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.
Read more about this topic: Ky Fan Inequality
Famous quotes containing the words fan, inequality, game and/or theory:
“Being a [Chicago] Cubs fan prepares you for life—and Washington.”
—Hillary Rodham Clinton (b. 1947)
“However energetically society in general may strive to make all the citizens equal and alike, the personal pride of each individual will always make him try to escape from the common level, and he will form some inequality somewhere to his own profit.”
—Alexis de Tocqueville (1805–1859)
“A man’s idea in a card game is war—cruel, devastating and pitiless. A lady’s idea of it is a combination of larceny, embezzlement and burglary.”
—Finley Peter Dunne (1867–1936)
“The theory of the Communists may be summed up in the single sentence: Abolition of private property.”
—Karl Marx (1818–1883)