Two players fight a duel as follows. Each has a (silent) gun with a single bullet. They begin 2n=10 paces apart. At a signal, each may fire. If either is hit or if both fire, the game ends. Otherwise, both advance one pace so they are now 2n-2 paces apart and again wait for the signal. The game ends in any case after at most n steps.
The probability of a shot hitting the opponent after the i-th step is (i/n)^2. The payoff is +1 to a player who survives or 0 to both if neither or both are hit. Note that since the guns are silent, neither knows whether his opponent has fired. You may assume that if they are both firing on the same step then both fire simultaneously.
Find the optimal strategy using each of the indicated pure strategies. Your values must be correct to within + or - 0.001 of the exact answers.
Shoot after 1 step with probability . ____
Shoot after 2 steps with probability .____
Shoot after 3 steps with probability .____
Shoot after 4 steps with probability .____
Shoot after 5 steps with probability .____
What are the probabilities after each step?
(Note: I think the probabilities after steps 1 and 4 is zero)