Multi-Population Aggregative Games

Abstract

In this article, we extend the theory of deterministic
mean-field/aggregative games to multipopulation games. We consider a set
of populations, each managed by a population coordinator (PC), of
selfish agents playing a global noncooperative 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 (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 the 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 nonexpansiveness condition, to
show convergence towards an
ε
-Nash equilibrium, where
ε
is inversely proportional to the population size.