Decision accuracy in complex environments is often maximized by small group sizes

Abstract

Individuals in groups, whether composed of humans or other animal species, often make important decisions collectively, including avoiding predators, selecting a direction in which to migrate and electing political leaders. Theoretical and empirical work suggests that collective decisions can be more accurate than individual decisions, a phenomenon known as the 'wisdom of crowds'. In these previous studies, it has been assumed that individuals make independent estimates based on a single environmental cue. In the real world, however, most cues exhibit some spatial and temporal correlation, and consequently, the sensory information that near neighbours detect will also be, to some degree, correlated. Furthermore, it may be rare for an environment to contain only a single informative cue, with multiple cues being the norm. We demonstrate, using two simple models, that taking this natural complexity into account considerably alters the relationship between group size and decision-making accuracy. In only a minority of environments do we observe the typical wisdom of crowds phenomenon (whereby collective accuracy increases monotonically with group size). When the wisdom of crowds is not observed, we find that a finite, and often small, group size maximizes decision accuracy. We reveal that, counterintuitively, it is the noise inherent in these small groups that enhances their accuracy, allowing individuals in such groups to avoid the detrimental effects of correlated information while exploiting the benefits of collective decision-making. Our results demonstrate that the conventional view of the wisdom of crowds may not be informative in complex and realistic environments, and that being in small groups can maximize decision accuracy across many contexts.

Keywords: collective behaviour; decision-making; information correlation; optimality.

Figure 1.

How an environment containing two cues affects the wisdom of crowds. (a) Diagram of the model scenario showing individuals in a group observing two environmental cues: one cue (left) has low correlation, whereas the other (right) has high correlation. (b) There exist two regimes of parameter space, one in which the wisdom of crowds is observed (grey), and another in which a finite optimal group size is observed (red). (c) The optimal group size across environmental and behavioural space. (d) The space of possible opinion states that a group can be in in a given decision trial, and the accuracy expected in that trial as a function of the group state. In the upper region, the majority of the group uses correct, low correlation information, so the group is guaranteed to make a correct decision. In the left region, the majority is not using correct, low correlation information, but there are sufficiently many individuals using the high correlation cue such that if that cue provided correct information, then there would be a majority using correct information. The group's accuracy is therefore r_H, because it is contingent on the high correlation cue. In the lower-right region, insufficiently many individuals use correct, low correlation information, and there are insufficiently many individuals using the high correlation cue such that the group will not make a correct decision, even if the high correlation cue is correct. An infinitely large group deterministically finds itself at the point (p, pr_L). (e) For finite-sized groups, the probability distribution within opinion space is described by a two-dimensional binomial distribution, which allows for enhanced accuracy if the additional probability of being in the upper region exceeds that of the lower-right region. N = 20, r_L = 0.55, p = 0.75. (f) Some examples of how collective accuracy varies with group size in the region where a finite group size is optimal. Values used are shown as points in figure 1b, with r_H = 0.5.

Figure 2.

Accuracy of optimally sized groups. Increase in accuracy (a) relative to infinitely large groups and (b) relative to solitary individuals, across voting strategies and environments where a finite group size is optimal. When r_H is low (left column), there is a substantial increase in collective accuracy compared with infinitely large groups, but only a minor improvement over solitary individuals. When r_H is high (right column), the converse is true.

Figure 3.

How an environment containing a fluctuating cue affects the wisdom of crowds. (a) Diagram of model scenario. There is one cue, which individuals independently sample, and which has reliability r. With probability g, all individuals have an increased probability r + ɛ of receiving correct information from the cue, and with probability 1 − g have a decreased probability r − ɛ of receiving correct information. Individual opinions are then aggregated into a consensus decision through simple majority rule, as in the previous model. (b) The wisdom of crowds is observed only if ɛ < r − 1/2; otherwise, an infinitely large group will achieve an accuracy of g. (c) The emergence of the wisdom of crowds depends on the environment, as in the previous model. Parameter values used are shown as points in figure 3b, with g = 0.5. (d) The optimal group size across environmental space. Compared with the previous model, the space in which moderate group sizes are optimal is small. Instead, for most of parameter space, the optimal group size is either infinitely large, or one.