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  "body": "\n## Statement\n\nLet $E$ be a real Hausdorff [[topological vector space|T32W553W95]], and let $C \\subseteq E$ be a nonempty [[compact|0R6EWFKRCF]] [[convex set|02AS45T3H9]].\nLet $f \\colon C \\times C \\to \\R$ be a function such that:\n\n1. for every $y \\in C$, the function $x \\mapsto f(x, y)$ is lower [[semicontinuous|RN4G8SGP6P]] on $C$;\n2. for every $x \\in C$, the function $y \\mapsto f(x, y)$ is quasiconcave on $C$, that is, $y \\mapsto -f(x, y)$ is [[quasiconvex|S4RMH7JQB8]].\n\nThen there is $\\bar x \\in C$ with\n$$\\sup_{y \\in C} f(\\bar x, y) \\le \\sup_{y \\in C} f(y, y).$$\nIn particular, if $f(y, y) \\le 0$ for all $y \\in C$, then there is $\\bar x \\in C$ with $f(\\bar x, y) \\le 0$ for all $y \\in C$.\n\n## Remarks\n\nIt suffices to prove the second form, applied to $f - c$ with $c = \\sup_y f(y, y)$ when $c$ is finite; the first form is trivial when $c = +\\infty$.\nFor each $y$ the set $S(y) = \\set{x \\in C : f(x, y) \\le 0}$ is closed by clause 1.\nIf a point $x$ of the convex hull of $y_1, \\dots, y_k$ lay outside every $S(y_i)$, clause 2 would give $f(x, x) \\ge \\min_i f(x, y_i) \u003e 0$, which is impossible.\nSo the sets $S(y)$ satisfy the hypothesis of the Knaster–Kuratowski–Mazurkiewicz lemma in Fan's form, and by compactness of $C$ they have a common point $\\bar x$.\n\nExample: let $\\Delta_k$ be the set of probability vectors in $\\R^k$, and let $A$ be a real $m \\times n$ matrix.\nThe payoff of the matrix game is $g(p, q) = p^T A q$ for $p \\in \\Delta_m$ and $q \\in \\Delta_n$, as in [[von Neumann's minimax theorem|GJ8VTBFNP2]].\nLet $C = \\Delta_m \\times \\Delta_n$, and for $z = (p, q)$ and $w = (p', q')$ in $C$ let $f(z, w) = g(p', q) - g(p, q')$.\nThis $f$ is continuous, affine in $w$, and $f(z, z) = 0$.\nThe point $\\bar z = (\\bar p, \\bar q)$ given by the theorem satisfies $g(p', \\bar q) \\le g(\\bar p, q')$ for all $p', q'$, so $(\\bar p, \\bar q)$ is a pair of optimal strategies.\n\nCompactness cannot be dropped: on $C = [0, \\infty) \\subseteq \\R$, the function $f(x, y) = y - x$ satisfies both clauses and $f(y, y) = 0$, but for every $x$ there is $y \u003e x$ with $f(x, y) \u003e 0$.\n\n## Explanations\n\n### L1 human\n\nThe inequality gives a point $\\bar x$ that no other point $y$ can beat, as measured by $f(\\bar x, y)$.\nIt asks only that comparing a point with itself gives no gain, and that $f$ has some mild continuity and convexity.\nMany existence results for equilibria have this form, so they all follow from this one statement.\n\n### L2 human\n\nKy Fan proved the inequality in 1972 as a common source of minimax theorems, fixed-point theorems and existence theorems for equilibria.\nFrom it one derives the minimax theorems of von Neumann and Sion, Nash equilibria of games with convex strategy sets, and solutions of [[variational inequalities|6QS7X5GVMY]] on compact convex sets.\nIts proof rests on the KKM lemma, which is equivalent to the Brouwer fixed-point theorem.\n",
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