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Asymptotics for the time of ruin in the war of attrition

Part of: Stochastic processes Game theory

Published online by Cambridge University Press:  26 June 2017

Philip A. Ernst  and
Ilie Grigorescu
Philip A. Ernst*
Affiliation:
Rice University
Ilie Grigorescu*
Affiliation:
University of Miami
*
* Postal address: Department of Statistics, Rice University, 6100 Main Street, Houston, TX 77005, USA. Email address: philip.ernst@rice.edu
** Postal address: Department of Mathematics, University of Miami, 1365 Memorial Drive, Coral Gables, FL 33146, USA.
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Abstract

We consider two players, starting with m and n units, respectively. In each round, the winner is decided with probability proportional to each player's fortune, and the opponent loses one unit. We prove an explicit formula for the probability p(m, n) that the first player wins. When m ~ Nx0, n ~ Ny0, we prove the fluid limit as N → ∞. When x0 = y0, z → p(N, N + z√N) converges to the standard normal cumulative distribution function and the difference in fortunes scales diffusively. The exact limit of the time of ruin τN is established as (T - τN) ~ N-βW1/β, β = ¼, T = x0 + y0. Modulo a constant, W ~ χ21(z02 / T2).

Keywords

War of attritiongambler's ruinevolutionary game theorydiffusive scalingnoncentered chi-squared

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 91A60: Probabilistic games; gambling
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 2 , June 2017 , pp. 388 - 410
Copyright
Copyright © Applied Probability Trust 2017 

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