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The Arbitrarian: Team Depth, and Correlates Thereof

September 11, 2008 in by with 1 Comment

David Sparks is the Arbitrarian. His stats column runs weekly here at HP. This week he discusses depth and its impact.

The survey responses to last week’s post were so interesting, I decided to do an immediate follow-up (if you haven’t read it, you may want to do so before continuing here). Last week, we focused on team rotation size, as measured by minutes played. Today, we will look at a very similar, but somewhat more interesting concept: team depth.

Depth and rotation are not necessarily the same. Since there must be five players on the court per team at all times, the theoretical minimum for rotation size is five, which you would see if a team played only five players, all game, every game. However, depth concerns not playing time, but production, and it is easy to imagine one of those five players contributing more than 20% of the team’s total production, while one or more of the others produces less than their share. (There is a metric, called the Valuable Contributions Ratio, which I use to measure players’ productive contributions relative to their floor time.)

If each player produced in proportion to their allocation of minutes, it would make no difference which players were on the floor, but obviously this is not the case. Rather, better players produce a greater proportion of their team’s production than their proportion of a team’s minutes played. This implies, of course, that a team’s rotation size will likely not be the same as its productive depth, and further, that depth will likely be smaller than rotation.

In fact, depth can be calculated in exactly the same way as rotation (see last week’s column), except instead of using minutes as the variable of interest, we use Model-Estimated Value (MEV), a productivity metric.

So many theories

Last week, I invited readers to speculate about the relationship between rotation size and team success. You submitted countless interesting ideas in response to this question, and made many other interesting suggestions about ways to assess rotation consistency, variations in rotation size by coach, and differences between regular-season and playoff play, among others. I hope, in time, to investigate some of these great ideas.

For now, let us turn to the relationship between rotation size and success. In response to my question, the plurality of respondents said that wins and rotation size would positively correlate, many noting that deeper rotations would probably enhance a team’s chances in the playoffs.

Others suggested that the relationship would be negative, due to the fact that poorer teams needed to give more playing time to younger, weaker players, to aid in their development.

A large minority of answers indicated that there should be no consistent relationship. Several of these claimed that rotation size was too idiosyncratic: a function of the coach, playing style, and available personnel, and successful teams could make any sort of rotation a winner.

Several others predicted a parabolic relationship, in which the smallest rotations would find success on the back of a few stars, the largest rotations succeed through roster flexibility, and those in the middle, by failing to follow either strategy, will not do well.

I must admit that I was intrigued by all of these arguments, especially the parabolic prediction. My personal hypothesis was that increased rotation size would lead to greater success, due to the positive effects of diversification, as in the stock market. With more diverse contributions, I thought, would come greater insurance that even if one player failed to show up, one or more of his teammates would pick up the slack and ensure victory.

There were a number of other interesting hypotheses: one was that since defense requires a greater exertion of energy and offense requires time to find a rhythm, defense would correlate positively, and offense negatively, with rotation size. Other noted that faster-paced teams may require longer rotations, due to greater energy expended per minute. Several others suggested that the age of the team would vary positively with rotation size, as younger players can typically play a greater number of minutes without hurting productivity.

The empirical evidence

Who was most correct? Well, first I should mention that part of the problem with my question last week was that rotation size was often conflated with depth, which I define as separate concepts. That said, after reviewing the graphical relationships, I must sadly rule out the parabolic hypothesis. The rest of the relationships (between all suggested variables), are depicted in the correlation matrix below:

     rotation  depth gameage   poss offeff defeff effdifrotation    1.000  0.412   0.016 -0.069 -0.057 -0.083  0.020depth       0.412  1.000  -0.007  0.079  0.375 -0.041  0.321gameage     0.016 -0.007   1.000 -0.085  0.069 -0.143  0.164poss       -0.069  0.079  -0.085  1.000  0.016  0.016  0.000offeff     -0.057  0.375   0.069  0.016  1.000  0.160  0.648defeff     -0.083 -0.041  -0.143  0.016  0.160  1.000 -0.648effdif      0.020  0.321   0.164  0.000  0.648 -0.648  1.000


Rotation and depth are measured as described previously. Game age is the playing-time-weighted age of the team. Possessions are a measure of pace. Offensive efficiency is a measure of a team’s scoring per possession, while defensive efficiency measures the same thing for their opponents (so better defensive teams have a lower defensive efficiency as constructed here). Efficiency difference is a measure of absolute quality, subtracting defensive from offensive efficiency.

Many of these results (the ones close to zero) indicate no relationship: Rotation size seems to be unrelated to anything but depth. However, depth appears to be positively correlated with offensive efficiency, and thereby, also positively correlated with efficiency differential–apparently teams with greater depth (at the per-game level) see improved efficiency differentials. One problem is that we cannot tell which direction causality moves in. Do deeper teams play better, or do teams who are winning by a lot give bench players increased minutes and thus increased time to produce?

To some extent, the likelihood of the second option can be tempered by the fact that rotation size has no real relationship with efficiency differential, but this question is still not definitively settled.

Expanding our scope

How have rotation sizes and depth changed over time? Which teams, historically, are the deepest? Due to data limitations, to investigate these questions, I must change the way I measure rotations and depth. Instead of assessing these at the per-game level, to make historical comparisons, I will measure at the season level, meaning that from this point on, rotation is best understood as the inverse of the concentration of minutes played over the course of the season, and depth is best understood as the inverse of the concentration of production over the course of the season. In general, these figures will be higher than each team’s mean per-game figures, due to changes in the roster and substitution patterns over the course of a season. However, error ought to be normally distributed, and so I will press forward using these slightly modified metrics, which are interesting enough in their own right.

As you can see in the plot above, both rotations and depth have increased over time. Rotation is denoted in red, and depth in cyan, and both are greater now than they were in the early years of the NBA. There could be any number of reasons for this–expansion, and the dilution of the talent pool, could be responsible; or merely a realization that heavy minutes’ loads may shorten player’s careers. Incidentally, I have scaled the size of each team-year marker to their winning percentage, but the relationship between depth, rotation, and winning is unclear in this depiction.

