The Ky Fan Inequality in Game Theory
A second inequality is also called the Ky Fan Inequality, because of a 1972 paper, "A minimax inequality and its applications". This second inequality is equivalent to the Brouwer Fixed Point Theorem, but is often more convenient. Let S be a compact convex subset of a finite dimensional vector space V, and let f(x,y) be a continuous function from S × S to the real numbers that is lower semicontinuous in x, concave in y and has f(z,z) ≤ 0 for all z in S. Then there exists x* ∈ S such that for all y ∈ S, f( x*, y ) ≤ 0 . This Ky Fan Inequality is used to establish the existence of equilibria in various games studied in economics.
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