The lockout shortened NBA season is officially rolling, and everyone is talking about the demanding, condensed schedule. The Lakers start was a case in point: three games back to back against the Bulls, Kings, and Jazz. They should have beaten the Bulls but blew a 5 point lead in the last minute. Then came the predictable, tired-legs loss to an inferior team in Sacramento.

The Lakers came into the season in disarray. The league denied their trade for Chris Paul, Kobe hurt his wrist (a full ligament tear that will likely get worse, not better, this season), they traded extremely disgruntled Lamar Odom for next to nothing (he was part of the Paul trade offer), and Pau Gasol remains somewhat disgruntled (also part of the Paul trade offer). To make matters worse, the Clippers swiped Chris Paul instead and beat the Lakers twice in the preseason. New coach Mike Brown felt it necessary to assure the media that the Lakers would make the playoffs before their third game against the Jazz.

After their loss to the Kings, a lot of people expected an 0-3 start for the Lakers. The Kings game was close until the final 1:30, so they should have been especially tired against the Jazz. Instead, the Lakers came out and demolished the Jazz.

I have a different theory for what happened to the Lakers. In most NBA seasons, teams play a few back to backs, and good teams lose some of those games and still make the playoffs. Even against a bad team, there’s not much incentive to try hard when you’re a little tired. Over half the league makes the playoffs. You can save your effort for another game when your legs are fresher. As a side note, I think effort is a big contributor to upsets in general in the NBA. Every shot is a probability event that contributes directly to the score line in basketball. If both teams try their best, all these individual probability events should reduce the variance in the score outcome (i.e., the better team should almost always win).

So, what’s new about back to back to backs? Well, after losing to the Kings, I think the Lakers realized they can’t slack off when they are tired. An 0-3 record was staring them in the face. They won’t make the playoffs if they lose all the back to back games in this year’s schedule. Maybe the condensed schedule will be good for the NBA in that way.

During the Jazz game, I also noticed something that bothers me in general about basketball, though I have never investigated it rigorously. In football, when a team is leading by a large margin in the fourth quarter, they will start to shorten the game by running the ball more and letting the play clock wind before snapping the ball. In the NBA, teams seem to pay no attention to clock management until the final 2 or 3 minutes. Let’s break down the Lakers game probabilistically. The Lakers had a 25 point lead going into the fourth quarter. Option A is to play as normal, aiming for 16 seconds per possession on average. Option B is to burn 15 seconds off the shot clock before starting an offensive play, using around 24 seconds per possession.

Assume that option A gives the Lakers a made basket with probability X, while option B gives the Lakers a made basket with probability Y. Option A will score a few more points in general, since it gives the Lakers time to change tactics if their first play doesn’t get a good look at the hoop (i.e., X > Y).

How many fewer possessions would the Jazz get with option B? Suppose the Jazz play quickly (regardless of what the Lakers do) and average 14 seconds per offensive possession. With option A, a possession for both teams runs off 30 seconds total. With option B, a possession for both teams runs off 38 seconds total. There are 12 minutes in the fourth quarter, so option A will generate 24 possessions for each team. Option B will generate about 19 possessions for each team. So, option B eliminates 5 possessions for the Jazz. Let’s give the Jazz a made basket with probability Z on each possession.

There’s a trade off for the Lakers between the two strategies. They will score more points on average with option A, but they also give the Jazz more opportunities to score. With more opportunities to score, the Jazz have more time to get the lucky run they need to come back from 25 points down.

Since the whole distribution of outcomes matters in determining how often the Lakers lose, the easiest way to figure out the right strategy is with some simulations. In the following figures, I simulate 500 fourth quarters with different values of X, Y, and Z. On each possession of each simulation, I flip a coin weighted by the relevant probability (X, Y, or Z) to determine whether the team scored a basket. At the end of each simulation, I add up the made baskets for each team and determine whether the Jazz overcame the deficit. I reduce the deficit to 20 points to give the Jazz a chance. To simplify things, I only allow two pointers.

Each figure holds the X and Y values constant and lets Z range from 0.4 to 0.6. The red bars show how many times the Lakers lose in 500 simulations when playing slowly (option B). The blue bars show the same when the Lakers play normally (option A). For example, the top left graph shows simulated Lakers losses if their probability of scoring a basket is 0.25 when playing option B and 0.4 when playing option A. With these parameters, if the probability of scoring a basket for the Jazz is 0.6, then the Lakers lose almost 100 times when playing option B and only about 40 times when playing option A (out of 500 simulations for each option).

The Lakers win more than they lose in all the simulations, since they start with a 20 point cushion. They lose more when the Jazz have a higher probability of making a basket (i.e., a higher Z) and lose less when they themselves have a higher probability of making a basket (a higher X for the blue bars and a higher Y for the red bars).

Which strategy minimizes the number of losses? If the Lakers shoot relatively well when playing normally (X > 0.45), then using option B has little value (see the graphs in the right column). They will make enough baskets relative to the Jazz that the Jazz will almost never catch up within 12 minutes, even if the Jazz have a slightly higher probability of scoring. Switching to option B increases the number of losses if there is a large difference between X and Y (i.e., if playing slowly drastically reduces the probability of scoring) and has little to no benefit if X and Y are similar.

If the Lakers shoot relatively poorly when playing normally (X < 0.5) *and* see little drop off when playing slowly (Y similar to X), then option B reduces the number of losses slightly. This is intuitive; a poor shooting team that is ahead should try to shorten the game if the cost to do so is low, since their poor shooting will undo them over the course of many possessions.

I was surprised to find that option A works better in general. My intuition was that the Lakers could use option B to essentially assure victory. For example, if the Lakers were up by 25 points with only 5 minutes remaining, they could run out the entire clock in about 12 possessions — not enough for the Jazz to realistically come back. All of this ignores fouling of course, but the bigger issue is that the time benefit from using option B is fairly low, while the cost of using option B in terms of shot probability is surely at least 5% (as I assumed above). Indeed, if I decrease the amount of time per possession in option A from 16 seconds to 12 seconds, option A worsens quite a bit when the Lakers shoot poorly (left column graphs). This still doesn’t change the right column results much, though, and I think the time parameters above are more realistic.

Excellent post. I wonder if NBA teams switch to “option B” at the right time, too early, or too late.

I’d like to pick up on your statement that “Every shot is a probability event that contributes directly to the score line in basketball. If both teams try their best, all these individual probability events should reduce the variance in the score outcome”. That’s a sentiment I agree with, and probably serves to explain why the NBA playoffs do a better job at determining the “best” team than any other sport, perhaps even better than a long regular season where effort can flag for long stretches. Similarly, this explains why soccer and hockey are especially prone to upsets – although it surprises me that hockey, at least anecdotally, seems to be more upset prone than soccer, since there are more shots taken in a hockey game. In those sports, the connection between quality play and scoring is more attenuated. Baseball is interesting because each positive action is more closely linked to the probability that you score than in soccer or hockey, although not quite as closely as in basketball, it’s just that the sample size is so small. Football operates much like baseball – plays have a strong impact on the chance that you will score on some subsequent play – but with a larger sample size (I think the frequency of “true upsets” in football is lower than people believe and “any given Sunday” is more nice marketing than objective reality). Rugby strikes me as straddling football and soccer/hockey.

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