Linearity of the Shapley value
= Linearity of the Shapley value
{title2=$\phi(av+bw)=a\phi(v)+b\phi(w)$}
The <Shapley value> is a linear map on real characteristic <functions> with zero value at the empty <coalition>. In each player <permutation>, the <marginal contribution> to $av+bw$ is $a$ times the marginal to $v$ plus $b$ times the marginal to $w$. Taking their <expectations> proves the formula for arbitrary real $a,b$. This directly proves real homogeneity as well as <cooperative-game additivity>.