Stochastic Contests with Linex Utility Functions

In this paper, we analyze the Nash equilibrium in a class of winner-takes-all stochastic contests among players with linear-exponential (linex) utility functions. In this contest, players are required to make upfront investments, which collectively determine their winning probabilities. We first show that a Nash equilibrium for such a contest exists and is unique, then set the equilibrium conditions, and study the properties of these conditions to gain insights into the structure of equilibrium. We show that the total equilibrium investment is bounded below and above, that the equilibrium has a cut characterization with respect to wealth, and that wealthier players invest more. The latter implies that richer is likely to get richer. For the special case with identical players, we show that an increase in the wealth or a decrease in the weight on the nonlinear component of the linex utility function results in an increase in the equilibrium investment