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Reach and speed of judgment propagation in the laboratory
Edited by Susan T. Fiske, Princeton University, Princeton, NJ, and approved March 7, 2017 (received for review August 8, 2016)

Significance
Individual judgments, feelings, and behaviors can spread from person to person in social networks, similarly to the propagation of infectious diseases. Despite major implications for many social phenomena, the underlying social-contagion processes are poorly understood. We examined how participants’ perceptual judgments spread from one person to another and across diffusion chains. We gauged the speed, reach, and scale of social contagion. Judgment propagation tended to slow down with increasing social distance from the source. Crucially, it vanished beyond a social horizon of three to four people. These results advance the understanding of some of the mechanisms underlying social-contagion phenomena as well as their scope across domains as diverse as political mobilization, health practices, and emotions.
Abstract
In recent years, a large body of research has demonstrated that judgments and behaviors can propagate from person to person. Phenomena as diverse as political mobilization, health practices, altruism, and emotional states exhibit similar dynamics of social contagion. The precise mechanisms of judgment propagation are not well understood, however, because it is difficult to control for confounding factors such as homophily or dynamic network structures. We introduce an experimental design that renders possible the stringent study of judgment propagation. In this design, experimental chains of individuals can revise their initial judgment in a visual perception task after observing a predecessor’s judgment. The positioning of a very good performer at the top of a chain created a performance gap, which triggered waves of judgment propagation down the chain. We evaluated the dynamics of judgment propagation experimentally. Despite strong social influence within pairs of individuals, the reach of judgment propagation across a chain rarely exceeded a social distance of three to four degrees of separation. Furthermore, computer simulations showed that the speed of judgment propagation decayed exponentially with the social distance from the source. We show that information distortion and the overweighting of other people’s errors are two individual-level mechanisms hindering judgment propagation at the scale of the chain. Our results contribute to the understanding of social-contagion processes, and our experimental method offers numerous new opportunities to study judgment propagation in the laboratory.
Social influence extends beyond people’s direct contacts. In recent years, a large body of research has demonstrated that judgments, feelings, and behaviors can “spread” from person to person in social networks, similarly to the propagation of infectious diseases. Social-contagion phenomena have been observed across a wide range of domains, such as public health (1⇓–3), altruism and cooperation (4), risk perception (5), violence (6), political mobilization (7), and emotional states (8). Social contagion has been modeled in laboratory experiments (9) and described in field studies on subjects ranging from hunter-gatherer villages in Tanzania (10) to Facebook users (7, 8) and online recommendation networks (11). Behavioral contagion also serves adaptive functions in the collective behavior of animal swarms (12).
Unlike the propagation of diseases or the diffusion of information, the propagation of judgments often requires more than a single interaction between a sender and a receiver. Rather, the receiver needs to be “won over” by the sender’s judgment before adopting and eventually spreading it further (13). Various social factors that reduce the receiver’s uncertainty about the quality of the sender’s judgment can thus facilitate judgment propagation (14). For instance, being exposed to the same judgment from different sources increases the probability of its adoption (13, 15, 16). Likewise, the sender’s reputation shapes the strength of social influence (17), with reputation being treated as a proxy for judgment quality. Another factor is the history of past interactions between the sender and the receiver. Specifically, being able to repeatedly observe the good performance of a sender can reduce the receiver’s uncertainty about the quality of the sender’s judgment and consequently enhance the sender’s social influence (18).
In a collective context, the repeated transmission of a judgment from person to person can give rise to global patterns of social contagion that can be described in terms of properties such as reach, speed, and intensity. In the context of innovation diffusion and viral marketing, simple mechanisms of influence can produce adoption cascades, in which the adoption of a product propagates across a large part of the population (11, 16, 19, 20). This form of social contagion, however, is restricted to environments in which people choose from a finite set of options and in which their choice is public and thus evident to all their neighbors. The spread of a judgment arises from a different and arguably more complex process, not least because judgments are often continuous and multidimensional. In addition, people might be only partially influenced by others’ judgments (5). In fact, global cascades of judgment propagation are rarely observed. According to the three-degrees-of-influence hypothesis, judgment propagation is inherently limited to a social distance of about three people (21). That is, a person’s influence on the judgment of neighboring individuals gradually dissipates with social distance and eventually ceases to have impact beyond three degrees of separation (1, 2, 4, 7). Several behavioral mechanisms have been proposed to explain this “wall,” including the progressive deterioration of the judgment due to information distortion during social transmissions, and the fact that the social ties connecting people may not be robust across time, meaning that propagation pathways are interrupted (21). In addition, the nature of the transmission method—active teaching or simple copying, for instance—can foster or hinder propagation efficacy, as has been demonstrated in the domain of human cultural evolution (22, 23).
