Multi-agent planning is a well-studied problem with various applications including disaster rescue, urban transportation and logistics, both for autonomous agents and for decision support to humans. Due to computational constraints, existing research typically focuses on one of two scenarios: unstructured domains with many agents where we are content with heuristic solutions, or domains with small numbers of agents or special structure where we can provide provably near-optimal solutions. By contrast, in this paper, we focus on providing provably near-optimal solutions for domains with large numbers of agents, by exploiting a common domain-general property: if individual agents each have limited influence on the overall solution quality, then we can take advantage of randomization and the resulting statistical concentration to show that each agent can safely plan based only on the average behavior of the other agents. To that end, we make two key contributions: (a) an algorithm, based on Lagrangian relaxation and randomized rounding, for solving multi-agent planning problems represented as large mixed integer programs; (b) a proof of convergence of our algorithm to a near-optimal solution. We demonstrate the scalability of our approach with a large-scale illustrative theme park crowd management problem.