A Consensual Linear Opinion Pool

An important question when eliciting opinions from experts is how to aggregate the reported opinions. In this paper, we propose a pooling method to aggregate expert opinions. Intuitively, it works as if the experts were continuously updating their opinions in order to accommodate the expertise of others. Each updated opinion takes the form of a linear opinion pool, where the weight that an expert assigns to a peer’s opinion is inversely related to the distance between their opinions. In other words, experts are assumed to prefer opinions that are close to their own opinions. We prove that such an updating process leads to consensus, \textit{i.e.}, the experts all converge towards the same opinion. Further, we show that if rational experts are rewarded using the quadratic scoring rule, then the assumption that they prefer opinions that are close to their own opinions follows naturally. We empirically demonstrate the efficacy of the proposed method using real-world data.

💡 Research Summary

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The paper addresses the problem of aggregating probabilistic forecasts from multiple experts into a single consensus distribution. Traditional linear opinion pools combine expert reports using fixed weights, but determining appropriate weights is non‑trivial, especially when past performance data are unavailable or when experts wish to keep their identities or raw opinions private. To overcome these limitations, the authors propose a dynamic, distance‑based weighting scheme that mimics continuous peer‑to‑peer learning among experts.

Model and Notation

• There are (z \ge 2) mutually exclusive outcomes (\theta_1,\dots,\theta_z).
• Each expert (i) initially reports a probability vector (f_i = (f_{i,1},\dots,f_{i,z})).
• The classic linear opinion pool is (T(f_1,\dots,f_n)=\sum_{i=1}^n w_i f_i) with non‑negative weights summing to one.

DeGroot’s Static Updating
DeGroot (1974) models opinion evolution as (f_i^{(t)} = \sum_{j=1}^n p_{i,j} f_j^{(t-1)}), where the matrix (P =