From the above literature, it appears that three main conditions should suffice for the emergence of long-distance judgment propagation: (i) the originator of the judgment consistently provides accurate judgments, so that observers have little uncertainty about the quality of his or her subsequent judgments; (ii) all individuals can accurately observe others’ judgments and their quality, meaning that transmission noise is minimal; and (iii) the connections between people are stable, so that propagation pathways are uninterrupted. Under these conditions, an observer B should eventually be prompted to adopt the judgment of an initiator A, who repeatedly makes accurate judgments. In the absence of information distortion, B should in turn start performing well. This should subsequently prompt a third individual C to adopt B’s—and thus indirectly A’s—judgment. If the same process is repeated across many adjacent individuals in a chain, then the judgment of A could, in principle, propagate from person to person over long social distances. However, any systematic distortion of A’s initial judgment over social transmission would make that judgment less accurate—and thus less persuasive. The increasingly distorted judgment would eventually cease to propagate. In this article, we examine whether the above three conditions are sufficient for the emergence of long-distance judgment propagation—and, if not, which other factors impede judgment propagation over long distances.
To this end, we present two studies. In study 1, we investigated—using a simple visual perception task—the conditions under which the judgment of an individual A can spread to another individual B, as B repeatedly observes the judgment and performance of A. The conventional experimental paradigm in the advice-taking literature consists of observing one-shot interactions between two strangers—the participants neither know nor repeatedly interact with one another (14, 24). In contrast, we investigated how repeated interactions shape the strength of social influence, as has been done in work on “reputation formation” (see, e.g., refs. 25 and 26). In study 2, we evaluated the collective dynamics of judgment propagation. We created unidirectional chains of six participants in which all individuals repeatedly interacted with their predecessor. That is, individual A initiated the chain, B repeatedly interacted with A, C repeatedly interacted with B, and so on up to individual F. Initially, all individuals were strangers. In each interaction round, they made an initial judgment, observed the judgment of their predecessor, potentially revised their judgment, and forwarded their final judgment to the next person in the chain. We then examined how far, under these favorable circumstances, A’s judgment would travel down the chain. The results of study 1 indicate that the influence of A on B increased gradually as the two got to know each other. However, as expected, this happened only if A performed considerably better than B. Study 2 showed that, whenever individual A (i.e., the originator of the chain) consistently rendered more accurate judgments than the rest of the chain members, waves of judgment propagation were triggered down the chain: round after round, A’s judgments traveled increasingly farther. Crucially, however, the influence of A vanished beyond a social distance of about three people. That is, A influenced individuals B, C, and D, but not E and F. By modeling the experimental results and running numerical simulations, we then show that the participants’ tendency to overweight their predecessor’s error, combined with information distortion, generates exponentially increasing delays in propagation. This, in turn, resulted in the gradual extinction of influence with social distance.
Results
Experimental Design.
We designed a visual perception task modeled on random-dot kinematograms (27). In each round, participants observed a set of 50 dots moving on a computer screen. Some dots moved consistently in a similar direction (“correlated dots”) and others in random directions. Participants were asked to determine the main direction of the correlated dots as accurately as possible (see details in Materials and Methods, Fig. 1, and Movie S1). The correct answer was a specific angle
Experimental design. (A) Participants were exposed to visual stimuli consisting of a set of 50 moving dots. A proportion pc of correlated dots moved in the direction
Schematic representations of the two experimental setups. (A) In study 1, two individuals, A and B, were paired for 15 consecutive rounds. In each round, they separately faced the visual perception task described in Fig. 1. The correct answer
Individual Behaviors.
In study 1, we paired an individual facing a task of difficulty level i with a partner facing a task of difficulty level j for 15 consecutive rounds. In each round, we measured the partner’s influence on the individual’s judgments. We tested six conditions by varying i in {2,3} and j in {1,2,3}. Each Si-Pj condition (individual facing a difficulty level i paired with a partner facing a difficulty level j) was repeated with 100 participants.
