Meritocratic Lotteries Under Intersectional Diversity Quotas

Meritocratic lotteries seek to implement prescribed selection probabilities reflecting merit or priority while meeting diversity quotas. We study this implementation problem on a hypergraph, with arbitrary target marginals and integer lower and upper quotas. Assuming consistency of the target marginals with the quotas and prescribed group size, we adapt existing decomposition and rounding results to establish two guarantees: (1) exact quota compliance when the traits admit a partition into two disjoint-set families; and (2) for general trait hypergraphs, a violation of each quota by at most D−1, where D is the maximum number of traits possessed by an agent. Both constructions preserve all marginals and the group size exactly, run in polynomial time, and produce explicit lotteries supported on at most as many selected groups as there are agents. The explicit lists of groups and their probabilities allow organisers to publish and inspect these lotteries before the draw. We then obtain trait-specific bounds when a small set of agents has high trait overlap and a concentration-based guarantee allowing small marginal errors. Applications to representative selection illustrate the trade-off between exact marginal implementation and exact quota compliance.