Ky Fan inequality (game theory)

Suppose that ${\displaystyle E}$ is a convex compact subset of a Hilbert space and that ${\displaystyle f}$ is a function from ${\displaystyle E\times E}$ to ${\displaystyle \mathbb {R} }$ satisfying

• ${\displaystyle x\mapsto f(x,y)}$ is lower semicontinuous for every ${\displaystyle y\in E}$ and
• ${\displaystyle y\mapsto f(x,y)}$ is concave for every ${\displaystyle x\in E}$.

Then there exists ${\displaystyle e\in E}$ such that

${\displaystyle \sup _{y\in E}f(e,y)\leq \sup _{y\in E}f(y,y).}$

• 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)