Kakutani Fixed-point theorem

The Kakutani Fixed-Point Theorem states that for a certain kind of mathematical function—called a set-valued function—that maps points in a compact, convex space to non-empty, closed, and convex sets, there is always at least one point where the point is contained within its own image. In simple terms, it guarantees the existence of a stable point where the choices or options represented by the function "balance out," which is crucial in economics, game theory, and optimization. It's an extension of fixed-point concepts like Brouwer's theorem, but applicable to more complex, multi-valued scenarios.