Enhanced or distorted wisdom of crowds? An agent-based model of opinion formation under social influence

We propose an agent-based model of collective opinion formation to study the wisdom of crowds under social influence. The opinion of an agent is a continuous positive value, denoting its subjective answer to a factual question. The wisdom of crowds states that the average of all opinions is close to the truth, i.e. the correct answer. But if agents have the chance to adjust their opinion in response to the opinions of others, this effect can be destroyed. Our model investigates this scenario by evaluating two competing effects: (i) agents tend to keep their own opinion (individual conviction $β$), (ii) they tend to adjust their opinion if they have information about the opinions of others (social influence $α$). For the latter, two different regimes (full information vs. aggregated information) are compared. Our simulations show that social influence only in rare cases enhances the wisdom of crowds. Most often, we find that agents converge to a collective opinion that is even farther away from the true answer. So, under social influence the wisdom of crowds can be systematically wrong.

💡 Research Summary

The paper presents an agent‑based model to investigate how social influence affects the “wisdom of crowds” (WoC) phenomenon. Each agent holds a continuous, positive opinion x_i(t) about a factual question with a true answer T. The opinion dynamics follow a Langevin‑type stochastic differential equation:
dx_i/dt = −β x_i + (1/N)∑j F{ij}(t) + S_i(t).
The term −β x_i represents an individual conviction that pulls the opinion back toward its initial value x_i(0). The stochastic term S_i(t)=β x_i(0)+A ξ_i(t) adds white‑noise fluctuations centered on the initial opinion. The interaction term F_{ij}(t) captures social influence and is defined in two distinct information regimes.

1. Full‑information regime: agents observe every other agent’s current opinion. The influence weight is
   w_{ij}=1/(1+e^{|x_j−x_i|/α}) / ∑k 1/(1+e^{|x_k−x_i|/α}),
   so that larger opinion differences reduce the weight. The social term becomes
   F{ij}=w_{ij}