It's True

This man has no gut.

Points on the Board

In the wake of the Lakers' mud-slogging Game 7 win in the NBA Finals over the Boston Celtics by the score of 83-79, some fans were incredulous that a team could shoot 32.5 percent (27 of 83) and still win. In fact, many of them felt that the Celtics lost the game, rather than the Lakers winning it. To me, that sounds a little silly, inasmuch as basketball is a head-to-head sport. If the Lakers were shooting that poorly, presumably the Celtics had something to do with that, and just as presumably, the Lakers were doing something else to win the game.

So what was that something? I'll give you a little hint. It begins with "offensive," and it rhymes with "rebounding."

In the unlikely event you haven't caught on, a major key to the Lakers' victory was their offensive rebounding; they won that battle 23-8 over the Celtics. To be sure, gathering 23 offensive rebounds is usually a dubious feat, for it requires the team to miss far in excess of 23 shots. So to a large extent, the dominance of the Lakers on the offensive boards was a reflection of their miserable 32.5 percent shooting clip.

However, the Celtics only gathered 32 defensive rebounds, meaning that of the 55 rebounds available after Lakers misses, the Lakers collected almost 42 percent of them. So not only did the Lakers get a lot of offensive rebounds, they got them at an stunning rate, and that doesn't depend on how many shots they missed. To give you an idea of just how stunning that is, the NBA league average is about 26 percent. The Lakers were more than half again as effective at getting offensive rebounds. By contrast, there were 38 rebounds available on the Celtics' offensive end, and they got only 8 of them, for an offensive rebounding rate of 21 percent, a bit lower than average.

That suggested the following little puzzle: All those offensive rebounds increased the Lakers' overall efficiency at the offensive end, by giving them extra shots at the basket on each possession. Can we express that increased efficiency in terms of shooting percentage—in effect, collapsing the two figures into one?

I believe we can. Suppose for the moment that we don't care about free throws, three-point shots, and all those aspects of scoring that in truth are rather important. We only care about the raw shooting percentage. The Lakers hit 0.325 of their shots. If their offensive rebounding rate was 0 percent, then the fraction of their shooting possessions (as opposed to possessions that end with a turnover, say) that they score on is 0.325.

However, in truth, they rebounded 0.42 of their misses. They miss 1 - 0.325 = 0.675 of the time, so out of all their shooting possessions, they end up with the ball again on 0.42 × 0.675 = 0.28 of the time. Then they'll score 0.28 × 0.325 = 0.09 of the time, and so on. If they miss, they can rebound again, which they'll do 0.28 × 0.675 × 0.42 of the time. And so on.

It's much more concise to put this symbolically, as follows:

Fraction of shooting possessions ending with a score = 0.325 + 0.675 × 0.42 × 0.325 + 0.675 × 0.42 × 0.675 × 0.42 × 0.325 + ...

Each time, there's an extra factor of 0.675 × 0.42, representing the Lakers missing and then picking up the rebound. Since this can happen an arbitrary number of times in a given possession, this equation can have an infinite number of terms (well, limited only by the length of the game). This is called a geometric series, and fortunately, there's a simple formula that allows you to calculate the sum without adding and multiplying an infinite number of terms. Therefore,

Fraction of shooting possessions ending with a score = 0.325 ÷ (1 - 0.675 × 0.42) = 0.45

That is to say, 45 percent of shooting possessions end in a score for the Lakers. Not to put too fine a point on it, that's still fairly awful. But not as awful as the original shooting percentage suggested.

But now, as I said, I'm going to try to combine the offensive rebounding and the shooting percentage into a single composite figure, by asking this question: Suppose the Lakers gathered only 26 percent of their misses as rebounds (the league average), instead of the 42 percent they actually gathered. How much better would their shooting have had to be in order to match that 45 percent per-possession efficiency? In symbolic terms, solve for x:

x ÷ (1 - (1- x) × 0.26) = 0.45

I'm not going to make you do that for homework; I'll just give you the answer: It turns out that x = 0.38. In other words, if the Lakers had crashed the offensive boards like an average team, they would have had to shoot 38 percent in order to score on 45 percent of their shooting possessions. Like I said: Bad, but not historically bad—not bad like 32.5 percent bad.

