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    Investigating the value of two-for-ones

    by Ken Pomeroy on Monday, July 1, 2013

    Normally in basketball, one needs to spend a possession on defense to earn the right to get the basketball, just as at the grocery store, you need to spend $4.99 to earn the right to consume a frozen pizza. When the opportunity presents itself to purchase two frozen pizzas for that same price, a rational human being jumps at it. Likewise, if one enjoys scoring points, the basketball version of the two-for-one would seem to be too good to ignore. Yet, coaches pass up the chance frequently.

    Some have offered reasoning to do so. Stan Van Gundy has gone on record saying that deviating from one’s normal offensive philosophy in these cases erodes the players’ trust in the basic underpinnings of good team basketball, and that jacking up a quick shot may harm the culture of the team. The former Vermont assistant also gets upset at his wife when she brings home two frozen pizzas, claiming that stashing one in the freezer will contaminate the pint of Karamel Sutra it will share space with. Something about pepperoni osmosis.

    In fact, my frozen pizza analogy is not really a fair one. In hoops, the two in the two-for-one involves a pair of shorter, possibly less efficient, offensive possessions. So it would be like getting two generic brand pizzas instead of one DiGiorno. If you value quality in your frozen pizza, you’re going to pay full price for one top-shelf pizza. I get that and can sympathize with SVG’s angst at his wife in that case. But in basketball, you are simply trying to pile up points, so even two below-average possessions should be more appealing than one full-quality chance with the ball.

    So far, I’ve been couching a lot of statements with qualifiers because to my knowledge, we really don’t have data regarding the real-world tradeoffs involved in a two-for-one. There are studies like this one which tell us what happens when everything goes perfectly. According to the authors, when a two-for-one occurs the team getting the additional possession outscores its opponent by an average of about one point. This shouldn’t be too surprising considering that’s about the average value of a basketball possession.

    But I don’t really want to know the benefits when a two-for-one actually occurs. What’s more valuable to me is what happens when there is a decision to make. If you can shoot quickly and attempt to earn two-for-one, should you? We can only know this by comparing cases where teams try for a two-for-one to cases where they don’t. In a fair analysis, we shouldn’t restrict the two-for-one attempts to situations where the two-for-one worked out perfectly.

    As with last February’s fouling-up-3 study, I will use the past four seasons of play-by-play data to investigate this issue. I must warn you, though, that this issue is a lot more complex than I thought before I started the research. It’s not as simple as examining the decision on whether to foul up 3. The most important thing learned from that work was that I would have been more successful trying to influence the audience’s opinion on abortion.

    The first order of business is determining what criteria will be used to identify a two-for-one opportunity. For the purposes of this study, a team has a decision to make when they take possession of the ball with between 0:55 and 1:05 remaining in the period. Use nearly all of the shot clock and the opposing team will be able to take the last possession of the half. But take an early shot and the opposing team will not have the last possession of the half.

    First, I looked at all cases in the last four seasons where a team took possession in a situation like this at the end of the first half where the margin was single digits. Then it was noted when the first action of the possession occurred, where “action” means either a shot attempt or a turnover. Finally, I aggregated the resulting data by the time of the first action, recording how many possessions occurred and how many points were scored by each team after a two-for-one attempt.

    So hey, let’s take a look at the data. I’ve organized it by five-second intervals based on the time remaining for the first action. Yes, this is a lot of data. I’ll simplify it eventually, but soak it in for a second.

    Two-for-one data, by time of first action
    Time     n   OP/n DP/n Diff  PF/n PA/n Diff   OE   DE
    50-54  1361  2.10 1.53 0.57  2.01 1.38 0.63  95.7 89.8
    45-49  1603  2.02 1.45 0.57  1.89 1.35 0.54  93.8 92.8
    40-44  1564  1.82 1.25 0.57  1.65 1.06 0.59  90.7 84.7
    35-39  1169  1.53 1.11 0.42  1.36 0.98 0.38  88.6 88.9
    30-34   850  1.40 1.06 0.34  1.25 0.95 0.30  89.4 90.2
    25-29   549  1.36 1.03 0.33  1.21 0.85 0.36  88.7 82.5
    20-24   239  1.33 1.03 0.30  1.28 0.99 0.29  96.2 96.3
    Time: time remaining of first action (shot or TO)
    n   : number of cases
    OP/n: offensive possessions per case
    DP/n: defensive possessions per case
    PF/n: points scored per case
    PA/n: points allowed per case
    OE  : average offensive efficiency per case
    DE  : average defensive efficiency per case

    Once you understand the assumptions of the study, the data on possessions-per-case isn’t too surprising. If the team’s first action occurs before 40 seconds are left, then it doesn’t really get a two-for-one, but something like a two-for-1.5.

    A team doesn’t always get a two-for-one when attempting it because there are going to be cases where it takes a shot early and there are offensive rebounds or fouls extending possessions. The same thing could happen defensively allowing a team that takes possession with more than 35 seconds left to run out the clock. And there will be cases where a team bungles the strategy altogether and commits a turnover, possibly leading to a breakaway for the opponent.

    So in a two-for-one attempt, a team can expect to get a smidge over a half-possession more than its opponent. That’s still 50% off of your second DiGiorno, so not a bad deal, really.

    Except. Even if a team waits past 0:40, it still gets a bonus, something like a third of a possession. Opposing teams are not always able to milk the very last second out of a possession when they get the ball back with the shot clock off. Or sometimes off a missed shot or turnover, the opponent may get a transition opportunity and take a shot early.

