The Bradley--Terry condition is $L_1$--testable

TL;DR

This paper introduces a constant-time algorithm that efficiently tests whether a weighted tournament can be modeled by the Bradley--Terry model or if the associated Markov chain is reversible, aiding in understanding pairwise comparison data.

Contribution

The paper presents the first constant-time algorithm for testing the Bradley--Terry condition and reversibility of Markov chains in weighted tournaments.

Findings

Algorithm distinguishes Bradley--Terry representability with high probability

Algorithm tests reversibility of Markov chains efficiently

Runs in constant time regardless of tournament size

Abstract

We provide an algorithm with constant running time that given a weighted tournament $T$, distinguishes with high probability of success between the cases that $T$ can be represented by a Bradley--Terry model, or cannot even be approximated by one. The same algorithm tests whether the corresponding Markov chain is reversible.