To put it another way, their tremendous offensive rebounding was worth 5.5 percentage points on their shooting. That's huge: 5.5 percentage points is usually worth about 10 points on the scoreboard by the end of the game.

We can turn this approach to the Celtics, too. They shot 41 percent (29 of 71), and picked up 21 percent of their offensive rebounds. That means that they ended 47 percent of their shooting possessions with scores:

0.41 ÷ (1 - 0.59 × 0.21) = 0.47

However, if they had just rebounded like an average team on their offensive end, they could have shot a bit worse and still matched that per-possession efficiency. Solve for y:

y ÷ (1 - (1- y) × 0.26) = 0.47

Again, I'll save you the algebra and give you a peek in the back of the book: y = 0.395. That is, if the Celtics were an average rebounding team, they would have achieved that efficiency by shooting just 39.5 percent. A bit better than the Lakers, but I think you'll agree that 39.5 to 38 is a lot closer than 41 to 32.5. Almost six times closer, even.

Now, the Lakers actually won, which means they must have done other things as well to get the win. For one, they turned the ball over somewhat less often, even with all the extra cracks at their offensive end: just 11 turnovers to Boston's 14. And the Lakers also visited the foul line more often (although some of those free throws were toward the end of the game, when the Celtics were fouling to stop the clock, and the Lakers shot poorly on their extra free throws just the same). Those two factors were enough to put the Lakers over the top. But the dominant factor in overcoming an awful shooting performance was their persistence in rebounding on the offensive end.

Game Theory and the Wing-Block Dynamic

In 2004, when the Lakers played the Pistons in the NBA Finals, a lot was made of Kobe Bryant continually jacking up outside jumper after outside jumper—none too efficiently, most of the time—while monster center Shaquille O'Neal was taking fewer shots, but making them much more efficiently. On the surface, it sure seemed as though Shaq should have been getting more shots, and of course Shaq, never a wallflower at the quietest of times, was not loathe to point this out.

In 2009, when the Lakers played the Magic in the NBA Finals, a lot was made of Kobe Bryant continually taking jumper after jumper—somewhat more efficiently than before—while his "newly tough" post player Pau Gasol was taking far fewer shots, but making them more efficiently. On the surface, it sure seemed as though Pau should have been getting more shots, and surprisingly Pau, generally a quiet fellow, pointed this out with a certain degree of mordacity.

Obviously, in retrospect, the two series turned out rather differently for the Lakers, which is why the former case was judged by many as the reason the Lakers lost the series, and the latter is just a footnote. Bryant's reputation as a ballhog, already in force before the 2004 Finals, was substantially bolstered by that series, and has only just faded within the last year or two. But is that fair? Is that the only possible interpretation for Kobe's shot-taking? Or could ballhoggery conceivably help a team?

Let me be clear here. There's no question in my mind that Kobe could stand to take fewer shots than he does (unless he's just red hot). The question isn't whether he should take as many shots as he does, but whether he should take shots even when he's shooting them at a lower percentage than the post players. And this really goes for any wing player who dominates the ball (e.g., LeBron, Wade, etc.). I just mention Kobe because I watch all the Lakers games.

I'm going to look at this from a game theory standpoint. Put into elementary game theory terms, Kobe and the Lakers have a set of tactical options, and the defenders have a set of tactical options. If each side optimizes its strategy with respect to the other side, then in the end, the game will reach what's called a Nash equilibrium: Neither side can improve its result by changing its strategy unless its opponent changes it too. (The equilibrium is not named after award-winning point guard Steve Nash of the Phoenix Suns, but John Nash, award-winning mathematician and subject of the award-winning book/movie, A Beautiful Mind.)

Suppose we simplify matters by assuming that the Lakers have just two options: Kobe shoots, or Kobe passes to the post, which then shoots. And the opponents likewise have just two options: double Kobe, or play man-to-man. And naturally, we assume that Kobe shoots a better percentage over man defense than over a double team, and the post shoots better when Kobe draws a double team than when the defense plays man-to-man.

