# The Wisdom of the Crowd and Higher-Order Beliefs

arXiv: https://arxiv.org/abs/2102.02666 · HTML: https://mlsanity.com/paper/2102.02666/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Feb 2021.

## Built on by (most cited first, 8 of 8)

- [Beyond Majority Voting: LLM Aggregation by Leveraging Higher-Order Information](https://mlsanity.com/paper/2510.01499/) (2025)
- [Eliciting Informative Text Evaluations with Large Language Models](https://mlsanity.com/paper/2405.15077/) (2024)
- [Robust Decision Aggregation with Second-order Information](https://mlsanity.com/paper/2311.14094/) (2023)
- [Surprisingly Popular Voting Recovers Rankings, Surprisingly!](https://mlsanity.com/paper/2105.09386/) (2021)
- [Dependence-Aware Label Aggregation for LLM-as-a-Judge via Ising Models](https://mlsanity.com/paper/2601.22336/) (2026)
- [Sample Complexity of Forecast Aggregation](https://mlsanity.com/paper/2207.13126/) (2022)
- [Surprisingly Popular Voting for Concentric Rank-Order Models](https://mlsanity.com/paper/2411.08367/) (2024)
- [SP-Rank: A Dataset for Ranked Preferences with Secondary Information](https://mlsanity.com/paper/2601.05253/) (2025)

## Abstract (arXiv)

We propose a new simple procedure called Population-Mean-Based Aggregation (PMBA) that enables a principal to "aggregate" information about an unknown state of the world from agents without understanding the information structure among them. PMBA only requires agents to communicate their beliefs about the state, and some agents to communicate their expectations of the population average belief. In a large population, for any finite number of possible states, and under weak assumptions on the information structure, allowing individual agents' beliefs to be misspecified, we show that PMBA infers the true state (in probability or almost surely under the stated conditions). We show how PMBA can be reinterpreted as a linear regression procedure, and how it can be used to aggregate information from a finite number of agents, allowing us to reuse existing results on inference in linear models. We conduct a novel experiment to show that the real-world performance of our procedure exceeds that of existing methods.