# characteristics of Shapley values

In Christoph Molnar's book (Interpretable Machine Learning), the characteristics of Shapley values are defined as Efficiency, Symmetry, Dummy and Additivity. But in Lundberg's paper, the characteristics are defined as Local accuracy, Missingness and Consistency. How do you explain this discrepancy?

I assume you mean the SHAP paper and are referring to this chapter: Shapley Values.

You can basically find the answer in the Supplemental Material of the Lundberg paper: http://papers.nips.cc/paper/7062-a-unified-approach-to-interpreting-model-predictions

I'll try to summarize it here.

Efficiency

Efficiency is defined as:

$$\sum_{i=1}^M \phi_i^* = f(x) - E_X(f(X))$$

Local Accuracy according to Lundberg is:

$$f(x) = \phi_0 + \sum_{i=1}^M\phi_ix_i'$$

By defining $$\phi_0^* = E_X(f(X))$$ and $$\phi^*_i=\phi_i x_i'$$ both local accuracy and efficiency are equivalent. They both say that the prediction must be fairly attributed to the feature values.