The conditions of the game do not require either side to do the same thing each time. Strategies can be mixed. So Kobe can shoot 60 percent of the time, and pass 40 percent of the time. The defense can double 70 percent of the time, and play man 30 percent of the time. The defense can even have partial strategies like a weak double versus a strong double. Under these simple assumptions, it's fairly straightforward to find the Nash equilibrium, where neither side can unilaterally improve their result. What's interesting about this Nash equilibrium is that both Kobe and the post should shoot exactly the same percentage.

Plainly, that doesn't happen very often. Very often, Kobe shoots a lower percentage than the post (even when factors such as free throws and the three-point line are taken into account); by comparison, it's relatively rare that it happens the other way around. Ostensibly, with Kobe shooting the ball so much, he's not adequately punishing the defense for doubling him. He should instead pass the ball into the post more, gradually causing the defense to double less and play more man defense, up to the point where his shooting percentage rises to match that of the post.

[EDIT: The rest of this post is largely different from what it used to be, because what follows totally swamps in significance what used to be here.]

Having said all that, I'm going to go back and suggest that that strategy actually isn't optimal. How can it be sub-optimal, if it's at the Nash equilibrium? Because the game doesn't stop when the ball hits the rim, so the game theory shouldn't, either.

When players shoot the ball against straight-up defense, the defense has the advantage on rebounding any misses, because they're usually between their man and the basket. However, when a perimeter player shoots against a double team, the rest of the players have a man advantage. In our scenario, this advantage plays out in the post, which means that (a) the chances are much improved for an offensive rebound, and (b) if an offensive rebound is gained, it usually leads to a high-percentage shot.

What effect does that have? Suppose that the man advantage on rebounding leads to an increase of 15 percent in the offensive rebound rate; for example, if the offensive used to get 20 percent of the rebounds, they now get 35 percent. And suppose also that this leads to a successful shot 60 percent of the time. If the wing player misses, let's say, 60 percent of his shots against a double team, and he faces a double team 50 percent of the time, the offensive rebounds effectively amount to an increase in shooting percentage of 0.5 × 0.6 × 0.6 × 0.15, or 2.7 percent. That doesn't sound like much, perhaps, but it's about a standard deviation's worth, the difference between a top-10 guard and a middle-of-the-road guard. And it's how much worse the wing should shoot than the post at the true optimal strategy.

Again, I'm not suggesting that this is how Kobe thinks (although I'm pretty sure he does think that his misses can lead to easy baskets for his team), or that Kobe shoots exactly as much as he ought to. But it might explain why, even if he's shooting a lower (true) percentage than his post players are, he shouldn't necessarily shoot it less.

Kobe, Once More Unto the Light

The bare facts: The Los Angeles Lakers dominated the Orlando Magic over the last three quarters to take Game 5 and clinch the NBA title, winning going away, 99-86. After a 16-0 run (capped by a nifty Lamar Odom reverse lay-up) took the Lakers from a 40-36 deficit to a 52-40 lead, the Magic never seriously threatened to take the game back, getting no closer than five points the rest of the way and spending most of the second half down by double digits.

Bryant was, I felt, the clear-cut MVP of this series, and of the playoffs, and even when his game was somewhat off in the middle three games of the series, he cast his enormous shadow over how the games were contested. Whether or not you thought he was over-dominating the ball, whenever he was on the floor, he set the tone for the other nine players.

In some sense, for most of his career, he has cast that same shadow on the NBA. For better or for worse (and there have been no shortage of those who see it for the worse), he has been the top talking point of the league. From his unbelievable moves on the court to his embarrassing personal problems in Colorado, his life trajectory thus far has been an eventful one. His triumphs and travails have galvanized public opinion like no other player, possibly in the history of the league. To Kobe haters, Kobe fans are as thin-skinned as their hero, reacting to any perceived slight as though it were heresy; to Kobe fans, Kobe haters seize any opportunity, twist any circumstance, and trample any logic to put the target of their envy in a negative light. Each group sees the other as the yin to its yang, a state of affairs that would be ludicrous with respect to any other player. But apparently it's de rigeur in Kobe's World.