Below, I plot team winning percentage (jittered) against team depth. The color scale indicates rotation size, going from small (red) to large (blue), so that if you see a blue team amongst several red ones, you know that that team has a relatively large rotation given its depth. I’ve also scaled markers by year, so that more recent teams stand out more.


Fullscreen Version

The first thing I notice is the outliers. The most concentrated teams appear to be several Chamberlain squads, in which he was an absolutely dominant producer, and carried his team more than any other player ever has on a consistent basis.

The least concentrated teams are several more recent, and fairly bad teams, topped by the 2002 Chicago Bulls, who were very deep with potential that had yet to develop into actuality.

As noted above, depth has increased over time, and so it is interesting to note the most concentrated teams in a more modern era (which I mark with the inception of the three-pointer, 1979-present). There are two very shallow Utah teams, lead by Malone and Stockton, and supported by almost no one else. The pre-Pippen Bulls show up here, as do the Kobe-only Lakers–teams with one star who did a substantial amount of the producing. We also see the ’87 Celtics, ’04 Timberwolves, and ’08 Hornets, each of which had a couple of extremely good players dominating the contributions to winning, and then filled the rest of the roster out with players who couldn’t hope to match the same level of productivity.

Among the very best teams, there is a decent variety of concentration, although it is interesting to see the ’08 Celtics at the high end of depth among this elite. Their big three may have gotten the headlines, but it the entire roster made important contributions. Further down and to the right, we see the ’08 Rockets, which put on the least likely 22-game winning streak in history, on the back of role players, a different one of which stepped up every night. This team was very successful, given its depth, and it will be interesting to see how this translates to future success.

What does it mean?

The overall trend is a slight but definite negative relationship between team depth and success, but it is unclear what conclusions can be drawn from this. Is this proof that a superstar (or a Big Two, or a Big Three) is key? Does it reflect the fact that it’s easier to field a team of equally poor players than a team of equally excellent players?

Since this graphic is based on season-level data, it may just mean that teams with less volatility in their rotation and minimal personnel turnover are more successful. However, I must admit to being unsure of what to make of these preliminary findings. Should teams dump their midlevel players (in salary and productivity terms), in pursuit of a bimodal roster of two stars and ten inexpensive warm bodies? Obviously, constructing a roster requires more than just collecting players at varying levels of talent–the interaction of their abilities is a key consideration–a team is more than the sum of its parts. I would love to hear your insight, explanations, and questions in the comments. Also, I would appreciate your taking the time to fill out the short survey below.

The Arbitrarian: Envisioning the Olympics

August 27, 2008 in by with 4 Comments

David Sparks is The Arbitrarian. He profiles his stastical work every Thursday here at Hardwood Paroxysm. David is glad to be back at school, especially with his new Trapper Keeper and abacus. This week he takes a look back at Olympic Basketball and the ramifications the numbers supply within.

Admittedly, it’s a little late for Olympic basketball coverage, given that the competition ended sometime around 4:00 am EST on Sunday morning, but Thursday is Arbitrarian day, and so today I’m going to try to tell the story of the US Men’s Olympic basketball team retrospectively, in statistics and graphics.

Predecessors

This most recent iteration of the US men’s basketball team was slated to “redeem” the American program in international competition. After several successive failures to dominate their competition, much was made about the degree to which the rest of the world had caught up to the level of American basketball and/or how the American players, because of would no longer be able to dominate in international competition. Several of the more recent US squads were derided as selfish, non-fundamentally sound, failing to take international competition seriously–the narrative was one of how hubris could lead even the mightiest to fall.

It has been said that during those dark years, the US was “just fielding all-star teams,” and that part of Jerry Colangelo’s plan for a return to dominance was to field carefully constructed teams, with role players and specialists–not just 12 guys who could score. To what extent is this true? How much credit does Colangelo’s craftsmanship deserve? As we like to do here, let’s take this subjective claim, and apply a little bit of rigor to see if it holds up without the patriotic feelings and stirring redemption narrative clouding our judgment. For answers, let us look to an application of the SPI style trichotomy:


(Note: If you turn captions on (second button from left on bottom), each diagram is labeled with its year. Also, hit pause and use the arrows to review each image at your own pace.)

Above is a series of graphics depicting the SPI styles (based on their NBA statistics) of each team fielded by the US in major international competition, from the Dream Team in 1992, to this year’s “Redeem” Team, with the exception of the 1998 World Championship team, which was largely composed of non-NBA players.

What differences can we identify in each team’s composition? Did Colangelo really put together a thoughtfully composed team? It appears to me that this was at least some part of the difference between this year’s team and those recent teams that ended in failure and disappointment. The main thing I notice, in comparing the 2002, ’04, and ’06 teams (although especially the first two) to each of the others, is a relative dearth in the pure perimeter region.

Each of these teams has an eclectic smattering of interior types–some years they appear more offensively-minded than others, and the 2008 Olympic team, interestingly has only three players classified as such in the SPI scheme. But look first at the 1992 team, which is stacked to the gills with players in the 10 o’clock to 12 o’clock range, meaning that their statistics indicate a focus on perimeter play, or an absence of focus on scoring, relative to the league. Such is the case, to a slightly lesser extent, with each of the other teams up through 2000.

In 2002, the perimeter appears to have become less of a priority, stocked with
Andre Miller, Davis, and young Jay Williams–good players, but not the “pure point” types which manned some of the other teams. Further, that team was full of Perimeter Scorer types, three of which (Reggie Miller, Finley, and Allen), are known more for their shooting than their all-around game.

2004 may have been an even more poorly-constructed team, with essentially no Pure Perimeter players. James and Wade are capable of facilitating, but this is not typically their primary role, and James played relatively few minutes anyway. Instead, that role was left mainly to Marbury and Iverson, who are known to look for their own shot as often as they pass–and this subjective reputation is backed up by the SPI analysis.

