{
  "id": "1807.06831",
  "version": "v1",
  "published": "2018-07-18T09:33:20.000Z",
  "updated": "2018-07-18T09:33:20.000Z",
  "title": "Family of chaotic maps from game theory",
  "authors": [
    "Thiparat Chotibut",
    "Fryderyk Falniowski",
    "Michal Misiurewicz",
    "Georgios Piliouras"
  ],
  "comment": "13 pages, 2 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "From a two-agent, two-strategy congestion game where both agents apply the multiplicative weights update algorithm, we obtain a two-parameter family of maps of the unit square to itself. Interesting dynamics arise on the invariant diagonal, on which a two-parameter family of bimodal interval maps exhibits periodic orbits and chaos. While the fixed point $b$ corresponding to a Nash equilibrium of such map $f$ is usually repelling, it is globally Cesaro attracting on the diagonal, that is, \\[ \\lim_{n\\to\\infty}\\frac1n\\sum_{k=0}^{n-1}f^k(x)=b \\] for every $x$ in the minimal invariant interval. This solves a known open question whether there exists a nontrivial smooth map other than $x\\mapsto axe^{-x}$ with centers of mass of all periodic orbits coinciding. We also study the dependence of the dynamics on the two parameters.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-18T09:33:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E05",
      "91A05"
    ],
    "keywords": [
      "chaotic maps",
      "game theory",
      "periodic orbits",
      "bimodal interval maps",
      "multiplicative weights update algorithm"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}