Through it all, Bryant was insouciant, an outwardly joyous 18-year-old rookie; then a driven talent, rising with center Shaquille O'Neal to dominate a trembling league; and then a fallen hero, commonly considered to have forced O'Neal and then coach Phil Jackson off the team. The haters had a field day watching Kobe try, and fail, to lead a ragtag crew to even the lower echelons of the playoffs, pride going before the fall. Jackson returned the following season, but the next two years were barely an improvement, with the Lakers falling to the Phoenix Suns each year in the first round. His undeniable skills on the court were only further testament, it seemed, to his failure to lead his team off it. Bryant himself appeared to adopt the demeanor of a flawed, secretive superhero with a dark past and a darker future, Batman to O'Neal's Superman. The 2007 off-season was the darkest yet, with Bryant railing to all within hearing range about the front office's inability to provide him with a sufficient supporting cast.

The next season brought a pleasant surprise, however, in the unexpected form of a contending team. And when rising young center Andrew Bynum went down with what turned out to be a season-ending knee injury, the beleaguered Lakers' front office obtained multi-talented Pau Gasol from the Memphis Grizzlies for a song, and the Lakers barely missed a beat. Bryant seemed readier than ever to share the ball with his teammates, making the team less predictable, more formidable. There was a regular season MVP for Bryant, his first, matching O'Neal's award from 2000. Even with Bynum out, the Lakers manhandled the rest of the Western Conference on the way to the NBA Finals. The Batsuit was ready to crack. But the Celtics sunk the Lakers in six games, trouncing L.A. by 39 in the clincher.

Back to the cave. Not alone, not to sulk, but this time with all his teammates, forging something of a defensive identity. Bryant and the Lakers were determined that this time would not be the monstrous disappointment of the previous season. There would be no MVP award this year. That would go to LeBron James, the new King. Bryant had no time for regular season plaudits anyway. He wasn't looking for redemption, either; he never felt he had anything he had to redeem himself for. What he was looking for, I like to imagine, was a lighter Kobe Bryant...

The Power of Flexibility

With 10.8 seconds left in regulation in Game 4 of the NBA Finals between the Lakers and the Magic, the Lakers had the ball out of a timeout after Dwight Howard had just missed two free throws to leave the Lakers down three, 87-84. Lakers coach Phil Jackson decided to take the ball near their own baseline (where the timeout had been called), rather than advance it to halfcourt. Trevor Ariza inbounded the ball to Kobe Bryant, who was immediately double-teamed. Bryant advanced the ball near halfcourt back to Ariza, who then cross-courted the ball to Derek Fisher. Fisher dribbled the ball up toward the three-point line on the right wing. Since Jameer Nelson was playing so far off Fisher, Fisher decided to hoist up the trey at that point and sank it to tie the game with 4.6 seconds remaining. The Magic failed to score in their final possession of regulation and the Lakers would go on to win the game in overtime to take a 3-1 lead in the series.

Tim Legler of ESPN later suggested that Jackson's decision made it easier on the Magic, because of the extra time that bringing the ball the length of the court would consume. I think this takes a narrow and unnecessarily time-centric view of the play.

In the first place, 10.8 seconds is a lot of time for a "last-second" play. It's nearly half of a full shot clock. The Phoenix Suns could probably run three whole plays in that amount of time. It's unlikely the Magic could delay the Lakers long enough to avoid giving them a decent look. Indeed, Fisher sank the three-pointer with 4.6 seconds left, but he actually released it with 6.2 seconds; the whole play took less than five seconds to execute.

Secondly, Legler underestimates the pressure that having to play full-court defense places on the Magic. If the Lakers had inbounded the ball at halfcourt, they would have had to set their offensive positions for the most part, showing their hand on the playcall and allowing the Magic to set their defense. Whether or not the decision to bring the ball up surprised the Magic, it concealed the Lakers' play from them and required them to cover a multitude of options.

