On Fri, Feb 10, 2012 at 7:34 PM, Henry Baker <hbaker1@pipeline.com> wrote:
Many sports are organized to magnify "luck" or small differences in ability into larger score differentials -- e.g., baseball. Once you get someone on base, it changes the nature of the defense, because the defensive players have to worry about base runners & so it actually becomes easier to get a "hit". Ditto with volleyball, where you first have to get the serve before you can score.
I've wondered about that. Does the "you only score a point if you are serving", actually increase the effect of skill? Of course, it does in a trivial sense, by making the match longer: a match to 15 points, scoring only when serving, has more points than a match to 15 scoring every point, and so the better team will have a better chance to win. But if we correct for this, and assume that the better team wins each point independently with probability p > 1/2, then which kind of game does the better team have the better chance of winning: A straight match to N, with all points scoring, or a match to M, with only points won by the server scoring, assuming we adjust N and M so that the expected game length in points is the same in both cases. For a slightly more realistic model, suppose that a team wins points that it serves with probability p, and points that the other team serves with probability q. In a game where every point counts, if we play that you must win by 2, the probability of winning a match depends only on p+q. To see this, observe that if we play our points in pairs, and only evaluate the winner after each pair, it never changes who wins. In an "only the server scores" match, there's an advantage to serving first, so assume that we flip a coin to determine who serves first. Is it still true that the chance of winning depends only on p+q, and not on p and q individually? I feel that there should be a symmetry argument that proves this, but I can't seem to find it. Andy