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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $S \\subseteq \\R^n$ be a nonempty [[compact|0R6EWFKRCF]] [[convex set|02AS45T3H9]].\nLet $\\varphi$ be a set-valued map that assigns to each $x \\in S$ a subset $\\varphi(x) \\subseteq S$, such that:\n\n1. $\\varphi(x)$ is nonempty and convex for every $x \\in S$;\n2. the graph $\\set{(x, y) \\in S \\times S : y \\in \\varphi(x)}$ is a [[closed set|0ZCK8DQYN1]].\n\nThen $\\varphi$ has a fixed point: there is $x \\in S$ with $x \\in \\varphi(x)$.\n\n## Remarks\n\nWhen each $\\varphi(x)$ is a single point $f(x)$, the graph is closed exactly when $f$ is continuous, and the theorem becomes the Brouwer fixed-point theorem.\nThe usual proof approximates $\\varphi$ by continuous maps on finer and finer triangulations of a simplex containing $S$.\nIt applies Brouwer's theorem to these maps and passes to a limit using the closed graph and the convex values.\n\nConvex values are needed.\nOn $S = [0, 1]$, let $\\varphi(x) = \\set{1}$ for $x \u003c 1/2$, $\\varphi(1/2) = \\set{0, 1}$ and $\\varphi(x) = \\set{0}$ for $x \u003e 1/2$.\nThe graph is $[0, 1/2] \\times \\set{1} \\cup [1/2, 1] \\times \\set{0}$, which is closed, but no $x$ lies in $\\varphi(x)$; the value $\\varphi(1/2)$ is not convex.\n\nExample: consider a two-player game with mixed strategy sets $S_1$ and $S_2$, which are simplices.\nLet $B_1(x_2) \\subseteq S_1$ be the set of best replies of player 1 to the strategy $x_2$ of player 2, and define $B_2(x_1) \\subseteq S_2$ in the same way.\nOn $S = S_1 \\times S_2$, let $\\varphi(x_1, x_2) = B_1(x_2) \\times B_2(x_1)$.\nIts values are nonempty and convex and its graph is closed, so it has a fixed point, which is a Nash equilibrium.\n\n## Explanations\n\n### L1 human\n\nBrouwer's theorem is about a map that sends each point to one point.\nKakutani's theorem allows each point to be sent to a whole convex set of points, and still finds a point that lies in its own image.\nThis is what one needs when a choice is not unique, such as the best reply to an opponent's move in a game.\n\n### L2 human\n\nNash used the theorem in his proof of the existence of equilibria in finite games, and it is basic in general equilibrium theory in economics.\nThe closed graph condition is a form of upper hemicontinuity, which is equivalent to it when the values are closed in a compact $S$.\nThe Kakutani–Glicksberg–Fan theorem extends the result to compact convex subsets of locally convex spaces, just as Schauder and Tychonoff extend Brouwer's theorem.\n",
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