Adolescents show collective intelligence which can be driven by a geometric mean rule of thumb

How effective groups are in making decisions is a long-standing question in studying human and animal behaviour. Despite the limited social and cognitive abilities of younger people, skills which are often required for collective intelligence, studies of group performance have been limited to adults. Using a simple task of estimating the number of sweets in jars, we show in two experiments that adolescents at least as young as 11 years old improve their estimation accuracy after a period of group discussion, demonstrating collective intelligence. Although this effect was robust to the overall distribution of initial estimates, when the task generated positively skewed estimates, the geometric mean of initial estimates gave the best fit to the data compared to other tested aggregation rules. A geometric mean heuristic in consensus decision making is also likely to apply to adults, as it provides a robust and well-performing rule for aggregating different opinions. The geometric mean rule is likely to be based on an intuitive logarithmic-like number representation, and our study suggests that this mental number scaling may be beneficial in collective decisions.

where R denotes one particular rule,   1 n i i c = is the set of observed discussion estimates agreed by each of the n groups, i r is the exact value that the rule takes for the -th i group, and i f was assumed to be a normal or log-normal with parameters i r and  . For each of the considered rules, we covered a wide range of possible values for  to test for the dependence of the likelihood on the level of noise considered. The geometric mean was found to be the most likely rule to be generating the experimentally observed group estimates (Fig 4), and the log-normal noise provided a higher log-likelihood value than the Gaussian noise.

The noisy geometric mean model
We modelled groups of three subjects reaching a consensus from their initial individual estimates. Specifically, we considered that, given three estimates 1 x , 2 x and 3 x , the group gave a consensus estimate c sampling from some probability density function: In the section 'Log-likelihood of simple aggregation rules' above, we show that of all the rules proposed, the one with a higher likelihood of producing the experimental results is the geometric mean (Fig 4): the geometric mean of the three initial estimates. The noise that produces the consensus value to deviate from the geometric mean can have a Gaussian form, Both methods provide similar results (discrepancies of less than 1%).
The set of simulated consensus estimates   for the case of log-normal noise. Note that uncorrelation between groups is assumed.
A third way of setting the noise would be to consider for each group a different standard deviation, estimated via the i  value defined in Eq. (7): but the previous two methods are in more agreement with the idea of a single 'noisy' rule generating the experimental results.

Confidence intervals for frequencies of the aggregation rules using the noisy geometric mean model
We estimated the  parameter to be used in Eq. (8) was computed. Then, the -th j rule was classified as followed by the -th The frequency j q of the -th j rule was not computed simply as the number of groups for which Eq. (12) was fulfilled by ij d , because for some groups Eq. (12) was fulfilled by not one but i n rules. Instead, the contribution of the -th i group to each of the rules was determined as and then the frequencies were actually computed as Note that, for each group, we did not sort the rules in ascending numerical order, but keeping always the same operation performed with the three pre-consensus values in the same position across groups. To turn the frequencies i q into probabilities i p , we divided each by the total number of rules considered: We repeated 10,000 times the process we have detailed, obtaining for each rule a sample distribution of 10,000 frequencies (and probabilities computed with Eq. (15) when required) compatible with the noisy geometric model. For each of these distributions, the mean and 2.5 and 97.5 percentiles were computed. This way, we obtained for each rule the mean of compatible probabilities (blue line in Fig 5a,c), and the limits that contain 95% of compatible probabilities (upper and lower limits of the shaded areas in Fig 5a,c).