The following duel problem is due to Ben Polak (maybe there's earlier origin, which I'll be glad to be informed about). The rule is as follows:

Two players 1 and 2 start a duel $N$ steps away from each other. They take turns to act. When it's somebody's turn, he must make one of the two choices:

**Choice A**: Shoot at his opponent with probability ${p}_{i}(d)$ of hitting the target, where $i=1,2$, and $d$ is distance (measured in steps) between them.

**Choice B**: Forsake the opportunity to shoot, and make one step forward toward his opponent.

Now the distribution of ${p}_{i}(d)$ is such that ${p}_{i}(0)=1$, and ${p}_{i}(d)>{p}_{i}(d+1)$, $d=0,1,2,...,N-1$, $i=1,2$. There're no other restrictions. Both players are assume to be rational and intelligent. ** A player's goal is to maximize his probability of killing the opponent**. Player 1 act first.

My question is: For all possible distributions of ${p}_{i}(d)$, $i=1,2$ described above, is there a simple and uniform decision rule according to which both players can make their choices at each distance?

(For example, the decision rule could be something like: "if ${p}_{1}(d)+{p}_{2}(d-1)>1$, then player 1 should shoot at distance d when it's his turn to move; otherwise step forward")

Edit: the original statement "a player's goal is to maximizing his surviving probability" is changed to "** A player's goal is to maximize his probability of killing the opponent**", due to Emil.