A Game of Random Variables

TL;DR

This paper studies a competitive game where players choose distributions with a fixed mean to maximize their maximum realization, revealing how the mean's position influences equilibrium strategies and outcomes.

Contribution

It characterizes the equilibrium strategies in a game where players select distributions with a fixed mean, highlighting the importance of the mean's position and the emergence of point masses at 1.

Findings

Above a critical mean, equilibrium involves a point mass at 1.

The critical mean threshold decreases as the number of players increases.

As the number of players grows, players place more weight on 1 at equilibrium.

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.