As it happens, the Magic decided to double Kobe, and the Lakers took advantage by quickly advancing the ball out of the double-team to give the Lakers a 4-3 man advantage on the rest of the court. The Lakers had used this ploy, a kind of basketball aikido, several times in the second half of Game 5 of the Western Conference Finals against the Denver Nuggets. In that game, the Nuggets decided to double team Kobe aggressively, pushing him all the way toward the halfcourt line. Kobe obliged them, drawing his two defenders so far away from the basket that by the time Kobe passed out of the double team, they were effectively out of the play, giving the Lakers a man advantage for long enough to get an easy shot. In hindsight, this strategic decision by the Nuggets was a main reason they lost the game and the series.

But even had the Magic chosen not to double Kobe, the Lakers still had a multitude of options to run, starting from the backcourt, and the Magic would have had to anticipate them all. Most options put the Lakers in a kind of semi-transition game, placing the Magic defense in jeopardy. Normally, teams run very unimaginative sets at the end of a period, and the Lakers are no different in this regard, typically putting the ball in Kobe's hands and letting him go 1-on-N. The fantastic play run by the Magic at the end of Game 2, freeing up Courtney Lee for an alley-oop attempt, was very much the exception rather than the rule. And in this game, Fisher still had to make the jumper. But Jackson's decision to bring the ball up the length of the court broke the usual mold and gave the Lakers their best chance at tying the game.

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.

The Infamous Fisher "0.4" Shot

Introduction

Perhaps no playoff shot has been dissected, debated, or deconstructed as much as the "0.4 Shot" made by Derek Fisher in Game 5 of the 2004 Western Conference Semifinals between the Los Angeles Lakers and the San Antonio Spurs. The Lakers did not win the title that year (they went on to be defeated by the Detroit Pistons in five games), but the closeness of the timing and the marquee nature of the two teams, who had combined to win the last five championships, conspired to focus unprecedented attention on the game-ending jumper.

Much speculation centered around whether Fisher could humanly have caught the ball, turned around, and released the ball, all in the 0.4 seconds available to him. Spurs partisans insisted that he couldn't possibly have done all of those things in so short a time; Lakers fans responded that Fisher didn't do all of those things sequentially, but combined them so that he could do them all. My own personal impression (possibly colored by my bias as a Lakers fan) was that the clock started somewhat late, but not substantially so.

Fortunately, there's no need to rely on anything so nebulous as whether Fisher's shot was plausible or not. Missing from all these speculations was an examination of the actual footage. Video from the game captures instants of the game that, for the live angle at least, are equally spaced in time. The video can therefore be used as a kind of "clock" to determine the interval of time that Fisher had possession of the ball. In assembling this particular look at the Fisher shot, I used a video file that was encoded at 25 frames per second (as I determined by stepping through frames at the end of each quarter). Unfortunately, this was not the native frame rate of the original broadcast, and this increases the random error involved in timing intervals between events. It should not, however, produce any systematic bias one way or the other.

By figuring out how many frames pass between the time that Fisher catches the ball and the time he releases it, and dividing by 25 frames per second, the elapsed time can be calculated. The bottom line, for those who are impatient or don't care about analysis: About five to six tenths of a second elapsed between the time that Fisher caught the ball and the time that he released it.

The Game

On May 13, 2004, the San Antonio Spurs played host to the Los Angeles Lakers in Game 5 of the Western Conference Semifinals. After leading most of the game by as many as 16 points, the Lakers went cold from the outside while the Spurs came steadily back, eventually going ahead 71-68 on a layup by Tony Parker with a little more than two minutes left in regulation.

After a timeout, Shaquille O'Neal responded with a turnaround eight-foot jumper in the lane to bring the Lakers to within a point. The teams traded empty possessions until Kobe Bryant sank a 19-footer from the left wing on a screen by Karl Malone, putting the Lakers ahead 72-71 with 11.5 seconds remaining.

