Fig. 1: Modeling collaboration under voluntary participation.

A Individuals can either opt in to group collaboration or choose an individual option outside of the group. When opting to collaborate, players are randomly assigned to groups of five members (if the total number of players opting in is not a multiple of five, any remaining players are assigned to the individual option). Group members can choose whether to cooperate (incurring the private cost of 10 points) on the group project. If the number of cooperators is equal to or more than the predetermined threshold value,q (q = 2, 4, or 5), then all the group members earn 30 extra points. When choosing the individual option, players are guaranteed to earn a smaller additional payoff of 10 points irrespective of other players’ choices. Expected payoff (B) and best response (C) as a function of subjective belief about how likely others are to cooperate within groups (i.e., \(\gamma\)). Each color represents a possible action under mandatory or voluntary participation: orange = cooperate (C), blue = defect (D), and green = leave (L). Recall that under mandatory participation, where leaving is not available, the best response is derived by comparing between the expected payoffs of cooperation and defection.