The collective misunderstanding syndrome

Consider a human collective whose members issue opinions about an event that can only be true of false. Individual reliability is p, ranging between zero and one. It is higher than 0.5 when the individual is right more often than wrong; equals 1 when they are always correct and 0 when they are always incorrect. John and Anna are two citizens with equally poor reliability, let us say 0.25, and they both coincide independently in their statements. This represents, therefore, a unanimity of just two individuals. We want to compare two probabilities: the probability that what Anna says is true and the probability that what Anna says is true when, in addition, John agrees with her independently. The probability that their statements coincide and are both true is 1/4 × 1/4 = 1/16. Similarly, the probability that they both coincide and both of them are wrong is 3/4 × 3/4 = 9/16. Therefore, the probability that Anna and John coincide (without taking into account whether or not they are correct) is 9/16 + 1/16 = 10/16. Finally, the probability that Anna’s statement is true when John coincides with her is the quotient of 1/16 divided by 10/16, that is, only 1/10. Look carefully at what this means: the unanimity of two unreliable people lowers the possibility that the judgement in which they coincide is true (from 1/4 to 1/10). Good heavens: does unanimity lower credibility?

The result of this simple example (Paulos, 2015) can be used to generalise for a collective of n individuals with p individual reliability (Wagensberg, 2016). If they are incorrect more than they are correct (p<0.5), then the probability that an unanimous opinion will be true decreases with the number of people in the unanimity. If, conversely, individuals are correct more than they are incorrect (p>0.5), the opposite occurs, and unanimity reinforces the probability that the coinciding judgement will be true, as it should be. The first case illustrates what we could call «collective misunderstanding syndrome» (CMS). Indeed, the probability of a group of independent and unreliable individuals being unanimously correct decreases with the size of such unanimity! The common phrase «if so many people claims it, it must be true» dissolves like a sugar cube. Annoying conclusion, because human collectives require unanimity when it comes to important decisions. Thus, unanimity is usually reserved for complex and sensitive high-level laws. It is not easy at all for a collective of independent people to randomly reach unanimity but, should it occur, then it is much less probable that a unanimity in the opposite direction appears. Can the CMS contribute something to our comprehension of social, political, or economic phenomena?

In order to answer this question, we must gain insight into two issues: When can we say that the members of a collective are independent from each other? What does it mean to have low individual reliability? Independence is the inevitable consequence of any unprecedented situation. It happens at the beginning of any crisis – economic, social, environmental, or cultural – before there is enough information, when no one understands anything yet – a stock market crash, an immigration boom, an artistic vanguard, a plague… In this early phase, all members of the group test their own theory, but they are not convinced enough to influence others through mutual dependence. At that moment, each citizen’s understanding depends only on their own individual reliability. A unanimity generated by cultured and rational citizens (p>0.5) will be interesting to follow; however, superstitious or dogmatic citizens (p<0.5) will produce dangerous unanimities, because the more people contribute to these, the lower the probability will be that their coinciding judgement is true. That is the most pure expression of the collective misunderstanding syndrome; perhaps the seed of many incomprehensible and irreversible historical tragedies.


Paulos, J. A. (2015). A numerate life. New York: Prometheus Books.

Wagensberg, J. (2016). The collective misunderstanding syndrome. Biological Theory, 11(4), 220–223. doi: 10.1007/s13752-016-0252-4

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Professor at the Department of Fundamental Physics. University of Barcelona (Spain).