Phase transitions in the Prisoner’s Dilemma game on scale-free networks

*BioPhysMath*, February, 1–9.

### Editorial introduction

Phase transitions in the Prisoner’s Dilemma game on scale-free networks, BioPhysMath, 2024:1, 9 pp.

In this short and to the point paper, the authors consider a population of N players changing under a stochastic dynamic process where payoffs arise from Prisoner’s dilemma games between neighbours on both Erdos-Renyi and Barabasi-Albert networks, where there is a cost γ to play the game. The payoffs thus follow the payoff matrix

CD

C 1−γ S−γ

D T−γ P−γ

which is the usual payoff matrix where the reward R is normalised to 1 and the cost γ is subtracted from all terms.

At each of a sequence of time points the individuals play games against all of their neighbours, receiving the sum of their rewards as their payoff. Then a randomly selected individual updates its strategy by comparing the strategies of itself and its neighbours, choosing the one with the highest value with probability 1−e, otherwise maintaining its current strategy.

Parameters were set at S=P=0, e=0.001. The population contained N=10,000 individuals, starting with a proportion 0.5 of cooperators; 100,000 simulation rounds were carried out, the process being repeated 50 times in each instance.

The Barabasi-Albert network is formed using preferential attachment, starting with m0 fully connected vertices. Subsequently the remaining N−m0 are added in turn, each being connected to m randomly selected vertices, with probabilities proportional to their current degree. The expected number of neighbours of a randomly selected individual is a function of N,m and m0 and denoted by α.

The authors proceeded to vary the size of the cost and see what effect this had on the evolution of cooperative behaviour in the population. They observed that there was a sharp transition when the cost hit a critical value for the Barabasi-Albert networks. For lower costs the population was comprised of almost all cooperators, whereas above the critical value there was coexistence between cooperators and defectors. For the above parameter values with T=1.5 and α=12 the critical value of γ was about 0.46, for example.

Spatial structure favours cooperation because it allows clusters of cooperators to form. The authors calculated an approximate theoretical formula for the critical value of the cost in terms of the expected number of neighbours and the variance of the number of neighbours; the greater this variability the greater the critical value, i.e. the higher the cost that can be borne before a population stops being fully cooperative. They found this in reasonably good agreement to their simulation results.

For the Erdos-Renyi network, each pair of players is connected independently with probability p, so that the expected number of neighbours for an individual is α=p(N−1). The transition described above does not occur for the Erdos-Renyi graphs, as the clustering described above does not happen to an equivalent degree, where instead a smooth decline in cooperative behaviour was observed.

The authors have found an interesting feature of the evolution of cooperation in this transition at a critical value, and provided interesting questions for further work.