Ky Fan inequality (game theory)

For Ky Fan inequality related to geometric mean arithmetic mean, see Ky Fan inequality.

In game theory, the Ky Fan inequality or Ky Fan minimax inequality^[1] is a result used to establish the existence of equilibria in various games, particularly in economics and mathematical analysis. It was introduced in Ky Fan's 1972 paper, "A minimax inequality and its applications",^[2] and is closely related to the Brouwer fixed-point theorem, though often more convenient for proving equilibrium results.

The inequality applies to functions defined on compact, convex subsets of vector spaces and provides a condition under which a minimax-type inequality holds. It is frequently used in the analysis of non-cooperative games, variational inequalities, and general equilibrium models.

Statement

Let ${\displaystyle E}$ be a convex compact subset of a Hilbert space, and let ${\displaystyle f:E\times E\rightarrow \mathbb {R} }$ be a function satisfying: For every ${\displaystyle y\in E}$, the function ${\displaystyle x\mapsto f(x,y)}$ is lower semicontinuous.

For every ${\displaystyle x\in E}$, the function ${\displaystyle y\mapsto f(x,y)}$ is concave.

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

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

This result generalizes the classic minimax theorem and is a key tool in fixed-point and equilibrium analysis.

References

1. Yu, J.; Yuan, G. X. -Z. (1998-05-01). "The study of Pareto equilibria for multiobjective games by fixed point and Ky Fan minimax inequality methods". Computers & Mathematics with Applications. 35 (9): 17–24. doi:10.1016/S0898-1221(98)00053-4. ISSN 0898-1221.
2. Fan, K. (1972) A Minimax Inequality and Applications. In: Sisha, Ed., Inequalities, Vol. 3, Academic, New York, 103-113.

• 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)