Approximately optimal distributed controls for high-dimensional stochastic systems with pairwise interaction through controls

TL;DR

This paper develops approximately optimal distributed control strategies for large-scale stochastic systems with pairwise control interactions, providing bounds on the performance gap between full-information and partial-observation scenarios.

Contribution

It introduces a novel analysis framework for pairwise interaction controls, extending existing models to high-dimensional stochastic systems with quantifiable performance bounds.

Findings

Established a bound on the difference between value functions of full and partial observation problems.

Reformulated control problems as Hamilton-Jacobi equations for analytical comparison.

Provided a method to compare distributions of optimal controls in complex stochastic systems.

Abstract

This paper investigates large-population stochastic control problems in which agents share their state information and cooperate to minimize a convex cost functional. The latter is decomposed into individual and coupling costs, with the distinctive feature that the coupling term is a pairwise interaction function between the controls. To address this setting, we follow closely (Jackson & Lacker, 2025): we introduce a related problem where each agent observes only its own state. We then establish a quantitative bound on the difference between the value functions associated with these two problems. We obtain this result by reformulating the problems analytically as Hamilton-Jacobi type equations and comparing their associated Hamiltonians. The main difficulty of our approach lies in establishing a precise comparison between the distributions of the corresponding optimal controls.