Thursday, June 4, 2009

The NBA Finals and the 2/3/2 Format

I'm not going to talk about these upcoming NBA Finals, actually. I want to, but I also want to avoid the wrath of the Gods of Woof.

Instead, I'm going to say a few words about the so-called "2/3/2 format" of the NBA Finals, which refers to the sequence of venues for the seven games. As in, the first two games and last two games are played at the court of the team with the better record, while the middle three games are played at the court of the team with the worse record. (Usual tiebreakers apply.) This is in contrast to the other 14 series in the NBA playoffs—and, for what it's worth, every series in the NHL playoffs—which use the 2/2/1/1/1 format. You'll excuse me if I don't spell out in gory detail how that format goes.

Each year, at around this time, we read and hear the same time-worn opinion pieces about how the difference in format (which is intended to minimize travel for the teams) affects the chances for the two teams. Some think that the difference favors the underdog, because if home-court advantage holds for the first five games, the favorite has to return to its own court having to win the last two games. Others think the difference favors the favorite, because to get to that point after five games, the underdog has to win three games in a row.

The first thing I want to do is dispense with this notion that the format confers any kind of inherent advantage to either team. Occasionally, one sees it pointed out that the odds for the two teams are unaffected by the difference in format. One relatively simple way to see this is that the result of the series is not changed at all if we somehow force both teams to play the full seven games, even if the series is already decided before that point. (I'm sure Madison Avenue is all for that.) Since the favorite hosts four games and the underdog hosts three, no matter which format is used, the odds should be the same.

It's important to note that this line of reasoning assumes that the game results are independent of each other. If the results of earlier games can statistically affect the results of later games, that argument loses force and it becomes quite possible that the series result could in fact be affected by the format.

In this light, one interesting observation that I haven't seen before (and it might just be that I haven't looked hard enough) is that the two formats are identical except for one small change: The venues for Games 5 and 6 are switched. Otherwise, Games 1, 2, 3, 4, and 7 are played in the same place in both formats. So let's restrict our analysis of the format to just those two games.

What are the possible situations going into Game 5? We can eliminate series sweeps, because in those cases, Game 5 never gets played. So either the series is tied 2-2, or else one team is up 3-1. And let's suppose that we believe in the notion that players tighten up under pressure (which we'll assume is the case if they're playing to stay in the series), lowering their winning percentage. What effect does the difference in format have under these assumptions?

If the series is 3-1, the team that's down is under the gun for both Games 5 and 6, and the difference in format doesn't have much effect at all. If, however, the series is tied 2-2, there's no more pressure on one team than the other in Game 5, but the loser of Game 5 has the pressure on them in Game 6. Now, look at this from the perspective of the underdog. Let h represent their winning percentage at home, v their winning percentage on the road, and µ the effect of pressure. In the 2/2/1/1/1 format, with Game 5 on the road for the underdog, the expected number of wins in Games 5 and 6 is then

E[wins] = v (1 + h + µ) + (1 - v) (h - µ) = v + h + 2µv - µ

In the 2/3/2 format, with Game 5 at home for the underdog, the expected number of wins is

E[wins] = h (1 + v + µ) + (1 - h) (v - µ) = h + v + 2µh - µ

Under these assumptions, the 2/3/2 format is better for the underdog, yielding on average 2µ (h - v) more wins for them when the series is tied 2-2 going into Game 5. In ordinary terms, playing Game 5 at home gives them a better chance of taking advantage of the pressure factor in Game 6, and a lower chance of suffering from pressure themselves. What's more, the win differential goes up in proportion with both the pressure factor µ and the home-court advantage (h - v).

Of course, you should take this with a sizable grain of salt. I'm using a very simplistic model of home-court advantage and pressure. If you like, you can extend the model to let the two teams have different pressure adjustments, or even have lack of pressure (instead of pressure itself) depress winning percentage. The more important thing to take out of this is the set of factors that bear into the difference between the two formats, because under most conditions, they really aren't as different as they're made out to be.

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