Leveling the playing field with math and statistics
March 28, 2007A recent blog entry by Brian Hayes, one of my favorite science writers, discusses the age-old ritual of dividing a group of kids into two teams:
The simplest algorithm has two captains, A and B, who take turns choosing players until everyone is assigned to one team or the other. Call this the ABAB algorithm. Donald B. Aulenbach suggested a very easy modification that produces more closely matched teams. Aulenbach’s proposal is the ABBA algorithm, where A gets the first pick in the first round but B goes first in the second round, and they continue alternating in successive rounds. (Another way of describing the same process is that A begins with a single turn and thereafter both captains take two turns in a row.)
Hayes then shows some simulation results confirming that the ABBA algorithm does indeed divide up the players more evenly than ABAB. He concludes,
Going back to my own childhood, I don’t think the kids in my neighborhood ever discovered the ABBA algorithm. We did recognize the inherent advantage of choosing first, and we compensated by adopting a separate ritual to decide who got the first pick. In baseball, this involved a hand-over-hand struggle for a grip on the bat. Sometimes I think the preliminaries were more fun than the game.
Another interesting mathletics tidbit was buried in a recent University Week article about Generation IX, a documentary on the University of Washington women’s volleyball team:
In the documentary, the viewer sees the team statistician toting up every play in practice as well as in games. As one of the players puts it, “If your numbers aren’t there, you know you’re not going to start.” The women say they like this objective system, because nobody can say, “Oh, it’s because [Coach McLaughlin] likes her better.”
The idea of determining playing time by objective statistical criteria is appealing to me. Could this approach be used in every sport at every level?
In disciplines like cross-country, of course, assessing performance is extremely straightforward — you just line up your runners and yell “Go!” and see who gets to the finish line first. But I wonder whether there are adequate statistics for quantifying the success of, say, football linemen or soccer fullbacks. And even if there are, what sample sizes would be necessary to determine that one person is “significantly” better than another? For example, if two shortstops are equivalent defensively, should you play the one with six hits in 20 at-bats over the one with four hits in 19 at-bats? I’m not so sure.
My Track Record 
"In disciplines like cross-country, of course, assessing performance is extremely straightforward — you just line up your runners and yell "Go!" and see who gets to the finish line first."– Well… this does not take repeat performace and injury over time, into the equation. It does make one-time, big-time wins seducing much like the lottery as in "If only I could win that one race one time, I'd have it made…." Is it sufficiently accurate to evaluate performance in cross country or track only in increments of 1 race? Without a clearly pre-established sampling period? Of races and training? Alison had a post possibly touching on this recently about the difference between DI and DIII environments. Maybe it could be objective statiscal criteria only indicates chance of "play", based also on "play" performance past, but in itself is not enough.As far as the two short-stops are concerned, it also could depends on who the opposition is… Beyond the math, one player might have some advantages over his colleague which are enhanced in certain conditions, by certain opponents… otherwise what should we do with the idea of "strategy"?As far as the kids are concerned, the palindromatic symmetry of the ABBA algo is readily visible if you assign number values: if, say the first pick has a value of 10, second pick 9, third pick 8, fourth pick 7…. and so on… after the first round A will have a value of 17 and B will also have a value of 17, instead of 18 and 16. But as far as teamwork is concerned, would you ever "certify" that the 1 + 4 = 5 has the same talent and capacity and "harmony" to play as well as the 2 + 3 = 5 team? They are equal, yes. But will they ever be the same?Anyway, I'm sure someone will correct the math 🙂
forgot to sign above.
That's OK, Corrado — I knew it was you.The beginning of your comment points out the obvious limitations of having a single qualifying race, which I wouldn't recommend under most circumstances (although the US Olympic Trials work that way). My point was simply that it's a lot easier to rank athletes in sports like cross-country running than in certain other sports. Your thoughts on the shortstops and kids are consistent with that idea, I think.
Greg, check out the book Moneyball, you are on to it.http://www2.wwnorton.com/catalog/spring04/032481.htm
"I knew it was you."Greg, sorry for being such a pain in the ass. It may seem like I am after yours, whereas, in fact, I am after Science's ass. Not for proprietorship. Not biased either way. Really just to keep agile, no more. But there is more: I am much intrigued by many of Science's B and C (even D) grade derivatives, often very unconvincingly applied in all sorts of fields. I'll go away if you want. But I hope you'll stay patient. It's good practice for parenthood :-)JL Philips: thanks for a great headsup on a definite future read!
Corrado, you are welcome to continue reading and commenting as often as you'd like. Though your comments often surprise and sometimes baffle me, I know that you offer them without malice.
ah, Greg, you understand… this is like… net-bonding… wow… 🙂