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February 2016 Gambling in contests with random initial law
Han Feng, David Hobson
Ann. Appl. Probab. 26(1): 186-215 (February 2016). DOI: 10.1214/14-AAP1088

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.

Citation

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Han Feng. David Hobson. "Gambling in contests with random initial law." Ann. Appl. Probab. 26 (1) 186 - 215, February 2016. https://doi.org/10.1214/14-AAP1088

Information

Received: 1 May 2014; Revised: 1 November 2014; Published: February 2016
First available in Project Euclid: 5 January 2016

zbMATH: 1335.60058
MathSciNet: MR3449316
Digital Object Identifier: 10.1214/14-AAP1088

Subjects:
Primary: 60G40
Secondary: 60J65, 91A05

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.26 • No. 1 • February 2016
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