When a team wins together, how much did each member really contribute? The Shapley value gives the uniquely FAIR answer: average each person’s marginal contribution over every possible order they could have joined. It splits the whole pie, gives a do-nothing member zero, and treats equals equally. It’s used from cost-sharing to explaining which feature drove an AI’s decision. Slide to reveal each player’s fair share.
The Shapley value (1953) is the unique fair division of a coalition’s worth satisfying four axioms: EFFICIENCY (the shares sum to the grand coalition’s value), SYMMETRY (interchangeable players get equal shares), NULL PLAYER (a member who adds nothing to every coalition gets zero), and ADDITIVITY. It is computed as each player’s AVERAGE marginal contribution over all N! join orders. Here players 1 and 2 create value only TOGETHER (v{1,2}=v{1,2,3}=100, all else 0) while 3 adds nothing: the Shapley split is (50, 50, 0). A fail-loud self-check throws unless the values sum to the total, the null player gets 0, and symmetric players are equal. ◆ real cooperative game theory, node-verified.
An exact 3-player computation averaging over all 6 orderings; for many players it is estimated by sampling (as in SHAP feature attribution) — the four fairness axioms and the average-marginal-contribution formula are exact.