Clarifying social norms which have robustness against reputation costs and defector invasion in indirect reciprocity

Abstract

The evolution of cooperation through indirect reciprocity is a pivotal mechanism for sustaining large-scale societies. Because third parties return cooperative behaviour in indirect reciprocity, reputations that assess and share these third parties' behaviour play an essential role. Studies on indirect reciprocity have predominantly focused on the costs associated with cooperative behaviour, overlooking the costs tied to the mechanisms underpinning reputation sharing. Here, we explore the robustness of social norms necessary to secure the stability of indirect reciprocity, considering both the costs of reputation and the resilience against perfect defectors. Firstly, our results replicate that only eight social norms, known as the 'leading eight,' can establish a cooperative regime. Secondly, we reveal the robustness of these norms against reputation costs and perfect defectors. Our analysis identifies four norms that exhibit resilience in the presence of defectors due to their neutral stance on justified defection and another four that demonstrate robustness against reputation costs through their negative evaluation of unjustified cooperation. The study underscores the need to further research how reputational information is shared within societies to promote cooperation in diverse and complex environments.

Keywords: Agent-based simulation; Evolution of cooperation; Indirect reciprocity; Social norm.

Figure 1

An outline illustration of the model: (1) A donor and recipient selected randomly from the population play the giving game. The donor chooses whether to cooperate or not in accordance with an action strategy. The behavioural strategy is defined in Table 1. (2) When the donor cooperates, he/she pays c, and the recipient receives b. When the donor does not cooperate, the payoffs of both remain unchanged. Donors other than perfect cooperators and perfect defectors pay a reputation cost $\gamma$ regardless of their behaviour. (3) The donor’s behaviour is evaluated in accordance with the social norm, and the donor’s reputation is updated. All members share a specific social norm.

Figure 2

Phase diagrams of cooperation ratios in population: x-axis represents the initial perfect defector ratio ($\eta$), and y-axis represents reputation cost ($\gamma$). In accordance with the values of $\eta$ and $\gamma$, there are two phases: one in which cooperation dominates (shown in blue), and another in which defection dominates (shown in red). The four norms in the upper row achieve cooperation robustly for $\eta$, while the four on the left side achieve cooperation robustly for $\gamma$. The social norm GBBK (L8) has the widest cooperation area.

Figure 3

A typical pattern of time evolution of cooperation and action strategies: Each panel shows a typical run of simulations with reputation costs and the initial ALLD population ($\eta =0.5$ and $\gamma =0.1$) x-axis represents the generations (log scale), and y-axis represents cooperation and population ratio. The dotted black line shows the cooperation ratio, and the solid coloured lines show the population ratio of each action strategy. Only GBBK (L8) and GRBK (L2) can maintain cooperation because they can invade into perfect defectors and never permit the invasion by perfect cooperators.

Figure 4

A typical pattern of time evolution of cooperation and action strategies in accordance with the values of $\gamma$: Each panel shows a typical run of simulations with various reputation costs ($\eta =0.0$ and $\gamma =0.1,0.3$ and 0.5, respectively) x-axis represents the generations (log scale), and y-axis represents cooperation and population ratio. In the case of panel (C), generations G is set to 10, 000; in other panels, $G=200$. The dotted black line shows the cooperation ratio, and the solid coloured lines show the population ratio of each action strategy. In all panels, the social norm is GBBK (L8).