Defectors’ intolerance of others promotes cooperation in the repeated public goods game with opting out

The research question centers on how opting-out rules influence cooperation levels in repeated multiplayer public goods games (PGGs). The classic single-round PGG, where individuals can choose to cooperate (contribute) or defect (not contribute), always results in a Nash Equilibrium (NE) of all players defecting, even though total group payoff is maximized when all cooperate. This presents a social dilemma. Repeated PGGs, with a fixed number of rounds, also lead to this all-defect outcome via backward induction. However, introducing probabilities of group disbandment between rounds or mechanisms for punishment or non-participation can lead to different NE, including cooperation. This paper investigates another method for promoting cooperation: allowing players to opt out of their group based on the contributions of its members. The authors simplify the strategy space by assuming each individual consistently cooperates or defects throughout the game, without allowing their actions to depend on past rounds' actions within their group. The decision to opt out is then solely a function of the number of cooperators in the group. The study's importance lies in exploring alternative mechanisms to resolve the social dilemma of the PGG, focusing on the role of individual preferences and intolerance in influencing group dynamics and cooperative behavior.

Existing research on opting out (or conditional dissociation) in social dilemmas primarily focuses on its effect on cooperation when used against defectors. This is intuitive because individuals are worse off the more opponents defect. However, this paper argues that other opting-out rules can be more effective. Previous studies have explored opting out in two-player repeated Prisoner's Dilemma (PD) games and two-player repeated PGGs. These studies demonstrate that opting out against defectors can promote cooperation by creating positive assortment of cooperators. However, the effectiveness of this approach in multi-player PGGs and the underlying reasons for its success are less clear. Other theoretical works have investigated the evolutionary stability of various opting-out rules by incorporating multiple rules into a single multi-strategy game, but this paper uses a simplified two-strategy framework for a clearer analysis.

The authors analyze a simplified repeated m-player PGG with opting out. Each individual is either a consistent cooperator (C) or defector (D) for the entire game. The opting-out rule for each individual depends on the number of cooperators in their group. If any member opts out, the group disbands; otherwise, it disbands with probability p (0<p<1). Individuals from disbanded groups are randomly reassigned to new groups. The paper simplifies the strategy space further by assuming all cooperators share the same opting out rule, as do all defectors. The payoff structure is based on the single-round PGG, where payoffs to a cooperator and defector are given by equations dependent on the number of cooperators (k) in the group, endowment (E), and enhancement factor (r). For a given opting-out rule, the fitness of a cooperator or defector is their expected payoff per round, calculated at the equilibrium group distribution. This fitness depends on m and r through the group distribution determined by the opting out rule and is a function of the number of cooperators (NC) and defectors (ND). The authors solve for Nash Equilibria (NE) in this evolutionary game, particularly focusing on stable NE using numerical methods for m>2. For m=2, they use analytical methods to determine NE. Two extreme opting out rules are analyzed: always opting out (equivalent to the single-round PGG) and never opting out (equivalent to the classic repeated PGG). The authors also analyze different opting out rules for m=2 and m=4, using numerical methods to analyze the NE for m=4 due to the computational complexity.

The study's central finding is that defectors' intolerance of cooperators significantly promotes cooperation. In the two-player game (m=2), analytical solutions reveal that an opting-out rule where players opt out if and only if their opponent does not share their strategy leads to the highest level of cooperation. This rule, and equivalent rules, prioritize homogeneity in groups. For multiplayer games (m>2), the analysis is numerical. The study finds that when homogeneous groups are expected to remain together for sufficiently many rounds, the best opting-out rule is one where heterogeneous groups disband immediately. This is observed across group sizes (m=2 to 10). For m=4, various threshold-based opting-out rules (where groups disband if the number of cooperators is below a threshold) are examined, and none consistently outperform all others in promoting cooperation. The numerical analysis reveals an intriguing symmetry between opting-out rules based on cooperator thresholds versus defector thresholds. Specifically, Theorem 1 illustrates a symmetry in the equilibrium group distribution and interior NE between these symmetric rules. Table 1 summarizes the best opting-out rules for different group sizes when the maximum number of rounds is 5, showing that opting out only when groups are heterogeneous consistently leads to the lowest enhancement factor (r) required for a stable coexistence NE. Table 2 shows the minimum number of rounds needed for the “opt-out-if-heterogeneous” rule to be optimal for different group sizes. The authors’ results strongly indicate that the focus of previous studies on opting out against defectors should be broadened to encompass other attributes of individual behavior, specifically the preference for like-minded individuals.

The findings address the research question by demonstrating that an intolerance of unlike individuals, rather than simply opting out against defectors, is a more effective way to promote cooperation in repeated PGGs. This is especially true when groups of like-minded individuals remain together for an extended period. The significance lies in highlighting the importance of individual preferences beyond simple payoff considerations in shaping cooperation. The “opt-out-if-heterogeneous” rule, generated by defectors' intolerance toward cooperators, outperforms rules based solely on single-round payoffs. This challenges the conventional wisdom that focusing solely on responses to defection is the optimal approach. The results suggest the importance of considering individual preferences for homogeneity in the design and analysis of mechanisms for promoting cooperation.

This study shows that defectors' intolerance of cooperators, leading to the immediate disbandment of heterogeneous groups, is the most effective way to promote cooperation in repeated public goods games with opting out, especially when homogeneous groups persist for many rounds. This challenges the prevalent focus on opting out against defectors and opens up new avenues of research into the role of individual preferences in collective action.

The study simplifies the strategy space, limiting individuals to consistent cooperation or defection throughout the game. It also assumes all cooperators and all defectors adopt identical opting-out rules. Future research could explore more complex strategy sets and opting-out rules to account for heterogeneity in behavior. The numerical analysis for multiplayer games, due to computational limitations, might not capture all possible opting-out rules or their dynamic properties.