When is the probabilistic serial assignment uniquely efficient and envy-free?

We study the problem of allocating objects using lotteries. For each economy, the serial assignment, the assignment selected by the (probabilistic) serial rule, is sd-efficient and sd-envy-free (“sd” stands for stochastic dominance) but in general, it is not the only such assignment. Our question is when the uniqueness also holds. First, we provide a necessary condition for uniqueness, termed top-objects divisibility. Exploiting the structure revealed by top-objects divisibility, we then provide two sufficient conditions: preference richness and recursive decomposability. Existing sufficient conditions are restrictive in that they are satisfied only if there are sufficiently many agents relative to the number of objects; and that they only focus on preferences, ignoring other aspects of the problem that are also relevant to uniqueness. Our conditions overcome these limitations and can explain uniqueness for a wide range of economies.