We consider a multi-agent system consisting of
several populations. The interaction between large populations
of agents seeking to regulate their state on the basis of the
distribution of the neighboring populations is studied. Examples
of such interactions can typically be found in social networks
and opinion dynamics, where heterogeneous agents or clusters
are present and decisions are influenced by individual objectives
as well as by global factors. In this paper, such a problem
is posed as a multi-population mean-field game, for which
solutions depend on two partial differential equations, namely
the Hamilton-Jacobi-Bellman equation and the Fokker-Planck-
Kolmogorov equation. The case in which the distributions of
agents are sums of polynomials and the value functions are
quadratic polynomials is considered. It is shown that for this
class of problems, which can be considered as approximations of
more general problems, a set of ordinary differential equations,
with two-point boundary value conditions, can be solved in
place of the more complicated partial differential equations
characterizing the solution of the multi-population mean-field
game.