Best-response dynamics in directed network games

We study public goods games played on networks with possibly non-recip-rocal relationships between players. Examples for this type of interactions include one-sided relationships, mutual but unequal relationships, and par-asitism. 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 and quadratic best-response potentials. Additionally, we show convergence in a second class of networks; those rescalable into networks with weak exter-nalities. We characterize the latter class by the spectral properties of the absolute value of the network’s weight matrix and by another best-response potential structure