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Math Help - Duel problem

  1. #1
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    Duel problem

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

    A and B start a duel 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:


    1. Shoot at his opponent with probability {p}_{i}(d) of hitting the target, where i=A or B, and d is distance (measured in steps) between them.
    2. Forsake the opportunity to shoot, and make one step forward toward his opponent.
    Now the distribution of {p}_{i}(d) is such that it's a strictly decreasing function of d, and that {p}_{i}(0)=1 for i=A, B. There's no other restrictions. Both players are intelligent enough.


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

    ------------------------------------------------------------------------------

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

    Maybe I'm wrong? Any ideas on this?
    Last edited by godelproof; September 9th 2011 at 08:31 PM. Reason: d>0
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