The 2006 team was much better–it is obvious that effort was made to compose a team of players of many different types–this is the only year in which there is at least one player from each sextant of the SPI plot. This is not necessarily a good thing for winning, but it indicates that thought was put into how each player would fit together into a whole. Further, two actual perimeter players were included, Paul and Hinrich, and this team performed substantially better than their Marbury- and Iverson-lead predecessors.

This year’s team sees a return to past glory, likely in no small part to a fully-stocked trio of Pure Perimeter players, able to push the ball up court and facilitate any of the able scorers on the team. Interior play was de-emphasized, as the team’s focus would be on a disruptive defensive style aimed at generating turnovers and leading to fast breaks–for this, speed, not size, was key.

In sum, it appears as though part of the credit for the USA’s Olympic success really might belong to Mr. Colangelo. Though it is the players on the floor who do the actual winning and losing, a large part of the results likely stemmed from what happened way before the opening tip.

Now that we have covered the pre-Olympic preparation phase, let us turn our attention to what actually happened in Beijing.

Assessing productivity in these Games

Due to limitations on the ease with which game-by-game data can be collected for the Olympic tournament, I will be discussing productivity (as measured by MEV) rather than value (as measured by MVP)–but here, the story is pretty clear.¹ Below is a list of each athlete, with their SPI factors, points- and MEV-per game numbers, and Valuable Contributions Ratio. I’ve also included what I call Points Per Points Possible (p4), which divides points scored by the number of points possible on each of their shot attempts (2 for all field goal attempts, plus an extra one on three-point attempts, plus one for each free throw attempt).


Many of the most productive individuals play professionally in the NBA. These numbers indicate that LeBron James was the most valuable to Team USA, but note that Wade was almost as productive in substantially fewer minutes (his VCR is the highest on the US team). As such, I have to name James the MVP (for the team and the whole tournament), but Wade is the US’s Most Efficient Player, which is exactly what the team needed from its first man off the bench.

How did contributions break down for each team? Below is a series of charts that plot the sources of production for each team, based on tournament-cumulative MEV. Each player is colored according to their SPI type, and players with negative MEV are zeroed out (because it’s hard to depict the area of a negative number):


Click here if you want a whole window full of these pie charts.

Among the best teams in the competition, Argentina was more highly dependent on their top-tier players than were Spain and the US. The two teams most reliant on a single player were China, anchored by Yao Ming, and Iran, lead by Ehadadi. Croatia appears to have had the most balanced contributions, although this is often a trait of weaker teams, because it is easier to field a team of equally poor players than one of equally excellent players.

What did each player produce individually? The table above gives the summary report of the points-value of each player’s production, in the form of MEV. Below, however, I have the complete breakdown of each player’s counting statistics for the Olympic tournament, as a percentage of the simple sum of these stats for that player. I have tried to arrange the graphs such that adjacent areas make for easy comparison of paired statistics–missed field goals is next to points, assists next to turnovers, offensive and defensive rebounds together, followed by the defensive statistics, etc. Players are sorted by MEV/gp. Coloration is of course derived from SPI type based on Olympic statistics.


Click here if you want a whole window full of these little pie charts.

Seeing these pie charts all together as small multiples allows us to easily compare two or more players. Note, for example, that Dwight Howard and Chris Bosh were almost perfect substitutes for one another: they have almost identical per-game MEVs, and their stat distributions look very similar–the only exception seems to be that Bosh seems to have grabbed relatively more defensive rebounds and turned the ball over slightly more, while Howard did a lot more fouling.

Carlos Delfino’s SPI color identifies him as a very tournament-representative player; that is, his relative distribution of scoring, perimeter, and interior statistics reflect that of all players collectively. The gray color indicates this league-relative neutrality, and he serves as a useful benchmark against which to compare others.

As is evidenced by his orange color and large segment devoted to pts and fgx, a large portion of Bryant’s statistical contributions came from scoring. However, these statistics likely do not give the full picture for Bryant, as his role for most of the duration of the tournament was to shut down the opposition’s best players, not unlike a “Doberman.”

Jason Kidd (very pale blue, about halfway down) is one of few players for whom pts is not the largest segment. Rather his defensive rebounds and assists took priority, although so too, unfortunately, did his turnovers and personal fouls.

Michael Redd (rusty color, much closer to the bottom of the list) offers an interesting example of the usefulness of such a visualization. The first thing one notices is that his pts sector is matched in size by his fgx sector–he missed almost as many baskets as he scored points. Tip for the uninitiated: this is not a productive way to play basketball.

Another way to look at the data is through parallel coordinate plots, which are useful for depicting the rank of an individual across multiple categories. Below, I present PC plots for each member of team USA, where the vertical axis indicates that individual’s rank in each of 9 metrics, relative to the entire pool of Olympic players. On each plot, for ease of comparison, I draw gray lines for the remainder of the US team, but highlight each player individually in their SPI color.


Click here if you want a whole window full of these parallel coordinate plots.

p4 is Points Per Points Possible, described above, AS:TO is the assist-to-turnover ratio, TR/min is total rebounds per minute, DEF:PF is (BK+ST)/PF, which is just an amateurish way of measuring defensive skill.

Looking at these plots, we can see that Wade performed very well. He is in the top four on the US team in each stat, and it is apparent that he is in the top half across the board among all Olympians. Redd, although he was called upon to provide a shooting spark off the bench, was mostly a dud, with a p4 among the lowest in the competition. Bryant was second lowest on the team, but his shooting efficiency looks to have been better than about a third of the Olympic players, and thus much better than Redd’s. Note that due to a small sample size, some of these ranks will appear odd, namely Redd’s high ranking on the DEF:PF statistic and Kidd’s high p4 rating. Neither of high rankings are what we would expect from these players, but Redd played relatively few minutes, and Kidd only took shots he couldn’t refuse to take, resulting in good ratings for in these areas over a small number of observations.

