Watching the Olympic Games this Summer I noticed that the volleyball competition has changed the scoring system from the old “sideout” system to what used to be called “quick score.”  (This change may have happened a long time ago, I don’t watch much volleyball.)  The traditional sideout scoring method increments the score only when the serving team wins a point.  When the serving team loses the point the serve is awarded to the other team (a “sideout”) but the score is unchanged.  This can lead to long drawn out games with repeated sideouts and little scoring.  As a stopgap, in the old days, volleyball matches would switch to the quick score system after a certain amount of time has elapsed.  In quick scoring a sideout earns a point for the team that gains the serve.

I always liked the sideout system, thinking of it as a characteristic volleyball rule that is compromised for expediency by the switch to quick score.  Instinctively it seemed that the fact you could only score when you are serving played a big role in volleyball strategy.  But when I was watching this summer it occurred to me that the two scoring systems are less different than it appeared at first.

The basic observation is that at any stage of the game sideout scores are just quick scores minus the number of sideouts.  And sideouts necessarily alternate between teams so the number you are subtracting differs by at most one across the two teams.  So I started to think if there was a way to characterize the mapping between scoring systems that would clarify precisely the strategic impact of the switch.  And I think I figured it out.

Quick scoring is defined as follows.  The team who wins a point has its score incremented by one, regardless of who was serving that point. (The serve switches when the receiving team wins a point just as in the sideout system.)  The winner of the game is the first team to have a score of at least 15 (or 25 in other cases) and at least a 2 point lead. (I.e. the game continues past 15 if neither team has a two point lead.)

Quick scoring is equivalent to the following system: 28 points will be played. After 28 points (let’s call it regulation) if the score is tied (14-14) then they continue to play until some team has a 2 point advantage.

This is in turn equivalent to side-out scoring with the following amended rules. Lets refer to the team that receives serve in the first point of the game as the receiving team.

1. A total of 28 ponts is played in regulation.
2. At the end of play if either team is ahead by 2 points then that team wins except if
3. the receiving team either scored the last point or earned a side-out in the last point and the receiving team is ahead by 1 point.  In this case the receiving team wins.

If none of these conditions are met then the game continues past regulation. We define the team that has the serve in the first point past regulation as team 1 and the other team as team 2. The score is reset to 0-0.  Play continues (with side-out scoring) until the first moment at which one of the following occurs.

1. Team 1 has a 2 point lead, in which case team 1 is the winner.
2. Team 2 has a 1 point lead, in which case team 2 is the winner.

The proof of this equivalence is below the jump. Here’s what it means. Quick scoring is not an innoccuous change in the rules to speed up play but its pretty close. Because a near identical outcome would obtain if instead of switching to quick score, we keep sideout scoring but cap the number of regulation points at 28. Its nearly, but not exactly identical because of the two scoring “epicycles” that have to be appended, namely #3 in regulation and #2 in overtime. Note that both of these wrinkles tend to benefit the receiving team. I don’t know the stats (anybody?) but it appears to me that the receiving team already has a large advantage in volleyball at the level of an individual point. You could say that an effect of sideout scoring is that it levels the playing field by giving a small overall advantage to the serving team. The switch to quick scoring eliminates that.

I wonder if there is a noticeable difference in the frequency with which the (initially) receiving team wins a volleyball game after the switch to quick scoring.

Suppose that the serving team (the team that served the first point) either won or earned a sideout in the last point of regulation.  Then there has been an even number of sideouts and thus each team has had the same number of sideouts.  That means that the score differential in sideout scoring is the same as in quick scoring.  Thus a team has (at least a) 2-point lead in quick scoring if and only if it has a 2-point lead in sideout scoring.  And since 28 is an even number, a team is ahead in quick scoring at the end of regulation if and only if it has a 2 point advantage.  Now suppose instead that it was the receiving team that either won the last point of regulation or earned a sideout.  Then the number of sideouts is odd and the receiving team’s lead under sideout scoring is one less than its lead in quick scoring.  The receiving team is ahead by at least 1 in sideout scoring if and only if it is ahead by at least 2 in quick scoring.  We have shown that conditions 1-3 are equivalent to the winning conditions for quick scoring in regulation.

In particular this means that if none of conditions 1-3 hold then the score is 14-14 in quick score.  Now according to quick scoring a team wins in overtime if and only if it takes a 2 point lead. We will first show team 1 takes a 2 point lead in quick scoring if and only if it takes a 2 point lead (starting from 0-0) in sideout scoring.  When team 1 takes a 2 point lead it has just won a point.  That means that the number of sideouts since regulation is even and so the point differential is the same for sideout scoring and quick scoring.  Next we will show that team 2 takes a 2 point lead in quick scoring if and only if it takes a 1 point lead in sideout scoring.  If team 2 takes a 1 point lead, it has just won a point and so the number of sideouts is odd and team 2’s lead in sideout scoring is one less than its lead in quick scoring.  We have shown that the winning conditions in overtime for quick scoring and the modified sideout scoring are equivalent.