how long will volleyball games last with side out vs. rally scoring: a Markov chain approach

I introduced a Markov chain to model volleyball scoring schemes in my course on probability models. I am old enough to remember side out scoring, where a team could score a point only if they served the ball. If the team did not score the point, there would be a side out, and the other team would serve the ball in recognition of winning the play. Games were played to 15 points. Under rally scoring, a team scores a point for every play they win regardless of who served. Games are generally played to 25 points. USA volleyball switched from side out scoring to rally scoring in 1999, and college and high school volleyball switched soon thereafter. More here.

Years after the switch, I met Phil Pfeifer from Darden. We discovered that we both played a lot of volleyball in graduate school. There was a push to rally scoring (then called Fin-30) when Phil was in graduate school well before the switch. He wrote a paper* in 1981 that modeled the two scoring schemes as a Markov chain to identify the situations under which rally scoring would leader to shorter or longer games. Because rec sports could be so lop-sided, it was hypothesized by some that rally scoring would lead to longer games. This is counter-intuitive. The Markov chain analysis confirms that this is true in some circumstances. The games tend to be longer under rally scoring when the serving team tends to hold the serve or when the teams are lopsided.

My slides from class are below. In my slides, I also examine the benefit of serving first under side out scoring. Since the serving team is the only team that can score points, the team that serves first is one step closer to winning. That is an enormous advantage if there is a high probability that the team that starts with the ball holds the serve.

* P. E. Pfeifer and S. J. Deutsch, 1981. “A probabilistic model for evaluation of volleyball scoring systems, Research Quarterly for Exercise and Sport 52(3), 330 – 338.

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