First, we measured the deviation
Participant behavior in study 1. (A) Illustration of the behavioral change between round 1 (black dots) and round 15 (white dots) for all 100 participants in condition S2-P1 (individuals, S, facing a medium difficulty level of 2 paired with partners, P, facing a low difficulty level of 1). The three background colors indicate the zones corresponding to the three strategies: ignore (red), compromise (yellow), and adopt (blue). Whereas participants mostly ignored their partner’s judgment at round 1 (72% of black points in the red zone), they tended to adopt it at round 15 (85% of white points in the blue zone). (B) The proportion of participants adopting each strategy across the 15 rounds in all experimental conditions. The detailed example shown in A corresponds to the first and last rounds in the Upper Left of B.
Modeling and comparison of adoption curves. Growth curve analysis was used to analyze the probability of adoption as a function of the number of the round and a participant’s difficulty level (for cases where the partner had a difficulty level of 1; see the two blue curves in Fig. 3B, left-most column). The probability of adoption was analyzed with a mixed-effects logistic regression model with fixed effects for the number of the round (cubic orthogonal polynomials), the participant’s difficulty level (2 vs. 3), and their interactions (all within participant). The model also included a random intercept for items (i.e., the true angle), participant random effects on all round-number terms, and Participant × Difficulty Level random effects on all round-number terms except the cubic trend. The model was estimated using Bayesian Monte Carlo Markov chains (MCMCs) simulations with default priors [function stan_glmer in the rstanarm R package]. (A) The proportion of adoption as a function of the number of the round and a participant’s difficulty level. Solid symbols indicate observed proportions. Solid lines show the medians of the posterior predictive distribution for the proportion of adoption (simulated using all random effects). The inner colored ribbons indicate the middle 50% of the posterior predictive distributions (i.e., first to third quartile), and the outer ribbons indicate the middle 90% (i.e., 5–95% quantile). The inline figure shows the posterior predictive distribution of the difference of proportion adopted (difficulty level 3 minus 2) per round. (B) Estimates for the fixed-effect parameters of the model: medians (points) and 80% and 95% credible intervals (thick red and thin black ranges, respectively). On the y axis, ot1, ot2, and ot3 are linear, quadratic, and cubic orthogonal polynomial terms, respectively; level3 is a dummy variable for difficulty level 3; that is, skill level 2 is the reference condition. Two sets of results stand out. First, the model indicates a positive linear (ot1) and cubic (ot3) trend as well as a negative quadratic trend (ot2) for the number of the round when participants operated at difficulty level 2. These positive linear and cubic trends were even more pronounced for difficulty level 3 (ot1:level3 and ot3:level3); the negative quadratic trend did not differ between difficulty conditions (i.e., ot2:level3 was not reliably different from zero). Second, across rounds, participants were more likely to adopt their partner’s estimate when individuals operated at difficulty level 3 compared with 2 [as indicated by the positive slope for level3; PMCMC(level3 > 0) = 1.000]; see also the inline figure in A.
Model.
Next, we modeled the above results using a computational model inspired by accounts of social reinforcement learning (28). The purpose was to establish a link between the mechanisms of influence within dyads observed in study 1 and the dynamics of judgment propagation that will be shown in study 2. For the model, we assumed that individuals assessed the performance or “quality”
Observed and predicted errors of final estimates
Comparison of the model performance with and without the overweighting of the partner’s errors (i.e.,
Observed error distributions and error distributions predicted by the model with and without the overweighting of the partner’s errors (i.e.,
Judgment Propagation.
In study 2, we examined the dynamics of judgment propagation in chains involving six people, A, B, C, D, E, and F (Fig. 2B). Each person in the chain was paired with their predecessor in the chain for 15 consecutive rounds. That is, in each round, the individual at position k observed the final estimate of the individual at position k ‒ 1 before entering her or his final estimate. This estimate, in turn, served as the advice given to the individual at position k + 1. The originator (individual A) had no partner and was thus never subject to social influence. A always experienced a low difficulty level of 1, whereas all subsequent individuals always experienced a high difficulty level of 3. At round 1, all individuals in the chain were strangers. However, we expected B to soon notice A’s good level of performance and to adopt A’s judgments. Consequently, we expected C to notice B’s good level of performance and to indirectly adopt A’s estimates. Round after round, we expected A’s judgments to propagate increasingly further down the chain, with F eventually performing as well as A.
