Gradual reputation dynamics evolve and sustain cooperation in indirect reciprocity

Abstract

Humans have achieved widespread cooperation, largely sustained by mechanisms such as indirect reciprocity, which relies on reputation and social norms. People are highly motivated to maintain a good reputation, and social norms play a critical role in reputation systems by defining acceptable behavior, helping prevent exploitation by free-riders. However, there is a gap between theory and experiment in handling reputation information, with experiments often failing to capture the complexity that theoretical models propose. Here, we address two key issues: what kind of information is needed to define reputation as a social norm and the appropriate level of granularity required for reputation information to function effectively. This paper combines scenario-based experiments and evolutionary game theory to investigate the social norms individuals adopt in real-world settings, aiming to uncover the stability of these norms. Our results show that reputations should be categorized into three levels good, neutral, and bad. Results suggest gradual reputation dynamics that increase and decrease gradually due to cooperation or defection. However, a person's reputation remains unchanged only when they defect against a bad reputation. Our experimental and theoretical results support critical insights into the dynamics of reputation and social norms within indirect reciprocity, challenging traditional binary reputational evaluations. The gradual nature of reputation updating and the use of nuanced evaluations provide a more realistic model of reputation dynamics.

Fig 1. Distribution of evaluations of donor’s behavior:

Violin plots show distribution of donors’ behavior evaluations for each of 18 scenes. Red and blue correspond to defection and cooperation, respectively.

Fig 2. State transition dynamics of reputation considering third-order information and multiple reputation values:

Reputation improves by one step for cooperation and deteriorates by one step for defection. However, when defection (justified defection) occurs against bad recipient, donor’s reputation remains unchanged.

Fig 3. Replicator dynamics on action rules and cooperation rate

Each panel represents generational consequences of changing the fraction of all action rules $A=a_0,a_1,\dots ,a_7$ and the cooperation rate. The horizontal axis represents generations that update strategies. Parameters are set to $(b,c)=(3,1)$ and $(e,\mu )$ is Left: $(2\%,2\%)$, and Right: $(5\%,1\%)$. The initial population is set to $p=(1/8,1/8,\dots ,1/8)$. The dotted black line shows the cooperation ratio, and the solid colored lines show the population ratio of each action rules.

Fig 4. Action rules that forms the majority in equilibrium:

As shown in Fig 3, the equilibrium state depends on the model parameters, particularly the error rate (e) and the mutation rate (μ). The system converges either to the a₃ regime, where the cooperation rate is nearly full and a₃ dominates the population (left panel of Fig 3), or to the a₂ regime, where the cooperation rate is approximately one-third and a₂ is most frequent (right panel of Fig 3). The figure indicates which regime emerges at equilibrium across different combinations of $(e,\mu )$ values, with yellow representing the a₂ regime and purple representing the a₃ regime. The parameters are set as $(b,c)=(3,1)$.