A distributed algorithm for average aggregative games with coupling constraints

We consider the framework of average aggregative games, where the cost function of each agent depends on his own strategy and on the average population strategy. We focus on the case in which the agents are coupled not only via their cost functions, but also via constraints coupling their strategies. We propose a distributed algorithm that achieves an almost Nash equilibrium by requiring only local communications of the agents, as specified by a sparse communication network. The proof of convergence of the algorithm relies on the auxiliary class of network aggregative games and exploits a novel result of parametric convergence of variational inequalities, which is applicable beyond the context of games. We apply our theoretical findings to a multi-market Cournot game with transportation costs and maximum market capacity.

💡 Research Summary

The paper addresses a class of average aggregative games (AAGs) in which each player’s cost depends on its own decision variable and on the average of all players’ decisions, while the collective decisions must also satisfy a set of linear coupling constraints. Such constraints naturally arise in many engineering contexts, for example capacity limits in power grids or transportation limits in logistics networks. The authors propose a fully distributed algorithm that requires only local information exchange over a sparse communication graph, yet provably converges to an ε‑Nash equilibrium of the original game.

The problem setting is formalized as follows. There are N agents, each with a private convex compact strategy set X_i ⊂ ℝⁿ and a continuously differentiable convex cost J_i(x_i,σ), where σ = (1/N)∑_j x_j denotes the population average. In addition, the strategies must satisfy a linear coupling constraint ˆAσ ≤ ˆb, where ˆA ∈ ℝ^{m×n} and ˆb ∈ ℝ^m. The game is denoted G^∞. The communication network is modeled by a weight matrix T ∈