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

Farzaneh Farhadi, Maria Chli

TL;DR

A constructive threshold rule is provided: 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) and the result is a general, scalable recipe for robust coordination in MAS with complementarities.

Abstract

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 $O(|Θ|\log|Θ|)$, 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.

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

TL;DR

A constructive threshold rule is provided: 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) and the result is a general, scalable recipe for robust coordination in MAS with complementarities.

Abstract

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 , 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.
Paper Structure (47 sections, 4 theorems, 28 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 47 sections, 4 theorems, 28 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Binary cooperation with complementarities.
  3. Limitations of classical information design.
  4. Robust implementation via smallest-equilibrium play.
  5. From LP characterization to constructive design.
  6. Empirical validation in MAS domains.
  7. Contributions.
  8. Related Work
  9. Why not use existing methods? From classical to robust information design.
  10. Outcome- vs. policy-level implementability under smallest play.
  11. MAS, network effects, and domain bridges.
  12. Problem Setting: Binary-Action Complementarities in MAS Domains
  13. Environment
  14. Agents' utilities and incentives
  15. Potential representation and convexity
  16. ...and 32 more sections

Key Result

lemma 1

For each state $\theta \in \Theta$, the potential function $F_\theta(n)$ defined in eq:potential is (discretely) convex in $n$.

Figures (3)

  • Figure 1: Robust information design under smallest-equilibrium play. Sequential obedience restricts implementability relative to classical persuasion. Under convex potential and welfare, the optimum is an extreme point, yielding a threshold policy computable in $\mathcal{O}(|\Theta|\log|\Theta|)$ time.
  • Figure 2: Technology adoption with nearly continuous states. Scores $S(\theta)$ induce a near-continuous threshold. The robust policy $\pi^{\mathrm{seq}\,\star}$ adopts only where coordination is sustainable, while BCE policy (orange) recommends broader adoption but collapses to no adoption under smallest-equilibrium play.
  • Figure 3: Welfare under Robust, BCE-Optimistic (classical obedience only), and BCE-Realized (same BCE policy evaluated under smallest-equilibrium). For medium costs, the BCE-Optimistic curve sits above Robust, but BCE-Realized drops to zero, revealing that the optimistic gains are not achievable under conservative play.

Theorems & Definitions (6)

  • lemma 1: Convexity of the potential
  • definition 1: Sequential obedience — cooperation
  • definition 2: Sequential obedience — non-cooperation
  • theorem 1: Characterization of implementable sequential policies
  • lemma 2: Feasibility
  • theorem 2: Perfect coordination is optimal