Figure 4.

Cluster expansion under different update rules. Payoffs are given in terms of the entries of the game matrix (4). The check marks indicate the individuals whose payoffs matter for cluster expansion. For Birth-Death (BD) updating, success is determined by comparing the payoffs of individuals at cluster boundaries. For Death-Birth (DB) updating, an individual at an interface must be compared to an individual of an opposing type in a cluster interior. For shift updating, success depends on the payoffs of all individuals. If the population size is large then most individuals have payoff a or d. These comparisons lead to the success conditions shown at right, which are valid in the limit of large population size, for the exponential payoff-to-fecundity mapping π ↦ e^δπ with any selection strength δ > 0.