This study extends the weak Prisoner’s Dilemma by introducing two proactive cooperative types—conservative cooperators and aggressive cooperators—together with defectors, and analyzes the resulting evolutionary dynamics. Conservative cooperators incur an ongoing identification cost to mitigate risk, whereas aggressive cooperators invest greater resources to enhance cooperative gains. The model is examined both for infinite well-mixed populations and for spatially structured populations implemented via cellular automata. Analytical stability conditions and numerical simulations show that lowering identification cost and raising cooperative benefit can promote cooperation in well-mixed settings, and that in structured populations the relative success of strategies depends on update rules and cost parameters. The presence of differentiated cooperative strategies helps sustain group cooperation and offers theoretical insight into cooperation evolution.
Cooperative behavior is a long-standing puzzle in evolutionary biology and social science. The classic Prisoner’s Dilemma frames the tension between individual incentive to defect and collective benefit from cooperation. Building on foundational work in evolutionary game theory and mechanisms identified by Nowak (kin selection, direct and indirect reciprocity, network reciprocity, and group selection), researchers have extended the canonical two-strategy models to include heterogeneous agent types, reputation effects, and dynamic networks. Such extensions help explain how cooperation can emerge and persist despite individual incentives to defect.
This paper situates itself within that literature by proposing two novel cooperative strategies—conservative cooperative and aggressive cooperative—that differ from previously studied quasi-cooperators, quasi-defectors, and defensive cooperators. Conservative and aggressive cooperators proactively evaluate expected profitability and bear fixed, ongoing costs regardless of opponents’ immediate actions, reflecting heterogeneity in risk preferences and resource endowment. The authors argue these types are particularly relevant to organizations with ex-ante decision frameworks, such as small and medium-sized enterprises with robust operational mechanisms.
The model uses the weak Prisoner’s Dilemma payoff structure with S = P = 0 and the usual ordering T > R > P > S and 2R > T + S. Under these baseline conditions, defection is the unique Nash equilibrium absent additional mechanisms. The population is assumed infinite and well mixed for part of the analysis; replicator dynamics are applied to track strategy frequencies over time.
Three strategies are defined: conservative cooperative (IC), aggressive cooperative (MC), and defection (D). Conservative cooperators are risk-averse and pay an identification cost when interacting strategically to reduce exposure to defectors. Aggressive cooperators pay a higher cooperation investment cost intended to increase cooperative returns through more effective decision-making or greater resource input. These costs are borne continuously during interactions rather than as contingent or reactive expenditures.
The analytical work derives conditions for equilibrium points and their stability in this three-strategy replicator system. The principal qualitative results reported are that reducing the identification cost and increasing cooperative benefit can, to some extent, promote the prevalence of cooperative strategies in the well-mixed setting, although the baseline payoff ordering of the weak Prisoner’s Dilemma frames defection as the default equilibrium absent these influences.
To capture spatial effects and local interactions, the authors implement a cellular-automaton-based structured-population model. Individuals occupy sites and interact with neighbors according to specified neighborhood structures. Strategy update rules include, among others, a random neighbor unconditional imitation rule, which the authors highlight as especially informative in their results.
In structured populations, local clustering and imitation dynamics can alter the relative success of strategies compared with the well-mixed case. The analysis tracks how varying identification cost and aggressive cooperation cost change the stage-dependent relative advantages among IC, MC, and D.
The paper reports derived stability conditions for equilibrium points in the replicator dynamics and then uses numerical simulations to illustrate the evolutionary trajectories under different parameter settings in both population types. Key findings include:
In infinite well-mixed populations, cooperation can be promoted by lowering identification cost and increasing cooperative benefit, consistent with the idea that lower proactive costs and higher returns make cooperation more favorable.
In structured populations, the success of conservative versus aggressive cooperation depends sensitively on the cost parameters and the imitation/update rule. Under the random neighbor unconditional imitation rule, the relative payoffs of the three strategies increase or decrease in a stage-dependent manner as identification and aggressive cooperation costs vary.
Overall, introducing differentiated cooperative strategies—proactive, cost-bearing cooperators—provides additional mechanisms that can sustain cooperation across contexts where two-strategy models predict defection.
The article presents multiple figures and a table supporting the simulations and stability analyses; all relevant data are reported as available in the manuscript files.
The conservative and aggressive cooperative strategies are framed as reflections of heterogeneity in cognitive ability, personality, risk preference, and capability endowments. The conservative cooperator is likened to a cautious actor who accepts ongoing identification costs to reduce exploitation risk; the aggressive cooperator resembles a proactive investor in cooperation who increases resource input to boost collective outcomes. The authors contrast these proactive types with defensive cooperators (reactive defenses) and quasi-cooperator/defector types (weakened variants), emphasizing that their proposed strategies are suited to settings with ex-ante decision-making and stronger institutional frameworks.
The study highlights implications for interpreting cooperation dynamics in biological, economic, and organizational contexts, and situates its contributions alongside classic cooperation mechanisms such as reputation and network reciprocity.
By incorporating conservative and aggressive cooperative strategies into a weak Prisoner’s Dilemma framework, analyzed in both well-mixed and spatially structured populations, the study shows that proactive, cost-bearing heterogeneity among cooperators can help sustain cooperation. The findings indicate that parameter choices—particularly identification cost and cooperation investment cost—shape the evolutionary outcomes, and that structured interactions under specific update rules can produce stage-dependent shifts in strategy advantage. These results offer new theoretical perspectives and model support for understanding cooperation evolution in realistic social and organizational settings.