Robust Information Design for Multi-Agent Systems with Complementarities: Smallest-Equilibrium Threshold Policies

We study information design in multi-agent systems (MAS) with binary actions and strategic complementarities, where an external designer influences behavior only through signals. Agents play the smallest-equilibrium of the induced Bayesian game, reflecting conservative, coordination-averse behavior typical in distributed systems. We show that when utilities admit a convex potential and welfare is convex, the robustly implementable optimum has a remarkably simple form: perfect coordination at each state: either everyone acts or no one does. We provide a constructive threshold rule: compute a one-dimensional score for each state, sort states, and pick a single threshold (with a knife-edge lottery for at most one state). This rule is an explicit optimal vertex of a linear program (LP) characterized by feasibility and sequential obedience constraints. Empirically, in both vaccination and technology-adoption domains, our constructive policy matches LP optima, scales as $O(|\Theta|\log|\Theta|)$, and avoids the inflated welfare predicted by obedience-only designs that assume the designer can dictate the (best) equilibrium. The result is a general, scalable recipe for robust coordination in MAS with complementarities.