Nucleolus

Definition

The nucleolus is a solution concept in cooperative game theory that focuses on distributing resources among players in a way that minimizes the dissatisfaction of the most dissatisfied player. It aims to find a stable allocation that ensures that no coalition of players would want to deviate from this distribution due to their dissatisfaction, thus addressing issues related to fairness and stability in resource allocation.

5 Must Know Facts For Your Next Test

1. The nucleolus is found by calculating the excess for each coalition and then minimizing the maximum excess, which results in a unique and stable allocation.
2. This solution concept is particularly useful in contexts where fairness and equity are important, such as cost sharing or voting power distributions.
3. Unlike the core, the nucleolus always exists and provides a unique solution for cooperative games, making it an attractive option for resolving disputes.
4. The nucleolus takes into account not just the average dissatisfaction but focuses on the worst-off player, ensuring that resources are allocated in a way that benefits those who are least satisfied.
5. In applications related to voting power, the nucleolus helps identify how power should be distributed among voters to minimize discontent and maintain stability.

Review Questions

• How does the nucleolus differ from other solution concepts like the core and Shapley value in cooperative game theory?
  • The nucleolus differs from other solution concepts primarily in its focus on minimizing the maximum dissatisfaction among players. While the core seeks allocations where no coalition has an incentive to deviate, it may not always exist or be unique. The Shapley value emphasizes fair distribution based on individual contributions rather than overall satisfaction. The nucleolus provides a unique and stable solution by addressing the needs of the worst-off player, making it particularly relevant in discussions of fairness.
• Discuss how the nucleolus can be applied in cost allocation scenarios to ensure fairness among participants.
  • In cost allocation scenarios, the nucleolus can help ensure fairness by focusing on minimizing dissatisfaction among all participants involved. By identifying how costs should be distributed such that the most dissatisfied participant has as little dissatisfaction as possible, organizations can create equitable solutions. This process involves calculating excess values for different coalitions and selecting an allocation that mitigates discontent, thus fostering cooperation and reducing potential conflicts over resource distribution.
• Evaluate the implications of using the nucleolus as a solution concept for power distribution in voting systems and its impact on political stability.
  • Using the nucleolus for power distribution in voting systems has significant implications for political stability. By ensuring that resources or votes are allocated in a way that minimizes dissatisfaction among voters, it creates an environment where all parties feel represented. This approach reduces incentives for coalitions to form against others since even the least satisfied voter finds their allocation acceptable. Consequently, this fosters cooperation among different factions, leading to a more stable political landscape where grievances are less likely to escalate into conflict.

Related terms

Core: The core is a set of feasible allocations in cooperative game theory where no subset of players can achieve a better outcome by forming a coalition and deviating from the proposed allocation.

Shapley Value: The Shapley value is a method for fairly distributing the total gains from cooperation among the players based on their contributions, taking into account the order in which they join the coalition.

Nash Bargaining Solution: The Nash bargaining solution is a solution concept that provides a unique allocation for two players negotiating over resources, focusing on maximizing the product of their utilities while respecting their reservation utilities.