Monday, January 5, 2015

Little's Result and the Baseball Hall of Fame

Today, in purely descriptive blog post titles...

Baseball Hall of Fame voters have been getting in their annual opportunity to gnash their teeth and/or practice their sanctimony, as a result of drugs that were first banned by the sport barely a decade ago.  It's become a thing, by which I mean that it is now possible to get all "meta" about it and write not only about the Hall of Fame itself, but also about the tooth-gnashing and sanctimony-practicing that goes on around the Hall of Fame.
Here is my "meta"; here are my two big thoughts about the Hall of Fame.

I have a principle about PEDs and the Hall of Fame that is conceptually simple but practically challenging. And that is, how do I think the player would have performed if he didn’t take PEDs? If I’m certain he didn’t take PEDs, then he would have performed as he actually did. If I’m certain he did take them, then I have to correct for how he would have performed without them. If I think there’s a chance he took them (but a chance he didn’t), there’s a correspondingly smaller correction.

With that in mind, I don’t think there’s a chance in hell (or anywhere, really) that, let's say, Barry Bonds isn’t a Hall of Famer. Even without PEDs, I feel confident that he’s a top-20 player. He might be better than that, but I don’t need to know that. That’s enough to put him in the Hall of Fame with room to spare.

Other decisions are harder than that, of course. But the most damning thing about the "no PEDs in the Hall of Fame" rule of thumb is the same thing that damns so-called "zero-tolerance rules": it relieves us of our need to make judgment calls. To think. If we deny ourselves of that, why even have human voters? Why not just set a machine to the task and leave it at that? And my answer to that is, because we want and crave human approval. Well, I don’t know about anyone else (that’s a lie), but I’d like my approval to come from humans who at least exercise a bit of thought and reason in the matter.

Here’s another thing, which just got called to my attention: the ten-year limit on player eligibility. When I was just a wee lad in late youth, I’d wonder why it was that the player vote percentage would inch up slowly year after year until they either made it in, or were ruled ineligible after ten years. The other possibility—that they would be removed from the ballot after not getting enough votes—that made sense to me. But the other one was mystifying, at first.

In time, of course, I figured it out. Because of the other limit, on the number of players one is allowed to vote for, there’s limited space in the pipeline, so to speak. I received my training for my day job in a fairly abstruse field called queueing theory. It’s essentially the study of waiting in lines, and although it has some applicability to computer networks (which is indeed why I took the course), it’s usually the class that people try to avoid taking.

Nonetheless, there’s a result of queueing theory which is extremely important, is broadly applicable to fields way outside computer science, and which ought to be known by anyone who tries to make things more efficient. It’s called Little’s Result (or Little’s Law), and it is usually taught within the first six weeks of queueing theory. It goes as follows:

In any system, at equilibrium, the average number of things in that system equals the average rate at which things enter the system, multiplied by the average time they spend there.

That’s it. And as evidence that it’s applicable to lots of things, I’ll apply it to Hall of Fame voting. Voters can vote for ten players, but in order to make it into the Hall of Fame, players must receive 75 percent of the vote. Roughly speaking, that means that each year’s class can contain no more than about 13 players, and that assumes that all of the 13 players receive almost exactly 75 percent of the vote (13 times 0.75 = 9.75, leaving about a quarter vote for the remaining eligible players). Each player can stay eligible for ten years at most.

That means that at best, there’s room for 13 times 10 = 130 players in the pipeline. Everyone else will be squeezed out. And it also explains why player vote percentages inch up; the voters have to vote for players earlier on in the pipeline, to get them out of the system before they can vote in the more recent players. They have to vote for the younger players just enough to keep them eligible. This is defensible, by the way; if you just allowed people to vote for however many players they wanted to, you’d have no control on the overall consistency of the selection. More generous voters would have disproportionately more influence on the result than more selective ones.

That’s theory. In practice, of course, it’s lower than 130; I’d be surprised if it was as high as 100. Well, you might say, that’s OK. If a hundred players was good enough for Ted Williams’s time, it ought to be good enough for ours, right? All other things remaining equal, to be sure.

The problem is, all other things have failed to remain equal! The biggest culprit is expansion. In Williams’s day, the league contained sixteen teams. (Williams was voted in in 1966, when there were twenty teams, but he didn’t compete against players active in 1966; he competed against players active much earlier.) The number of players retiring to become eligible for the Hall of Fame was probably about a hundred a year—again, a number you can ballpark with Little’s Result.

Today’s league contains thirty teams, nearly twice the number in Williams’s day. A smaller but still not neligible factor is the increasing specialization in the league. There are more players playing a significant role on teams (especially with the pitching staff). The number of retiring players is therefore about twice was it was before, about 200. But here we are, trying to shoehorn all those extra players into the same pipeline we had back when the league was much smaller.

Little’s Result also tells us what you need to do to expand the pipeline. If you want to scale it to the size of the league, then you just need to expand the vote limit and the time limit enough to double the pipeline. You could expand the number of votes to 20. Or you could expand the time limit to 20. Or you could expand them to 12 and 15, and make up the difference by reducing the vote requirement to 70 percent. But keeping them the same artificially raises the bar for entry into the Hall of Fame, unless you think today’s league draws from a talent pool no larger than before, despite baseball’s growing internationalism and the world population boom.

I still care about the Hall of Fame, too. I’d rather care about a better product, but I’m human and can’t help myself: I’ll probably always care about the Hall of Fame.

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