After a non-shooting foul by Derek Fisher, the Spurs inbounded the ball in their frontcourt with 5.4 seconds left. Manu Ginobili passed the ball into Tim Duncan, and tried to cut to the basket for a return pass, but got tangled up with O'Neal and was out of the play. With no other clear options, Duncan faked one way, then dribbled the other toward the top of the key, taking a blind fadeaway jumper that touched nothing but net. The clock read just 0.4 seconds.

The Lakers called timeout. Dejected and weary players trudged slowly back to the bench, none wearier than Bryant, who was exhausted not only by the 47 minutes he had played in the game, but also by the constant jetting back and forth between the team and his legal troubles in Colorado. The Lakers' play out of the timeout called for the players to stand in a stack near the top of the key, in an attempt to break out one of their stars, O'Neal or Bryant, for a quick shot or a tip-in. But before the Lakers could inbound the ball, the Spurs called a timeout. They had seen enough, they hoped, in order to defend the play well.

After the timeout, the Lakers came out in a different set, with the players scattered across the halfcourt. Players cut, especially Bryant, but with Robert Horry doubling on Bryant rather than playing Payton inbounding the ball, Payton couldn't find an open teammate and had to call the Lakers' final timeout.

When the ball was brought into play for the final time, the Lakers returned to their original set. Bryant broke out from the stack toward halfcourt, tailed by Horry and Devin Brown. O'Neal curled toward the basket, while Malone drifted toward the top of the key. Finally, Fisher broke toward Payton.

Payton tossed the ball, leading Fisher toward a spot about 18 feet from the basket on the left wing. Fisher began angling his body for the turn before catching the ball in mid-air, then coiled on his legs and prepared to shoot over Ginobili's outstretched arms. At seemingly the same instant, Fisher released the ball, the game horn sounded, and the backboard's red light came on. Nineteen thousand people held their collective breath. The ball arced upward and came down; Fisher thought he had pushed it off too hard, but it was offset just enough by his backward motion from the basket, and the ball fell perfectly through.

A hush fell over the crowd as Fisher ran down the court in celebration, eluding his mobbing teammates and streaming down the tunnel toward the locker rooms. Rasho Nesterovic and Kevin Willis waved their hands to indicate the shot got off late. Duncan stood unmoving, hoping they were right. The referees, who had called the shot good when it happened live, convened at the scorer's table to examine the video of the play from the ABC cameras. A few tense minutes passed before the referees confirmed their initial call was correct: The shot was good. The Lakers had won Game 5, 74-73, and returned home to trounce the Spurs in Game 6 to win the series, 4-2.

The Aftermath

Writing on May 14, the day after Game 5, Dusty Garza, the editor of Spurs Report, relayed news that the Spurs had filed a formal protest with the league office, claiming that the clock started too late after Fisher touched the ball, and that the shot should not have counted. The league denied the protest that same day.

Garza also offered his personal opinion that Fisher's shot did not get off in time—indeed, could not have gotten off in time—based on the notion that human reaction time is, on average, three-fourths of a second (750 milliseconds). Since the clock couldn't have started any faster than that, Garza wrote, Fisher could have had anywhere up to a bit more than a second to shoot the ball.

This seems an unreasonable conclusion. In the first place, Garza contends that the average human reaction time is three-fourths of a second, then says that unless the referees are superhuman, they couldn't possibly have pushed the button less than three-fourths of a second after Fisher touched the ball. Well, if the three-fourths of a second is an average, wouldn't half the human population be able to do it faster (assuming negligible skew in the distribution)? And presumably NBA referees are trained to be a bit faster than average.

Secondly, research indicates (Laming 1968, Welford 1980) that simple reaction time—the time required to do something simple like push a button after a visual stimulus—is more like one-fifth of a second (200 milliseconds), rather than three-fourths.

What's more, it's unclear that human reaction time is involved here at all. Bennett Salvatore once said, speaking to Henry Abbott of ESPN's TrueHoop blog, that NBA referees don't anticipate calls; they only observe the game. However, that can't possibly be literally true all the time. When Payton passes the ball in-bounds, it is immediately evident that the ball will be caught (or at least touched) first by Fisher. It is human nature to anticipate this first contact, and act accordingly. But what does it mean to "act accordingly"?

