How to interpret the Shapley formula

62 Views Asked by Joe At 11 October 2022 at 04:19

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 ??

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

There are 1 best solutions below

Avocado

Avocado On 16 March 2023 at 12:59 BEST ANSWER

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.