{
  "id": "2006.12993",
  "version": "v1",
  "published": "2020-06-19T22:49:13.000Z",
  "updated": "2020-06-19T22:49:13.000Z",
  "title": "Mean Field Games of Controls: on the convergence of Nash equilibria",
  "authors": [
    "Mao Fabrice Djete"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we investigate a class of mean field games where the mean field interactions are achieved through the joint (conditional) distribution of the controlled state and the control process. The strategies are of $open\\;loop$ type, and the volatility coefficient $\\sigma$ can be controlled. Using (controlled) Fokker-Planck equations, we introduce a notion of measure-valued solution of mean-field games of controls, and through convergence results, prove a relation between these solutions on the one hand, and the $\\epsilon_N$--Nash equilibria on the other hand. It is shown that $\\epsilon_N$--Nash equilibria in the $N$--player games have limits as $N$ tends to infinity, and each limit is a measure-valued solution of the mean-field games of controls. Conversely, any measure-valued solution can be obtained as the limit of a sequence of $\\epsilon_N$--Nash equilibria in the $N$--player games. In other words, the measure-valued solutions are the accumulating points of $\\epsilon_N$--Nash equilibria. Similarly, by considering an $\\epsilon$--strong solution of mean field games of controls which is the classical strong solution where the optimality is obtained by admitting a small error $\\epsilon,$ we prove that the measure-valued solutions are the accumulating points of this type of solutions when $\\epsilon$ goes to zero. Finally, existence of measure-valued solution of mean-field games of controls are proved in the case without common noise.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-19T22:49:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mean field games",
      "nash equilibria",
      "measure-valued solution",
      "mean-field games",
      "player games"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}