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

Public goods games (PGG) exemplify social dilemmas: in single-round m-player PGG with enhancement factor r, defecting is the only Nash equilibrium when 1 < r < m, although groups earn the highest payoff when all contribute their endowment E. By backward induction, the same holds for repeated PGG with a fixed, common number of rounds. Allowing exogenous disbanding with probability p (so expected rounds 1/p) can introduce other equilibria, especially with punishment or non-participation. This study investigates a different mechanism: voluntary opting out based on group composition. Individuals are fixed types (Cooperator C or Defector D) for the entire game; their opting-out decision depends only on the number of cooperators in the current group. If any member opts out, the group disbands; otherwise it continues and disbands stochastically with probability p per round (expected τ=1/p rounds). The research questions are whether opting out promotes cooperation in m-player repeated PGG, which opting-out rules do so, and why such rules would be adopted.

Prior work shows conditional dissociation promotes cooperation in repeated two-player dilemmas, including the Prisoner’s Dilemma (PD) and two-player PGG, primarily via positive assortment when players opt out against defectors (e.g., Aktipis 2004, 2011; Izquierdo et al. 2010; Zhang et al. 2016; Deer & Bayer 2016). However, cooperative tendencies in PD may not carry over to PGG (Mullett et al. 2020). Multi-strategy approaches have examined evolutionary stability of opting-out rules where later-round choices depend on earlier actions (Fujiwara-Greve & Okuno-Fujiwara 2009; Kurokawa 2019). Experiments often find greater opting out against defectors, but homophily and intolerance toward out-groups are also well-documented behavioral drivers. This study builds on analytical results for two-player games with interaction times and extends to multi-player PGG using a simplified two-strategy framework with composition-based opting-out rules.

• Game structure: Repeated m-player PGG with two fixed strategies (C or D). In each round, cooperators contribute E; defectors contribute 0. Total contributions are multiplied by r (>1) and shared equally. Single-round payoffs: in a group with k cooperators (k=0,…,m), per-round payoffs are π_C(k) = (k r E)/m − E and π_D(k) = E + (k r E)/m.
• Opting-out rules: For each k, groups either disband immediately after one round (τ_k=1) or, if no one opts out, continue for an expected τ rounds (τ_k=τ=1/p). Cooperators share a common rule; defectors share a common rule. This induces a vector (τ_0,…,τ_m) with entries in {1, τ} for most analyses; some comparisons consider general 1 ≤ τ_k ≤ τ.
• Group formation dynamics: A large population (size N, multiple of m) forms N/m groups. After each round, groups with expected duration τ_k disband at rate 1/τ_k. Singles randomly reform groups; the distribution of new groups follows a binomial with respect to current frequencies. This yields discrete-time dynamics for the number of groups with k cooperators, n_k, leading to an equilibrium group distribution n_k* that depends on (τ_0,…,τ_m) and the frequency of cooperators.
• Fitness: Defined as expected payoff per round at the equilibrium group distribution. For cooperators, Π_C = (1/N_C) Σ_k k n_k* π_C(k); for defectors, Π_D = (1/N_D) Σ_k (m−k) n_k* π_D(k). These fitnesses define a symmetric two-strategy evolutionary game.
• Equilibria and stability: Nash equilibria satisfy Π_C = Π_D (interior NE) or boundary conditions (all C or all D). Stability is assessed via standard evolutionary dynamics (e.g., replicator): stable NE have Π_C > Π_D just below and Π_C < Π_D just above the equilibrium cooperator frequency.
• Analytical results (m=2): Closed-form equilibrium group distributions and NE conditions are derived. Key assortment condition depends on τ_1^2 relative to τ_0 τ_2; positive assortment arises when τ_1^2 < τ_0 τ_2.
• Numerical methods (m>2): For m≥3, interior NE and equilibrium group distributions are found numerically (Mathematica) by solving Π_C = Π_D together with the equilibrium conditions for n_k* across N_C = 0,…,N. Extensive searches over all binary opting-out vectors (τ_k ∈ {1, τ}) identify best rules under specified τ and m. Stability is determined by sign changes of Π_C − Π_D around candidate equilibria.
• Symmetry theorem: For any (τ_0,…,τ_m), replacing τ_k by τ_{m−k} while swapping C and D yields mirrored interior NE at 1−N_C with opposite stability; Π_C and Π_D transform affinely, establishing symmetry between ‘threshold on cooperators’ and ‘threshold on defectors’ rules.

