Wiser than the Wisest of Crowds: The Asch Effect Revisited under Friedkin-Johnsen Opinion Dynamics

TL;DR

The paper investigates how interconnected agents governed by the generalized Friedkin-Johnsen model produce a wisdom-of-crowds estimate and how a small set of stooges can bias the collective outcome. It proves that selecting up to $k$ stooges to optimize $MSE$ or polarization is NP-hard and shows the absence of submodularity and supermodularity, motivating a greedy heuristic. The proposed method alternates between approximating equilibrium opinions and a lazy greedy selection over stooges, and it scales to large networks while delivering strong performance on synthetic and real-world networks, including Twitter data on the Ukraine war. The results reveal that maximizing polarization can substantially alter the wisdom of crowds and that even a few stooges can meaningfully shift macro-level estimates, with practical implications for information spread and policy design in connected societies.

Abstract

In 1907, Sir Francis Galton independently asked 787 villagers to estimate the weight of an ox. Although none of them guessed the exact weight, the average estimate was remarkably accurate. This phenomenon is known as wisdom of crowds. In a clever experiment, Asch employed actors to demonstrate the human tendency to conform to others' opinions. The question we ask is: what would Sir Francis Galton have observed if Asch had interfered by employing actors? Would the wisdom of crowds become even wiser or not? The problem becomes intriguing when considering the inter-connectedness of the villagers, which is the central theme of this work. We examine a scenario where $n$ agents are interconnected and influence each other. The average of their opinions provides an estimator of a certain quality for some unknown quantity. How can one improve or reduce the quality of the original estimator in terms of the MSE by utilizing Asch's strategy of hiring a few stooges? We present a new formulation of this problem, assuming that nodes adjust their opinions according to the Friedkin-Johnsen opinion dynamics. We demonstrate that selecting $k$ stooges for maximizing and minimizing the MSE is NP-hard. We also demonstrate that our formulation is closely related to maximizing or minimizing polarization and show NP-hardness. We propose an efficient greedy heuristic that scales to large networks and test our algorithm on synthetic and real-world datasets. Although MSE and polarization objectives differ, we find in practice that maximizing polarization often yields solutions that are nearly optimal for minimizing the wisdom of crowds in terms of MSE. Our analysis of real-world data reveals that even a small number of stooges can significantly influence the conversation on the war in Ukraine, resulting in a relative increase of the MSE of 207.80% (maximization) or a decrease of 50.62% (minimization).

Figures (19)

• Figure 1: An example graph that shows that our objective is not submodular. We use $\overline K_\ell$ to denote the empty graph on $\ell$ nodes. Resistances and innate opinions of all nodes are as stated in the figure (for convenience, we allow negative innate opinions). We use a sufficiently large $\ell$ such that the contribution of $v$ and $w$ to the polarization vanishes.
• Figure 2: Maximization (left) and minimization (right) of the MSE (top) and polarization (bottom) on the Vax instance.
• Figure 3: Innate opinions ($s$), equilibrium opinions before ($x^\star$) and after adding stooges for optimizing the MSE ($x^\star(\mathrm{MSE})$) and the polarization ($x^\star(\mathrm{Polarization})$). We show maximization (left) and minimization (right) on the Baltimore-Follow dataset.
• Figure 4: Maximization (left) and minimization (right) of the MSE on the Vax dataset. We show $\mathrm{Bias}^2 = (\hat{\theta} - \frac{1}{|V|}\sum_u x^\star_u)^2$, variance (polarization), and the MSE.
• Figure 5: Comparing the set of stooges chosen for optimizing the MSE and Polarization. We report the Jaccard similarity on both sets for an increasing number of stooges. The left plot shows the maximization and the right plot the minimization for different random graphs. We report mean and standard deviation across five runs.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof