# Ranking Items from Discrete Ratings: The Cost of Unknown User Thresholds

arXiv: https://arxiv.org/abs/2510.01871 · HTML: https://mlsanity.com/paper/2510.01871/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 13 references that are in the map.
- First posted on arXiv 2 Oct 2025.

## Builds on (most cited first, 6 of 6)

- [Estimation from Pairwise Comparisons: Sharp Minimax Bounds with Topology Dependence](https://mlsanity.com/paper/1505.01462/) (2015)
- [Bridging the Gap: A Survey on Integrating (Human) Feedback for Natural Language Generation](https://mlsanity.com/paper/2305.00955/) (2023)
- [Your 2 is My 1, Your 3 is My 9: Handling Arbitrary Miscalibrations in Ratings](https://mlsanity.com/paper/1806.05085/) (2018)
- [Approximate Ranking from Pairwise Comparisons](https://mlsanity.com/paper/1801.01253/) (2018)
- [Perceptual adjustment queries and an inverted measurement paradigm for low-rank metric learning](https://mlsanity.com/paper/2309.04626/) (2023)
- [Designing Optimal Binary Rating Systems](https://mlsanity.com/paper/1806.06908/) (2018)

## Abstract (arXiv)

Ranking items is a central task in many information retrieval and recommender systems. User input for the ranking task often comes in the form of ratings on a coarse discrete scale. We ask whether it is possible to recover a fine-grained item ranking from such coarse-grained ratings. We model items as having scores and users as having thresholds; a user rates an item positively if the item's score exceeds the user's threshold. Although all users agree on the total item order, estimating that order is challenging when both the scores and the thresholds are latent. Under our model, any ranking method naturally partitions the n items into bins; the bins are ordered, but the items inside each bin are still unordered. Users arrive sequentially, and every new user can be queried to refine the current ranking. We prove that achieving a near-perfect ranking, measured by Spearman distance, requires Theta(n2) users (and therefore Omega(n2) queries). This is significantly worse than the O(nlog n) queries needed to rank from comparisons; the gap reflects the additional queries needed to identify the users who have the appropriate thresholds. Our bound also quantifies the impact of a mismatch between score and threshold distributions via a quadratic divergence factor. To show the tightness of our results, we provide a ranking algorithm whose query complexity matches our bound up to a logarithmic factor. Our work reveals a tension in online ranking: diversity in thresholds is necessary to merge coarse ratings from many users into a fine-grained ranking, but this diversity has a cost if the thresholds are a priori unknown.