  The effect of population heterogeneity in multi-agent learning is practically
relevant but remains far from being well-understood. Motivated by this, we
introduce a model of multi-population learning that allows for heterogeneous
beliefs within each population and where agents respond to their beliefs via
smooth fictitious play (SFP).We show that the system state -- a probability
distribution over beliefs -- evolves according to a system of partial
differential equations akin to the continuity equations that commonly desccribe
transport phenomena in physical systems. We establish the convergence of SFP to
Quantal Response Equilibria in different classes of games capturing both
network competition as well as network coordination. We also prove that the
beliefs will eventually homogenize in all network games. Although the initial
belief heterogeneity disappears in the limit, we show that it plays a crucial
role for equilibrium selection in the case of coordination games as it helps
select highly desirable equilibria. Contrary, in the case of network
competition, the resulting limit behavior is independent of the initialization
of beliefs, even when the underlying game has many distinct Nash equilibria.
