Bradley-Terry-Luce models to incorporate within-pair order effects: Representation and uniqueness theorems

The paper presents different representation theorems for the Bradley-Terry-Luce (BTL) models of Beaver and Gokhale and of Davidson and Beaver. In particular, algorithms that can be used in constructing BTL scales are provided. The uniqueness theorems show that the Davidson-Beaver model should be preferred to the Beaver-Gokhale model since the multiplicative order effect parameter is uniquely determined whereas the additive effect parameter is merely a ratio scale. Finally, a relationship to the simple BTL model is established. Let p(a, b) denote the probability that a is chosen when (a, b) is presented in a fixed order. Then the probabilities p(a, b) satisfy the Beaver-Gokhale model if and only if the balanced probabilities p(b)(a,b):= 1/2(p(a, b) + 1 - p(b,a)) satisfy the simple BTL model.