This is a problem from Ben Polak's Yale open course lectures (lecture 16):

A and B start a duel $\displaystyle N$ steps away from each other. The rule is as follows:

A and B take turns to act. When it's somebody's turn, he must make one of the following choices:

Now the distribution of $\displaystyle {p}_{i}(d)$ is such that it's a strictly decreasing function of d, and that $\displaystyle {p}_{i}(0)=1$ for i=A, B. There's no other restrictions. Both players are intelligent enough.

Shoot at his opponent with probability $\displaystyle {p}_{i}(d)$ of hitting the target, where i=A or B, and d is distance (measured in steps) between them. Forsake the opportunity to shoot, and make one step forward toward his opponent.

Question: When should a player shoot? When should he move forward?

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Now Ben Polak's solution is, when distance is d, player A should choose to shoot iff $\displaystyle {p}_{A}(d)+{p}_{B}(d-1) \geq 1$ (for all$\displaystyle d>0$), with player B's strategy defined similarly. I'm really suspicious of this solution! Just think of a case where $\displaystyle N=2$. Direct calculation seems to suggest Ben's solution to be incorrect.

Maybe I'm wrong? Any ideas on this?