{
  "id": "2212.14388",
  "version": "v1",
  "published": "2022-12-29T17:43:32.000Z",
  "updated": "2022-12-29T17:43:32.000Z",
  "title": "From the binomial reshuffling model to Poisson distribution of money",
  "authors": [
    "Fei Cao",
    "Nicholas F. Marshall"
  ],
  "comment": "19 pages, 5 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We present a novel reshuffling exchange model and investigate its long time behavior. In this model, two individuals are picked randomly, and their wealth $X_i$ and $X_j$ are redistributed by flipping a sequence of fair coins leading to a binomial distribution denoted $B \\circ (X_i+X_j)$. This dynamics can be considered as a natural variant of the so-called uniform reshuffling model in econophysics [2,14]. As the number of individuals goes to infinity, we derive its mean-field limit, which links the stochastic dynamics to a deterministic infinite system of ordinary differential equations. The main result of this work is then to prove (using a coupling argument) that the distribution of wealth converges to the Poisson distribution in the $2$-Wasserstein metric. Numerical simulations illustrate the main result and suggest that the polynomial convergence decay might be further improved.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-29T17:43:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82C22",
      "91B80",
      "60J28"
    ],
    "keywords": [
      "binomial reshuffling model",
      "poisson distribution",
      "main result",
      "long time behavior",
      "deterministic infinite system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}