Kakutani Fixed Point Theorem

Definition

The Kakutani Fixed Point Theorem states that every upper semi-continuous and convex-valued multifunction defined on a compact convex subset of a Euclidean space has at least one fixed point. This theorem is significant because it extends the classical Brouwer Fixed Point Theorem to cases involving multifunctions, which are useful in various fields including game theory and economic equilibrium. The theorem's applications range from demonstrating the existence of equilibria in economic models to providing insights in optimization problems.

5 Must Know Facts For Your Next Test

1. The Kakutani Fixed Point Theorem applies to multifunctions, which can map a single point to multiple outputs, unlike traditional functions.
2. It guarantees the existence of fixed points in settings where standard function fixed point results might not apply, particularly in economics and game theory.
3. The theorem requires the multifunction to be defined on a compact convex subset, emphasizing the importance of these mathematical structures.
4. One common application of the theorem is in demonstrating the existence of Nash equilibria in non-cooperative games.
5. The conditions of upper semi-continuity and convexity are crucial for ensuring that the fixed point exists, linking it closely to other areas of mathematics.

Review Questions

• How does the Kakutani Fixed Point Theorem extend the concepts found in the Brouwer Fixed Point Theorem?
  • The Kakutani Fixed Point Theorem builds on the Brouwer Fixed Point Theorem by addressing multifunctions instead of single-valued functions. While Brouwer guarantees a fixed point for continuous functions from compact convex sets to themselves, Kakutani applies this principle to situations where one input can yield multiple outputs. This makes Kakutani's theorem particularly useful in contexts such as economics and game theory, where decisions often depend on multiple players' strategies.
• Discuss how upper semi-continuity plays a role in the application of the Kakutani Fixed Point Theorem.
  • Upper semi-continuity is crucial in the Kakutani Fixed Point Theorem because it ensures that small changes in input do not lead to drastic changes in output for the multifunction. This condition allows for a more stable environment where fixed points can be guaranteed. In practical terms, this stability means that potential solutions or equilibria derived from economic models using this theorem are likely to be robust against small perturbations in parameters or strategies.
• Evaluate the significance of the Kakutani Fixed Point Theorem in modern economic theory and game theory.
  • The significance of the Kakutani Fixed Point Theorem in modern economic theory and game theory cannot be overstated. It provides a foundational result that underpins many equilibrium concepts, particularly Nash equilibrium in non-cooperative games. By establishing that solutions exist under certain conditions, it helps economists and theorists model complex interactions among agents. The theorem’s implications extend beyond pure mathematics, offering insights into strategic decision-making and market behavior, which are essential for understanding real-world economic dynamics.

Related terms

Brouwer Fixed Point Theorem: A fundamental result in topology stating that any continuous function from a compact convex set to itself has at least one fixed point.

Upper Semi-Continuity: A property of a multifunction where the preimage of an open set is open; this ensures that small changes in input do not lead to large jumps in output.

Convex Set: A set in which, for any two points within the set, the line segment connecting them lies entirely within the set.