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

TL;DR

This paper extends a classic symmetric game involving risky and cautious strategies from two players to N players, deriving an exact mixed-strategy Nash equilibrium and analyzing its behavior as N grows large.

Contribution

It provides the first explicit analytic expression for the mixed-strategy Nash equilibrium in an N-player extension of the classic game, including the large N limit.

Findings

Existence of a mixed-strategy Nash equilibrium for N players.

Analytic expression for the equilibrium strategy.

Mean-field behavior in the large N limit.

Abstract

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.