Consider an economy with equal amounts of N types of goods, to be allocated to agents with strict quasi-convex preferences over lotteries. We show that ex-ante, all feasible and Pareto efficient allocations give almost all agents a binary lottery. Therefore, even if all preferences are the same, some identical agents necessarily receive different lotteries. Our results imply that many of the popular allocation mechanisms used in practice are not ex-ante efficient. Assuming the reduction of compound lotteries axiom, social welfare deteriorates by first randomizing over these binary lotteries. Full ex-ante equality can be achieved if agents satisfy the compound independence axiom.