Paired comparison models with strength-dependent ties and order effects

Paired comparison models, such as the Bradley and Terry (1952) model and its variants, are widely used to measure relative strength or merit in competitive settings, including games, sports and other domains where two entities are compared. Extensions have been proposed to account for order effects (e.g., home-field advantage) as well as the possibility of a tie as a distinct outcome. We propose a novel paired comparison model that accounts not only for ties and order effects, but also recognizes two phenomena not addressed in commonly used models. First, the probability of a tie may be greater when the two items or competitors being compared are of higher quality or merit. Second, order effects may be more pronounced among higher-quality items or competitors. While the model is motivated by tournament chess game outcomes, the framework applies more broadly to other competitive and evaluative contexts, such as other sports, product testing or judged competitions. We demonstrate the approach using results from the US Chess Open tournaments held from 2006 to 2019, large events featuring players of wide-ranging strengths.