Figure 3

Configurations of cooperators and defectors on the Frucht graph, a 3-regular graph with 12 vertices and no non-trivial symmetries; see ref. 85.

Panels (a–f) show the effects on the critical benefit-to-cost ratio of adding additional cooperators to the initial state. Panel (e) shows the global minimum of , which is achieved by just (e) and its conjugate; adding additional cooperators to the configuration in (e) only increases . The configuration of (e) is ‘optimal’ for cooperation in the sense that if selection increases the fixation probability of cooperators in some state, then it does so in state (e) as well. Relative to all possible initial states, selection can increase the fixation probability of cooperators in (e) under the smallest b/c ratio. Panels (g–i) show that when cooperators are added in a different order (starting with just a single cooperator), the critical benefit-to-cost ratio can actually be increased. Each of these three configurations has isolated cooperators, and (i) gives the global maximum of , which is achieved by just (i) and its conjugate. Since N₀ = 3, (i) is a maximal isolated configuration. The initial state in (i) is least conducive to cooperation in the sense that, relative to all other initial configurations, (i) requires the largest b/c ratio for selection to increase the fixation probability of cooperators. If selection increases this fixation probability when starting from state (i), then it does so when starting from any other mixed initial configuration.