For years, the clock was operated manually, by the timekeeper, based on the rules of the game and the whistles of the referees. The system worked well most of the time, but placed a lot of reliance on the alertness of the timekeeper.

In 1999, the NBA installed a new system developed by Mike Costabile, an NCAA referee who previously officiated in the NBA. Each referee carries a small transmitter attached to his or her belt, with a button. When the clock is to start, each referee pushes the button at the exact instant at which he or she believes the ball to be in play. The first button push triggers an automatic start to the clock. The system also includes a microphone that is sensitive to the particular frequency of the whistles used by NBA referees, and stops the clock when the whistle is blown.

In order to activate the clock, at least one of the referees must push a button at the instant he or she believes the ball to be first touched. Obviously, this generally doesn't happen right at the moment the clock is "supposed" to start. There are two potential delays here: reaction time, and execution time (the time it takes the finger to actually push the button).

To understand the relationship between these two, and how the actual delay is affected by context, suppose I ask you to clap your hands as soon as one of the following happens:

• A basketball I drop from four feet hits the floor.
• I clap my hands.
• I move my hands at all.

In the first case, it takes about half a second for a basketball released from four feet up to hit the floor. That is enough time for you to react and execute the act of clapping your hands at the precise moment the ball hits the floor. In the second case, both of us need our execution times to clap our hands, but you have to react to the start of my clapping motion. In counting the delay, your execution time is cancelled out by my execution time, leaving just your reaction time. And in the last case, I can move my hands without warning, meaning that the delay is your reaction time plus your execution time.

In the case of the play in question, it took about half a second for the ball to pass from Payton's hands to Fisher's hands. For a referee who is ordinarily alert, this is plenty of time to predict the path of the ball and press the button almost immediately upon contact. Even if we accept that referees do not anticipate events, they must at least be prepared for the potential event of contact between Fisher and the ball; there is no reason, at any rate, for the delay to be anywhere near three-fourths of a second.

But let's not be too hard on Dusty Garza. He was writing in the heat of the moment, and from honest feeling. Besides, let any of us without team favoritism cast the first stone. Let's get down to brass tacks: Garza sincerely believes that the video shows that Fisher took about a second to get the shot off. Did he?

The Video

The video file I used in putting this article together is encoded at 25 frames per second. (I determined this by advancing the video frame by frame, 200 frames, at the end of each of the four quarters, when the clock is counting down at tenths of a second. Each time, 200 frames corresponded to an interval of exactly 8 seconds, so the video must be progressing at 200/8 = 25 frames per second.) Therefore, each frame represents 1/25 = 0.04 second. This is not the frame rate of the original broadcast, which would probably have been 30 frames per second. As part of the re-encoding process, frames would lose some definition, increasing the error involved in estimating precisely when events happen. Since these errors do not accumulate, however, they add a small random error, but they do not systematically bias estimates of interval lengths.

One problem in reviewing this particular game video is that only the live shot actually keeps time accurately. In all subsequent replays, the ABC crew slowed the video at a variable rate, in order to allow Al Michaels and Doc Rivers to comment on it. But the live shot has the camera in line with Fisher and Payton, making the determination of the instant Fisher first caught the ball difficult. Here, for instance, are three successive frames of the live shot, obtained using the snapshot function of the xine video player. Which one of these frames do you think shows Fisher actually catching the ball?

I think it's pretty clear that this angle can't be used to reliably determine when Fisher touched the ball (which would have started the clock). Fortunately, we can still make use of other camera angles. This has precedent in the NFL, in which "composite" video reviews are conducted. This allows the referee (and video replay official) to assess multiple angles in order to come to a firm conclusion, even when no single angle provides all of the information necessary. This isn't to say that the NFL uses fancy three-dimensional visualization tools (à la The Matrix), since they can't do that in time, and neither is it necessary here. We'll just combine the angles mentally.

