# Mixed-strategy Nash equilibrium for a discontinuous symmetric N -player game

We consider a game in which each player must find a compromise between more daring strategies that

carry a high risk for him to be eliminated, and more cautious ones that, however, reduce his final

score. For two symmetric players this game was originally formulated in 1961 by Dresher, who modeled

a duel between two opponents. The game has also been of interest in the description of athletic

competitions. We extend here the two-player game to an arbitrary number N of symmetric players. We

show that there is a mixed-strategy Nash equilibrium and find its exact analytic expression, which

we analyze in particular in the limit of large N , where mean-field behavior occurs. The original

game with N = 2 arises as a singular limit of the general case.