### 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 ^{-β} W ^{1/β}, β = 14;, T = x 0 + y 0. Modulo a constant, W ~ χ^{2} 1(z 0 ^{2} / T ^{2}).

Original language | English (US) |
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Pages (from-to) | 388-410 |

Number of pages | 23 |

Journal | Advances in Applied Probability |

Volume | 49 |

Issue number | 2 |

DOIs | |

State | Published - 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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## Cite this

Ernst, P. A., & Grigorescu, I. (2017). Asymptotics for the time of ruin in the war of attrition.

*Advances in Applied Probability*,*49*(2), 388-410. https://doi.org/10.1017/apr.2017.6