Title: A Game of Random Variables,
Abstract: This paper analyzes a simple game with $n$ players. We fix a mean, $\mu$, 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 $\mu$, 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.