# Consistency of weighted majority votes

arXiv: https://arxiv.org/abs/1312.0451 · HTML: https://mlsanity.com/paper/1312.0451/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Dec 2013.

## Built on by (most cited first, 11 of 11)

- [A Worker-Task Specialization Model for Crowdsourcing: Efficient Inference and Fundamental Limits](https://mlsanity.com/paper/2111.12550/) (2021)
- [Online Mixture of Experts: No-Regret Learning for Optimal Collective Decision-Making](https://mlsanity.com/paper/2510.21788/) (2025)
- [Optimal Fair Aggregation of Crowdsourced Noisy Labels using Demographic Parity Constraints](https://mlsanity.com/paper/2601.23221/) (2026)
- [Stability of Weighted Majority Voting under Estimated Weights](https://mlsanity.com/paper/2207.06118/) (2022)
- [Scheduling to Learn In An Unsupervised Online Streaming Model](https://mlsanity.com/paper/2112.01576/) (2021)
- [Recovering Top-Two Answers and Confusion Probability in Multi-Choice Crowdsourcing](https://mlsanity.com/paper/2301.00006/) (2022)
- [Spectral Clustering for Crowdsourcing with Inherently Distinct Task Types](https://mlsanity.com/paper/2302.07393/) (2023)
- [Entry-Specific Matrix Estimation under Arbitrary Sampling Patterns through the Lens of Network Flows](https://mlsanity.com/paper/2409.03980/) (2024)
- [Weak Supervision for Improved Precision in Search Systems](https://mlsanity.com/paper/2503.07025/) (2025)
- [A Model for Imbalanced Label Aggregation: A Focus on Minority-Class Detection](https://mlsanity.com/paper/2607.24622/) (2026)
- [On the Calibration of Crowdsourcing Aggregators](https://mlsanity.com/paper/2610.16815/) (2026)

## Abstract (arXiv)

We revisit the classical decision-theoretic problem of weighted expert voting from a statistical learning perspective. In particular, we examine the consistency (both asymptotic and finitary) of the optimal Nitzan-Paroush weighted majority and related rules. In the case of known expert competence levels, we give sharp error estimates for the optimal rule. When the competence levels are unknown, they must be empirically estimated. We provide frequentist and Bayesian analyses for this situation. Some of our proof techniques are non-standard and may be of independent interest. The bounds we derive are nearly optimal, and several challenging open problems are posed. Experimental results are provided to illustrate the theory.