Monday, June 8, 2009

Points are Points, Sort Of

In the wake of last night's Finals Game 2 between the Lakers and the Magic, which the Lakers won in overtime, 101-96, a lot of attention was focused on various plays that the Lakers made down the stretch and the Magic didn't. Now, obviously, in a game that close, there were plays—even down the stretch, at least in regulation—that the Magic made and the Lakers didn't, and if the game had gone the other way, we'd be talking about those plays. But this just by the way.

Some folks pointed out that although those plays late in the game are magnified in our mind, they aren't worth more on the scoreboard than plays earlier in the game, even in the first quarter. A clutch shot made with the game clock running down is not given more points than an identical shot made in the opening minutes. So undue attention, it is claimed, is placed on, say, Courtney Lee's missed lay-up with 0.6 seconds left in regulation, a shot that would have given Orlando the win and a split in the first two games of the series. (I was watching the game, by the way, and I somehow totally missed the play developing: the jab step, the perfect screen from Rashard Lewis, everything. Thank goodness!) If the Magic simply make one of the two-pointers they missed earlier in the game, it doesn't come down to Lee's make or miss at the end of the game.

It all sounds reasonable, doesn't it? It did to me, too, at first.

Except how do we square this line of thought with the end of Game 1, which the Lakers won going away? At the very end of the game, the Lakers are leading 97-75, and they inbound the ball with the game clock showing ever so slightly more time than the shot clock. If it had been the other way around, it is one of the Great Unbreakable Rules of the game that you are not supposed to shoot, and just let time run out. But for some reason, if the shot clock isn't turned off, you get to shoot with impunity. Never mind that the Magic couldn't possibly have fired off a 22-point shot with only a couple of seconds left in the game. Anyway, with time running out, end-of-the-bench Lakers forward Josh Powell dribbles to his left and hoists up a three-pointer that amazingly goes in. It is the first three-pointer of his entire career, playoffs or regular season.

So, I don't think you'd have any problem convincing anyone that this shot was meaningless. It turned a 97-75 blowout into a 100-75 blowout. It almost certainly didn't mean much in Vegas: I'm sure the Lakers beat the spread, pretty sure that this kept the game in the under.

The problem is, if this shot is meaningless, and three points is three points, then isn't every other shot the Lakers made similarly meaningless? Are we supposed to think this shot was almost meaningless? Perhaps, if we add up enough "almost meaningless" shots, we actually get a meaningful result. Personally, I don't buy that. In terms of the actual game and series (in other words, ignoring Vegas, which probably had some incredibly tangential bet involving Powell and a trey at the end of the game), this shot was not just mostly meaningless, it was entirely meaningless.

What I'm going to propose is a kind of probabilistic importance—the idea being that points matter to the extent that the game is in doubt at the moment, to the extent that they bear on the result of the game. I've seen, as a kind of experimental thing with the NFL on some sports Web sites, a play-by-play measure of the winning probability for the team that makes the play. If the Baltimore Ravens score a touchdown, it increases their chance of winning from, let's say, 43 percent to 59 percent. And so on.

Now, imagine the same gadget being used for basketball. How much do you suppose a two-point basket is worth in the opening moments of the game, when the winning probability for both of two evenly matched teams is 50 percent? Actually, more than you might think. Suppose the standard deviation on scoring difference between the two teams is 10 points, and that teams score about a point per possession, close enough. A two-point basket is an increase of one point over what was expected for that possession, and a single point—0.1 standard deviations—is worth about 3.6 percent. In other words, that two-point basket would increase the winning percentage from 50 percent to 53.6 percent. If, on the other hand, the shot was missed, the winning percentage would drop from 50 percent to 46.4 percent. That shot is a swing of 7.2 percent, believe it or not.

Now let's consider the same shot in the closing seconds of the game. The team with the ball is down one, and is holding for the final shot. Obviously, if they make the shot, their winning probability is 100 percent; if they miss it, it's 0 percent. The percentage swing here is 100 percent, and clearly 100 >> 7.2.

But this huge swing is counteracted by the fact that in most cases, the game doesn't come down to that. Most of that time, that shot would be worth 0.4 percentage points, or 1.1, or something like that. At the very end of the game, it would be worth 0 most of the time. On average, that two-pointer would be worth 7.2 percent, just like the earlier shot was. It's sort of like the lottery: Would you rather have 35 cents, or a lottery ticket that gives you a one in 100,000,000 chance of winning 35 million dollars? On average, they're both worth 35 cents. But I think you'd have a hard time convincing yourself they're exactly the same.

So, I guess, I'm not letting Courtney Lee off the hook. Make the shot, and the winning probability swings from 50 percent (overtime) to 100 percent (game over, Magic win). Two points is two points, but I think people's intuition is right: When the points happen matters, and matters a lot.

EDIT: I corrected some of the above exposition to account for the fact that the hypothetical early-game two-pointer can be missed, which is one point lower than expected for the possession.

Secondly, here's a more self-contained example of this kind of probabilistic importance. Suppose that the two teams are evenly matched—50/50 to win each game, home or away. In a seven-game series, the swing for the series win in a Game 7 is obviously 100 percent: The team that wins Game 7 wins the series. However, Game 7 only gets played when the series goes 3-3, which happens about 31.2 percent of the time. In contrast, Game 1 gets played 100 percent of the time. However, it isn't as pivotal as Game 7: It can be shown that the Game 1 winner's odds of winning the series go from 50 percent to 65.6 percent, and the losing team's odds from 50 percent to 34.4 percent. That's a swing of 31.2 percent. So Game 1 swings the odds by 31.2 percent, 100 percent of the time, whereas Game 7 swings the odds 100 percent, 31.2 percent of the time. They therefore have exactly the same average importance, but Game 7 is obviously more important when it does get played.


  1. This is a great way to put some numbers on an intuition we all have but couldn't well-justify (at least I couldn't). I wonder, as an extension of your logic, if there could be some kind of way to actually compute a statistically meaningful "probability to win" at any point during a game. To get started, you'd want to determine the mean and standard deviation of, say, points per minute (and possibly you'd want these numbers for each team separately). Given how many minutes are left in a game and the current score, you have some distribution of possible final scores for each team, and then it's like a signal detection problem - you could do an ROC analysis to get the probability to win.

  2. There are two ways (or maybe three) that you could handle this. You could do it based on historical evidence, if there are enough games to match up with all conceivable circumstances, as identified by a triple: (score differential, time remaining, team with ball). Or, you could do it the way Hollinger does with his playoff simulator, which is to have a discrete event simulation of each play in the game. Or, you could do a hybrid, by running the simulator until you get to a good renewal point (such as end of quarter).

    I'm not sure how meaningful such numbers would be, although perhaps you could get something insightful out of the *delta* in winning probability (as opposed to the actual estimated winning probability).