June 2022 Optimal full ranking from pairwise comparisons
Pinhan Chen, Chao Gao, Anderson Y. Zhang
Author Affiliations +
Ann. Statist. 50(3): 1775-1805 (June 2022). DOI: 10.1214/22-AOS2175


We consider the problem of ranking n players from partial pairwise comparison data under the Bradley–Terry–Luce model. For the first time in the literature, the minimax rate of this ranking problem is derived with respect to the Kendall’s tau distance that measures the difference between two rank vectors by counting the number of inversions. The minimax rate of ranking exhibits a transition between an exponential rate and a polynomial rate depending on the magnitude of the signal-to-noise ratio of the problem. To the best of our knowledge, this phenomenon is unique to full ranking and has not been seen in any other statistical estimation problem. To achieve the minimax rate, we propose a divide-and-conquer ranking algorithm that first divides the n players into groups of similar skills and then computes local MLE within each group. The optimality of the proposed algorithm is established by a careful approximate independence argument between the two steps.

Funding Statement

The first and second authors were supported in part by NSF CAREER award DMS-1847590 and NSF Grant CCF-1934931.
The third author was supported in part by NSF Grant DMS-2112988.


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Pinhan Chen. Chao Gao. Anderson Y. Zhang. "Optimal full ranking from pairwise comparisons." Ann. Statist. 50 (3) 1775 - 1805, June 2022. https://doi.org/10.1214/22-AOS2175


Received: 1 February 2021; Revised: 1 October 2021; Published: June 2022
First available in Project Euclid: 16 June 2022

MathSciNet: MR4441140
zbMATH: 07547950
Digital Object Identifier: 10.1214/22-AOS2175

Primary: 62F07

Keywords: Bradley–Terry–Luce model , divide-and-conquer , Full ranking , minimax risk

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 3 • June 2022
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