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.

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