Shapley Value

Definition

The Shapley value is a solution concept in cooperative game theory that assigns a unique distribution of payoffs to players based on their individual contributions to the total value created by a coalition. It considers all possible coalitions and calculates each player's marginal contribution, ensuring that players receive compensation proportional to their contributions while promoting fairness among all participants in the game.

5 Must Know Facts For Your Next Test

1. The Shapley value was developed by Lloyd Shapley in 1953 and is widely used in economics, political science, and game theory to analyze cooperative scenarios.
2. It is computed by considering every possible order in which players can join a coalition and averaging their marginal contributions across all orders.
3. The Shapley value satisfies several key properties, including efficiency, symmetry, and additivity, which ensure fair distribution of payoffs among players.
4. It provides a way to resolve disputes over how much each player should receive when the total payoff from a coalition is known, promoting cooperation among players.
5. In practical applications, the Shapley value can be used in various fields, such as cost allocation problems and fair division issues, demonstrating its versatility in economic contexts.

Review Questions

• How does the Shapley value ensure fairness among players in a cooperative game?
  • The Shapley value ensures fairness by calculating each player's contribution to the total value generated by different coalitions. It takes into account every possible coalition formation and averages the marginal contributions of each player across these formations. This approach guarantees that players are compensated proportionally to their contributions while promoting equity, as it treats all players equally regardless of their order of entry into the coalition.
• Discuss how the properties of efficiency and symmetry apply to the Shapley value in cooperative game theory.
  • The efficiency property of the Shapley value means that the total payoff distributed among players equals the total value created by the coalition. In contrast, the symmetry property states that if two players contribute equally to every coalition they are part of, they should receive equal payoffs. These properties work together to ensure that the distribution of payoffs is not only fair but also reflective of the true contributions made by each player within the cooperative framework.
• Evaluate the implications of using the Shapley value for cost allocation in a collaborative project involving multiple stakeholders.
  • Using the Shapley value for cost allocation in collaborative projects allows stakeholders to fairly distribute costs based on their individual contributions to the project's success. This approach fosters trust and cooperation among participants as they see a transparent method for compensation. However, it requires accurate assessment of each stakeholder's marginal contributions, which can be complex and contentious. Ultimately, employing the Shapley value can lead to more sustainable collaborations and reduce conflicts over financial responsibilities.

Related terms

Coalition: A group of players who come together to achieve a common goal and share the rewards from their collective effort.

Nash Equilibrium: A concept where no player can benefit from changing their strategy while other players keep theirs unchanged, representing a state of stability in non-cooperative games.

Marginal Contribution: The additional value that an individual player brings to a coalition, which is crucial for calculating the Shapley value.