Asymptotics for the time of ruin in the war of attrition

Philip A. Ernst, Ilie Grigorescu

Research output: Contribution to journalArticle

1 Citation (Scopus)

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

Fingerprint

Attrition
Normal cumulative distribution function
Fluid Limits
Unit
Normal distribution
Distribution functions
Modulo
Explicit Formula
Directly proportional
Converge
Fluids
Standards

Keywords

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

ASJC Scopus subject areas

  • Statistics and Probability
  • Applied Mathematics

Cite this

Asymptotics for the time of ruin in the war of attrition. / Ernst, Philip A.; Grigorescu, Ilie.

In: Advances in Applied Probability, Vol. 49, No. 2, 01.06.2017, p. 388-410.

Research output: Contribution to journalArticle

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