• Core result: The opting-out rule that best promotes cooperation is to keep only homogeneous groups together (all C or all D) for many rounds while all heterogeneous groups disband after one round. Behaviorally, this arises if defectors are completely intolerant of cooperators (defectors opt out whenever any cooperator is present), causing all mixed groups to dissolve immediately.
• Two-player case (m=2):
  • Opting out against defectors (groups disband if any D) yields positive assortment and a stable coexistence of C and D when τ is sufficiently large; for r<2, an interior stable NE exists when τ > (r/(r−1))^2.
  • The opt-out-if-heterogeneous rule (disband unless the pair is CC or DD) dominates all other rules by multiple criteria: lowest r threshold for coexistence at fixed τ, lowest τ threshold at fixed r, largest domain of attraction, and higher cooperation level at the stable NE among rules that admit coexistence.
• Multi-player cases (m>2):
  • Analytical solutions are not available; numerical analyses (notably for m=4) reveal that threshold rules based on single-round payoff (e.g., disband unless at least k* cooperators) can produce interior coexistence NE for both rm, but no single payoff-based threshold dominates across criteria.
  • Rules based on homogeneity (disband if heterogeneous) are superior when continuing groups persist sufficiently long (large τ). Symmetric rules based on thresholds in the number of defectors yield mirrored NE due to the symmetry theorem.
• Best rules under τ=5 (maximum expected rounds):
  • Identified via exhaustive search over τ_k ∈ {1,5} for m=2,…,10. Minimum enhancement factor r at which a stable coexistence NE appears: • m=2: best rule ‘515’ achieves r≈1.2; another strong rule ‘5115’ has r≈1.5. • m=3: ‘51115’, r≈2. • m=4: ‘511155’ and its symmetric ‘551115’, r≈2.43. • m=5: ‘5111555’ and ‘5551115’, r≈2.88. • m=6: ‘51115555’ and ‘55551115’, r≈3.34. • m=7: ‘511155555’ and ‘555551115’, r≈3.80. • m=8: ‘5111555555’ and ‘5555551115’, r≈4.26. • m=9: ‘51115555555’ and ‘55555551115’, r≈4.72. • m=10: ‘555555551115’ (best rule listed; minimum r implied by search).
  • Interpretation of codes: sequences indicate τ_k pattern with τ=5 and 1’s, where homogeneous groups (all C or all D) have τ_k=5 and heterogeneous groups mostly have τ_k=1; for m≥5, a further threshold within heterogeneous compositions refines which mixed groups disband immediately.
• Dependence on τ (number of rounds staying together):
  • For m up to 6, there exists a minimum τ such that, for all larger τ, the best rule is to disband if and only if the group is heterogeneous. The minimum τ (as expected rounds) grows with m (e.g., m=2,3: τ≥2 suffices; m=4: τ≥5; m=5: τ≥17; m=6: τ≥50).
• Boundary NE: As in single-round PGG, all D is strict NE when r<m; all C is strict NE when r>m. With the simplified two-strategy setup (action not contingent on history), all C cannot be achieved for r≤m, underscoring the role of opting-out assortment rather than contingent strategies in promoting cooperation.

The findings demonstrate that assortment induced by homogeneity-driven opting out—especially defectors’ intolerance of cooperators—can transform the evolutionary outcome in repeated PGG. By forcing mixed groups to disband immediately, the population distribution shifts toward homogeneous groups, increasing the relative success of cooperators at coexistence equilibria. This mechanism differs from payoff-based opting out (more likely to leave when many defectors) that, while intuitive, is not optimal for fostering cooperation. The symmetry between cooperator- and defector-focused thresholds clarifies how mirrored rules yield mirrored equilibria with opposite stability. Behaviorally, the optimal rule aligns with homophily or intolerance toward unlike-minded individuals, which is empirically observed in social contexts. The model predicts that such intolerance by defectors can paradoxically increase overall cooperation by preventing persistent exploitation within mixed groups. The results also clarify why classic repeated PGG with history-dependent strategies can reach higher cooperation: here, with only two fixed strategies, cooperation emerges via structural assortment from group persistence rules. The work suggests experimental tests: allowing participants’ continued interaction to depend solely on group homogeneity should elevate cooperation relative to payoff-based opt-out criteria, particularly when non-disbanding groups persist for many rounds.

This study introduces and analyzes a simplified repeated m-player PGG with composition-based opting-out rules and shows that cooperation is fostered most when only homogeneous groups persist between rounds. The mechanism is strongest when defectors are intolerant of cooperators, causing all mixed groups to disband after one round, which maximizes positive assortment and yields stable coexistence with elevated cooperation. Analytical results for m=2 and extensive numerical results for m≥3 identify the best rules and thresholds across group sizes and persistence levels. Future work should: (1) test these predictions experimentally, especially the homogeneity-based rule; (2) extend models to allow richer strategy sets with history-dependent actions and heterogeneous opting-out rules; (3) explore finite-population and stochastic effects; (4) examine institutional or platform designs that implement homogeneity-based continuation to promote cooperation in real-world group interactions.

• Strategy simplification: Individuals are fixed as C or D for the entire game; no contingent or history-dependent actions are allowed, limiting potential for full cooperation when r≤m.
• Homogeneous rules: All cooperators share one opt-out rule and all defectors another; individual-level heterogeneity is not modeled.
• Analytical tractability: Closed-form results are only available for m=2; for m>2, equilibria are determined numerically and may depend on parameter grids and computational precision.
• Outcome scope: Focus is on Nash equilibria and their evolutionary stability; dynamic transients and finite-population stochastic effects are not fully analyzed.
• External validity: Behavioral assumptions (e.g., intolerance) are motivated by empirical observations but not tested here; experimental validation is needed.
• Model constraints: Only two τ levels (1 or τ) are mostly considered in searches; broader τ_k variation is only touched upon conceptually.