How Shapley Values Work#

Stage 5 · 📈 Regression · Lesson 37 of 56 · intermediate

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Important

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A fair division problem#

The feature-importance question has a surprisingly deep answer borrowed from economics. Imagine several players cooperating to earn a joint payout — how should the winnings be split fairly, according to each player’s real contribution? Lloyd Shapley solved this in 1953, and the Shapley value he defined turns out to be exactly what is needed to attribute a model’s prediction to its features.

Marginal contributions#

The idea rests on marginal contribution. A player’s contribution to a group (a coalition) is how much the payout grows when they join it: \(v(S \cup \{i\}) - v(S)\) for a coalition \(S\). But that depends on who is already there — a player may add a lot to a small group and little to a large one. The Shapley value resolves this by averaging a player’s marginal contribution over every possible coalition (equivalently, every order in which players could join).

The formula#

Written out, the Shapley value of feature \(i\) among \(n\) features is that weighted average:

\[\phi_i = \sum_{S \subseteq N \setminus \{i\}} \frac{|S|!\,(n - |S| - 1)!}{n!}\,\bigl[v(S \cup \{i\}) - v(S)\bigr].\]

The weights count the orderings, so each coalition is credited correctly. What makes the Shapley value special is that it is the unique scheme satisfying four fairness axioms: efficiency (the parts sum to the whole), symmetry (equal contributors get equal shares), dummy (a feature that changes nothing gets zero), and additivity.

Explaining predictions#

In machine learning the analogy is exact: the features are the players and the prediction is the payout. The Shapley value of a feature is how much it pushed this prediction above or below the average prediction — a fair, model-agnostic attribution that works for any model, regression or classification. The popular SHAP framework (2017) builds on this, decomposing a prediction into feature contributions that sum to the output. The catch is cost: exact values require all \(2^n\) coalitions, so in practice they are approximated. Unlike the greedy selection of earlier lessons, Shapley values weigh every feature fairly against all others.

Hint