{
  "id": "2411.00633",
  "version": "v1",
  "published": "2024-11-01T14:43:09.000Z",
  "updated": "2024-11-01T14:43:09.000Z",
  "title": "Pasting of Equilibria and Donsker-type Results for Mean Field Games",
  "authors": [
    "Jodi Dianetti",
    "Max Nendel",
    "Ludovic Tangpi",
    "Shichun Wang"
  ],
  "categories": [
    "math.OC",
    "math.PR"
  ],
  "abstract": "This paper studies the relation between equilibria in single-period, discrete-time and continuous-time mean field game models. First, for single-period mean field games, we establish the existence of equilibria and then prove the propagation of the Lasry-Lions monotonicity to the optimal equilibrium value, as a function of the realization of the initial condition and its distribution. Secondly, we prove a pasting property for equilibria; that is, we construct equilibria to multi-period discrete-time mean field games by recursively pasting the equilibria of suitably initialized single-period games. Then, we show that any sequence of equilibria of discrete-time mean field games with discretized noise converges (up to a subsequence) to some equilibrium of the continuous-time mean field game as the mesh size of the discretization tends to zero. When the cost functions of the game satisfy the Lasry-Lions monotonicity property, we strengthen this convergence result by providing a sharp convergence rate.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-01T14:43:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "equilibrium",
      "donsker-type results",
      "multi-period discrete-time mean field games",
      "continuous-time mean field game models",
      "single-period mean field games"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}