{
  "id": "1607.07636",
  "version": "v1",
  "published": "2016-07-26T10:53:46.000Z",
  "updated": "2016-07-26T10:53:46.000Z",
  "title": "Asymptotics for the Time of Ruin in the War of Attrition",
  "authors": [
    "Philip Ernst",
    "Ilie Grigorescu"
  ],
  "comment": "31 pages",
  "categories": [
    "math.PR"
  ],
  "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, $W \\sim \\chi^{2}_{1}(z_{0}^{2}/T^{2})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-26T10:53:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G40",
      "91A60",
      "60C05"
    ],
    "keywords": [
      "asymptotics",
      "standard normal cdf",
      "first player wins",
      "probability proportional",
      "opponent loses"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}