Table 1.
Default parameter values.
Fig 1.
Asymmetric effects of voluntary participation on the long run average frequency of cooperation.
Dotted lines show the simulation results of 100 runs of the compulsory participation scenario, and solid lines the results of the voluntary participation scenarios. Asymmetric effects are found in two dimensions, i.e., nonparticipants’ payoff and group size. First, only within a certain range of nonparticipants’ payoff (between 0.8 and 1.0), voluntary participation leads relatively large groups (n = 80, 100, and 120) to achieve higher frequencies of cooperation than compulsory participation. Note that the highest frequency of cooperation is observed when nonparticipants’ payoff (Ω) is 0.95 which is the same as a contributor’s payoff in a group where the faction of cooperators is 30% (x = 0.3); and the frequency of cooperation drops sharply when nonparticipants’ payoff increases to 1.05 which is the same as a contributor’s payoff in a group where the fraction of cooperators is 50% (x = 0.5). Second, such positive effects of voluntary participation are not observed in relatively small groups (n = 20, 40, and 60) regardless of changes in nonparticipants’ payoff. The data used in this figure are provided in S1 Data.
Fig 2.
The long run average frequencies of participation (cooperation and defection) and nonparticipation.
(A) Cooperation (including contributors and punishers); (B) Defection; and (C) Nonparticipation. The dotted lines indicate the long run results of the compulsory participation scenario, and the solid lines the long run results of the voluntary participation scenarios. Three types of markers (triangle, square, and diamond) of the solid lines represent the low level of nonparticipants’ payoff (0.8), the moderate (0.925), and the high (1.05), respectively. N/A signifies compulsory participation. The data used in this figure are provided in S1 Data.
Fig 3.
Differences in the long run average frequencies of behaviors across group sizes.
Given the moderate level of nonparticipants’ payoff (0.925), each bar represents the difference calculated by subtracting the frequency of each behavior in the compulsory participation scenario from that in the voluntary participation scenario. Three different colors of bars represent group sizes. For instance, the bottom white bar shows that voluntary participation makes the long run average frequency of cooperation higher than compulsory participation, given that group size is large (n = 120) and the nonparticipants’ payoff is moderate. The data used in this figure are provided in S2 Data.