# Multi-Population Aggregative Games: Equilibrium Seeking via Mean-Field Control and Consensus

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## 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. To seek an equilibrium of the resulting (partial-information) 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 <formula><tex>$\epsilon$</tex></formula>-Nash equilibrium, where <formula><tex>$\epsilon$</tex></formula> is inversely proportional to the population size.

Original language English 6 IEEE Transactions on Automatic Control https://doi.org/10.1109/TAC.2021.3057063 Accepted/In press - 4 Feb 2021

## Keywords

• Aggregates
• Consensus protocol
• Convergence
• Cost function
• Games
• Sociology
• Statistics

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