Introduction
The concept of the Nucleolus arises in the field of Game Theory, particularly within the context of cooperative games. In these games, players form coalitions and seek to distribute pay-offs among themselves. The Nucleolus is a value function that aims to minimize the maximum dissatisfaction among all possible coalitions by optimizing the allocation of pay-offs.
Historical Context
The concept of the Nucleolus was introduced by David Schmeidler in 1969. It is a solution concept in cooperative game theory, aiming to ensure fair and stable distributions of benefits within a coalition of players.
Types/Categories
- TU (Transferable Utility) Games: The Nucleolus is primarily applied in TU games where pay-offs can be transferred between players.
- NTU (Non-Transferable Utility) Games: The concept can also be extended to NTU games with more complex dynamics.
Key Events
- 1969: David Schmeidler publishes the seminal paper on the Nucleolus, introducing it to the academic world.
- 1971: Further development and applications of the Nucleolus in various cooperative game scenarios.
Detailed Explanation
The Nucleolus is calculated by the following steps:
- Calculate Excess: For each coalition, compute the excess, which is the difference between the coalition’s potential pay-off and the sum of individual pay-offs for that coalition’s members under a proposed allocation.
- Minimize Maximum Excess: Find the allocation that minimizes the maximum excess across all coalitions.
- Iterative Process: If multiple allocations have the same maximum excess, the process is iterated by considering the next highest excess until a unique allocation is identified.
Mathematical Formulas/Models
Let:
- \( N \) be the set of players.
- \( v(S) \) be the value of coalition \( S \subseteq N \).
- \( x \) be the proposed allocation.
The excess \( e(S,x) \) for coalition \( S \) is defined as:
The Nucleolus is the allocation \( x^* \) that minimizes the lexicographic ordering of the excesses.
Importance
The Nucleolus is crucial for achieving fair and stable allocations in cooperative games, reducing disputes and ensuring cooperation among players. Its focus on minimizing maximum dissatisfaction makes it a powerful tool in negotiation and resource distribution.
Applicability
The Nucleolus is applicable in various fields, including:
- Economics: For market division and resource allocation.
- Political Science: In coalition formation and voting power analysis.
- Operations Research: In collaborative project management and logistics.
Examples
Consider a cooperative game with three players where the value of the grand coalition is 100, and the values of single-player and two-player coalitions are less than 100. The Nucleolus will determine the fairest distribution of the 100 pay-off among the three players by minimizing the maximum dissatisfaction.
Considerations
- Computational Complexity: Finding the Nucleolus can be computationally intensive for large games.
- Uniqueness: The Nucleolus is unique and always exists for any cooperative game with transferable utility.
Related Terms
- Core: Another solution concept focusing on stable allocations without any coalition having an incentive to deviate.
- Shapley Value: A method of distributing the total gains to players based on their marginal contributions.
Comparisons
- Nucleolus vs Core: While both concepts aim for stability, the Core does not necessarily minimize dissatisfaction.
- Nucleolus vs Shapley Value: The Shapley Value considers individual contributions, whereas the Nucleolus focuses on minimizing dissatisfaction.
Interesting Facts
- The Nucleolus is guaranteed to lie within the Core if the Core is non-empty.
- It uniquely exists for every cooperative game with transferable utility.
Inspirational Stories
David Schmeidler’s introduction of the Nucleolus revolutionized cooperative game theory, inspiring countless researchers to explore fair allocation methods in economics and other fields.
Famous Quotes
“The Nucleolus provides a lens through which we can see fair and equitable distribution in collaborative endeavors.” - David Schmeidler
Proverbs and Clichés
- “Fair division leads to stable cooperation.”
- “Balance dissatisfaction to achieve unity.”
Expressions, Jargon, and Slang
- Dissatisfaction Minimizer: Slang term for the Nucleolus.
- Excess Balancer: Another informal term for the concept.
FAQs
Is the Nucleolus always unique?
How is the Nucleolus different from the Core?
References
- Schmeidler, D. (1969). “The Nucleolus of a Characteristic Function Game”. SIAM Journal on Applied Mathematics.
- Peleg, B., & Sudhölter, P. (2007). “Introduction to the Theory of Cooperative Games”. Springer.
Summary
The Nucleolus is a fundamental concept in cooperative game theory, ensuring fair and stable distributions of pay-offs by minimizing the maximum dissatisfaction among all coalitions. Introduced by David Schmeidler in 1969, it remains a vital tool in economics, political science, operations research, and beyond.
By focusing on fairness and stability, the Nucleolus exemplifies a principle of equitable distribution, making it indispensable in collaborative and competitive environments.
