A Game of Random Variables
For game theorists and economists, this provides a novel characterization of equilibrium in a simple game, though the problem is niche.
This paper analyzes a zero-sum game where players choose random variables with a fixed mean to maximize the probability of having the highest realization. It finds that above a crucial threshold, the unique equilibrium includes a point mass on 1, with the cutoff decreasing in the number of players.
This paper analyzes a simple game with $n$ players. We fix a mean, $μ$, in the interval $[0, 1]$ and let each player choose any random variable distributed on that interval with the given mean. The winner of the zero-sum game is the player whose random variable has the highest realization. We show that the position of the mean within the interval is paramount. Remarkably, if the given mean is above a crucial threshold then the unique equilibrium must contain a point mass on $1$. The cutoff is strictly decreasing in the number of players, $n$; and for fixed $μ$, as the number of players is increased, each player places more weight on $1$ at equilibrium. We characterize the equilibrium as the number of players goes to infinity.