I understand Shapley value in game theory is a means to capture the average marginal contribution of a player. Obviously the way to do it would be to consider all possible coalitions of $|N|$ players excluding a particular player, say $i$.
So the Shapley value can be $\phi_i(N)=\sum_{s \subseteq N \setminus i} k\ \left(v(s \cup \{ i \})-v(s)\right)$. The value of $k$, the factor used for averaging is found to be $\frac{|s|!(|N|-|s|-1)!}{|N|!}$. I can see that the value of $k$ is such that the solution concept is allocatively efficient (and satisfies a few other axioms, besides), but is there any direct intuitive way to understand the formula?