Variational Inequality Gap-Based Equilibrium Metrics and a Simplified Multi-Agent Reinforcement Learning Scheme for Multi-Commodity Trade Networks

Abstract

This paper formulates a multi-commodity trade network as a finite-dimensional monotone variational inequality (VI) and introduces a normalized equilibrium metric derived from the classical VI gap function. For the box-constrained feasible set considered here, the gap can be evaluated exactly from the network state, and the resulting metric vanishes if and only if the state is a VI solution. We then study a projected stochastic approximation scheme whose joint state updates can be interpreted as a simplified VI-aware learning dynamic. Under the convex-potential structure of the trade-network operator, unbiased noise with bounded conditional variance, and Robbins–Monro step sizes, we prove that the iterates converge almost surely to a VI solution and that the gap-based equilibrium metric tends to zero. Unlike heuristic equilibrium-proximity indices used in recent MARL–VI architectures, the proposed metric is derived directly from the VI formulation and is accompanied by explicit stochastic-approximation guarantees. The framework is illustrated on a 32-dimensional trade network constructed from 2022 CEPII BACI bilateral trade data for cereals and oilseeds among the United States, China, Germany, and Brazil. The experiments report the VI gap, normalized equilibrium metric, projected residual, and potential gap in addition to the distance to a reference solution. These diagnostics decrease consistently over the finite computational horizon. The results are largely insensitive to the tested noise levels but show a stronger dependence on the initial step size, illustrating numerical behavior consistent with the theoretical analysis.