Modelling outcomes in volleyball

A Markov Chain model is applied to volleyball to calculate win probabilities and mean lengths with the associated variances, conditional on both the scoreboard and the server. A feature of this model is that it predicts outcomes conditional on both the scoreboard and the serving team. The inclusion of the serving team in the event space is an essential requirement of this model, and arises from the rule in volleyball that the winner of each point in a set must serve on the following point. The average probability of a team winning a point on serve is less than 0.5, and so rotation of serve is commonplace. The key to the analysis of an evenly contested set is the observation that, from the situation where the scores are level (after at least 46 points have been played), the team that wins the set must eventually win two successive points. If the two points are shared then the score is level once again, although a rotation of server has occurred. This scoring structure, combined with the method of rotating the serve, distinguishes volleyball from other racket sports such as tennis, squash, badminton and table tennis. Results from the model indicate that it is advantageous to be the receiver on the opening point of a set and the team that wins the toss at the start of the fifth set (if the set score reaches 2-all), has an advantage for the remainder of the match. However, due to the rotation of serve after each set, there is no advantage for either side in being server or receiver at the start of the match