I would be very interested to hear any more insights you glean from the above displays–feel free to copy any of the charts for your own use, just also please provide a link back to HP.

Olympic style

We’ve seen the NBA styles of the players that make up Team USA, we’ve seen their SPI factors and even their specific statistical breakdown. Now, we turn to a full SPI Spectrum graphic depicting each Olympic competitor, and their type, based solely on their production in the Olympics. Player names are scaled according to their MEV totals, so that the most productive players are the easiest to spot.


Fullscreen Version

Several things stand out to me. First, I am impressed by the degree to which this Olympics-based diagram matches up with the NBA-based diagram, for players who appear in both. Redd, Bryant, Williams, Kirilenko, Howard, Yao and Boozer all played similar styles in these Olympic games as they did in the 07-08 NBA.

Even more enlightening are the differences: Louis Scola played much more of a scoring role for Argentina than he does for the Rockets (understandably so). Dwyane Wade and Chris Paul shifted their focus away from scoring, relative to their NBA style, likely because they were not required on this team to carry their team’s point production. Anthony’s purported focus on rebounding is reflected in his shift from a somewhat perimeter-biased Scorer to an Interior Scoring type. Jason Kidd became an even more extreme Scorer’s Opposite, eschewing shooting opportunities whenever possible.

The most significant shift, however, might be seen in the play of LeBron James. Last season in the NBA, James lined up at about 12 o’clock on the diagram; the style with which he most closely aligned was Perimeter Scorer. In these Olympics, however, James’ style reflects his commitment to doing whatever was needed by the team. His minty-green color and placement at a little before 11 o’clock reflect his Pure Perimeter style, though his relative proximity to the center of the diagram indicates that his fit here is not perfect. Rather than being the primary scorer for this team, as he is accustomed to being in Cleveland, James stepped up the defensive intensity, leading his team in blocks (with eight), and finishing second in the tournament in steals (with 19!), not to mention leading the tournament, by a landslide, in menacing scowls. Further, he was second on the US team in assists (30; Paul had 33), his assist-to-turnover ratio was a respectable 1.76, and he finished in the tournament top ten in total rebounds. To put it in perspective, the role James filled for this US team was similar to that played by Magic Johnson on the showtime Lakers, which is quite a niche, indeed.

Conclusion

In sum, we can see that at least some of the hype is true. There has been some well-placed cynicism regarding the extent to which the “Redeem Team,” and our collective impression thereof, is a product of marketing. I have no doubt that at least some of what we believe about this team and its players is fabricated for the purpose of generating a positive image, and greater sales. However, at least two claims made about this team can be empirically verified, and I have tried to do that here.

The first claim is that this team is different from the failures which came before. Using NBA statistics and the SPI Typology, I am inclined to believe that in construction, this team is different than its three previous iterations, and more similar in design to the Dream Teams of the 1990s.

The second claim is that the players on this team changed their styles to accomodate each other, to better fit together as a team. Comparing SPI positions in the Olympics to SPI positions in the NBA, we can see which players had similar statistical distributions, and those which modified their style. Each player on the US team was either accustomed to or able to lead their NBA teams in scoring on any given night, and in Olympic competition, this ability to rely on others to score allows (at least theoretically) unselfish play. The question was always whether or not this team of able shooters would be able to “put aside their egos” and fill a specific role for this team, which may or may not include a substantial amount of offensive production. By and large, it appears as though the players asked to do so have responded positively. Though several US team members played with styles similar to their NBA styles, this reflected the reported desire of the coaching staff and management of the team (i.e. Michael Redd is supposed to be a shooter). Other players saw drastic shifts in their style of play, especially movement away from a focus on scoring, as a universally capable offense permitted each individual to do less of the shooting than may be required on their NBA squads. Based on this graphical evidence, I am willing to advance a tentative rejection of the null hypothesis that the players did not fill the roles they were asked to. Rather, it appears as though they played as a cohesive unit, maximizing their strengths and possibly sacrificing for the team.

I hope this late coverage was worth waiting for. I would be very interested in hearing your reactions to any of the ideas I’ve put forward, and I would especially like to know if you see any interesting relationships jump out in any of the SPI diagrams. I haven’t even begun here to discuss the interesting similarities between several of the international players and those from our own NBA in the Olympics, I suppose I will leave that to you. As usual, I’d love to hear from you in the comments, and in the survey, and please Buzz this up!







¹ If you are particularly interested in game-by-game contributions and value, I did track a modified version of MVP for team USA throughout the Olympics and pre-Games warmups. You can see the per-game and cumulative results here.

The Arbitrarian: A generalized continuous typology of playing styles

August 14, 2008 in by with 2 Comments

The Arbitrarian column is written weekly by David Sparks. You can read more of his work at his own blog. This week’s head asploding column is on player positions versus styles, and what they’re all about. Note to feed subscribers: The graphics in today’s post won’t come across in the feed, so you might want to click through to the original at HP.

What does it mean to be a Point Guard? Typically, point guards are expected to carry the ball up the court, set up the offense, make passes, and take few shots, at least relative to other players on the court. But how much can the term “point guard” actually mean if it applies to both Jason Kidd and Baron Davis? Further, what does it mean to be a “small forward” if Dominique Wilkins, LeBron James and Shane Battier all fall into that category? What do Vlade Divac and Amare Stoudemire have in common, aside from both being called “centers”?

The obvious point is that traditional position classifications, while they mean something, still convey relatively little information about a player’s function on the court. As observers of the game, we attempt to compensate for this by adding any number of modifiers to these position descriptions: combo guard, pure point guard, defensive center, swingman, etc. Each of these is used to more accurately specify a player’s style or role on the team, yet each is still somewhat definitionally ambiguous and subjective by design. One Tom Ziller has done some work in attempting to statistically classify guards on a continuum between “small two-guards” and “pure points,” but this is only a small first step in the right direction. I present here a generalized methodology for structuring a playing style spectrum, and identifying each player’s position within the continuum. By looking at actual statistics produced, we may eschew fuzzy descriptors of position and style in favor of a very specific, yet still highly flexible system of style identification–which provides us with an improved vocabulary with which to describe, among many other things, player types and team styles.

