Abstract
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. Using network-flow integrality and mean-preserving rounding, we give an explicit lottery satisfying every quota when the traits admit a partition into two disjoint-set families. For general trait hypergraphs, a floating-variable construction preserves all marginals and the group size exactly, while violating each quota by at most \( D-1 \), where \( D \) is the maximum number of traits possessed by an agent. The construction runs in polynomial time and outputs at most as many selected groups as there are agents. We further 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. The resulting framework combines explicit lottery construction, overlap-sensitive bounds, and fixed-cardinality selection.