Here are video stills from the opposite baseline. It shows the ball and Fisher approaching one another. In this first frame, it's not very easy to see the ball, but you can see it superimposed on referee Joe Forte's right foot. At this point, the ball is still a couple of feet from Fisher's outstretched hands.

The frame below shows Fisher and the ball considerably closer to one another, but there still appear to be several inches in between them.

This third frame shows Fisher's hands possibly touching the ball for the first time. They don't clearly touch, but this is the first frame from this angle where contact is plausible.

Note the positions of the other players. The positioning of the left foot of Duncan (guarding Malone at the free-throw line) is especially revealing. That foot covers the free-throw line, as seen from this camera angle. Duncan's foot must therefore be in reality at least as far out as the free-throw line. It could be beyond it, if it's above the floor, but it certainly cannot be between the basket and the free-throw line. This crucial piece of video detail shows that Fisher does not touch the ball until the second of the three live video frames above. Below are successive frames from the live video, starting from the one where Fisher first touches the ball, and running until he releases it.

Frame 1:

Frame 2:

Frame 3:

Frame 4:

Frame 5:

Frame 6:

Frame 7:

Frame 8:

Frame 9:

Frame 10:

Frame 11:

Frame 12:

Frame 13:

Frame 14:

Frame 15:

In my opinion, Frame 14 (which shows the clock switching from 0.1 to 0.0) shows Fisher apparently having released the ball—about as apparently as he touches it in Frame 1. If we take those two frames as the endpoints of Fisher's possession of the ball, then he has the ball for 14 minus 1, or 13 frames in all. At 25 frames per second, that works out to 13/25 = 0.52 seconds. The method I've used to produce our composite review I estimate to have an error of a frame or two in either direction, which works out to plus or minus 0.06 seconds; add another 0.02 seconds for the video re-encoding at 25 frames per second.

In addition, I should account for my status as a Lakers fan. (Who else would go through this much trouble for Fisher's shot?) I remember sitting in bed, having twisted my ankle in my own basketball game earlier that afternoon, and feeling pretty good about the Lakers until the fourth quarter, then anxious, then frustrated, then angry, and finally elated. It is sensible, to account for this possible systematic bias, to add a frame's worth of time to the figure above to yield 0.56 seconds. Note that the ball has clearly left Fisher's hands in Frame 15, which also shows the red light on the backboard going on for the first time.

To summarize, this video shows that Fisher had possession of the ball for about 0.5 to 0.6 seconds. One corollary of this finding is that the referees started the clock approximately 0.1 to 0.2 seconds after he caught the ball. This is entirely typical and in line with usual execution times; it would be unreasonable to claim the clock was started "late." It's certainly shorter than the three-quarters of a second that Garza claimed was necessarily human reaction time; after all, Fisher executed his entire possession in less time than that.

Final Thoughts

Some Lakers fans pointed out, in the aftermath of the series, that prior to Fisher's game-winner, Duncan's shot swished through the hoop with considerably more than 0.4 seconds left on the clock. See, for instance, this video frame:

If so, claimed Lakers fans, the Lakers should have had more time on the clock, possibly rendering the above dispute moot. NBA rules stipulate that the clock should be stopped at the moment the ball exits the bottom of the basket, including the nylon, not when it enters the basket. The frame above shows the ball exiting the basket with the clock switching from 0.8 to 0.7 seconds. By that token, it must have taken somewhere between 0.3 and 0.4 seconds for the referees to whistle the clock stopped after the shot swished through. Why did it take longer for the clock to stop after Duncan's shot than it did for it to start after Fisher's contact?

It's impossible to state for certain, but one possibility is that because it's less predictable that Duncan's shot will exit the bottom of the basket than it is that Fisher will touch the ball, the referees had to wait longer to be sure that the basket was made before whistling the clock stopped. Then, too, it's a whistle blow that stops the clock, as opposed to a button press that starts it again, and those two actions may well have different execution times. But it seems plausible that had perfect timekeeping prevailed in the final seconds, Fisher's shot would have been good by about 0.1 seconds.