of hitting is very high.

To help model this game, we will denote the chance player i (i is either 1 or 2) hits when distance d away is p_i(d). You can think that p_i(40) is pretty low, while p_i(5) is close to 1. Again, for simplicity, we will let the predesignated distance be 1, and the chance of success at that distance is 100%.

The two players do not have the same ability.

We will call the better shooter player 1.

Thus p₁(d) ≥ p₂(d) for all d.

The fact that the chance of success at d=1 is 100% can be written as

p₁(1) = p₂(1) =1.

Let player 1 be the one on the left and thus the one who gets to move first on each turn.

To figure out what each party will do at the start when they are forty paces, we look at the potential end game when they are only 1 pace apart.

For Question Below there are 2 options shoot or wait please check my answer

Assume neither player 1 nor 2 has shot and they are now only 1 pace apart. It is player 1's turn. What should player 1 do? 

shoot

In the period before, the two parties are two paces apart and it is player 2's turn. What should player 2 do? Note that at this distance, player 2's chance of success is less than 100%.

shoot

In the period before that, the two parties are still two paces apart and it is player 1's turn.  (Each side gets a turn to shoot or not at each distance.) What should player 1 do?  Specifically what should Player 1 do if p₁(2) = 0.6 and p₂(2) = 0.55

shoot

The same question as just above, but now what what should Player 1 do if p₁(2) = 0.6 and p₂(2) = 0.3

shoot