  We study public goods games played on networks with possibly non-reciprocal
relationships between players. Examples for this type of interactions include
one-sided relationships, mutual but unequal relationships, and parasitism. It
is well known that many simple learning processes converge to a Nash
equilibrium if interactions are reciprocal, but this is not true in general for
directed networks. However, by a simple tool of rescaling the strategy space,
we generalize the convergence result for a class of directed networks and show
that it is characterized by transitive weight matrices. Additionally, we show
convergence in a second class of networks; those rescalable into networks with
weak externalities. We characterize the latter class by the spectral properties
of the absolute value of the network's weight matrix and show that it includes
all directed acyclic networks.
