Shapley Value

Game Theory · Axiom Academy

Unit 6 - Cooperative Game Theory: A fair solution to the division problem Imagine three players form a coalition. Together they create value, but each player contributes differently. How should we fairly divide the total payoff? The Shapley value answers this by considering: What does each player contribute when they join? The marginal contribution of player i to coalition S is the value they add when joining: where v(S) is the characteristic function giving the value of coalition S . This value changes depending on which players are already present! The Shapley value averages marginal contributions across all possible orderings. For player i : where the sum is over all coalitions S that don't contain i , and the weight captures how many orderings lead to that coalition forming before i joins. Let's calculate the Shapley value for a 3-player game with players A, B, C . Step 1: Enumerate all coalitions and compute values v(S) Step 2: For each player, compute marginal contribution to each coalition Step 3: Weight each contribution by the coalition's probability Step 4: Sum weighted contributions to get the Shapley value 5. Interpretation as Expected Marginal Contribution The Shapley value can be interpreted probabilistically: imagine all n ! orderings of players joining are equally likely. The Shapley value is the expected value of a player's marginal contribution when they join in a random position.

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