Brouwer's Fixed Point Theorem

Definition

Brouwer's Fixed Point Theorem states that any continuous function mapping a compact convex set to itself has at least one fixed point. This means that there exists a point in that set such that the value of the function at that point is equal to the point itself. This theorem is crucial in understanding equilibrium in economics, as it guarantees the existence of equilibrium points in certain conditions.

5 Must Know Facts For Your Next Test

1. Brouwer's Fixed Point Theorem applies to continuous functions defined on a compact convex set, like a closed disk in 2D space.
2. This theorem is fundamental in proving the existence of Nash equilibria in non-cooperative games.
3. The theorem demonstrates that if a firm has a continuous reaction function, there will be an equilibrium price and quantity.
4. Brouwer's theorem has implications beyond economics, being utilized in fields like topology and differential equations.
5. The theorem was first proved by mathematician L.E.J. Brouwer in 1911 and has since become a foundational result in fixed-point theory.

Review Questions

• How does Brouwer's Fixed Point Theorem ensure the existence of equilibrium in economic models?
  • Brouwer's Fixed Point Theorem ensures the existence of equilibrium by stating that if a continuous function maps a compact convex set to itself, there is at least one point where the function equals the input. In economic models, this translates to finding an equilibrium price and quantity where supply equals demand. The conditions required by the theorem align well with many economic scenarios, making it a powerful tool for proving that an equilibrium exists.
• What role do compact and convex sets play in Brouwer's Fixed Point Theorem within economic contexts?
  • Compact and convex sets are essential to Brouwer's Fixed Point Theorem because they provide the necessary conditions for the theorem to hold. In economics, these sets often represent feasible allocations or strategies within which agents operate. By ensuring that the underlying set is both compact and convex, economists can apply the theorem to guarantee that equilibrium points exist for various market models, such as those involving consumer choice or production.
• Evaluate how Brouwer's Fixed Point Theorem contributes to advancements in economic theory, particularly concerning Nash equilibria.
  • Brouwer's Fixed Point Theorem significantly contributes to advancements in economic theory by providing a mathematical foundation for finding Nash equilibria in game theory. As it confirms that continuous functions on compact convex sets have fixed points, it allows theorists to assert that under certain conditions, stable strategies will exist where no player has an incentive to deviate unilaterally. This insight not only enhances our understanding of strategic interactions among agents but also aids in formulating predictions about behavior in competitive markets.

Related terms

Fixed Point: A fixed point of a function is a point where the value of the function at that point equals the point itself.

Compact Set: A compact set is a set that is closed and bounded, which implies it contains all its limit points.

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