Shapley Value

A unique fair allocation in cooperative games satisfying efficiency, symmetry, dummy, and additivity.

Ask the Game theory assistant 1 min read · Updated September 9, 2026

Definition

$\phi_i(v)=\sum_{S\subseteq N\setminus\{i\}}\dfrac{|S|!\,(|N|-|S|-1)!}{|N|!}\bigl(v(S\cup\{i\})-v(S)\bigr).$

Equivalently an average of player $i$ 's marginal contribution over all orderings of $N$ .

Each player is paid their expected marginal contribution if players join in a uniformly random order.

It is the unique payoff vector satisfying four natural fairness axioms.

For the majority game $v(S)=\mathbf 1\{|S|\ge 2\}$ with $N=\{1,2,3\}$ , $\phi=(1/3,1/3,1/3)$ .

Airport cost-sharing: incremental runway costs yield Shapley values that equal the average cost-per-user.

Shapley (1953) characterized $\phi$ uniquely via efficiency ( $\sum\phi_i=v(N)$ ), symmetry, dummy/null player, and linearity/additivity.

For convex games, the Shapley value lies in the core (Shapley, 1971).

Cost allocation, feature attribution in ML (SHAP values), power indices in voting, and collaborative learning.

Interpretable ML tool (SHAP) that has become standard practice.

Assembled from the ReLU.chat curated knowledge base. These explanations are concise on purpose; check the sources for anything important.