Abstract
“Wisdom of crowds” refers to the phenomenon that the average opinion of a group of individuals on a given question can be very close to the true answer. It requires a large group diversity of opinions, but the collective error, the difference between the average opinion and the true value, has to be small. We consider a stochastic opinion dynamics where individuals can change their opinion based on the opinions of others (social influence α), but to some degree also stick to their initial opinion (individual conviction β). We then derive analytic expressions for the dynamics of the collective error and the group diversity. We analyze their long-term behavior to determine the impact of the two parameters (α,β) and the initial opinion distribution on the wisdom of crowds. This allows us to quantify the ambiguous role of social influence: only if the initial collective error is large, it helps to improve the wisdom of crowds, but in most cases it deteriorates the outcome. In these cases, individual conviction still improves the wisdom of crowds because it mitigates the impact of social influence.
Highlights
The idea to establish social science in the spirit of mathematics and physics dates back to the first half of the 19th century, when Auguste Comte (1798–1854) launched sociology based on the belief that the society follows general laws very much like the physical world
Later developments in sociophysics [1] tried to adhere to these two approaches: derive a general dynamics applicable to societies, and analyze social data to find universal laws
With these expressions we have completely described the dynamics of the collective error
Summary
The idea to establish social science in the spirit of mathematics and physics dates back to the first half of the 19th century, when Auguste Comte (1798–1854) launched sociology based on the belief that the society follows general laws very much like the physical world. Later developments in sociophysics [1] tried to adhere to these two approaches: derive a general dynamics applicable to societies, and analyze social data to find universal laws. We do not enter the controversial discussion to what extent sociophysics has really contributed to the understanding of social systems. A few conceptual frameworks from physics have inspired the discussion about how to formalize social dynamics. Williard Gibbs (1839–1903), is the problem of how the microscopic dynamics of system elements is linked to the dynamics of macroscopic system variables. This question is of paramount importance for the description of social and of economic systems. How do the opinions of individuals contribute to the public opinion? How do decisions by individual consumers influence the market dynamics?
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