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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?

Joe
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    It's the other way around: `S` is a subset of `F` (the superset goes on the right-hand-side of the **⊆** symbol). – Dai Oct 11 '22 at 04:24
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    `F\{i}` is [set difference, or relative complement](https://en.wikipedia.org/wiki/Complement_(set_theory)#Relative_complement), a set of all elements of `F` except `i`. `S` is subset of or equal to that set, not `F`, because of precedence: `S⊆(F\{i})`. For example, if `F` is `{1, 2, 3}` and `i` is `2`, then `S⊆{1, 3}`. However, this is not on-topic for Stack Overflow, as it has nothing to do with programming: this is mathematics. – Amadan Oct 11 '22 at 04:24
  • @Amadan, thanks for your answer. I looked for the Mathematics forum but only saw "mathematics of teachers". I thought that the `Shapley` topic will allow my question to fall in Stack Overflow. You are correct though. Is there a Mathematics forum in Stackoverflow except that one for Mathematics teachers? Thanks again! – Joe Oct 11 '22 at 04:42
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    [maths.SE]: "Mathematics Stack Exchange is for people studying mathematics _at any level_ and professionals in related fields." (emphasis mine) – Amadan Oct 11 '22 at 14:23

1 Answers1

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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.
Avocado
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