Figure 1.
The initial configuration of the system in the absence of social influence.
(A) Initial distribution of opinions for one representative example question (see Fig. S1 for an overview of all 32 items). The normalized answer corresponds to the estimate of the participants divided by the true value (i.e., 660°C for this question). The red curve shows the best fit of a lognormal distribution. The green dots at the top indicate the location of estimates associated with high confidence levels (). One of them constitutes an outlier. (B) Accuracy of participants’ answers as a function of their confidence level, as determined from the complete dataset (32 items).
Figure 2.
Effects of social influence on the wisdom of crowds (A), and the relevance of the confidence cue (B). The error is the deviation from the true value as a percentage. (A) Before any social influence occurs, the arithmetic (Arith.) mean is sensitive to single extreme opinions and does not appear as a relevant aggregating method. The median and geometric (Geo.) mean are more robust to outliers. When social influence occurs, however, the distributions are skewed to the right and the three indicators are more likely to generate high error values. (B) In the absence of social influence (SI), a clear and continuous trend is visible, where individuals with high confidence () constitute a good indicator of the quality of the answer. When social influence is injected in the system, however, the distribution becomes noisier and less predictable. Overall, social influence generates unpredictability in the observed trends.
Figure 3.
(A) The influence map extracted from our experimental data and (B) a simplified representation of it as implemented in the model. The color coding indicates the heuristic that is used by a majority of people, as a function of the difference in confidence and the distance between the normalized opinions
. Positive values of
indicate that the focus subject is more confident than the influencing individual (called feedback), whereas negative values indicate that the focus subject is less confident. White zones in (A) indicate the absence of sufficient data. Although the majority of people prefer to keep their initial opinion when they are more confident than their partner (i.e. the blue strategy dominates for
), a zone of strong influence is found at an intermediate distance with
. (C) The decision tree describing the decision process with three different outcome strategies. The individual first looks at the distance between opinions
, then looks at the difference of confidence
, and finally chooses a strategy accordingly.
Figure 4.
The probability of increasing (red), decreasing (blue), or maintaining (green) the confidence level after social influence.
Changes in confidence are indicated according to the opinion distance classes as defined in the influence map (Fig. 3): (A) near when , (B) intermediate when
, and (C) far when
. A tendency to increase confidence is visible in the near and intermediate zones when participants interact with a more confident subject. Confidence can also decrease in the far zone, when
.
Figure 5.
Three representative examples of the collective dynamics observed in the computer simulations.
For each example, the initial opinion map is shown on the left-hand side (experimental data), and the final opinion map after N = 300 rounds of simulations on the right-hand side. The opinion maps represent the proportion of individuals with a given opinion (x-axis) and a given confidence level (y-axis). As in Fig. 1, the normalized opinion is the actual opinion divided by the true value. The correct answer is represented by the red dashed lines (corresponding to a value of 1). Outliers with normalized opinion greater than 2 are not shown. The arrow maps represent the average movements over both opinion and confidence dimensions during simulations. Examples 1, 2, and 3 correspond to the questions “What is the length of the river Oder in kilometers? ”, “How many inhabitants has the East Frisian island Wangerooge?”, and “How many gold medals were awarded during the Olympics in China in 2008?”, respectively. The final convergence point may be determined by a dense cluster of low confidence individuals, as illustrated by Example 2 (majority effect), or by a few very confident individuals as in Example 3 (expert effect).
Figure 6.
Which attractor dominates when the majority effect and the expert effect apply simultaneously?
(A) The evolution of collective opinion when varying the relative proportion of experts pExp, holding an opinion Oexp and a high confidence level Cexp = 6, and the proportion of people in the majority group pmaj holding an opinion Omaj and a low confidence level randomly chosen in the interval Cmaj = [1 3]. Here, the number of neutral individuals is fixed to pNeut = 0. (B) Phase diagram showing the parameter space where the majority or the expert effects applies, when increasing the proportion of neutral individuals pNeut holding a random opinion and a low confidence level randomly chosen in the interval Cuni = [1 3]. The schematic regions delimited by black or white dashed lines show the zones where the collective opinion converges toward the majority or the expert opinion, respectively. In the transition zone, the collective opinion converges somewhere between Oexp and Omaj. In some rare cases, the crowd splits into two groups or more.