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.

Coke, Currency, and Contagion

Recently, there was a report, from the American Chemical Society, that about 90 percent of U.S. currency in circulation has detectable traces of cocaine on it. Apparently, the middle currencies—from Lincoln on up through Jackson—are the most susceptible. I guess Washington and Franklin don't rate. Also, not surprisingly, the percentage varies according to the community. Rural areas are less hit by cocaine-laden dollar bills, but in major metropolitan centers, essentially every piece of currency has coke on it. What's more, the percentage appears to be rising. In 1985, a study found that anywhere from a third to a half of bills had cocaine on them; in 1995, the proportion was three in four; and in 1997, it rose to four in five. Now it's nine in ten.

No need to panic, though. First of all, the traces are generally tiny, much smaller than a grain of sand, and not enough to get any kind of buzz from. And secondly, probably much, though apparently not all, of this increase has to do with the improved sensitivity of the cocaine sniffing tools.

The question is, how does cocaine get on all these bills? Certainly not all of the bills get cocaine on them because they were directly around the stuff, either during deals or during use. A small number do, of course, but the vast majority get them through contamination. But is that really plausible? Can so many bills be contaminated so quickly?

Well, let's take a look at that. Suppose that, initially, some small fraction of all the dollar bills have detectable cocaine on them; these are the initial set that get cocaine on them through direct contact with bulk quantities of the drug. Let's call this proportion p. The money isn't discarded, generally; it's put back into circulation (let's not get into how they get put back into circulation). Once that happens, those bills come into contact with other bills, which pick up some proportion of the drug. Apparently, there's an attraction between the drug particles and the green ink used to print U.S. currency.

When I use a bill, and it goes somewhere else, it now comes into contact with, let's say, one new bill. If a contaminated bill comes into contact with another contaminated bill, nothing happens to p, of course; both bills were already contaminated. Same thing holds true if an uncontaminated bill comes into contact with another uncontaminated bill.

But if the bill I had was contaminated and its new companion wasn't, or vice versa, then one new bill gets contaminated. The probability of this happening depends on the current value of p; specifically, it must be proportional to p (1 - p), since we need a contaminated bill and an uncontaminated one. We can put this in terms of a differential equation:

dp / dt = kp (1 - p)

The constant of proportionality k indicates how quickly bills come into contact with one another, and can be eliminated by setting the unit of time equal to the mean time it takes for a bill to be used (and therefore find a new neighbor). I don't have any hard figures, but from my own, non-cocaine-related currency use, it seems to be about a week or so. We can then set k = 1 and solve this equation fairly straightforwardly to yield the formula

p = C e ^t / (1 + C e ^t )

where C is closely related to the initial proportion of contaminated bills. (To be exact, C = q / (1 - q), where q is the initial proportion. Where q is very small, as in most cases, the two are almost exactly the same.) As t increases, C e ^t gets large pretty quickly, and p very quickly approaches 1. If, for instance, q = 0.000001—that is, one bill in a million is contaminated directly by the drug—then it takes a bit more than three months for the fraction of contaminated bills to exceed one-half. But because of the rapid growth of the exponential function, it takes only one more week for the proportion to exceed three-fourths. By the end of the fourth month, the fraction of uncontaminated bills is less than one percent. (Click to enlarge.)

That exceeds even the ACS's report. Why? Well, for one thing, even today's instruments are not perfectly sensitive; there still remain bills with undetectable traces of cocaine, surely. And after a while, there just isn't enough cocaine to go around (for the bills, that is). If, for the sake of argument, we assume that the initial fraction is one in a million, then the ACS's estimate of 90 percent contamination indicates that that first direct contamination can only be split about twenty times before it drops below undetectability.

But a second reason is that bills don't stay in circulation forever. According to the U.S. Treasury, currency stays in circulation, on average, for about 20 months—about 85 to 90 weeks. This makes the dynamical solution to the differential equation a bit more complicated. Let's simplify matters and only look at the equilibrium solution. At equilibrium, the contaminated dollar bills being taken out of circulation each week equal those being contaminated by new contact each week. That is,

p (1 - p) = rp

which yields an equilibrium solution of p = 1 - r, where r is the fraction of bills being taken out of circulation each week (about 1/85 to 1/90). So even with this new influx of bills, if detection tools were perfect, they'd detect traces of cocaine on about 99 percent of bills. Apparently, we still have a few rounds of "alarming" reports about cocaine contamination of currency to look forward to.

OK, here's a less overblown concern. The same model can essentially be used to analyze long-lived infections (such as oral herpes, which infects about 60 to 70 percent of all people worldwide). Such infections are removed from the population only when a person dies. As the above models show, if people were immortal, they'd eventually all be infected with such diseases (and in fairly short order, too). Of course, such diseases couldn't incapacitate their hosts too much, because otherwise they'd fail to be transmitted.