The Annals of Applied Probability

Gambling in contests with random initial law

Han Feng and David Hobson

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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.

Article information

Ann. Appl. Probab., Volume 26, Number 1 (2016), 186-215.

Received: May 2014
Revised: November 2014
First available in Project Euclid: 5 January 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60J65: Brownian motion [See also 58J65] 91A05: 2-person games

Gambling contest Nash equilibrium Skorokhod embedding Seel–Strack problem


Feng, Han; Hobson, David. Gambling in contests with random initial law. Ann. Appl. Probab. 26 (2016), no. 1, 186--215. doi:10.1214/14-AAP1088.

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