Fixed-point theorem

Definition

A fixed-point theorem states that under certain conditions, a function will have at least one point where the value of the function at that point equals the point itself. This concept is crucial in various areas of mathematics and economics, particularly when analyzing solution concepts such as the core, Shapley value, and nucleolus. By demonstrating the existence of equilibria, fixed-point theorems help explain how cooperative solutions can be achieved in games involving multiple players.

5 Must Know Facts For Your Next Test

1. The most famous fixed-point theorem is Brouwer's Fixed-Point Theorem, which states that any continuous function mapping a compact convex set to itself has at least one fixed point.
2. Fixed-point theorems are often used to show the existence of stable allocations in cooperative games, which helps in identifying core solutions.
3. The Shapley value relies on concepts from fixed-point theorems to distribute payoffs fairly among players based on their contributions to the total payoff.
4. The nucleolus focuses on minimizing dissatisfaction among players and can be established using fixed-point results to identify stable outcomes.
5. In economics, fixed-point theorems can help illustrate the convergence to equilibrium in various markets and scenarios involving multiple agents.

Review Questions

• How does Brouwer's Fixed-Point Theorem apply to understanding stable allocations in cooperative games?
  • Brouwer's Fixed-Point Theorem demonstrates that under specific conditions, there exists a point where a function will equal its output within a certain space. In cooperative games, this theorem ensures that stable allocations can be achieved since it guarantees at least one outcome where players can agree on the division of resources. This fundamental result underpins many solution concepts, showing how cooperative solutions are possible even when players have conflicting interests.
• Discuss how the Shapley value utilizes fixed-point theorems to ensure fair distribution of payoffs among players.
  • The Shapley value is a method for fairly distributing payoffs based on each player's contribution to the total outcome. Fixed-point theorems are instrumental in this process as they establish the conditions under which a unique distribution exists. By proving that a continuous mapping related to player contributions has a fixed point, we can derive an allocation that reflects fairness, ensuring all players receive their due based on their input into the coalition.
• Evaluate the significance of fixed-point theorems in advancing our understanding of equilibria in cooperative game theory and their implications for economic behavior.
  • Fixed-point theorems play a vital role in understanding equilibria within cooperative game theory by confirming that stable solutions exist under specific conditions. Their significance lies in facilitating deeper insights into how coalitions can form and allocate resources effectively. The implications for economic behavior are profound; they suggest that even in complex multi-agent environments, players can reach stable agreements that lead to efficient outcomes, ultimately guiding policy-making and market behavior in real-world scenarios.

Related terms

Nash Equilibrium:

A situation in a game where no player can benefit by changing their strategy while the other players keep theirs unchanged.

Cooperative Game Theory:

A branch of game theory that studies how groups of players can form coalitions and share payoffs among themselves.

Bargaining Set:

A solution concept in cooperative game theory that consists of outcomes which are stable against objections from coalitions.