Linearity of the Shapley value ($\varphi (av+bw)=a\varphi (v)+b\varphi (w)$)

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.