One of the speakers thought the Dodgers should have done

*something*at least. He based his assertion on the notion that there is such a thing as a championship window, and that many teams, including the Dodgers, don't pay enough attention to that, but instead meander from season to season, doing their best to maintain the best team they can within the strictures of their finances. He felt that the Dodgers should instead opportunistically go "all in" for a season or two, to maximize their chances of winning a title within that window, and pay the cost of mediocrity (or worse) down the road, rather than maintain respectability on a continual basis at the cost of never winning a title.

Actually, I rather think he overplayed the extent to which teams are unaware of their championship windows, the way that he was describing them. I tend to believe the Dodgers are perfectly aware that there is a finite window for them, since that is true for everyone. (Even the Yankees.) Nonetheless, let's take a look at the championship window, and maybe there's something interesting to be divined from it.

Normally, when people think of a championship window, they tend to think of it as having a certain length—of time, that is, usually measured in years. The Miami Heat have had a window of about five years, during which they went to the NBA Finals four times and won two championships. It appears to be closing, given the departure of LeBron James, but it hasn't shut entirely, I think most people would say.

The fact that people do think of a championship window as having gradations of openness suggests that there's a second dimension to the championship window: its height, which we might conceive of as representing a team's probability of winning a championship during any given year. For instance, if the Dodgers have, let's say, a 15 percent chance of winning the World Series any time in the next three years, we might say that the window is three years long—or wide, perhaps it's better to say—and 15 percent tall.

The sports talk host's opinion might then be construed as being that the Dodgers should have made some kind of deal that might shorten the window to two years, but increase its height to 22 percent, or 25 percent. Would that be worth it? Well, let's think about that a bit. If you start off with a 15 percent chance of winning a title in each of three consecutive years, that means that at the end of the window, you'll have won 0.45 titles on average.

If, instead, you have a two-year, 25-percent window, you'll win an average of 0.50 titles. On that basis, we might consider that kind of deal to be worth making (if you can make it). On the other hand, if you have a two-year, 22-percent window, you'll win an average of 0.44 titles, which would seem to make that deal just barely not worth making.

The average title count isn't all that matters, however. Extra importance is attached to the first title; there's a much bigger jump perceived from zero titles to one title than there is from one title to two titles (or, conceivably, to any larger number of titles). We might evaluate championship windows based on the probability of winning

*at least*one title during that window.

A three-year, 15-percent window wins at least one title about 38.6 percent of the time, a two-year, 25-percent window wins one about 44.8 percent of the time, and a two-year, 22-percent window wins one about 39.2 percent of the time, which would (according to this standard) make that deal just barely worth making.

Of course, a window need not be uniformly high. Maybe the Dodgers could make a deal that would put their title probability up to 30 percent in 2014, but have it drop to 10 percent in 2015, and just 5 percent in 2016. That would yield an average title count of 0.45—same as the initial situation—but now the probability of winning at least one title would be 40.2 percent.

At this point, it occurred to me that there's one aspect of championship windows that people don't talk about a lot, and when they do, it's not really couched in terms of the window. The fact is that multiple teams can have championship windows at the same time, and when they do, they tend to squeeze against each other. Imagine a top-heavy league in which two teams each have a three-year, 40-percent window, and the remaining 20 percent of a title is parceled out to all the rest of the teams. Those two teams would each win, on average, 1.20 titles in the next three years, and each would have a whopping 78.4 percent chance of winning at least one title during that time.

Now, suppose that one of those two teams can make a deal that, in isolation, would front-load their window, raising it to 65 percent this year, but dropping it to 40 percent the following year, and only 15 percent the year after that. The average title count would remain at 1.20, but the probability of winning at least one title goes to 82.2 percent. Seems like a marginally better deal, right?

But what if the other team could make the same deal? Worse yet, what if the other team could make the same caliber of deal, but an entirely different one, so that both deals could be made at the same time? They can't

*both*win a title this year with a probability of 65 percent; the best they can do is win one with 50 percent. And in fact, it would very likely be less than that—let's say, 45 percent. Perhaps, as a result of both front-runners making that deal, they would win the following year at 40 percent each, and the year after that at 20 percent each.

That yields an average title count of "only" 1.05, and a probability of winning at least one title of "just" 73.6 percent. In other words, both teams are still good, but somehow worse off now than if

*neither*of them had made a deal. On the other hand, it's also the case that either team would be better off making the deal, whether or not the other team made their deal, which makes this situation a little Prisoner's Dilemma-ish. (This reminds me that I've never written a post on the Prisoner's Dilemma, and I really should get to that at some point.) It intrigues me that two of the also-ran teams could screw the front-runners up by conspiring to offer them both "good" deals.

In practice, of course, you probably couldn't force the two front-runners to pull the trigger on the deals at the same time. One deal would almost certainly make the news before the other. At that point, it's not clear what the other team would do. The

*rational*thing to do might be to search for some other deal that has the same general impact, but perhaps

*back-loading*it so that the championship windows dovetail with each other, rather than squeezing each other out. My intuition, though, suggests that the other team would probably try to engage in the arms race, so perhaps my Prisoner's Dilemma-ish scenario would still play out.

*Yes, I realize that they are technically the Los Angeles Angels of Anaheim. You'll pardon me if I refuse to employ that ungainly circumlocution.

(Also, this post would probably benefit from some figures. I'll try to add them at some point.)