### 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) |
---|---|

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 |

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### 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

*Advances in Applied Probability*,

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

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

Research output: Contribution to journal › Article

*Advances in Applied Probability*, vol. 49, no. 2, pp. 388-410. https://doi.org/10.1017/apr.2017.6

}

TY - JOUR

T1 - Asymptotics for the time of ruin in the war of attrition

AU - Ernst, Philip A.

AU - Grigorescu, Ilie

PY - 2017/6/1

Y1 - 2017/6/1

N2 - 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 ~ χ2 1(z 0 2 / T 2).

AB - 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 ~ χ2 1(z 0 2 / T 2).

KW - Diffusive Scaling

KW - Evolutionary Game Theory

KW - Gambler's Ruin

KW - Noncentered Chi-Squared

KW - War Of Attrition

UR - http://www.scopus.com/inward/record.url?scp=85021359942&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85021359942&partnerID=8YFLogxK

U2 - 10.1017/apr.2017.6

DO - 10.1017/apr.2017.6

M3 - Article

VL - 49

SP - 388

EP - 410

JO - Advances in Applied Probability

JF - Advances in Applied Probability

SN - 0001-8678

IS - 2

ER -