Very rudimentary factor and cluster analysis I performed a long time ago indicated that there are distinctions in the data between players who tend to try to score a lot, those who play a “smaller” game, and those who play like “big men.” In terms of the NBA’s tracked counting statistics, this translates to a differentiation between those who specialize in points and field goal attempts, rebounds and blocks, and steals and assists. I have chosen to call each of these three tendencies Scorer, Perimeter, and Interior, and collectively they form the SPI Style Trichotomy.

Calculation

To identify each player’s style is conceptually simple, but computationally somewhat more complex. Essentially, one sums each player’s fga + tr + bk + as + st, and determines what percentage of the total each SPI factor constitutes:

  • Scorer percentage = fga / (fga + tr + bk + as + st)
  • Perimeter percentage = (as + st) / (fga + tr + bk + as + st)
  • Interior percentage = (tr + bk) / (fga + tr + bk + as + st)

These numbers are interesting on their own, but for the calculation of an index of style, they require further manipulation. In the league as a whole, the Scorer percentage is around 50%, the Perimeter percentage around 20%, and Interior 30%. Thus, if using these percentages, the vast majority of players would appear to be very scoring-centered. My concern here, in constructing a useful index, is to identify player propensities relative to other players, and for that, I calculate the percentile of each player’s percentages.

  • Scorer index = percentile(Scorer percentage)
  • Perimeter index = percentile(Perimeter percentage)
  • Interior index = percentile(Interior percentage)

Thus, even though the maximum Scorer percentage in a season might be close to 75% while the maximum Perimeter percentage is closer to 25%, the players with the highest percentages in the sample under consideration will be assigned an index value of 1. Players with median values on a percentage will have an index value of 0.5, and so on. The percentilization normalizes across style tendencies and player subpopulations, and has the added virtue of scaling from 0 to 1.

Interpretation

Thus we have a set of three numbers for each player which can be used to characterize his playing style. The numbers easily translate to more qualitative descriptions. A player with a SPI triple of (0.8, 0.2, 0.7) is an interior scorer, without much perimeter production. A player with this triple (0.1, 0.7, 0.75) is anything but a scorer, sometimes called a “glue” guy. Someone at (0.5, 0.5, 0.5) produces the league median of each type, which is different from a player whose percentages are 33%, 33% and 33%. Such a player would have a relatively lower Scoring index, for example.

Since each individual is characterized by three variables, their SPI type can be plotted in three dimensions. Unfortunately, three dimensions are difficult to convey on a computer screen, so here is a plot which depicts Perimeter indices along the X-axis, Interior indices on the vertical axis, and Scoring indices as the size of the point.

(Click to enlarge)

Historical application note: Since steals and blocks have not been kept for the entirety of the history of professional basketball, players from earlier eras may have slightly skewed SPI values. While percentages and indices can still be calculated based only on fga, tr, and as, it is not difficult to see that leaving out blocks and steals, in comparison to eras in which those defensive statistics are included, will tend to skew players from an earlier era more toward the Scoring type. Unfortunately, without substantial era-specific correction, this effect is unavoidable. However, the sorting still manages to work well, especially if this detail is kept in mind when making certain cross-temporal comparisons.

Presentation

One of the advantages of using three sub-indices to construct the overall SPI Trichotomy is the convenient translation of index values to color. The three primary colors of light are Red, Green and Blue, and when combined in certain proportions, it is possible to generate infinite gradations of color (see Wikipedia). This means that each SPI triplet for each player can be represented as a single color. This aids understanding and comparison, as it is much easier to keep in mind that a certain player is a deep red than that his SPI triplet is (0.9, 0.1, 0.2), or that a player is a medium grey than that his triplet is (0.45, 0.53, 0.55). Further, a greenish-blue player is easily identified with another greenish-blue player, without having to specifically compare each of the players’ three index values. The human eye is capable of extremely high-resolution discernment, and using a single color to represent three numerical values takes advantage of this.

Here is the above plot, with color added according to RGB values derived from each player’s SPI indices, as you can see, “blueness” increases from bottom to top, “greenness” from left to right, and “redness” varies with the size of the point. The top-right corner is aqua or cyan, while the bottom left is mostly reddish, due to an absence of green and blue.

(Click to enlarge)

Unfortunately, this presentational format leaves a lot to be desired. Since each player can be represented by just one color, can we do better than a pseudo-3-dimensional plot? The answer is yes and no: No, because to ensure that the hue, saturation, and value of each color are captured, we still require three variables (see Wikipedia); yes, because most of what we are interested in here is hue–the underlying color for each player, red, yellow, green, aquamarine, vivid tangerine, indigo, etc. The other two components of HSV color space, saturation and value, allow us to see how “pure” the hue is, which in our basketball application, translates to how “pure” an individual’s playing style is.

Playing style as a continuous spectrum

Using polar coordinates, we can plot each player’s position in a continuous spectrum of playing styles. Each individual may be represented as a vector, with Hue translating to direction/angle and Saturation+Value translating to magnitude/distance. The angle of the vector indicates the player’s style, and the magnitude of the vector indicates the “fit” of that player to that style–that is, since it is unlikely any given player’s statistical profile will assign him perfectly to a given category, there is a level of fitness that captures the extent to which they do. Very rarely will a player have some assists and steals, but no blocks, rebounds or field goal attempts, which would give them a P index of 1, but S and I indices of 0. Because of this, rarely will any player be a pure green, or pure blue or red. The degree to which they are a mixture of styles/colors is captured somewhat by their fit.

