  Allocating multiple scarce items across a set of individuals is an important
practical problem. In the case of divisible goods and additive preferences a
convex program can be used to find the solution that maximizes Nash welfare
(MNW). The MNW solution is equivalent to finding the equilibrium of a market
economy (aka. the competitive equilibrium from equal incomes, CEEI) and thus
has good properties such as Pareto optimality, envy-freeness, and incentive
compatibility in the large. Unfortunately, this equivalence (and nice
properties) breaks down for general preference classes. Motivated by real world
problems such as course allocation and recommender systems we study the case of
additive `at most one' (AMO) preferences - individuals want at most 1 of each
item and lotteries are allowed. We show that in this case the MNW solution is
still a convex program and importantly is a CEEI solution when the instance
gets large but has a `low rank' structure. Thus a polynomial time algorithm can
be used to scale CEEI (which is in general PPAD-hard) for AMO preferences. We
examine whether the properties guaranteed in the limit hold approximately in
finite samples using several real datasets.
