Imputation Set

Definition

An imputation set is a collection of payoff distributions in cooperative game theory that ensures all players in a game receive a minimum payoff, while the total payoff does not exceed the value assigned to each coalition. This concept is crucial in understanding how resources can be fairly allocated among players based on their contributions to coalitions. The imputation set thus serves as a foundation for determining stable and equitable distributions of payoffs in cooperative scenarios.

5 Must Know Facts For Your Next Test

1. The imputation set is defined within the context of a cooperative game by ensuring that the sum of payoffs allocated to players does not exceed the value of the grand coalition.
2. All imputations must satisfy individual rationality, meaning that no player receives less than their marginal contribution to any coalition they belong to.
3. The imputation set can be empty if there is no allocation that satisfies both the efficiency and individual rationality conditions for all players involved.
4. In many cases, finding a point within the imputation set involves using mathematical programming techniques or graphical methods to visualize the distributions.
5. The boundaries of the imputation set can be analyzed using geometric representations, often leading to insights about stability and fairness in resource allocation.

Review Questions

• How does the imputation set relate to the core of a cooperative game, and why is this relationship important?
  • The imputation set contains all possible payoff distributions that satisfy individual rationality and efficiency, while the core represents those distributions where no group of players would benefit by forming their own coalition. The relationship is important because if an allocation lies within the core, it is stable against deviations by coalitions. Understanding this relationship helps determine not only fair distributions but also which allocations are sustainable in cooperative scenarios.
• Discuss how the characteristic function plays a role in determining the imputation set within a cooperative game.
  • The characteristic function provides essential information about the value each coalition can achieve, which directly influences the construction of the imputation set. By knowing the worth of different coalitions, one can identify feasible allocations that fall within the total value generated by players working together. This ensures that any distribution within the imputation set respects the limits set by the characteristic function, helping to maintain fairness and equity among participants.
• Evaluate the implications of an empty imputation set for players in a cooperative game, and analyze potential strategies they might employ to achieve cooperation.
  • An empty imputation set indicates that there is no feasible distribution satisfying both individual rationality and efficiency, leaving players without a stable outcome. This situation can lead players to seek alternative strategies, such as forming smaller coalitions that may yield better payoffs or renegotiating terms to find common ground. Additionally, players might explore external agreements or contracts to ensure more favorable conditions, thereby facilitating cooperation despite inherent challenges in payoff distribution.

Related terms

Core: The core is the set of feasible allocations in a cooperative game where no subset of players would prefer to deviate and form their own coalition, as they would receive a better payoff.

Characteristic Function: A characteristic function assigns a value to each coalition of players, representing the maximum worth that coalition can achieve together.

Nash Bargaining Solution: The Nash Bargaining Solution is a solution concept in cooperative game theory that provides a fair outcome based on the players' utility functions and their disagreement points.