riddle@emory.UUCP (Larry Riddle) (10/11/86)
Last fall I posted a request for "real world" applications of stochastic processes that I could use in a course I was going to teach. Many people sent me suggestions, one of which was an example in Kemeny and Snell's book on finite Markov chains in which they consider a game of tennis as a simple random walk (when the score reaches deuce). Well, the students really liked this example, so this summer, just for fun, I spent some time expanding upon their model and wrote a short paper comparing the four common scoring systems currently in use in tennis - the original rules (win by two games), the 9 point tiebreaker, the 12 point tiebreaker, and the Van Alen No-Ad scoring systme. My basic assumption was that each player has a fixed probability of winning a point when that player serves. I then derived formulas for computing the probabilities of winning a set under the four scoring systems. Of course, if each player has the same probability of winning a point when serving, then the probability of winning a set should be 1/2 for each (it makes a difference with the 9 point tiebreaker which player serves first, so there I average the probabilities obtained for when each player is the first server of the set). However, if one player has a small advantage over the other in serving efficacy, then this advantage gets magnified when considering the entire set. Which scoring system produces the least magnification? Interestingly, the 9 point and the No-Ad systems are "best" in this respect. If anyone is interested in seeing a copy of the paper, just sent me mail and I will send you one.
desj@brahms (David desJardins) (10/23/86)
In article <1715@emory.UUCP> riddle@emory.UUCP (Larry Riddle) writes: >[...] However, if one player has a small advantage over the other in >serving efficacy, then this advantage gets magnified when considering >the entire set. Which scoring system produces the least magnification? >Interestingly, the 9 point and the No-Ad systems are "best" in this >respect. This has nothing to do with mathematics, but isn't the system that produces the *greatest* magnification "best"? Presumably the objective is to determine the better player, so it seems desirable to have the most sensitive possible tool for doing that. As for the statistics, I hope that you took into account the length of the sets resulting from the various scoring systems. Otherwise, all you have discovered is that in longer sets the inferior player's chance of winning is reduced (hardly a revelation!). The question you probably want to ask is something like, "Given a certain average number of points to be played, which scoring system best uses those points to discriminate between the players?" Figuring out exactly what question to ask and how to answer it is actually a fairly interesting problem -- I'm not a statistician, so I'll leave it to them to discuss if they wish. -- David desJardins
wjh@wayback.UUCP (Bill Hery) (10/23/86)
> Last fall I posted a request for "real world" applications of stochastic > processes that I could use in a course I was going to teach. > Many people sent me suggestions, one of which was an example in > Kemeny and Snell's book on finite Markov chains in which they consider > a game of tennis as a simple random walk (when the score reaches > deuce). Well, the students really liked this example, so this summer, > just for fun, I spent some time expanding upon their model and wrote > a short paper comparing the four common scoring systems currently in > use in tennis - the original rules (win by two games), the 9 point > tiebreaker, the 12 point tiebreaker, and the Van Alen No-Ad scoring > systme.... > .................Which scoring system produces the least magnification? > Interestingly, the 9 point and the No-Ad systems are "best" in this > respect. > I would think that the best is the one that produces the MOST magnification; i. e., the best way to score tennis is the way in which when player A is only a little better than player B, then A has a very high probability of winning. The best method is thus the best dicriminator between players. If I remember correctly, Kemeny and Snell also did an analysis for the world series format (win 4 out of 7), and showed that it was a much worse discriminator than even a single game of tennis with the regular deuce rules. Also, please email me a copy of the paper you wrote. I couldn't reach you via the paths you posted. Bill Hery ihnp4!bonnie!wayback!wjh