We can describe a player’s style by their SPI indices, or by their color, but we can also describe them according to their angle, which is most easily communicated by referring to positions on a clock. In the graphic below, the top of the circle can be thought of as 12 o’clock, the far right translates to 3:00, the bottom is 6 o’clock, etc. This is yet another way to describe style more easily than by referring to the player’s SPI triple, but more accurately and consistently than by descibing color. Finally, I have assigned arbitrary descriptive names to each of six major “spokes” on the diagram, which should help the uninitiated translate commonly-used adjectives into positions on the clock. Here is a listing of SPI indices, fit, clock positions, and shorthand labels for each player in the 07-08 season, as well as 500 all-time greats.


Graphical Display

Below is a graphical depiction of the SPI Playing Style Spectrum, with the positions of 250 of the NBA’s all-time best.


Fullscreen Version

As you can see, the SPI typology encompasses Mr. Ziller’s point guard continuum, and much more. “Small two-guards” (exemplified by Barbosa, Ellis, Terry and Iverson) line up at about 1 o’clock; “Combo guards” mostly fall between 11:30 and 12:30; “Pass-first points” even more to the left; “Pure point guards” are seen at about 11 o’clock. The spectrum continues, however, to more defensive/bigger guards, more well-rounded perimeter players, point-forwards, glue guys, defensive stoppers, big men, widebodies, power forwards, pure scorers, and back to shooting guards.

One interesting use of the spectrum graphic is to make comparisons. Unsurprisingly, Kevin Johnson and Steve Nash have similar styles; Kobe Bryant and Michael Jordan are in close proximity; and Tim Duncan and David Robinson filled almost exactly the same role for the same team. It’s also interesting to make comparisons across eras: Dennis Rodman/Bill Russell, Vince Carter/Rick Barry, Michael Jordan/Jerry West, Magic Johnson/Jason Kidd, etc. It’s also possible to identify stylistic opposites: Chris Paul-David West, Shaquille O’Neal-Kobe Bryant, Allen Iverson-Marcus Camby, etc.

Here is a SPI plot for just the 2007-08 Season (note that player names are represented in abbreviated form):


Fullscreen Version

Thus far, the SPI typology is useful mostly as a classification system, but if you’re interested, I’ve spent some time looking into the relative value of certain types, as well as their interactions. There’s much more to be done in this vein, but some of the initial findings have been interesting. (APBRmetrics discussion)

Conclusions

Evidently, it’s possible to develop a comprehensive classification system of playing styles using statistics alone. Now that the SPI color scheme has been introduced, you might find it interesting to refer back to the graphics I presented last week, in which I’ve applied the scheme. It adds a dimension of information to the season and team history graphics. I’d be very interested in hearing your thoughts in the comments, as well as in the obligatory survey below.

NEW! I’ve just created desktop wallpaper-sized All-Time Great SPI Graphics. Download them and enjoy! [1024 x 768] [1280 x 1024]

The Arbitrarian: Individual Contributions To Team Success

August 7, 2008 in by with 17 Comments

Note for those viewing this post in a feedreader: today’s Arbitrarian is very graphics-heavy, and the images won’t show up in the feed version of the post. If you want the full experience, I suggest clicking through to see the original at HP.

Last week, we looked at player productivity, as based on box score output. Today, we’re going to look at value, which, as you will see, is somewhat different than productivity.

The MVP is not necessarily the best player in the league, nor the most efficient. Many times, the MVP award goes to the player widely considered to be the best player on one of the better teams, but when it comes to arbitrating between comparing the best players on several of the best teams, there appears to be no hard-and-fast rule, and subjectivity enters into play. Today, I will propose that value ought to be quantified in terms of individual contributions to team success, where success is measured in wins.

If we can estimate the number of wins for which each player is responsible, we can do away with the arbitrary focus on only the best few teams. It is theoretically possible, for example, for the most valuable player to be an absolutely dominant but lonely contributor on a middling team, while the better teams each have enough decent players that no single one can be credited with a large portion of their success. We are still, however, left with the problem of objectively measuring each player’s contribution to team wins. To do this, I’d first like to explore…

A non-basketball thought experiment

Imagine a lemonade stand owned and staffed by Xavier, Yvette, and Zach. They make money by selling home-brewed lemonade at the end of their cul-de-sac, and only one of them staffs the stand at any given time. After their first month in business, they look at their lemonade sales revenue, and try to figure out which salesperson deserves what part of the income. One option would be to split the revenue into thirds–three employees, three parts. Zach claims that such a distribution is unfair because he worked over half of the total number of hours, while Yvette and Xavier worked about a quarter of the hours each. He claims that the distribution should thus be more like (1/4, 1/4, 1/2).

Xavier points out, however, that if they are trying to assess each employee’s value, they should try to find a more specific measure of actual revenue generated by each seller. He suggests that, since revenue is generated by lemonade sales, revenue generation should be measured in terms of the number of lemonades sold by each employee. Since they kept detailed records of such numbers, this is easy to calculate: Xavier sold 2/5 of all glasses, Yvette 1/2, and Zach just 1/10. Zach is disappointed that his pay-per-hour gambit was foiled, but must concede that this arrangement is more just–Yvette and Xavier are much better salespersons, and did more to help the company make money, while Zach mostly daydreamed during his hours on the job.

Back to basketball

What I have in mind is the application of a similar methodology to basketball. We have an excellent estimator of aggregate value–team wins; and credit for these wins can be apportioned to the players who work for those wins. There are a plethora of ways this could be done–we could arbitrarily estimate credit for each player on each team: The superstar might get 50% of the credit, the rest of the starters get 10% of the credit each, while the remainder is split amongst the bench players. Perhaps we could look at minutes played–after all, ceteris paribus, removing a mediocre player, and replacing him with a better player for the same number of minutes, should result in a greater number of team wins. Similarly, increasing the number of minutes played by a good player (to a point) should increase wins, while increasing minutes played by a bad player should lead to fewer wins.

