Scoring markets: Theory and application in sports economics

Some organizations can be analyzed as if they were markets, even if they do not precisely meet the canonical definition of one. In this article, we introduce and theoretically define the concept of scoring markets, referring to organizations in which outcomes are determined by a scoring system. We show that such organizations possess a distinct feature: the set of attainable configurations is discrete and discontinuous. This characteristic allows for the straightforward assignment of probabilities to feasible configurations, as well as the application of numerical methods to identify particular outcomes. Scoring markets arise in regulated markets, contests, elections, and sports events. Sports economics provides a compelling context for illustrating and advancing the understanding of scoring markets, especially through bilateral football competitions. Drawing from our theoretical framework, we identify potential levels of competitive balance in the group stage of the Union of European Football Associations Champions League. We compute the probabilities of these outcomes and compare the theoretical predictions to the actual results observed across each of the eight groups in every season from 1999/2000 to 2023/2024. This comparison reveals a general trend toward greater point concentration, corresponding to less probable theoretical configurations and indicating a broader decline in competitive balance.