Asymptotics for the Time of Ruin in the War of Attrition

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, $W \sim \chi^{2}_{1}(z_{0}^{2}/T^{2})$.