Excess

Definition

Excess refers to the amount by which a certain quantity surpasses a given limit or expectation. In the context of cooperative game theory, it is particularly relevant in evaluating how much value can be gained or lost when players form coalitions, especially when considering solutions like the core, Shapley value, and nucleolus. Understanding excess helps in analyzing whether a coalition can achieve a better outcome than what they could individually.

5 Must Know Facts For Your Next Test

1. In cooperative games, the excess of a coalition is calculated as the difference between the value generated by the coalition and the total payoffs received by its members.
2. If a coalition has a positive excess, it means that there are gains to be made by forming that coalition rather than acting independently.
3. A core allocation eliminates excess for all coalitions, ensuring stability as no group has an incentive to deviate.
4. The nucleolus seeks to minimize the maximum excess across all coalitions, representing a fair compromise among players.
5. In calculating the Shapley value, excess plays a role in determining how much each player contributes to the overall success of the coalition.

Review Questions

• How does excess inform the stability of coalitions within cooperative game theory?
  • Excess directly impacts the stability of coalitions since it measures whether players can gain more by working together or if they would fare better independently. If a coalition has a positive excess, members are incentivized to stay united as they can achieve better outcomes together. Conversely, if excess is negative or zero, players may reconsider their participation in the coalition, potentially leading to its dissolution.
• What role does excess play in defining the concept of the core in cooperative games?
  • The core is defined by allocations where no coalition can achieve a better outcome on their own; hence, excess is critical in this context. An allocation that lies within the core results in zero excess for every possible coalition, indicating that no group can improve their situation by breaking away. This ensures that all players are satisfied with their payoff given the cooperation within their coalition.
• Evaluate how the concept of excess influences the computation of the Shapley value and nucleolus in cooperative games.
  • Excess significantly influences both the Shapley value and nucleolus by providing insights into individual contributions and fairness within coalitions. The Shapley value allocates payoffs based on each player's marginal contribution, effectively addressing excess by ensuring fair compensation. On the other hand, the nucleolus aims to minimize maximum excess across coalitions, seeking an equitable distribution that satisfies all players as closely as possible. Together, these concepts rely on understanding and addressing excess to ensure stability and fairness in cooperative outcomes.

Related terms

Coalition: A group of players who come together to achieve a common goal or maximize their collective payoff in a game.

Core: The set of allocations where no coalition can improve upon its outcome, meaning there is no excess for any coalition.

Shapley Value: A solution concept that assigns a unique distribution of a total value generated by the coalition among its members based on their marginal contributions.