Ky Fan inequality (game theory)

In mathematics, there are different results that share the common name of the Ky Fan inequality. The Ky Fan inequality presented here is used in game theory to investigate the existence of an equilibrium. Another Ky Fan inequality is an inequality involving the geometric mean and arithmetic mean of two sets of real numbers of the unit interval.

Statement

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Suppose that   is a convex compact subset of a Hilbert space and that   is a function from   to   satisfying

  •   is lower semicontinuous for every   and
  •   is concave for every  .

Then there exists   such that

 

References

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  • Aubin, Jean-Pierre (1998). "Optima and Equilibria". Graduate Texts in Mathematics. 140. Springer-Verlag. doi:10.1007/978-3-662-03539-9. {{cite journal}}: Cite journal requires |journal= (help)