Fixation Probabilities for Any Configuration of Two Strategies on Regular Graphs

Abstract

Population structure and spatial heterogeneity are integral components of evolutionary dynamics, in general, and of evolution of cooperation, in particular. Structure can promote the emergence of cooperation in some populations and suppress it in others. Here, we provide results for weak selection to favor cooperation on regular graphs for any configuration, meaning any arrangement of cooperators and defectors. Our results extend previous work on fixation probabilities of rare mutants. We find that for any configuration cooperation is never favored for birth-death (BD) updating. In contrast, for death-birth (DB) updating, we derive a simple, computationally tractable formula for weak selection to favor cooperation when starting from any configuration containing any number of cooperators. This formula elucidates two important features: (i) the takeover of cooperation can be enhanced by the strategic placement of cooperators and (ii) adding more cooperators to a configuration can sometimes suppress the evolution of cooperation. These findings give a formal account for how selection acts on all transient states that appear in evolutionary trajectories. They also inform the strategic design of initial states in social networks to maximally promote cooperation. We also derive general results that characterize the interaction of any two strategies, not only cooperation and defection.

Evolution of Cooperation for Multiple Mutant Configurations on All Regular Graphs with N ≤ 14 Players

“…Also, for a constant number of coplayers, σ max increases with N , which is the number of players. The increase, however, gets gradually smaller and converges for N → ∞ to a constant, which is σ(π, G) → σ = (k + 1)/(k − 1) [6,21]. For instance, for k = 3, the structure coefficients converge to σ(π, G) → σ = 2.…”

“…Players may update their strategies in an updating process, for instance death-birth (DB) or birth-death (BD) updating [1,24]. Recently, it was shown by Chen et al [6] that strategy π i = 1 = C is favored over…”

“…Compared with these findings, the problem of multiple cooperators (or more than one mutant) is far less studied. One approach uses configurations and structure coefficients [6] and has shown that cooperation is favored over defection under conditions which can be linked to spectral graphs measures and cooperator path length [27,28].…”

“…Cooperators cluster on these blocks and serve as a mutant family that may invade the remaining graph. The study presented here uses structure coefficients, which have been derived for birth-death and death-birth processes [6]. However, as the structure coefficients solely depend on the distribution of cooperators and defectors on the evolutionary graph, they could be, at least in principle, also calculated for other strategy updating processes as long as these processes are not completely random.…”

Relationships Between Dilemma Strength and Fixation Properties in Coevolutionary Games

“…By converting the payoff into accumulable fitness, repeating the interaction and allowing players to change strategies depending on the accumulated fitness, the long-term effect of strategy selection becomes visible [2,4,6,17]. A frequently studied question of considerable biological relevance is whether or not one strategy is favored over another depending on the values of the payoff matrix and on the structure of the interaction network specifying who-plays-whom [5,6,11,12]. Recently, and independent from each other, two proposals have been made to formalize strategy selection and payoff allocation, on the one hand, and strategy distribution over interaction networks, on the other hand.…”