Re: [math-fun] Modelling human skill ...
I know more about athletic games than about non-athletic games. I think that "levels" seem to involve some sense of fair play. When team A can beat team B 99 times out of 100, most people would say they aren't in the same league. Perhaps 1% may be to great a gap, but I would think that the usual gap would be in the single digit percentile range. Age group, weight class, and gender class athletics have been extremely well studied. For example, women runners are approx. 10% slower than male runners in the elite (Olympic) class events. This is actually surprisingly good, since in other kinds of events -- e.g., where strength is important, the differences are far greater -- e.g., rowing events. Lightweight rowing is particularly interesting, since the physics indicate that lightweights should be capable of competing on an equal basis with heavyweights; the problem seems to be that the weight of the boat, oars and coxswains is much heavier on a relative basis for the lightweights than for the heavyweights. For a 4 oared shell without a coxswain, the difference is amazingly small -- perhaps only 4 seconds in a ~6 minute race. In most sports, the difference in skill level is logarithmic -- for each increment in skill level, the probability of winning falls by an order of magnitude. If you've ever run a marathon, you realize this when the winners are already home & finished showering (or are on the airplane home), and you're still running. 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. In rowing, once you get further back than a couple of boat lengths, you have to deal with wash from the other boat(s), which will magnify the margin of victory. There was a lot of study at the end of the 19th century about the skill of factory workers & it appears that some factory workers were an order of magnitude more productive than others. This sort of differential was highly embarrassing in the Soviet Union. In the modern professional economy, differences in productivity can be astoundingly different. One singer or composer can produce 100x or 1000x another singer or composer in the quality & quantity of work. One software engineer can be 100x more productive than another software engineer.
Speaking of sports, I've always wondered: how many superbowl games would one need to be, say, 95% sure that one team really is better than the other? Somehow, one game doesn't seem like enough. Same question for the baseball world series. Bob
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
participants (3)
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Andy Latto -
Henry Baker -
Robert Baillie