Ex-Ante Stable Lotteries

We study the allocation of indivisible objects (e.g. school seats) to agents by lotteries. Agents have preferences over different objects and have different priorities at different objects. The priorities can contain indifferences, some agents may have the same priority at some object. A lottery is ex-ante stable if there does not exist an agent-object pair such that we can increase the probability of matching this pair at the expense of agents, who have lower priority at the object, and of objects which are less preferred by the agent. As a first result, we show that this fairness condition is very demanding: Only few agent-object pairs have a positive probability of being matched. The number of pairs in the support depends on how many indifferences in the priorities the lottery exploits. In the extreme case where no object is matched with positive probability to two equal priority agents, the lottery is almost degenerate. Otherwise, the size of the support is completely determined by the size of the lowest priority classes of which agents are matched to the respective objects. We interpret our result as an impossibility result. With ex-ante stability one cannot go much beyond randomly breaking ties and implementing a (deterministically) stable matching with respect to the broken ties. As a second result, we derive a new characterization of the set of lotteries that can be decentralized by a pseudo-market with priority-specific pricing as introduced by He et al. (2015). These allocations coincide with the ex-ante stable lotteries that do not admit a strong stable improvement cycle.