Deterministic and stochastic cooperation transitions in evolutionary games on networks

TL;DR

This paper explores how network structure and decision-making processes influence cooperation transitions in evolutionary games, revealing conditions for abrupt or gradual changes and identifying optimal social temperatures for cooperation levels.

Contribution

It develops a theoretical framework combining master equations and simulations to analyze cooperation transitions, including the effects of deterministic and stochastic decision-making.

Findings

Discontinuous cooperation changes occur at zero temperature with deterministic decisions.

Continuous and discontinuous phase transitions are identified at higher temperatures.

Optimal social temperatures can maximize or minimize cooperation density.

Abstract

Although the cooperative dynamics emerging from a network of interacting players has been exhaustively investigated, it is not yet fully understood when and how network reciprocity drives cooperation transitions. In this work, we investigate the critical behavior of evolutionary social dilemmas on structured populations by using the framework of master equations and Monte Carlo simulations. The developed theory describes the existence of absorbing, quasi-absorbing, and mixed strategy states and the transition nature, continuous or discontinuous, between the states as the parameters of the system change. In particular, when the decision-making process is deterministic, in the limit of zero effective temperature of the Fermi function, we find that the copying probabilities are discontinuous functions of the system's parameters and of the network degrees sequence. This may induce abrupt changes in the final state for any system size, in excellent agreement with the Monte Carlo simulation results. Our analysis also reveals the existence of continuous and discontinuous phase transitions for large systems as the temperature increases, which is explained in the mean-field approximation. Interestingly, for some game parameters, we find optimal "social temperatures" maximizing/minimizing the cooperation frequency/density.