Multipopulation Aggregative Games:Equilibrium Seeking via Mean-Field Control and Consensus

Abstract

In this paper, we extend the theory of deterministic mean-field/aggregative games to multi-population games. We consider a set of populations, each managed by a population coordinator (PC), of selfish agents playing a global non-cooperative game, whose cost functions are affected by an aggregate term across all agents from all populations. In particular, we impose that the agents cannot exchange information between themselves directly; instead, only a PC can gather information on its own population and exchange local aggregate information with the neighboring PCs. To seek an equilibrium of the resulting (partialinformation) game, we propose an iterative algorithm where each PC broadcasts a mean-field signal, namely, an estimate of the overall aggregative term, to its own population only. In turn, we let the local agents react with a best response and the PCs cooperate for estimating the aggregative term. Our main technical contributions are to cast the proposed scheme as a fixed-point iteration with errors, namely, the interconnection of a Krasnoselskij-Mann iteration and a linear consensus protocol, and, under a non-expansiveness condition, to show convergence towards an ε-Nash equilibrium, where ε is inversely proportional to the population size.

Tracking-Based Distributed Equilibrium Seeking for Aggregative Games

“…with λ ∈ R m being the multiplier associated to the coupling constraints, and ρ > 0 a constant. We therefore address the v-GNE seeking problem by obtaining a saddle point of (27) through the discrete-time dynamics:…”

“…Note that all the quantities involved in the agent's calculations are purely local, thus making Algorithm 2 fully distributed. Differently from customary primal-dual schemes, (33b) does not need the projection over the positive orthant R m + due to the chosen augmented Lagrangian functions L i (27). We only need to initialize λ 0 i ≥ 0 for all i ∈ I, and choose δ and ρ appropriately so that we avoid situations where λ t i ≥ 0 implies λ t+1 i < 0.…”

A Survey on Noncooperative Games and Distributed Nash Equilibrium Seeking over Multi-Agent Networks

“…To be specific, agents send the latest equilibrium strategy to the central scheduler, which performs centralized operation and broadcasts the aggregation strategy to all agents in the network. Paccagnan et al [100] presented the existence and uniqueness conditions of Nash equilibrium based on the variational theory, and then proposed a semi-decentralized equilibrium learning method based on projection gradient. Moreover, the theory of variational inequality was used to study the -Nash equilibrium of the game [46,57,100] , including existence, uniqueness and sensitivity.…”

“…Paccagnan et al [100] presented the existence and uniqueness conditions of Nash equilibrium based on the variational theory, and then proposed a semi-decentralized equilibrium learning method based on projection gradient. Moreover, the theory of variational inequality was used to study the -Nash equilibrium of the game [46,57,100] , including existence, uniqueness and sensitivity. In addition, the properties of Wardrop equilibrium of aggregative game are discussed, and the operator splitting and fixed point iterative approximation method are used to realize the Nash equilibrium seeking under the semidecentralized communication structure.…”

Mean field games of energy storage devices: A finite difference analysis

“…The MFG model based on a controlled Langevin equation is considered in [24]. The authors of [25] extend the theory of deterministic mean-field/aggregative games to multipopulation games. Based on the MFG theory, competitive interaction among electrical vehicles or hybrid electrical vehicles is investigated in [26,27].…”