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 Pareto efficient allocations give almost all agents lotteries over at most two outcomes. Therefore, even if all preferences are the same, some identical agents necessarily receive different lotteries. Our results provide a simple criterion to show that many popular allocation mechanisms are ex-ante inefficient. Assuming the reduction axiom, social welfare deteriorates by first randomizing over these binary lotteries. Efficient full ex-ante equality is achieved if agents satisfy the compound independence axiom.