Asymptotics for the Time of Ruin in the War of Attrition

TL;DR

This paper analyzes the asymptotic behavior of the time until ruin in a stochastic game of attrition, providing explicit formulas, fluid limits, and distributional results for large initial fortunes.

Contribution

It derives explicit formulas for winning probabilities and characterizes the asymptotic distribution of the ruin time in the war of attrition model.

Findings

Explicit formula for winning probability p(m,n)

Fluid limit behavior as initial fortunes scale with N

Distribution of ruin time converges to a scaled chi-square distribution

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 $m\sim Nx_{0}$, $n\sim N y_{0}$, we prove the fluid limit as $N\to \infty$. When $x_{0}=y_{0}$, then $z\to p(N,N+z\sqrt{N})$ converges to the standard normal CDF and the difference in fortunes scales diffusively. The exact limit of the time of ruin $\tau_{N}$ is established as $(T-\tau_N) \sim N^{-\beta}W^{\frac{1}{\beta}}$, $\beta=\frac{1}{4}$, $T=x_{0}+y_{0}$. Modulo a constant, $W \sim \chi^{2}_{1}(z_{0}^{2}/T^{2})$.