Important properties

1. If two explanatory variables \(j\) and \(k\) are interchangeable, then their Shapley values are equal

\[ \varphi(\underline{x}_*,j) = \varphi(\underline{x}_*,k) \]

2. If an explanatory variable \(j\) does not contribute to any prediction for any set of explanatory variables, then its Shapley value is equal to 0:

\[ \varphi(\underline{x}_*,j) = 0. \]

3. If model \(f()\) is a sum of two other models \(g()\) and \(h()\), then the Shapley value calculated for model \(f()\) is a sum of Shapley values for models \(g()\) and \(h()\).

4. The sum of Shapley values is equal to the model’s prediction.

\[ f(\underline{x}_*) - E_{\underline{X}}\{f(\underline{X})\} = \sum_{j=1}^p \varphi(\underline{x}_*,j), \]