This method isn’t foolproof, however: certain high-minute players might be daydreamer-types like Zach in the example above, while others might be feverishly productive. Consider, for example, Matt Carroll versus Yao Ming in 2007-08. Both played roughly the same number of minutes (2016 and 2044), but Yao was substantially more productive (by almost any measure) than was Carroll in that amount of time. Estimates of their value should reflect this difference.

Instead of minutes, I have chosen to use Model-Estimated Value (or MEV, discussed here) as an estimate of player productivity. There are several advantages to this choice, but two stand out. First, as discussed previously, MEV is a good estimator of per-game productivity, and so is more helpful to us than looking at, say, games played, minutes played, or points scored alone.

The second advantage comes from the fact that MEV does not perfectly capture player value. If it did, then team-level MEV would correlate perfectly with team wins, and we would not need separate measures for productivity and value. Rather, since some aspects of player value are omitted from the box score–things like defense, effort, intensity, etc–we may scale our MEV productivity estimates by team success, which does implicitly measure all of each player’s contributions.

Bruce Bowen, for example, had a 07-08 per-game MEV of 5.60, which put him below, among others, Wally Szczerbiak. Many would cite this as an example of the failings of MEV–its inability to fully measure defense (not to mention a lack of adjustment for playing time and pace) leads to an undervaluation of players like Bowen. However, Bowen’s defense does show up in the Spurs’ success–no small part of their winning can be attributed to his contributions. Similarly for Szczerbiak–his contributions are reflected in the success had by Cleveland and Seattle–that is, relatively little success. Thus, by crediting players for team success, using MEV as our measure of productivity, we may get closer to measuring each players’ actual value.

This method is still not perfect. We might still be undervaluing Bowen’s relative contribution to the Spurs, and overvaluing Szczerbiak’s contribution to his teams. However, given two players with idential MEV numbers on teams with otherwise identical rosters, the player whose MEV is “worth more” will help his team win more games.

Calculation and results

The measurement of each player’s value to their team is straightforward. Merely take each player’s season total MEV for a given team, and divide it by that team’s season total MEV. This gives us a metric I call Percent Valuable Contributions, or PVC. For Kevin Garnett in 07-08, this calculation takes his season total MEV (1,534.19) and divides it by that of the Celtics as a whole (8,282.62), resulting in a percentage (expressed as a decimal): 0.185. This means that Garnett is responsible for 18.5% of the Celtics’ success, which is a rather large portion, indeed. From here, estimation of value is very easy. Simply take this PVC number, and multiply it by team wins. This gives you each player’s BoxScores (BXS), their individual contribution to team success. The first tab on the table below depicts the numbers that go into the BoxScores calculation for each player in the 2007-08 season.



I’ve sorted each player by their PVC for the sake of comparison. In terms of value to their team, the top three players are James, Paul, and Jefferson. All three are good players, but certainly Jefferson is a step below the other two. Since BoxScores accounts for team success, we can clearly see Jefferson’s actual value is much less than that of James and Paul–he might be the most valuable player on the Timberwolves roster, but such is not a high distinction.

For more insight, note the series of players whose PVC comes it at around 0.185: Steve Nash, Joe Johnson, Kevin Garnett, Richard Jefferson, and Carlos Boozer. Even the casual fan knows that these players are not all equally valuable, though they may be equally valuable to their respective teams. Note that Nash and Boozer generated a much higher MEV total than the other three–this is largely because, as the table also shows, Phoenix and Utah had much greater team MEV totals, thanks to a faster-paced playing style. Team wins complete the picture–Richard Jefferson was responsible for 18.5% of his team’s 34 wins. His value is thus estimated at 6.29 BXS. Kevin Garnett was responsible for 18.5% of his team’s 66 wins. His value is thus 12.23 BXS. As you can see, by accurately measuring productivity (MEV), and accounting for team success (wins), we are able to objectively assess each player’s value (BXS).

An aside into rated productivity

Another useful measure, especially for comparing players on poor teams, or those who played limited minutes, is what I call the Valuable Contributions Ratio (VCR). This is a pace- and playing time- adjusted metric of productivity assessed at the per-minute level. As above, this calculation is straightforward and intuitive. Merely take each player’s PVC (MEV/team MEV) and divide it by each player’s percent of team minutes played (min/team min). Thus, we are dividing a percentage by another percentage (which is why I call it a ratio–units are somewhat meaningless). This statistic controls for team pace and playing time, and is independent of team quality–it captures productivity relative to the time allowed for production.

This is useful for comparing bench players, players who miss a substantial number of games, and rookies. Bench players get a “fair shake” by this statistic, because they often have less time on the floor in which to accumulate MEV toward a larger cumulative share of team success. Same for injured players–Andrew Bynum did not play very many games for the Lakers in 07-08, and as such was less valuable in terms of team wins. However, when he did play, he produced very efficiently, with a VCR of 1.36. (This means that he was responsible for 1.36% of his team’s production for every 1% of team minutes played–which is very efficient.) VCR is useful for comparing rookies, as well, since they often play relatively few minutes, and since their teams often win very few games. Rookies with high BXS are the most impressive, but more often than not, rookies don’t produce many wins. Rather, they may produce MEV efficiently, and we can see this in VCR. Among rookies with substantial playing time in 07-08, Carl Landry produced the most efficiently, with a very respectable VCR of 1.39. (The Arbitrary Rookie of the Year, Kevin Durant, was 7th among rookies by VCR, and 5th on the rookie BXS list after Scola, Horford, Moon, and Thaddeus Young. He did, however, lead all rookies in points per game. WoW Club!)

BXS MVPs and all-time greats

Look again at the table above, but this time, select the second sheet, titled “07-08 BXS.” This lists each player from the 07-08 season, on each team for which he played, and includes measures of productivity (PPG, and MEV/gp), efficiency (VCR), and value in terms of BXS. As you can see, the obvious most valuable player in 2007-08 was Chris Paul, who was responsible for over a quarter of his team’s 56 wins, for a BXS of 15.41. Kobe Bryant, this year’s Arbitrary MVP, had a good showing as well, but was responsible for almost three fewer wins than was Paul. Kevin Garnett, who was thought to be a more subjective favorite for MVP, acquitted himself nicely in objective terms, by generating 12.23 wins even while missing eleven games.

