Learning Lotteries with Minimal Violations from Membership Queries

We study how to choose a probability distribution, or lottery, over $m$ alternatives for $n$ agents when each agent's acceptable lotteries form an unknown linear halfspace of the simplex. The learner observes only binary accept/reject membership queries. Prior work asks whether there exists a lottery accepted by every agent; we study the relaxation that minimizes the number of agents who reject the lottery. We first show that this problem is APX-hard even when all halfspaces are explicitly known and scores are binary. We then give exact algorithms under additional structure: fixed $m$, and a margin condition under which an optimal lottery can be replaced by one supported on $\mathcal{O}(\log n/\gamma^2)$ alternatives. With limited queries, uniform sampling of agents gives additive population guarantees after recovering the sampled agents' normalized halfspaces. For arbitrary $m$, we give a polynomial-time LP based on a one-sided margin penalty and prove a population guarantee depending on the number of true violations and the number of agents near their thresholds. Finally, we prove a tight deterministic lower bound: without a stochastic link between queried and unqueried agents, querying only $K$ distinct agents cannot guarantee additive gap below $n-K$, and this gap is achievable.