Shapley value denominator in a simple game ($n!{\varphi }_i(v)\in \mathbb{Z}$)

In an $n$-player simple cooperative game, every marginal contribution along a player permutation is zero or one. The Shapley value is its average over the $n!$ equally likely permutations. Thus $n!{\varphi }_i(v)$ counts the orders in which player $i$ changes a losing coalition into a winning one and is an integer. In lowest terms, the coordinate's denominator divides $n!$. This is a necessary constraint on a proposed Shapley value vector, not a sufficient one.