“The Shapley decomposition iterates the cumulative approach for every possible order (permutation) of variables. With n variables, we need to make n! calculations, with each calculation based on another order for including new variables. The Shapley value implies that taking the average of the n! estimated contributions for every variable, yields the true contribution of each variable. As a result, the Shapley decomposition has three major advantages. First of all, the decomposition is perfect, meaning that the sum of the impacts, allocated to each of the explanatory variables, equals the observed change in the decomposed variable. One does not need to make any assumptions or effort to allocate the residual, as the solution is free from residuals. Secondly, the Shapley decomposition is symmetric (or anonymous): the factors are treated in an even-handed manner, without making any further theoretical assumptions. Thirdly, the Shapley decomposition allows for very complex decompositions that would otherwise be troublesome because of very high residuals and subsequent interpretation problems. We will illustrate the Shapley decomposition of a more complex identity in the next section.“

Johan Albrecht^， Delphine François^， and Koen Schoors（2001）

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