We collected data for 20 independent chains, each composed of six unique participants. In each round, we measured the influence of individual A on all subsequent individuals (B to F). Social influence s was measured as follows:
Judgment propagation down the chain (study 2). Observed intensity of social influence averaged over 20 experimental chains. The color coding indicates the influence of the originator’s judgment on the final estimates of all other individuals in the chain, as a function of their social distance from the originator (x axis) and the number of the round (y axis; Fig. 2B). Individuals located one degree of separation from the originator (i.e., social distance = 1) rapidly adopted the originator’s judgments (social influence approached 1 as early as round 2). Individuals located two degrees of separation from the originator were influenced by the originator after about four rounds of interaction. Participants located at a social distance greater than three were rarely influenced by the originator.
To better understand the mechanisms underlying this propagation pattern, we conducted simulations of the same experimental setup using our model (calibrated with the fitted parameter values from study 1). We first evaluated the impact of each component of the model separately (Fig. 5A). Compared with a simple contagion model, the inclusion of information distortion curtailed the range of propagation to about 10 degrees of separation. This is obviously insufficient to explain the much shorter propagation ranges observed. However, combining information distortion with the overweighting of the partner’s errors drastically limited the propagation range to around three to four degrees of separation, consistent with our data (see the analytical calculations in Supporting Information).
Dynamic features of judgment propagation in study 2. (A) The intensity and range of social contagion after 15 rounds, assuming different propagation mechanisms (model previously calibrated in study 1): full and immediate adoption of the partner’s judgment (in black), distortion of information only (in blue), and distortion of information and error overweighting (in red). Circles correspond to experimental data. (B) Intensity and range of social contagion over 100 rounds of interactions as predicted by the model. The range of social contagion initially increased with time but eventually plateaued at a social distance of four. (C) The time delay required for social influence to travel through the chain increased exponentially with social distance. The dots indicate the model predictions (for 1,000 simulation runs), and the lines show the best exponential fit. For distances larger than four persons, the time delay exceeded 100 rounds before a weak influence (s = 0.2, in blue) appeared. Strong influence (s = 0.8, in red) could not propagate beyond a social distance of three persons because information distortion rendered the (indirect) adoption of the originator’s judgment increasingly impossible. The Inset shows the same figure with a logarithmic transformation along the y axis, confirming the exponential growth of the time delay.
Finally, we used the model to explore the dynamics that take place over an extended time horizon. Fig. 5B shows the social influence of originator A over 100 rounds of interactions. The propagation range initially increased with the number of rounds but eventually plateaued at a social distance of approximately three degrees of separation for strong influence (s > 0.7) and four degrees of separation for moderate influence (s > 0.4). At five degrees of separation, social influence was no longer visible. We then calculated the time required until the originator’s influence traveled different social distances. As Fig. 5C shows, the time required increased exponentially with social distance. That is, the time needed to reach the next individual in the chain was almost three times longer than the time needed to reach the previous one. The contagion wave thus slowed down with social distance, and traveling further than four individuals from the source thus required an excessively large number of rounds (see the propagation speed curves in Fig. S5). An analytical exploration of the model (presented in Supporting Information) revealed that each component of the model changed the functional form of the propagation delays. In the absence of information distortion and error overweighting, propagation time increased linearly with social distance. However, the inclusion of information distortion resulted in a polynomial increase in propagation times, and the addition of error overweighting resulted in the propagation time growing exponentially, consistent with our observations (Figs. S6 and S7). Both of these factors impacted the functional form of the propagation delays.
Speed of judgment propagation as a function of the social distance from its source. The colors indicate the strength of social influence: weak (s = 0.25, in blue), intermediate (s = 0.5, in red), and high (s = 0.75, in yellow). Waves of social contagion slow down with increasing distance from the judgment source.
Propagation mechanisms. Social influence of the originator when assuming (A) a simple contagion effect, (B) the distortion of information, and (C) the addition of error overweighting. The color-coding indicates the strength of social influence of the originator as a function of the social distance (y axis) and the round number (x axis). Results are averaged over 1,000 simulation runs.
