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 $m\sim N{x}_0$, $n\sim N{y}_0$, 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, $W\sim {\chi }_1^2(z_0^2/T^2)$.

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