Competitive Balance Beyond Standings: Bayesian Inference for Latent Tiers and Structural Heterogeneity

Abstract

Competitive balance has long been a central concern in sports economics, tied to outcome uncertainty, fan engagement, and the way leagues design their policies. Most existing measures of competitive balance are built from standings or winning percentages, which say little about how overall parity relates to the way teams sort into latent competitive tiers. We propose a Bayesian framework based on the Bradley-Terry model that jointly estimates the number of competitive tiers and team memberships. To capture the overall degree of competitive heterogeneity, we introduce a structural balance parameter defined on the unit interval within a hierarchical model, and we establish theoretical results giving this parameter a clear interpretation:it is directly tied to the separation between tiers, and overall heterogeneity depends on both the size and frequency of cross-tier differences. Importantly, structural balance is distinct from the information content of the schedule itself, so leagues with different numbers of teams and games can be compared on equal footing. Inference is carried out with an efficient posterior sampling algorithm that jointly handles team abilities, tier structure, home advantage, and structural balance. Simulation studies validate the proposed method, showing accurate recovery of the latent competitive structure and correct performance of the structural-balance testing procedure. Applied to the NBA and NFL, it reveals that the two leagues exhibit similar global structural balance despite very different competitive organizations: the NBA has three latent tiers, while the NFL has two broader ones, and the NFL's sparser schedule produces considerably more posterior uncertainty. Overall parity, it turns out, can mask quite different forms of competitive stratification-something standings-based measures alone cannot detect.