Indirect reciprocity under opinion synchronization

Abstract

Indirect reciprocity is a key explanation for the exceptional magnitude of cooperation among humans. This literature suggests that a large proportion of human cooperation is driven by social norms and individuals' incentives to maintain a good reputation. This intuition has been formalized with two types of models. In public assessment models, all community members are assumed to agree on each others' reputations; in private assessment models, people may have disagreements. Both types of models aim to understand the interplay of social norms and cooperation. Yet their results can be vastly different. Public assessment models argue that cooperation can evolve easily and that the most effective norms tend to be stern. Private assessment models often find cooperation to be unstable, and successful norms show some leniency. Here, we propose a model that can organize these differing results within a single framework. We show that the stability of cooperation depends on a single quantity: the extent to which individual opinions turn out to be correlated. This correlation is determined by a group's norms and the structure of social interactions. In particular, we prove that no cooperative norm is evolutionarily stable when individual opinions are statistically independent. These results have important implications for our understanding of cooperation, conformity, and polarization.

Keywords: conformity; cooperation; evolutionary game theory; indirect reciprocity; social norms.

Fig. 1.

Schematic illustration of the models considered in this paper. We consider a population of $N$ players (here, $N=6$). At each time step, randomly chosen donor-recipient pairs play the donation game. In each game, the donor decides whether to cooperate ($C$) or to defect ($D$) according to its action rule $P$. Other players may observe the donor’s decision and update its reputation according to their assessment rule $R$. The resulting reputations can be represented by an image matrix. It records how each player assesses every other at a given time. Here, we represent the image matrix by a square with six rows and six columns. Black entries indicate that the respective row player thinks negatively of the column player. White entries indicate positive opinions. In the following, we revisit four classical models that differ in how these image matrices are updated. On the Left, there is the solitary observation model. Here, each interaction is observed by a single player. As a result, the rows of the image matrix turn out to be independent. In the Top-Middle, there is the simultaneous observation model, where each action is observed by any given player with probability $q$. As the observation probability $q$ approaches $1/N$, the model simplifies to the solitary observation model. In the Bottom-Middle, there is the gossiping model by Kawakatsu et al. (39). In this model, players share their opinions with other population members. When the gossip duration $\tau$ approaches zero, the model becomes equivalent to the solitary observation model. When $\tau \to \infty$, the model approaches the public assessment model depicted on the Right. In that model, opinions are perfectly synchronized.

Fig. 2.

Relationship of the cooperation levels between residents and mutants. As resident norms, we consider Simple Standing, Stern Judging, Image Scoring, and Shunning, as defined in Table 1. Dashed lines are theoretical predictions obtained from Eq. 5. Points are obtained from numerical simulations for $N=100$ and ${\mu }_a=0.02$ (see Materials and Methods, Numerical Simulations for details). Circles indicate results for unconditionally cooperating mutants with cooperation probability $p_{\mathrm{mut}\to \mathrm{res}}\in \{0,0.2,0.4,0.6,0.8,1.0\}$, respectively. Triangles are the results for deterministic second-order mutants. We observe that regardless of the mutant strategy, a resident’s average cooperation rate toward the mutant, $p_{\mathrm{res}\to \mathrm{mut}}$, is a linear function of the mutant’s cooperation rate $p_{\mathrm{mut}\to \mathrm{res}}$.

Fig. 3.

(A and B) The relationship between $h$ and $h_G$ for the norms L6 (Stern Judging) and L3 (Simple Standing). The gray dashed lines indicate the case $h=h_G$, which is shown as a reference. For both norms, $h$ increases as $h_G$ increases. The leftmost $h_G$ is the minimal value obtained for the solitary observation model; here, $h_G=h$. The rightmost $h_G$ corresponds to the public assessment model $h_G=1$. Results for the simultaneous observation model with $N=100$ and $q=1$ are shown as red triangles. Purple crosses indicate the gossiping model with $\tau =0.1$, $0.3$, $1$, and $3$, from left to right. (C and D) The blue area indicates the benefit-to-cost ratios $b/c$ for which a conditionally cooperative action rule is stable. As $h_G$ increases, the stable range of $b/c$ expands. We use an assessment error rate of ${\mu }_a=0.02$.