discussion_of_duel

# discussion_of_duel - Discussion of\Duel Econ 159a/MGT522a...

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Discussion of \Duel" Econ 159a/MGT522a Ben Polak November 1, 2007 The argument below is a little more formal (and a little less direct) than the one we used in class, but the gist of the message is the same. Some of you may °nd it useful. The Rules. There are two players, each with a wet sponge. They start at either side of the room. The players have alternate turns. Each turn, the player can either shoot or take a step forward. If she shoots and hits then she wins. If she misses, then the game continues so e°ectively she loses (since the other player can wait and shoot point blank). The Strategic Decision. When to shoot? Extra structure to make the analysis tractable. Assume that the abilities of the players are known. In particular, ° let P 1 ( d ) be player 1’s probability of hitting if she shoots at distance d ; and ° let P 2 ( d ) be player 2’s probability of hitting if she shoots at distance d ; In addition, let us make two plausible assumptions: ° P 1 (0) = P 2 (0) = 1; that is, they hit for sure from point blank. ° Both P 1 and P 2 are decreasing; that is, the probabilities of hitting go down with distance. Notice we do need not assume that the players are equally good shots, or even that one is better than the other at all distances, just that their abilities are known. Speculations. We might think that a better shot will shoot ±rst since she has a better chance of hitting. Alternatively, we might think that a worse shot will try to pre-empt the better shot.

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