Evolution of honesty in higher-order social networks

Sender-receiver games are simple models of information transmission that provide a formalism to study the evolution of honest signaling and deception between a sender and a receiver. In many practical scenarios, lies often affect groups of receivers, which inevitably entangles the payoffs of individuals to the payoffs of other agents in their group, and this makes the formalism of pairwise sender-receiver games inapt for where it might be useful the most. We therefore introduce group interactions among receivers and study how their interconnectedness in higher-order social networks affects the evolution of lying. We observe a number of counterintuitive results that are rooted in the complexity of the underlying evolutionary dynamics, which has thus far remained hidden in the realm of pairwise interactions. We find conditions for honesty to persist even when there is a temptation to lie, and we observe the prevalence of moral strategy profiles even when lies favor the receiver at a cost to the sender. We confirm the robustness of our results by further performing simulations on hypergraphs created from real-world data using the SocioPatterns database. Altogether, our results provide persuasive evidence that moral behavior may evolve on higher-order social networks, at least as long as individuals interact in groups that are small compared to the size of the network.

Figure 1

Schematic of different aspects of our work. (a) The hypergraph considered for our simulations—the hyperring. The one shown in the figure consists of 11 nodes with each hyperedge containing 4 nodes. Panels (b) and (c) denote two possible scenarios which exemplify the need to incorporate group interactions—suppose a sender sends a deceitful message to a group of receivers, which could possibly lead them to disregard the social distancing norms in the pandemic. The situation described in panel (b) depicts a scenario where none of the receivers were deceived (all marked in green), whereas panel (c) represents a scenario where a receiver who was not deceived (center, marked in green) is surrounded by receivers who were deceived. It is clear that the central receiver in panel (c) should obtain a lower payoff than the one in panel (b) as the pandemic is being facilitated (by the deceived receivers) and restrictions will be extended for everyone. This group effect is not captured by the pairwise models considered in literature.

Figure 2

Final densities of the four strategy profiles, $(T,B), (T,N), (L,B)$, and $(L,N)$, in well-mixed populations for (a) $k_R=0$ and $k_W=10$, (b) $k_R=0$ and $k_W=0.25$, (c) $k_R=10$ and $k_W=0$, and (d) $k_R=0.25$ and $k_W=0$. We show that the presence of group interactions can lead to evolution which is quite different from the case where only pairwise interactions are considered (compare with Fig. S1 in Ref. [54], and Fig. 2 of Ref. [40]).

Figure 3

Final densities of the four strategy profiles, $(T,B), (T,N), (L,B)$, and $(L,N)$, in the hyperring where each hyperedge has size 4, for (a) $k_R=0.25$ and $k_W=0$ and (b) $k_R=0$ and $k_W=0.25$. We also provide a comparison between $(k_R,k_W)=(0.25,-0.25)$ for (c) well-mixed populations and (d) hyperrings. Here we note that there is an emergence of the moral profile on the hyperring, which is not present in well-mixed populations for ($s>0, r<0$) and ($s<0, r>0$).

Figure 4

Stationary frequencies of the four strategy profiles for parameters $s=0.2, r=-0.3, k_R=0.25$, and $k_W=-0.25$ as a function of number of elements in a hyperedge. We show that the emergence of the moral profile due to the spatial correlations provided by the hypergraph structure is only significant when group sizes are small, and the effect dies out as we increase the group size—the relative density of $(T,B)$ in the figure is the highest when each group contains four elements, and monotonically decreases as the group sizes increase.

Figure 5

Persistence of honest signaling and believing behavior $(T,B)$, demonstrated on four hypergraphs generated using real-world interaction patterns, namely, primary school (a), conference (b), high school (c), and workplace (d) datasets (descriptions in text) for $k_R=0.25, k_W=-0.25, s=0.2$, and $r=-0.3$. These simulations provide evidence for our observation on hyperrings that higher-order interactions, coupled with spatial correlations, allow for the emergence of the moral profile $(T,B)$.