Definition

Define the Shapley value as the solution concept Sh⁡\operatorname{Sh}, for coalitional game (N;v)(N; v) where NN is the set of players and v(N)v(N) is the worth of the grand coalition NN,

Shi⁡(N;v):=1n!∑π∈Π(N)(v(Pi(π)∪{i})−v(Pi(π))),∀i∈N\operatorname{Sh}_i(N;v) := \frac{1}{n!}\sum_{\pi \in \Pi(N)} (v(P_i(\pi) \cup \{i\}) - v(P_i(\pi))), \quad \forall i \in N

Equivalently,

ϕi(v)=∑S⊂Ni∈S(|S|−1)!(n−|S|)!n![v(S)−v(S−{i})]\phi_i(v) = \sum_{\substack{S \subset N\\ i \in S}} \frac{(|S|-1)!(n-|S|)!}{n!}[v(S)-v(S-\{i\})]

Notes

References

  1. M. Maschler, E. Solan, and S. Zamir, Game Theory, 1st ed. Cambridge University Press, 2013. doi: 10.1017/CBO9780511794216, pp. 748-755.
  2. T. S. Ferguson, Class Lecture, Topic: "Part IV. Games in Coalitional Form". Math 167, School of Computer Science, Carnegie Mellon University, Pittsburgh, Winter 2000. https://www.cs.cmu.edu/afs/cs/academic/class/15859-s05/www/ferguson/coal.pdf
  3. https://shap.readthedocs.io/en/latest/example_notebooks/overviews/An introduction to explainable AI with Shapley values.html