For a more historical perspective, see the third tab, “BXS Seasons.” This lists the same information, but for the 500 most valuable seasons from the population of every professional basketball season since the beginning of the NBA, even including the ABA. Unsurprisingly, Chamberlain tops the list, although his most valuable season was not his most productive in MEV terms. Shaquille O’Neal’s dominating performance for the incredible 67-win 99-00 Lakers is the most valuable season in recent memory, followed closely by Jordan’s post-first-retirement 72-win season in Chicago. Perhaps surprising, but perhaps not to those who have always appreciated the Big Ticket, is Kevin Garnett’s extremely high value in 03-04. Always a valuable player to his team, Garnett and the Timberwolves finally put it together for one great year, and Garnett’s relative value (PVC) translated to absolute value (BXS).

The final tab, “BXS Careers,” accumulates the performance of 500 NBA greats. The table is sorted by BXS82, which is the number of wins each player would be expected to produce in 82 games played, given his career performance. There may be a few surprises, but they are instructive: The first is Alex Groza, who was a great player in the early years of professional basketball, but whose career was cut short. The second surprise might be the ordering of Michael Jordan, relative to Magic Johnson and Tim Duncan. Many fans and observers would identify Jordan one of the most, if not the most, valuable player ever, and here he ranks sixth at a per-game level. The first thing to note is that Duncan has always been more valuable than his box score statistics might indicate, and this is reflected in his BXS measure. Secondly, Duncan has not yet seen his productivity or value decline substantially due to aging. For the most part, Duncan’s 13.73 BXS/82 average comes from the peak of his career. Johnson retired after a relatively short career, and his comeback in 1996 was brief. Jordan, by comparison, had a second comeback for Washington during which he played 142 games of much less valuable basketball. If you exclude Jordan’s Washington years, his career BXS82 becomes 14.62, which puts him solidly above the other two.

Visualizing value

Now for the first graphical visualizations in the life of this young column. Since BXS is derived by multiplying player contributions (PVC) by team success (team wins), we can envision BXS itself as the area of a rectangle with sides of PVC and Wins. This lends itself to graphical expression, with the league as a rectangle, 41*30= 1230 wins wide, and 100% high. Partitions may be made on the horizontal axis for each team, scaling each section by that team’s number of victories. Within each team’s segment, further divisions can be made for each player, according to their contribution to team success. The best explanation is an example, displayed below:


Fullscreen Version

You may use the controls at the top left to zoom in and pan across the graphic to see more detail, or an expanded overview, as you wish. The “Fullscreen Version” link directs you to a much larger version of the same display, which may be easier to grok. The graphic above displays BXS for the 2007-08 season, with team success increasing from left to right, and player contributions increasing from bottom to top. Colors are derived from statistically-derived playing style, where red indicates a propensity for scoring, green denotes perimeter play, and blue highlights interior-play tendencies. Much more will be said about this classification scheme next week.

Several things I’d like to point out in the graphic above to get you started: First, note how tall the rectangles of Chris Paul, LeBron James and Al Jefferson are–this is because height is scaled according to PVC, and these three players were the most responsible for their teams’ success.

Looking across the top row of the graphic, we can identify each team’s Most Valuable Player. For the lowly Heat, Dwyane Wade was most valuable, despite injury. Calderon was most valuable in Toronto (7.1 BXS), though Bosh was a very close second (7.0 BXS). The most valuable player on the best team was Kevin Garnett, but since he had a very supportive team behind him, his individual value was somewhat less than that of Chris Paul, whose supporting cast drops off substantially in terms of contributions after Stojakovic.

One useful perspective granted by displaying contributions in this manner, is that it is easy to compare units across teams. For example, Boston was famed for the Big Three of Garnett, Pierce and Allen. Using the scale on the right of the graphic, we can see that together, these three accounted for almost half of Boston success. Looking across the graphic from the lowest part of Allen’s rectangle, however, we can see that the big three that was most valuable to their team can actually be found in New Orleans, where Paul, West and Chandler can be credited with almost 60% of the Hornets’ success. On the other side of the coin, Detroit, Houston, and Chicago all got a fairly balanced set of contributions, as their subjective reputations might have suggested. I would be very interested to hear about your own observations, as well as your opinions as to how well this graphic meshes with your subjective opinions, in the comments.

I’ve also developed an interactive presentation of the graphic above, with even more detailed statistics. Just follow the link below to the Interactive BoxScores Explorer page. The league-wide graphic has been scaled to fit in your browser window, and players’ statistical details pop up on mousover. Try it–it’s somewhat addictive.

Make sure to click around a little bit–I’ve created “player cards” for each individual, which display even more detailed statistical information, including their playing style, most and least similar players, the mean and standard deviation of their “counting” statistics, and a season-long sparkline of their productivity. Feel free to use them in any application you wish.

The player cards are a quick and easy way to quickly assess any player. For fantasy purposes, for example, if you’re comparing two players with similar averages in assists, you might want to pick the player with the smaller standard deviation about that mean, as indicated by the error bars in the middle section of the card. Alternatively, if you are interested in whether a certain player tends to produce more as the season goes on, the seasonal trends should give some insight into this, as well as how long it takes the player to recover fully from injury, or how much they produce when their minutes go up. Also, I’ve included each player’s most- and least- similar match, based on the 07-08 season, which can help give you an idea of the niche they fill on their team.

Historical BXS franchise timelines

Another excellent use of this BXS area diagramming visualization is to display franchise histories. Better years are represented by wider segments, and the best players rise to the top with tall rectangles. Eras can by identified by patterns in color. Here are two examples, the first depicting the LA Lakers franchise, and the second, