Evolution of cooperation under a generalized death-birth process

TL;DR

This paper investigates how introducing a focal weight factor in a generalized death-birth process affects cooperation evolution, revealing a critical tipping point where cooperation becomes impossible, supported by theoretical analysis and simulations.

Contribution

It generalizes the death-birth process by incorporating a focal weight factor and analyzes its impact on cooperation using evolutionary graph theory.

Findings

Focal weight always hinders cooperation under weak selection.

A non-trivial tipping point exists where cooperation success shifts dramatically.

Results are robust across various graph structures and game settings.

Abstract

According to the evolutionary death-birth protocol, a player is chosen randomly to die and neighbors compete for the available position proportional to their fitness. Hence, the status of the focal player is completely ignored and has no impact on the strategy update. In this work, we revisit and generalize this rule by introducing a weight factor to compare the payoff values of the focal and invading neighbors. By means of evolutionary graph theory, we analyze the model on joint transitive graphs to explore the possible consequences of the presence of a weight factor. We find that focal weight always hinders cooperation under weak selection strength. Surprisingly, the results show a non-trivial tipping point of the weight factor where the threshold of cooperation success shifts from positive to negative infinity. Once focal weight exceeds this tipping point, cooperation becomes unreachable. Our theoretical predictions are confirmed by Monte Carlo simulations on a square lattice of different sizes. We also verify the robustness of the conclusions to arbitrary two-player prisoner's dilemmas, to dispersal graphs with arbitrary edge weights, and to interaction and dispersal graphs overlapping arbitrarily.