Propagation delays. The time delay required for social influence to travel through the chain, assuming the three mechanisms presented in Fig. S6 is shown (for 1,000 simulation runs). Although a simple contagion mechanism yields linear propagation delays (in black), the inclusion of information distortion yields a polynomial increase (in blue), and the addition of error overweighting yields exponential increase. For the red curve, the propagation delays for social distances 6 and 7 equal 168 rounds and 696 rounds, respectively.
Discussion
Social-contagion phenomena are difficult to study because they root in two intertwined layers of complexity, namely (i) that induced by the network structure and (ii) that induced by the behavioral processes that operate within the network. In this article, we disentangled both layers and examined, in detail, the behavioral processes that drive judgment propagation for a simple network structure, namely, a transmission chain. Our experimental design enabled us to trigger waves of judgment propagation down the chain and study the behavioral factors influencing propagation.
For the time window of 15 interactions and a linear, unidirectional network structure examined here, we found that judgments rarely propagated beyond a distance of about three individuals. Our numerical and analytical results suggest that judgments could propagate farther, but the time necessary to do so increases exponentially with social distance. Propagation over distances greater than three to four individuals would require (i) a consistently more accurate originator, (ii) an error-free observation of others’ performance, and (iii) a static chain structure to be maintained during several hundreds of interactions.
Our results reveal some of the factors impeding spatial propagation beyond this range. First, judgments become progressively more distorted over successive social transmissions. Such distortion may even occur when individuals are able to accurately observe others’ judgment and performance. In our behavioral data, distortion stemmed from imprecisions in imitating the partner’s judgment or from slight deliberate alterations to the partner’s judgment. Consequently, judgments tend to become gradually less accurate—and therefore less influential—as they travel further from their source. The second factor that impairs long-range propagation is the overweighting of other people’s errors [also called “egocentric discounting” (25)]. This effect has previously been described for pairs in the advice-taking and reputation formation literature (25); we have shown that it is also key to judgment propagation or lack thereof. In a pair, overweighting the partner’s errors slows down the adoption of judgments because each error committed needs to be compensated for by several good performances. In a chain, the delay is amplified at each further position: the perceived quality of individual B in the eyes of individual C decreases as long as B performs poorly and thus as long as B has not yet adopted the judgment of A. For each subsequent step down the chain, the same process repeats with longer time delays. Overweighting other people’s errors therefore contributes to the observed exponential growth of the time delays.
These results raise a number of interesting issues. Most importantly, future research should investigate how these behavioral dynamics may change in more complex network topologies (29). It is conceivable that multiple simultaneous sources of influence, as rendered possible in a more complex network, convey different judgments and may operate as sources of “noise.” If so, such noise, in turn, is likely to impair propagation. However, clustered networks with redundant ties could also expose individuals to a converging judgment originating from different pathways and thus provide social reinforcement for its adoption (15). Our experimental design is versatile. It could be adapted to examine how judgments propagate through more complex network topologies and the extent to which presently observed limitedness of judgment propagation generalizes.
When combined with recent techniques assessing the structure of a social network (30), our approach could offer an experimental microworld that eventually can help to predict where and when the results of a promotion campaign will be most visible. Clusters of people separated by fewer than three individuals are likely to respond similarly to social interventions, giving rise to the clustering of judgments within regions of social networks (19). Additional simulations that we conducted show that good performers engender longer influence pathways (Fig. S8). This “human factor”—that is, the initiator’s past performance—influences the dynamics in addition to other structural factors related to the target’s surrounding network structure, such as the node’s degree or centrality (e.g., ref. 15). In sum, the complex interplay between human factors and the structure of the social network is key to better understand the dynamics of judgment propagation.
Impact of the initiator’s initial performance. (A) The initiator’s cumulated social influence
Materials and Methods
Experimental Design.
The experimental task was to estimate the direction (i.e., angle) in which 50 dots, displayed in a circular area, moved on a computer screen (Fig. 1). The task was developed using HTML/JavaScript and SmartFoxServer. A set of 25 unique angles equidistantly distributed between 0° and 360° served as true angles. The animation was displayed at a rate of 20 frames per s. While the dots were still moving, participants indicated their estimate by using the computer mouse to place a black arrow in the circular area. Once they had confirmed their first estimate (by pressing a mouse button), the movement of the dots was interrupted and a blue arrow indicating the partner’s estimate was displayed. Participants could then use the mouse to revise the angle of their black arrow and pressed a button to confirm their final estimate. Participants who did not wish to revise their estimate could immediately press the confirmation button. Finally, the correct angle was displayed as a red arrow. Furthermore, the number of points p scored by the participant in that round was displayed. The number of points awarded was an exponentially decreasing function of the error
Procedure.
