Kakutani's Fixed Point Theorem

Definition

Kakutani's Fixed Point Theorem states that any upper semi-continuous and convex-valued multifunction defined on a convex compact set has at least one fixed point. This theorem is significant in the study of equilibrium in economic models, as it extends the Brouwer Fixed Point Theorem to situations with multiple decision-makers or agents, making it crucial for understanding the existence and uniqueness of equilibrium in markets.

5 Must Know Facts For Your Next Test

1. Kakutani's theorem is particularly useful in game theory, as it applies to scenarios with multiple players, each with their own strategy sets.
2. The conditions for Kakutani's theorem involve both upper semi-continuity and convexity, which are key properties that ensure the existence of fixed points.
3. This theorem provides a powerful tool for proving the existence of equilibria in markets with many interacting agents, especially when preferences or strategies are represented as multifunctions.
4. Kakutani's Fixed Point Theorem guarantees at least one fixed point but does not necessarily provide uniqueness; multiple equilibria can exist depending on the structure of the problem.
5. Applications of Kakutani's theorem extend beyond economics into various fields like game theory, differential equations, and optimization problems.

Review Questions

• How does Kakutani's Fixed Point Theorem build upon Brouwer's Fixed Point Theorem in economic contexts?
  • Kakutani's Fixed Point Theorem generalizes Brouwer's Fixed Point Theorem by allowing for multifunctions instead of single-valued functions. This means that in economic contexts where multiple agents are making decisions simultaneously, each with potentially different outcomes depending on others' actions, Kakutani's theorem can be applied. It provides a framework to analyze equilibria in complex situations where agents have strategy sets that depend on each other, ensuring that at least one equilibrium exists.
• Discuss the importance of upper semi-continuity and convexity in ensuring the existence of fixed points through Kakutani's theorem.
  • Upper semi-continuity and convexity are crucial conditions for Kakutani's theorem because they ensure that small changes in strategies or inputs do not lead to large deviations in outputs. Upper semi-continuity guarantees stability in response to slight adjustments by agents, while convexity allows for combinations of strategies to also yield valid strategies. Together, these properties create an environment where fixed points can be reliably found, which is essential for establishing equilibria in economic models with multiple decision-makers.
• Evaluate how Kakutani's Fixed Point Theorem can be utilized to demonstrate the existence of equilibrium in a specific market model.
  • To evaluate how Kakutani's Fixed Point Theorem can show equilibrium existence, consider a market with multiple consumers whose demand functions depend on the prices set by firms. By representing consumer demand as a multifunction that takes into account preferences and available goods, and ensuring that this function satisfies upper semi-continuity and convexity, we can apply Kakutani's theorem. This approach reveals that there exists at least one price vector where supply equals demand across all consumers, thus confirming an equilibrium exists within this market setup. This illustrates how theoretical constructs become practical tools in economic analysis.

Related terms

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

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

Upper Semi-Continuity: A property of a function where, roughly speaking, small changes in the input do not lead to large jumps in the output, allowing for more stable behavior in economic models.