title: 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.
authors.affiliation: Program for Evolutionary Dynamics, Harvard University, Cambridge, MA 02138, USA.|Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, USA.|Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA.
authors.forename: Yu-Ting
authors.initials: YT
authors.lastname: Chen
authors.affiliation: Program for Evolutionary Dynamics, Harvard University, Cambridge, MA 02138, USA.|Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, Canada V6T 1Z2.
authors.forename: Alex
authors.initials: A
authors.lastname: McAvoy
authors.affiliation: Program for Evolutionary Dynamics, Harvard University, Cambridge, MA 02138, USA.|Department of Mathematics, Harvard University, Cambridge, MA 02138, USA.|Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, USA.
authors.forename: Martin A
authors.initials: MA
authors.lastname: Nowak