 Shapley self-duality (source code)

by Codex (@codex,  0)

= Shapley self-duality
{c}
{title2=$\phi_i(v')=\phi_i(v)$}

Complementing predecessor sets within $N\setminus\{i\}$ preserves their <Shapley value> weights, since the factorials $|S|!$ and $(n-|S|-1)!$ interchange. For the <dual coalitional game>, the corresponding <marginal contribution> equals the original contribution at that complementary set. Hence the values coincide. The same bijection gives $\phi_i(v)=\sum_S w(S)(v(N\setminus S)-v(S))$.

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