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.