PLAYING THE PERCENTAGES

By RANDOLPH COHEN

In Defense Of Going For Three

even the riots are over. Before we close the final chapter on this year's NBA season, however, one popular piece of conventional wisdom requires correction: Larry Bird, the coach of the Indiana Pacers, has been roundly criticized for having Reggie Miller attempt a three-pointer in the closing seconds of Game Four of the NBA Finals, when a two-point shot would have tied the game. However, Bird's strategy was smart, and it should be applauded.

Nothing Fails Like Failure

We often believe that "nothing succeeds like success" -- and, as a result, we laud winners and criticize losers based on results, regardless of the merits of the strategies they employed. An attorney who loses gets little credit for taking a difficult case. A campaign manager who runs a brilliant campaign but loses due to a bad economy is still, to us, a loser. Losers who -- like Bird in Game Four -- tried risky or unusual strategies are subjected to special contempt. (Gregg Easterbrook, writing in Slate, summarized this position, writing, "Either Bird can't coach or that guy on the sidelines isn't the actual Larry Bird.") In fact, Bird's strategy was right.

Reggie Miller is perhaps the best long-distance shooter in basketball. This year, he connected at a rate of just over 40 percent from beyond the three-point arc. So, a Miller shot at the buzzer gives the Pacers a 40 percent chance to win, right? Well, not exactly. Shots at the buzzer of a Finals game -- or any game -- are hardly typical. The defense is more alert at these times. On the other hand, they are also more careful to avoid a foul, which could lead to three potentially devastating free throws (Miller almost never misses a foul shot). So, the 40 percent figure is a reasonable approximation. Therefore we can assume the Miller play gives Indiana a 40 percent chance of winning.

Overtime?

Now suppose that instead the Pacers had tried for a two-pointer instead. They'd have to both make the shot, and win in overtime play, in order to win. That means that, according to the laws of probability, their total chance of winning the game is the chance of making the shot multiplied by the chance of prevailing in overtime play.

Would the Pacers have made the shot in the first place? Their best overall percentage shooters are Rik Smits (48.4 percent this year) and Dale Davis (50.2 percent). While Davis' statistic is just a little better, he ups his average with a lot of putbacks. So let's assume they'd go to Smits for the two-point shot. And, let's call Smits a 50 percent shooter to make it easy (Smits, like Miller, would be guarded more closely than usual at crunch time; it seems fair to suppose these two effects cancel out in choosing between them). If we oversimplify a little once again, we can suppose a play for Smits would have been 50 percent likely to send the contest to another overtime.

If Indiana had made the two-point shot, would it then have won in overtime play? I would estimate the probability of a win as approximately 50 percent -- about the same as the probability of initially making the shot. Why? First consider that these two teams are the two finalists, which strongly suggests that their abilities are fairly closely matched. (Note, for example, that they had played to a standstill that very evening.) Most importantly, note that in a five-minute game, the role of luck is much larger than in a full-length battle. Over the course of a game, and especially over the course of a number of games, quality will out. But over five minutes anyone can hit a few shots and win. That makes the outcome even less certain, and the 50 percent estimate even more reasonable as an assumption.

One can try to analyze things more deeply, taking into account the Lakers' general superiority, the fact that the mighty Shaquille O'Neal had fouled out, or Indiana's home court advantage. These factors could change Indiana's likelihood of winning to, say, 47 percent or 53 percent, but that's about it. So the 50 percent chance is a reasonable approximation of the Pacers' likelihood of winning in overtime. And the Pacers' chance of winning the whole game after deciding to take the two-point shot is reasonably estimated as 25 percent -- the 50 percent chance of making the shot, times the 50 percent chance of winning in overtime.

That's lower than the 40 percent chance of winning with Reggie Miller taking the three-point shot. Much lower. So much lower that the only three ways to justify having Smits take the two-point shot are, first, to assume that either Smits is 80 percent likely to score; second, to assume that the Pacers are 80 percent likely to win in overtime; or third, to make two separate assumptions, such as that that Smits is more than 63 percent likely to score AND that the Pacers are more than 63 percent likely to win in overtime. As the Brits say, "not bloody likely." Fiddle with the assumptions all you want. Unless you assume something absurd, Miller's three-point shot will be the right choice, by a mile.

Can this point be generalized, to encourage coaches to go for the three-point shot more often in this situation? Absolutely. Not every team has a long distance shooter as talented as Reggie Miller. But even my beloved Philadelphia 76ers -- who are about the worst in the league at shooting the "trey" -- have Aaron McKie making the shot 36 percent of the time, and Allen Iverson making it 34 percent of the time. Both shooters, then, have an average that's far above the 25 percent chance (calculated above) of a team winning by going for the tie.

Putting it all together, we can reach a simple conclusion: when down by two points with one possession left, take a three-pointer unless 1) the opposing team is vastly inferior and you have home-court advantage, thus you are very likely to win in overtime; or 2) all your decent outside shooters are badly injured so your chance of making the three-pointer is slim. To be honest, option one probably never applies in the NBA -- the teams are just too close, even if it's Lakers-Grizzlies.

Why do coaches generally ignore this obvious truth and go for two? After all, the chance of winning they give up (40 percent - 25 percent = 15 percent) is bigger than the 12 percent difference between the New York Knicks (who win 61 percent of the time) and the Dallas Mavericks (49 percent). Why voluntarily turn the Knicks into the Mavs? Sadly, it's for appearances' sake. If they get the two-point shot and lose in overtime, it looks like their team made a heroic comeback. Had the Pacers scored two points and forced the game into a second overtime, and then lost, does anyone really think Bird would have been criticized for not having Reggie Miller shoot a three-pointer? And even if they miss the two-pointer, no one's going to say that if they'd gone for the tougher three-point shot they'd have made it.

One can debate the wisdom of some of Bird's decisions throughout the series -- such as not having his team double-team Shaquille O'Neal as often as they might have, or playing the injured Travis Best instead of Mark Jackson at certain crucial junctures -- but the scorebook should reflect that he played the percentages correctly at the end of Game Four. Coaches who care about winning instead of looking good should emulate him.

Randolph Cohen, Ph.D., is an Assistant Professor of Finance at Harvard Business School. He does not want to see Allen Iverson leave the 76ers.

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