Search code examples
shapley

How to interpret the Shapley formula

For the Shapley equation \begin{equation} \phi_{i} = \sum_{S\subseteq F\backslash {i}}\dfrac{|S|!(|F| - |S| - 1)!}{|F|!}\left[f_{S\cup{i}}(x_{S\cup{i}})-f_{S}(x_{S})\right] \end{equation}

How to read enter image description here ??

I know the first part says F is a subset of S, but what does the \{i} mean?

Solution

  • We want to sum over all coalitions where agent i is not present. In this notation it is denoted by F \ {i}.

    The interpretation of the Shapley value:

    • Take a coalition where agent i is not present.
    • Add agent i and calculate the increase of the coalition value: value of the coalition after agent i joined minus value of the coalition when agent i was not in the coalition. This is agent i's marginal cost/value of joining this coalition.
    • Multiply this by the specified weight.
    • Do all these steps for all coalitions where agent i is not present and sum the weighted marginal costs/values: hence the notation you are asking about.