The Wisdom of the Crowd and Higher-Order Beliefs

In a new theoretical economics paper, Yi-Chun Chen, Manuel Mueller-Frank, and Mallesh M. Pai propose Population-Mean-Based Aggregation (PMBA), a simple procedure that allows a principal to extract accurate information from a crowd without needing to understand the complex information structure among agents. The method only requires agents to communicate their own beliefs about an unknown state of the world, and for some agents to also communicate their expectation of the average belief across the population. Remarkably, PMBA works even when individual agents' beliefs are misspecified — meaning they may be systematically wrong — as long as the population is large enough. The authors prove that, for any finite number of possible states, PMBA infers the true state with probability approaching 1 (or almost surely under stronger conditions).

Beyond its theoretical elegance, PMBA is practically useful: it can be reinterpreted as a linear regression procedure, making it straightforward to apply with finite populations and leveraging existing statistical inference tools. The authors also conduct a novel experiment demonstrating that PMBA's real-world performance exceeds that of existing aggregation methods. This work bridges theoretical economics and machine learning, offering a scalable, assumption-light way to harness the wisdom of the crowd — even when the crowd is biased or the principal is ignorant of how information flows between agents.