RUNNING HEAD: NETWORK DYNAMICS OF THE WISDOM OF CROWDS

A longstanding problem in the social, biological, and computational sciences is to determine how groups of distributed individuals can form accurate judgments. Since Galton’s discovery of the “wisdom of crowds” [Galton F (1907) Nature 75:7] theories of collective intelligence have suggested that in order for a group to be accurate, individuals within the group must be either independent, with uncorrelated beliefs, or diverse, with negatively correlated beliefs [Page, S (2007) The Difference]. Previous experimental studies argued that social influence undermines the wisdom of crowds, showing that individual estimates became more similar when subjects observed each other’s beliefs, reducing diversity without a corresponding increase in group accuracy [Lorenz J et al (2010) Proc Natl Acad Sci 108:22]. In contrast, we find general network conditions under which social influence can improve group estimates, even as individual opinions become more similar. We present theoretical predictions and large scale experimental results showing that in decentralized communication networks, group estimates become more accurate as a result of exposure to social information. We also present results showing that in centralized networks, the influence of central individuals dominates the collective estimation process, and group estimates are as likely to increase in error as they are to become more accurate. NETWORK DYNAMICS OF THE WISDOM OF CROWDS 3 Significance Statement Since the discovery of the wisdom of crowds over 100 years ago, theories of collective intelligence have held that group accuracy requires either statistical independence or informational diversity among individual beliefs. Empirical evidence suggests that allowing people to observe the beliefs of others leads to increased similarity of individual estimates, reducing independence and diversity without a corresponding increase in group accuracy. As a result, social influence is expected to undermine the wisdom of crowds. We present theoretical predictions and experimental findings demonstrating that in decentralized networks, social influence generates learning dynamics that reliably improve the wisdom of crowds. We identify general conditions under which influence, not independence, produces the most accurate group judgments. Main Text Since Galton’s discovery of the “wisdom of crowds” over 100 years ago (1), results on crowdsourcing (2,3), prediction markets (4), and financial forecasting (5,6) have shown that the aggregated judgment of many individuals can be more accurate than the judgments of individual experts (1,7-9). Statistical explanations for this phenomenon argue that group accuracy relies on estimates taken from groups where individuals’ errors are either uncorrelated or negatively correlated, thereby preserving the diversity of opinions in a population (10). Thus, while individuals may have estimates both far above and far below the true value, in aggregate these errors cancel out, leaving an accurate group judgment (2,10,11). NETWORK DYNAMICS OF THE WISDOM OF CROWDS 4 Recent experimental evidence has suggested that the wisdom of crowds may be undermined by processes of social influence, in which people exchange information about their estimates and revise their judgments to align with one another (12-14). When social influence leads to correlated errors, both independence and diversity are reduced, which has been argued to compromise the reliability of the group judgment (10,12-19). In direct contrast with these results, however, theoretical models of social learning (20-22) have suggested that the effects of social influence on collective decisions vary based on the structure of the interaction network, predicting that under the right conditions, social learning can lead a group’s median judgment to improve (21-25). This prediction derives from the assumption that when people learn about the beliefs of others, they revise their own beliefs to become more similar to their social referents (12,13,26,27). Following the DeGroot model of social learning, this theory suggests that each individual’s revisions are based on a weighted average of their own belief and the beliefs of their social referents (20). Thus, an individual’s revision is determined in part by the amount of weight they place on their own belief relative to social information. When this “selfweight” is independently and identically distributed throughout a population, and the population is embedded in a decentralized social network (i.e., one in which everyone is equally connected), this model predicts that belief distributions will converge on the statistical mean of the initial, independent beliefs (SI Appendix). Thus, if the initial group mean is accurate, exposure to social influence will lead individuals in the group to become more accurate, improving the accuracy of the group’s median even as the group mean remains fixed (SI Appendix). NETWORK DYNAMICS OF THE WISDOM OF CROWDS 5 We build on the DeGroot model to generate novel theoretical predictions for how social influence can affect the accuracy of group judgments. We show that if individuals’ selfweights are not identically and independently distributed in the population, but are instead correlated with individual accuracy, then social influence may not only improve the median judgment by bringing it toward the mean, but may also result in the mean of the population estimate becoming more accurate. This prediction builds on DeMarzo, Vayanos, and Zweibel’s (21) notion of “social influence weight,” which identifies the amount of influence that each individual in a network has on the collective belief. Because self-weight contributes to social influence weight, a correlation between accuracy and self-weight means that more accurate individuals will be more influential in the group estimate, resulting in a direct improvement in the accuracy of the group mean. Our predictions also show that this process can go awry if some individuals in a population are more prominent than others, giving them disproportionate levels of social influence in the population. Theoretical results suggest that when networks are highly “centralized” in this way, instead of efficiently aggregating all available information, populations are biased toward the beliefs of the central individuals (21), which can significantly influence the accuracy of the collective judgment (SI Appendix). This effect of centralization on group estimates has been predicted by a variety of social learning models, including both fixed (21,24) and growing (22,25) networks, as well as models of both discrete choice (24,25) and continuous estimation (21,22). We test these theoretical predictions using a Web-based experimental design. We recruited subjects to participate in a series of large group estimation tasks, and compared the effects of social influence in both centralized and decentralized networks to a control NETWORK DYNAMICS OF THE WISDOM OF CROWDS 6 condition in which there was no social influence. Consistent with previous work, our theoretical results predict that centralized networks will exhibit a bias toward the beliefs of central individuals. However, in contrast to prior work showing that social influence undermines group accuracy (12-19), we predict that social influence in decentralized networks will improve the accuracy of the group median (SI Appendix). Moreover, we also predict that social influence can produce systematic improvements in the accuracy of the group mean if the individual revision process is not identically and independently distributed, but is correlated with individual accuracy. As described below, our experimental design permits a direct test of these theoretical predictions based on our extensions of the DeGroot model. Theoretical Model We build on DeGroot’s (20) formalization of local information aggregation, in which subject i updates their response estimate, i t R , , after being exposed to the estimates of their network neighbors, i N j t R  , We define a subject’s revision process with three components: their own estimate; the estimates of network neighbors; and “self-weight,” or the amount of weight they place on their own estimate relative to the estimates of their network neighbors. In this model, each subject responds to social information by adopting a weighted mean of their own estimate and the estimates of their neighbors, according to the rule: i N j t i i t i i t R R R  +  − +  = , , , 1 ) 1 (   , where the value i t R , indicates the response of subject i at time t; i  indicates the self-weight a subject places on their own initial estimate; ( 1i  ) indicates the weight they place on the NETWORK DYNAMICS OF THE WISDOM OF CROWDS 7 average estimate of their network neighbors; and i N j t R  , indicates the average estimate of subject i’s network neighbors at time t. Outcomes are therefore determined by three parameters: the communication network (i.e., who can observe whom), the distribution of initial estimates, 1 R , and the distribution of self-weights, αi. At the population level, this model states that the group estimate after t revisions can be calculated as a function of a weighted, directed network of social influence (20). In this social influence network, a tie exists from node i to node j if i can observe j in the communication network. The tie has a numeric value that indicates the weight that node i places on the estimate of node j which is determined by αi. For any given node i, the sum of the outgoing tie weights equals (1 – αi). Consistent with previous implementations of this model (20-22), we represent the self-weight that node i places on its own estimate ( i  ) as a “self-tie” from i to i. The set of each node’s outgoing tie weights (including their self-tie) sums to 1. The sum of each node’s incoming ties (including their self-tie) is proportional to their overall influence in the network during each r