Data were collected between November 2015 and January 2016. The experimental procedure was approved by the Ethics Committee of the Max Planck Institute for Human Development. All participants gave informed consent to the experiment. Fifteen participants were recruited for a preliminary study in which we collected the partners’ estimates, and 100 participants were recruited for the main phase comprising studies 1 and 2 (58 males, 42 females; Mage = 25.4 y; SD = 3.7). They received a flat fee of €10 for their participation, in addition to a monetary bonus depending on performance. In the preliminary phase, we collected independent estimates from 15 participants for each of the 25 true angles, and for each of the three difficulty levels (Fig. 1 for the implementation of the difficulty levels). Thus, each participant in the preliminary study provided one estimate for each true angle at each difficulty level (75 estimates in total). Partners’ estimates in the main experiment (i.e., studies 1 and 2) were drawn from this pool of data.
In the main experiment, after receiving instructions, participants first completed five individual training rounds at each difficulty level. For study 1, they were then paired with a partner for 15 consecutive rounds (Fig. 2A). The partner was chosen randomly among the participants in the preliminary study (see above). The 15 true angles and the corresponding estimates of the chosen partner were randomly drawn from the data provided by that partner during the preliminary phase. In each round, participants first estimated the angle, then observed their partner’s estimate, and then indicated their final estimate. Subsequently, the correct angle was revealed, the participant’s total score was updated, and a new round started. After 15 rounds, participants were informed that they would be paired with a new partner and another series of 15 rounds started. This procedure was repeated six times, corresponding to the six conditions Si-Pj with i in {2,3} and j in {1,2,3}, where i is the difficulty level faced by the participant and j is the difficulty level faced by the partner. As a manipulation check, we implemented a seventh condition S1-P3, in which the individual faced a low difficulty level and the partner faced a high difficulty level. As expected, participants in this condition systematically disregarded their partner’s judgment because they could easily provide an accurate answer of their own. The results of this condition are not reported in the manuscript. For study 2, each participant participated in another series of 15 rounds in the chain setup (Fig. 2B). A new chain was started every five participants (i.e., at participant numbers 1, 6, 11, and so on). In 15 consecutive rounds, the first participant in each chain (e.g., participant number 1) was paired with a partner (the originator) randomly drawn from the participants in the preliminary study. The 15 final estimates given by that participant served as the 15 pieces of advice that the next participant (participant 2) received. This procedure was repeated until the end of the chain was reached (e.g., until participant 5 was paired with participant 4) and a new chain started (e.g., participant 6 was paired with a random partner from the preliminary study). In this way, we collected a total of 20 independent chains. All participants in the chains experienced the highest difficulty level (except for the originator of the chain, who was drawn from the preliminary study and experienced a low difficulty level). In total, participants took part in eight series of 15 rounds: seven corresponding to the seven Si-Pj conditions in study 1, and one corresponding to the chain setup in study 2. The order of the eight conditions was randomized, that is, the position of each condition in the experiment was counterbalanced across participants. Finally, all participants completed eight additional series of 15 rounds for another study, in which they faced two or three partners at the same time. The results of this study are not reported in the present paper. The experiment lasted about 60 min in total.
Analytical Calculation of the Propagation Delays
In this section, we examine analytically and numerically the role of (i) the distortion of information and (ii) the error overweighting (also called “egocentric discounting” in ref. 26) on the dynamics of judgment propagation and their impact on the nature of the propagation delays.
Our experimental results have shown that individuals assess the performance of their partner relative to their own. That is, individual B will adopt individual A’s judgment if B believes that A performs better than her with a sufficiently high degree of certainty. Formally, we have described this process as follows: at the end of each round
In the following, we will rely on this model to examine information distortion and the error-overweighting effect impact on the nature of the propagation delays in a chain where the first individual (the “originator”) faces an easy task and all proceeding individuals face a difficult task (i.e., as in our chain setup in study 2).