    The bottom line is that a two-for-one attempt nets you an extra quarter-possession over what you’d get from waiting. One way of thinking about this is that trying for the two-for-one gets you an additional possession about 55% of the time. But even not trying for it gets you an extra possession 30% of the time. A quarter possession is nothing to sneeze at, but small enough where I guess SVG can play the culture card and have a case.

    Not that I am siding with the portlier Van Gundy brother. What matters more is how many points result from the occasional extra possession. The extra quarter-point reaped in these cases would turn an otherwise average offense into the nation’s best for the closing sequence of the half. Compare that to the case of a coach calling a timeout for the final possession of a half. It’s doubtful this timeout is worth a quarter point, yet it’s a time-honored practice. Using a timeout in a potential two-for-one case should provide at least as much benefit.

    However. There are a few issues with this analysis. For one thing, you’ll notice that shooting in the 50-54 second range results in the best offensive efficiency, which is counterintuitive. Shooting as quickly as possible is probably not the path to a more efficient offense in these cases. Some of these early shots are fast-break chances that typically yield more points per possession than half-court sets. We really don’t want these possessions in our analysis since it’s not like a team is consciously thinking about taking advantage of a two-for-one in these cases.

    One way to do this is to restrict cases to those where the two-for-one opportunity occurs after an opponent’s made shot. Yes, it’s still possible to have transition after a made shot, but it’s much less likely than after a steal or even a missed shot. We lose a lot of cases in doing this, but it’s important to see the difference.

    Two-for-one data, after opponent’s made shot only
    Time    n   OP/n DP/n Diff  PF/n PA/n Diff   OE   DE
    50-54  370  2.14 1.55 0.59  1.86 1.37 0.49  87.0 88.5
    45-49  683  2.02 1.48 0.54  1.85 1.37 0.48  91.7 92.9
    40-44  830  1.83 1.27 0.56  1.64 1.10 0.54  89.5 87.1
    35-39  679  1.53 1.10 0.43  1.29 0.97 0.32  84.6 87.7
    30-34  509  1.40 1.05 0.35  1.31 0.99 0.32  93.7 93.8
    25-29  297  1.35 1.03 0.32  1.18 0.84 0.34  87.7 82.0
    20-24  165  1.30 1.01 0.29  1.25 0.99 0.26  96.3 97.6

    As expected, the offensive efficiency is somewhat worse when something happens in the 50-54 range. The efficiency values in the other bins are changed to a lesser degree and now we seem to have a sweet spot in the 40-44 second range for implementing the strategy. Let’s simplify the data into two groups – one where the team shoots (or turns it over) before 40 seconds and cases where something happens after 40 seconds.

    Time    n   OP/n DP/n Diff  PF/n PA/n Diff   OE   DE
    40-54 1883  1.96 1.40 0.56  1.76 1.25 0.51  89.8 89.6
    20-39 1650  1.43 1.06 0.37  1.27 0.95 0.32  88.9 89.5

    Ignoring potential transition cases ends up decreasing the benefit of a two-for-one to about 0.20 points. Still, the two-for-one would appear to be something worth taking advantage of. An average offense still ends up being close to the best in the land for one possession.

    But. There are still plenty of things left to consider. A biggie is that the two bins imply we’re comparing philosophies of “two-for-one” and “one-for-one”. In reality the competing philosophies are “two-for-one” and “eh, let’s do what we normally do for the sake of culture and all that”.

    With the latter, we’ll occasionally see some shots taken early, and one could argue those will normally be pretty good looks. Part of the tradeoff of forcing a two-for-one is that the first shot is rushed enough to reduce the chances of scoring a bit.

    And we don’t see it in the data. Here’s how the efficiency on the first possession breaks down in each group:

    Time   Poss  
    40-54  94.6  
    20-39  97.5  

    There’s about a 3% penalty for taking a shot early when there’s a two-for-one opportunity. Now if that’s the true penalty for taking advantage of a two-for-one, it would be clear it’s a good idea from a numbers standpoint, because that’s a tiny price to pay in exchange for the possibility of getting another possession. But I think it’s a reasonable assumption that the true cases of forcing a quick possession to get a two-for-one are going to be less efficient than when a team just runs its regular offense and happens to take a shot early.

    Unfortunately, we have no way of knowing which of those early cases are true two-for-one attempts and which aren’t. Given the prevailing opinion is that few coaches take advantage of the two-for-one, one might guess that a significant chunk of the early cases are not of the forced variety. One might also assume that the efficiency in those unforced cases would be similar to the efficiency in late cases. I’m not going to work through the math at this point, but any assumptions one applies along these lines further reduces the observed benefit of consciously attempting a two-for-one.

    What does all this mean? The two-for-one is a complicated issue, and it generally doesn’t provide as much benefit as one might think. Like the fouling-up-3 conundrum, if the strategy is executed perfectly, a large benefit is likely. But players aren’t robots, and all of the imperfect acts that can disrupt the strategy eat away at the potential benefit. Assuming the average gain is a fifth of a point, that’s worth slightly less than one percent in terms of win probability at the end of a half. A coach implementing this strategy will win one extra game out of 100 - and that’s out of 100 games where a two-for-one opportunity exists!

    Things are different in the NBA where there are potentially three opportunities for a two-for-one each game, so I’m not going to let Van Gundy completely off the hook. When there’s an easy way to add a percent or two to your chances of winning, one should think carefully about whether the culture issue is a valid reason to ignore that opportunity.

    However, in the college game, the two-for-one at the end of the half doesn’t present a huge advantage. It’s an advantage, sure, and I’d bet coaches worry about things that give them even smaller edges. But after going through this analysis, I’m going to move the two-for-one down on my list of strategic outrages.