This is a very simple and practical problems, well known among turtles and other creatures.
Here is the description given by Mathflow:
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 pi(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 [sic] pi(d) is such that pi(0)=1 and pi(d) > pi(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 [sic] pi(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 p1(d) + p2(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.
It is usually presented in the literature in a different way, but don't worry about this. We shall discuss this matter later. Our main goal is to examine how DP attack this problem.
For the time being, you are welcome to to attend the famous Turtle Bladers show.