Kakutani fixed-point theorem

A fixed-point theorem stating that an upper hemicontinuous correspondence from a nonempty compact convex subset of Euclidean space to its nonempty compact convex subsets has a fixed point.

Set-valued function: A function that assigns each input a set of possible outputs. The theorem applies to correspondences rather than ordinary single-valued functions.
Brouwer fixed-point theorem: A theorem stating that every continuous self-map of a nonempty compact convex subset of Euclidean space has a fixed point. Brouwer handles single-valued continuous maps; Kakutani extends the guarantee to correspondences.
Nash equilibrium: A strategy profile in a game where no player can improve their payoff by changing strategy alone. Kakutani can establish equilibrium existence through the players' best-response correspondence.
Shizuo Kakutani: A Japanese-American mathematician known for work in analysis, probability, and fixed-point theory. Kakutani published the correspondence fixed-point theorem that bears his name.
Upper hemicontinuity: A continuity condition on a correspondence that prevents nearby inputs from producing outputs far outside the current value. This regularity condition is essential to the theorem's fixed-point guarantee.
Closed graph theorem for correspondences: A result relating closed graphs to upper hemicontinuity for correspondences under suitable compactness conditions. It offers a practical route to checking one of Kakutani's key hypotheses.
Best response: An action or set of actions that maximizes an agent's payoff given others' choices. Best responses form the correspondence whose fixed point is a Nash equilibrium.
John von Neumann: A Hungarian-American mathematician and polymath whose work shaped game theory, computing, and mathematical physics. Kakutani's theorem later supplied a fixed-point route to von Neumann's minimax result.
Compact set: A topological space in which every open cover has a finite subcover. Compactness of the domain and values supports the theorem's existence conclusion.
Hausdorff metric: A metric measuring how far two nonempty compact subsets of a metric space are from one another. It provides a way to express continuity for set-valued maps, distinct from upper hemicontinuity.

Linked from 7 pages