Best-response dynamics in directed network games

TL;DR

This paper investigates how certain learning processes in public goods games on directed networks converge to equilibrium, identifying specific network structures that guarantee convergence through rescaling techniques.

Contribution

It introduces a rescaling method to extend convergence results to directed networks with transitive or weak externality properties, including all directed acyclic networks.

Findings

Convergence is guaranteed for networks with transitive weight matrices.

Networks rescalable into those with weak externalities also ensure convergence.

Directed acyclic networks are included in the class where convergence occurs.

Abstract

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.