Page 13 of 21 by which receivers integrate received information impacts cooperation. In addition, our peer-to-peer model assumes that players can exchange gossip with anyone (i.e., a complete gossip network), and as a result the population approaches full consensus as gossip duration increases. But under what conditions (e.g., modular or dynamic gossip networks) would the population split into different camps that view a focal individual differently, and how would polarization in reputations impact cooperation? How does the number of sub-groups in a populationas well as their relative sizes and internal structuresaffect the rate of convergence to consensus and stability of cooperation, and how do differential rates of gossip within and between sub-groups modulate these effects? These questions remain open for future research.

One limitation of gossip is that reputations are not always transmitted faithfully (Seki and Nakamaru, 2016; Wu et al., 2021; Dores Cruz et al., 2021). Noise during transmission can either be unbiased (e.g., accidental errors) or biased (e.g., intentional misrepresentation). We have shown that unbiased noise tends to undercut the benefits of gossip. This result is perhaps unsurprising because, much like errors in assessment or execution (Hilbe et al., 2018), transmission noise impedes agreement in the population. It is notable, however, that biased gossip can sometimes stabilize cooperation relative to unbiased gossip or relative even to noiseless gossip. This is true for all norms studied when gossip is biased toward positive reports. In addition, under the Stern Judging norm, a strong negative bias can also stabilize cooperation, compared to noiseless gossip. While this is potentially good newscooperation does not necessarily unravel even when gossip is biasedour analysis has been li

An important extension for future research is to study heterogeneity in how bias is applied. For example, in a population with group structure, individuals may have different levels of bias when gossiping about in-group versus out-group members (e.g., in-group > 0 and out-group < 0).

We have assumed that the propensity to gossip is both uniform across the population and exogenously fixed. But competition between gossip strategies may complicate the picture: for example, previous work has found that dishonest gossip strategieswhere gossipers deterministically transmit false information (i.e., pure bias, akin to u = 1 or v = 1)can outperform honest gossip strategies under certain conditions, although dishonest gossip tends to undermine cooperation (Wu et al., 2016a; Nakamaru and Kawata, 2004; Seki and Nakamaru, 2016). A natural question, then, is whether the amount and quality of gossip that stabilizes cooperation would naturally evolve if individuals were allowed to adjust the frequency and fidelity with which they transmit information. A recent study on the role of empathy in indirect reciprocity has found that populations can evolve empathetic evaluation under the right conditions (Radzvilavicius et al., 2019)which implies that single-source gossip (Q > 0) can als