Asymptotics for the time of ruin in the war of attrition

Philip A. Ernst, Ilie Grigorescu

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

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 ~ Nx 0, n ~ Ny 0, we prove the fluid limit as N → âž. When x 0 = y 0, 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 τ N is established as (T-τ N) ~ N W 1/β, β = 14;, T = x 0 + y 0. Modulo a constant, W ~ χ2 1(z 0 2 / T 2).

Original languageEnglish (US)
Pages (from-to)388-410
Number of pages23
JournalAdvances in Applied Probability
Volume49
Issue number2
DOIs
StatePublished - Jun 1 2017

Keywords

  • Diffusive Scaling
  • Evolutionary Game Theory
  • Gambler's Ruin
  • Noncentered Chi-Squared
  • War Of Attrition

ASJC Scopus subject areas

  • Statistics and Probability
  • Applied Mathematics

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