Gambling in contests with random initial law

TL;DR

This paper extends a contest model involving Brownian motions by allowing initial values to be random variables with a common law, establishing a unique symmetric Nash equilibrium characterized by a target law with specific properties.

Contribution

It introduces a variant of the contest model with random initial states and proves the existence and uniqueness of the equilibrium with a detailed characterization of the target law.

Findings

Existence and uniqueness of symmetric Nash equilibrium.

The equilibrium target law is greater than or equal to the initial law in convex order.

The target law has a specific structure with an atom at zero and a decreasing density on positive values.

Abstract

This paper studies a variant of the contest model introduced in Seel and Strack [J. Econom. Theory 148 (2013) 2033-2048]. In the Seel-Strack contest, each agent or contestant privately observes a Brownian motion, absorbed at zero, and chooses when to stop it. The winner of the contest is the agent who stops at the highest value. The model assumes that all the processes start from a common value $x_0>0$ and the symmetric Nash equilibrium is for each agent to utilise a stopping rule which yields a randomised value for the stopped process. In the two-player contest, this randomised value has a uniform distribution on $[0,2x_0]$. In this paper, we consider a variant of the problem whereby the starting values of the Brownian motions are independent, nonnegative random variables that have a common law $\mu$. We consider a two-player contest and prove the existence and uniqueness of a symmetric Nash equilibrium for the problem. The solution is that each agent should aim for the target law $\nu$, where $\nu$ is greater than or equal to $\mu$ in convex order; $\nu$ has an atom at zero of the same size as any atom of $\mu$ at zero, and otherwise is atom free; on $(0,\infty)$ $\nu$ has a decreasing density; and the density of $\nu$ only decreases at points where the convex order constraint is binding.