What do the Shapley axioms - efficiency, symmetry, dummy and additivity - guarantee?

## Why axioms at all There are many ways to split a prediction into per-feature pieces, and most of them are ad hoc: drop a feature and see what happens, read the coefficients, look at the gradient. The reason Shapley attribution occupies the position it does is that it is the *unique* rule satisfying a short list of properties that most people would agree an attribution ought to have. If you accept the four axioms, you have accepted the Shapley value - there is no second method to argue about. Throughout, write `v(S)` for the model's output when the coalition `S` takes this row's values and everything else comes from a background, and `phi_i` for feature `i`'s attribution. ## Efficiency (local accuracy) ``` sum over all features of phi_i = v(all features) - v(empty set) ``` The attributions account for the whole gap between this prediction and the baseline prediction, with nothing left over and nothing invented. This is the axiom you can audit with arithmetic, and you should. A food-delivery ETA model predicts 38 minutes for an order; the background row predicts 26. Then the per-feature attributions for that order must sum to exactly 12 minutes: perhaps +7 for distance, +4 for restaurant prep backlog, +2 for rain, -1 for the courier already being nearby. If your numbers sum to 9 or 15, either the baseline you are quoting is not the one the attributions were computed against, or the values are approximate and you are seeing sampling error, or the attribution method is not Shapley at all. Efficiency is also what makes the explanation *addable*: you can group features into themes and the theme totals still sum to 12. ## Symmetry (equal treatment) If two features `i` and `j` satisfy `v(S + i) = v(S + j)` for every coalition `S` containing neither, then `phi_i = phi_j`. The operational version of this is duplicated columns. Suppose a subscription-price model is fed the same signal twice - `plan_price` and `plan_price_copy`, identical values in every row. The pair jointly carries whatever contribution that signal makes, and symmetry forces the two columns to receive exactly half of it each. Nothing about the ordering of the columns, the order in which they were engineered, or which one the model happened to split on first changes the split. That is a feature of the rule, not a bug: it is what stops attributions from depending on arbitrary implementation details. It does, however, mean that a duplicated signal *looks* half as important as it is, which is a real reporting trap when a stakeholder reads a single row's top attributions. ## Dummy (null player) If a feature never changes any coalition's value - `v(S + i) = v(S)` for all `S` - then `phi_i = 0`. Exactly zero, not merely small. This is the axiom that makes an attribution report readable. A column the model genuinely ignores, an ID field that survived into the feature set, a constant - all of them come out at zero, so a non-zero attribution is real signal about this model on this row. Note that in a *sampled* Shapley estimate the same feature will show a small non-zero value, purely as Monte Carlo noise; the axiom holds for the exact value, and near-zero-with-noise is the expected symptom. ## Additivity (linearity) For two models `f` and `g`, the attributions for the model `f + g` equal the attributions for `f` plus those for `g`, feature by feature. The practical consequence is compositional: for an ensemble that averages or sums the outputs of many base models, you can attribute each member separately and combine, instead of treating the ensemble as an opaque whole. It is also what makes attributions behave sensibly under a model that is a sum of a base score and a correction. ## The uniqueness result Shapley's theorem is that exactly one attribution rule satisfies efficiency, symmetry, dummy and additivity: the average-marginal-contribution value. This is the strongest argument available for the method, and it is what an interviewer is usually fishing for. The natural follow-up is what the axioms do *not* buy you. They say nothing about whether the explanation is causal, whether the baseline is a sensible reference point, whether the coalition values were computed over plausible inputs, or whether the numbers are stable under resampling. The axioms constrain the *split*, given a value function; they do not validate the value function. ## What to watch for A common confusion is reading efficiency as a global statement - it is per row, and the sum target changes for every row because the prediction changes. Another is expecting symmetry to spread credit across *correlated* features; symmetry only bites when two features are interchangeable in every coalition, which duplicates satisfy and merely correlated features generally do not.