Evolutionary games on the lattice: death-birth updating process

TL;DR

This paper investigates how spatial structure influences evolutionary game dynamics, demonstrating that space can promote cooperation and lead to different outcomes compared to well-mixed populations, especially in the death-birth updating process on lattices.

Contribution

It provides a detailed comparison between spatial lattice models and non-spatial replicator equations, identifying conditions under which space alters strategic dominance and coexistence.

Findings

Large interaction range supports coexistence on the lattice.

Space can favor cooperation over defection.

Discrepancies between spatial and non-spatial models are identified.

Abstract

This paper is concerned with the death-birth updating process. This model is an example of a spatial game in which players located on the~$d$-dimensional integer lattice are characterized by one of two possible strategies and update their strategy at rate one by mimicking one of their neighbors chosen at random with a probability proportional to the neighbor's payoff. To understand the role of space in the form of local interactions, the process is compared with its non-spatial deterministic counterpart for well-mixing populations, which is described by the replicator equation. To begin with, we prove that, provided the range of the interactions is sufficiently large, both strategies coexist on the lattice for a parameter region where the replicator equation also exhibits coexistence. Then, we identify parameter regions in which there is a dominant strategy that always wins on the lattice whereas the replicator equation displays either coexistence or bistability. Finally, we show that, for the one-dimensional nearest neighbor system and in the parameter region corresponding to the prisoner's dilemma game, cooperators can win on the lattice whereas defectors always win in well-mixing populations, thus showing that space favors cooperation. In particular, several parameter regions where the spatial and non-spatial models disagree are identified.