Wisdom of crowds proposed by Golub and Jackson (2010) is a study on the convergence to the truth of individual beliefs. Agents receive independent noisy signals about the true value of a variable and update their beliefs by communication determined by a social network. It extends the classic model of DeGroot (1974) about reaching a consensus in which agents repeatedly update their beliefs by taking weighted averages of neighboring beliefs. Golub and Jackson's (2010) main finding is that the crowds is wise (all beliefs converge to the truth) if and only if individual influence vanishes as the network grows. This seems to support the idea that opinion formation in large social networks is eventually infallible.
We show that the appealing wisdom of crowds phenomenon heavily relies on the - unfortunately - unrealistic assumption that agents always update their belief in exactly the same way. This crude assumption reflects the usual way social network data is collected, where the strength or weight Pij of a connection between two agents i and j is a combined frequency of interaction and influence. We refer to this approach as superposition, e.g. the weighted interaction matrix P is a (linear) combination of (different) matrices P = a X + (1-a) Y. For instance, consider data collected from online communication taken over one year in order to estimate the connections in a given group of users. Alternatively, consider splitting the period into two half years or four quarter years. Obviously, the estimated networks will be different while providing the same superpositioned aggregated data over one year. Consequently, there is aleatoric uncertainty about the actual pattern of belief updating. We show that wisdom of crowds is an illusive concept and it bares the danger of mistaken consensus for truth.