Evolutionary Games on Graphs and Discrete Dynamical Systems

TL;DR

This paper rigorously analyzes evolutionary games on graphs, focusing on attractors and update rules, providing new theoretical insights into cooperation dynamics across various graph structures.

Contribution

It formalizes dynamical systems concepts in evolutionary games on graphs and characterizes attractors for different update rules and graph types.

Findings

Characterization of attractors for complete graphs under synchronous and sequential updates

Sufficient conditions for full cooperation and defection attractivity on k-regular graphs

Examples showing these conditions are not necessary

Abstract

Evolutionary games on graphs play an important role in the study of evolution of cooperation in applied biology. Using rigorous mathematical concepts from a dynamical systems and graph theoretical point of view, we formalize the notions of attractor, update rules and update orders. We prove results on attractors for different utility functions and update orders. For complete graphs we characterize attractors for synchronous and sequential update rules. In other cases (for $k$-regular graphs or for different update orders) we provide sufficient conditions for attractivity of full cooperation and full defection. We construct examples to show that these conditions are not necessary. Finally, by formulating a list of open questions we emphasize the advantages of our rigorous approach.