# Error Bounds for Statistical Estimators in BTL Model with Parametric Multivariate Utility Functions

arXiv: https://arxiv.org/abs/2609.26326 · HTML: https://mlsanity.com/paper/2609.26326/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 80 references that are in the map.
- First posted on arXiv 22 Sep 2026.

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

- [When is the estimated propensity score better? High-dimensional analysis and bias correction](https://mlsanity.com/paper/2303.17102/) (2023)
- [Finite-sample performance of the maximum likelihood estimator in logistic regression](https://mlsanity.com/paper/2411.02137/) (2024)
- [Ranking from Pairwise Comparisons in General Graphs and Graphs with Locality](https://mlsanity.com/paper/2304.06821/) (2023)
- [When is it worthwhile to jackknife? Breaking the quadratic barrier for Z-estimators](https://mlsanity.com/paper/2411.02909/) (2024)
- [Modified Polyhedral Method for Elicitation of Shape-Free Utility and Conservatism Reduction in Robust Optimization](https://mlsanity.com/paper/2503.23269/) (2025)
- [Coordinate-wise Polyhedral Method for Eliciting Multivariate Linear Utility and Univariate Nonlinear Utility Functions](https://mlsanity.com/paper/2606.13418/) (2026)

## Built on by (most cited first, 1 of 1)

- [Graph-monotone entrywise guarantees for MLE and Rank Centrality on general comparison graphs](https://mlsanity.com/paper/2610.09030/) (2026)

## Abstract (arXiv)

We study preference elicitation under the Bradley-Terry-Luce (BTL) model where the true partworth vector is unknown and has to be estimated as a parameter with elicited preference information. The set of selected pairwise queries is non-uniform, deterministic, and arbitrary over a collection of alternatives, provided that it satisfies a joint identifiability condition. We focus on understanding when the canonical maximum likelihood estimator (MLE) is finite and admits sharp error bounds without explicit compactness constraints on the feasible set or external regularizers. To this end, we derive minimax lower bounds under the standard bounded dynamic range condition, and find that the same Fisher-information geometry in the classic Cram'er-Rao lower bounds underpins the finite-sample difficulty of the estimation problem. By combining a non-asymptotic expansion of the likelihood score equation with a fixed-point localization argument, we identify a design-dependent sample size threshold above which the unconstrained canonical MLE exists and is unique with high probability. The same expansion yields a decomposition of the estimation error into a linear stochastic term, an explicit second-order bias, and a higher-order remainder. A refined analysis gives sufficient sample size conditions under which the canonical MLE attains the minimax rates up to logarithmic and constant factors. These results provide a unified non-asymptotic theory for parametric utility elicitation and reveal when the inference is determined by response data alone rather than by external regularization. Preliminary numerical results are consistent with the theoretical findings.