Simple Contagion.
In this first, simple scenario, we assume no information distortion and no error overweighting. That is, the parameter
For instance, if the individual B needs
Consequently, in this simple contagion model, if a judgment needs
Distortion of Information.
Let us now examine the impact of the distortion of information (while still assuming no error overweighting, i.e.,
At the first transmission in the chain (i.e., between A and B), the difference
Once B has adopted A’s judgment, the error of B remains higher than the original error of A due to the information distortion. Thus, at the next transmission between B and C, the difference
Analogously, the judgment that C adopts from B has undergone two successive distortions. Once C adopts B’s judgment, the perceived quality
More generally, the judgment will need
Error Overweighting.
Finally, let us examine the impact of the error overweighting. As we observed in our experiment (and is established in extant literature; see ref. 26), people tend to overweight their partner’s errors relative to their own. Based on the empirical observations, we estimated the error-overweighting coefficient to be
In other words, if B needs
Formally, let
In conclusion, each component of the model has a different qualitative impact on the nature of the propagation delays, which is confirmed by numerical simulations in Figs. S6 and S7.
Enhancing Judgment Propagation
In this section, we explore numerically some additional properties of judgment propagation based on the characteristics of the originator of the judgment.
In contrast to our experimental setup, which assumed that the members of a chain are complete strangers at the beginning of the process, real-world communication chains are mostly composed of people who already know each other well because they share a common history of past interactions. Real-world social networks are thus comparable to the state of our experimental chains after many rounds of interactions. To account for this fact, we created chains of six individuals (denoted A, B, C, D, E, and F) and “trained” the chains during 50 consecutive rounds (corresponds to a delay after which the adoption curves have flattened; Fig. 5B in the main text). During this training phase, each individual in the chain could observe and possibly adopt the judgment of the preceding individual (i.e., as in our previous simulation setup). In such a way, we produced situations where the individuals know their predecessor well, which is comparable to real-world situations. After this training phase (i.e., at round 51), we injected a signal in A and measured how it impacted the judgments of B, C, D, E, and F.
First, we tested the extent to which the performance of A during the training phase impacted the reach of A’s judgment on the chain. For this, we varied the estimation error
The results are shown in Fig. S8A. The higher the error of A during the training phase, the weaker the overall social impact of the intervention targeted at individual A on the rest of the chain. In practice, this result implies that targeting an individual who is perceived by her peers as being competent in a particular domain would boost the propagation of the desired judgment over a larger social horizon (even influencing those who do not know the target person directly).
It is also interesting to examine the interplay between the originator’s performance and its degree of connectivity (i.e., how many other persons are directly connected to A). In fact, a well-known result from the literature suggests that targeting individuals with a high degree of connectivity is an efficient propagation strategy (see, e.g., ref. 17). In fact, if A has direct influence on two individuals, B and B′, then his or her judgment should propagate along two chains of influence simultaneously: to C and D, but also to C′ and D′, thus influencing twice as many individuals. Although high connectivity diffuses the judgment through many chains in parallel, high performance diffuses the judgment farther in each chain. Clearly, the influence of A is stronger when both criteria are maximized. However, a compensation effect exists: high connectivity with low performance might be as inefficient as high performance with low connectivity, which is indeed what additional simulations show (Fig. S8B).
In sum, we have shown that the reach of judgment propagation is enhanced when the originator of the judgment is recognized by her peers as being competent. Although this effect can be counterbalanced by a low degree of connectivity, the opposite is also true: targeting an individual with a high degree of connectivity is only efficient if that individual also exhibits a history of good performances.
Acknowledgments
We thank Larissa Conradt and Winnie Poel for fruitful discussions and early contributions to the project. We are grateful to Anita Todd and Susannah Goss for editing the manuscript. This research was supported by a grant from the German Research Foundation as part of the priority program on “New Frameworks of Rationality” (SPP 1516) awarded to R.H. (HE 2768/7-2). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Footnotes
- ↵1To whom correspondence should be addressed. Email: moussaid{at}mpib-berlin.mpg.de.
Author contributions: M.M., S.M.H., J.E.K., and R.H. designed research; M.M., S.M.H., and J.E.K. performed research; M.M. and S.M.H. analyzed data; and M.M., S.M.H., J.E.K., and R.H. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1611998114/-/DCSupplemental.
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