Catalyst strategy for the evolution of cooperation in indirect reciprocity

Indirect reciprocity, as a primary mechanism for cooperation between unrelated individuals, evaluates individuals' behavior and assigns reputation labels based on social norms. Since evaluating reputation is challenging in practice, unlike previous studies, we do not introduce the reputation evaluation rule but only record two recent action information as individuals' labels, including the most recent actions of the individual and his partner. A new type of strategy set is constructed, in which each strategy is represented by a quadruplet, with each element corresponding to an action when facing a different label partner. We explore the invasion and competition of these strategies in a population when playing donor games in both noiseless and noisy scenarios. By conducting pairwise invasion simulations between all sixteen strategies, we find that in the noiseless case, cooperative strategies with evolutionary stability can identify reasonable cooperative behaviors, tolerate unreasonable cooperative behaviors, and adopt a harsh attitude toward unreasonable defection. In the noisy case, tolerant cooperative strategies that can distinguish unreasonable defection are evolutionary stable. Furthermore, by simulating multiple strategy combinations, we find a key strategy which we call the catalyst strategy that can act as a refuge for cooperators. Although the strategy is not evolutionary stable, it can help conditional cooperative strategies resist the intrusion of defection and help the system reach a fully cooperative steady state. We also verify that the system's evolutionary outcome is robust to noise intensity and that the presence of noise cannot lead to qualitative changes in the system. Action errors have a greater impact on the average cooperation rate and average payoff of the population relative to label errors. In addition, we find that the catalytic potential of the key strategy is fully realized in larger populations. Conversely, smaller population sizes diminish the efficacy of the catalyst strategy in promoting cooperation, and this detrimental effect is exacerbated in the noisy scenario.