{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $A$ be a real $m \\times n$ matrix.\nFor $k \\ge 1$ let $\\Delta_k = \\set{p \\in \\R^k : p_i \\ge 0 \\text{ for all } i, \\ p_1 + \\cdots + p_k = 1}$ be the standard [[simplex|YX1DJK2QAG]] of probability vectors.\nThen\n$$\\max_{x \\in \\Delta_m} \\min_{y \\in \\Delta_n} x^T A y = \\min_{y \\in \\Delta_n} \\max_{x \\in \\Delta_m} x^T A y.$$\n\n## Remarks\n\nThe inequality $\\max_x \\min_y \\le \\min_y \\max_x$ holds for every function on a product.\nThe reverse inequality is the content of the theorem.\nOne proof writes both sides as linear programs that are dual to each other, and uses the strong duality theorem of linear programming.\nAnother proof separates a convex set from a point with a hyperplane.\nThe common value is the *value* of the game, and the theorem says that $x^T A y$ has a saddle point on $\\Delta_m \\times \\Delta_n$.\n\nExample: matching pennies, with $A = \\begin{pmatrix} 1 \u0026 -1 \\\\ -1 \u0026 1 \\end{pmatrix}$.\nFor $x = y = (\\tfrac12, \\tfrac12)$ one gets $x^T A y' = 0$ for every $y'$ and $x'^T A y = 0$ for every $x'$, so the value is $0$.\nMixing is necessary: over pure strategies, that is vertices of the simplices, the row player can only guarantee $\\max_i \\min_j a_{ij} = -1$, while $\\min_j \\max_i a_{ij} = 1$.\n\nThe theorem extends to a continuous function $f(x, y)$ on $X \\times Y$ with $X, Y$ compact convex sets in $\\R^m$ and $\\R^n$, concave in $x$ and convex in $y$.\nSion's minimax theorem weakens these hypotheses further.\n\n## Explanations\n\n### L1 human\n\nIn a two-player zero-sum game, one player's gain is the other's loss.\nIf the players may randomize, it does not matter who has to announce a strategy first.\nThe best guaranteed payoff of the first player equals the smallest loss the second player can guarantee.\nThis common number is the value of the game.\n\n### L2 human\n\nThe theorem is the founding result of game theory, proved by John von Neumann in 1928.\nIt is equivalent to the duality theorem of linear programming, and the value and optimal mixed strategies of a matrix game can be computed by the simplex algorithm.\nThe general minimax theorems for convex-concave functions, and Sion's version for quasiconvex-quasiconcave functions, all rest on the same convexity.\nThey are the model for saddle point formulations of duality in convex optimization, such as saddle points of a Lagrangian.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "statement",
    "msc": [
      "91A05",
      "49J35",
      "90C47"
    ],
    "name": "von Neumann's minimax theorem",
    "role": "theorem",
    "sources": [
      {
        "locator": "Bilinear functions and zero-sum games",
        "url": "https://en.wikipedia.org/wiki/Minimax_theorem"
      }
    ],
    "tag": "GJ8VTBFNP2",
    "uses": [
      "YX1DJK2QAG"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/GJ8VTBFNP2/",
  "links": [
    {
      "field": "uses",
      "target": "YX1DJK2QAG"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/GJ8VTBFNP2.md",
  "msc": [
    "91A05",
    "49J35",
    "90C47"
  ],
  "name": "von Neumann's minimax theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GJ8/GJ8VTBFNP2.md",
  "status": null,
  "tag": "GJ8VTBFNP2",
  "uri": "m2://GJ8VTBFNP2",
  "url": "https://tamnd.github.io/math2/wiki/von-neumanns-minimax-theorem/",
  "writing": {
    "tag": "GJ8VTBFNP2",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}