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Shapley value
3 key takeaways
Copy link to section- The Shapley value provides a fair distribution of the total payoff among players in a cooperative game, reflecting each player’s contribution to the overall success.
- It is calculated by considering all possible coalitions of players and the marginal contributions of each player to these coalitions.
- The Shapley value has applications in economics, political science, machine learning, and various fields requiring fair allocation of resources or benefits.
What is the Shapley value?
Copy link to sectionThe Shapley value, named after Lloyd Shapley who introduced it in 1953, is a method used in cooperative game theory to determine the fair distribution of payoffs among players who have contributed to a cooperative effort. It assigns a unique distribution of the total gains to each player based on their marginal contributions across all possible coalitions. The Shapley value ensures that each player is compensated fairly for their individual contributions to the collective outcome.
Calculation of the Shapley value
Copy link to sectionThe Shapley value for each player is calculated by considering the value of all possible coalitions that the player can be part of and the marginal contribution of the player to each of these coalitions. The formula for the Shapley value of a player ii in a game with NN players is:
The formula for the Shapley value of a player i in a game with N players is:
φ_i(v) = Σ [ |S|!(|N|-|S|-1)! / |N|! ] * ( v(S ∪ {i}) – v(S) )
Where:
- v is the value function that assigns a worth to each coalition of players.
- S is a subset of the player set N that does not include player i.
- |S| is the number of players in coalition S.
- v(S ∪ {i}) – v(S) is the marginal contribution of player i to coalition S.
This formula considers all possible ways to form coalitions and averages the marginal contributions of each player, weighted by the number of different orders in which players can join the coalitions.
Properties of the Shapley value
Copy link to sectionThe Shapley value possesses several key properties that make it a desirable solution concept in cooperative game theory:
- Efficiency: The total payoff is distributed entirely among the players. The sum of the Shapley values of all players equals the total value of the grand coalition (the coalition of all players).
- Symmetry: Players who contribute equally to all coalitions receive the same Shapley value.
- Dummy player: A player who does not contribute to any coalition (a dummy player) receives a Shapley value of zero.
- Additivity: For any two games with value functions vv and ww, the Shapley value of the combined game v+wv + w is the sum of the Shapley values of vv and ww.
These properties ensure that the Shapley value provides a fair and consistent method for distributing payoffs among players.
Applications of the Shapley value
Copy link to sectionThe Shapley value is widely used in various fields to solve problems related to fair distribution and allocation. Some notable applications include:
- Economics: The Shapley value is used to allocate costs and benefits in cooperative ventures, such as joint ventures, partnerships, and cost-sharing arrangements.
- Political science: It helps in analyzing power distribution in voting systems, such as determining the influence of different parties in a coalition government.
- Machine learning: The Shapley value is used to interpret the contributions of individual features in predictive models, providing insights into feature importance.
- Network design: In telecommunications and transportation networks, the Shapley value helps allocate the costs of shared infrastructure among users.
These applications demonstrate the versatility and usefulness of the Shapley value in addressing various real-world problems.
Examples and case studies
Copy link to sectionConsider a simple example with three players A, B, and C in a cooperative game with the following value function v:
- v({A}) = 0
- v({B}) = 0
- v({C}) = 0
- v({A, B}) = 100
- v({A, C}) = 150
- v({B, C}) = 200
- v({A, B, C}) = 300
The Shapley value for each player can be calculated by considering their marginal contributions to all possible coalitions:
For Player A: φ_A = (1/6) * (300 – 200) + (1/6) * (300 – 150) + (1/3) * (150 – 0) + (1/3) * (100 – 0) = 100
For Player B: φ_B = (1/6) * (300 – 200) + (1/6) * (300 – 100) + (1/3) * (200 – 0) + (1/3) * (100 – 0) = 100
For Player C: φ_C = (1/6) * (300 – 150) + (1/6) * (300 – 100) + (1/3) * (200 – 0) + (1/3) * (150 – 0) = 100
Thus, each player receives a Shapley value of 100, reflecting their fair contribution to the total value of the grand coalition.
The Shapley value is a powerful tool in cooperative game theory for fair distribution of payoffs based on individual contributions. Its properties and applications make it a widely used concept in economics, political science, machine learning, and beyond, helping to solve complex allocation problems in various fields.
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Sources & references

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