A unified theory for bivariate scores in possessive ball-sports: The case of handball

Abstract

We present a unified theory that posits three fundamental models as necessary and sufficient for modelling the bivariate scores in possessive ball-sports. These models provide the basis for perhaps more complicated models that can be used for prediction, experimentation, and explanation. First is the Poisson-match, for when goals are rare, or when goals are frequent but the restart after a goal is contested. Second is the binomial-match, for when goals are frequent and the restart uses the alternating rule. Third is the Markov-match, for when the restart uses the catch-up rule. We describe in detail the new model amongst these, the Markov-match, which is complementary to rather than competing with the binomial-match. The Markov-match is a bivariate generalisation of the Markov-binomial distribution. Its structure (catch-up restart) induces a larger correlation between the scores of competitors than does the binomial-match (alternating restart) but slightly more tied outcomes. The Markov-match is illustrated using handball, a high-scoring sport. In our analysis the time-varying strengths of 45 international handball teams are estimated. This poses some mathematical and computational problems, and in particular we describe how to shrink the strength-estimates of teams that play fewer games in tournaments because they are weaker. For the handball results, the Markov-match gives a better fit to data than the Poisson-match.