An exact analysis of stable allocation

Shapley and Scarf introduced a notion of stable allocation between traders and indivisible goods, when each trader has rank-ordered each of the goods. The purpose of this note is to prove that the distribution of ranks after allocation is the same as the distribution of search distances in uniform hashing, when the rank-orderings are independent and uniformly random. Therefore the average sum of final ranks is just $(n+1)H_n-n$, and the standard deviation is O(n). The proof involves a family of interesting one-to-one correspondences between permutations of a special kind.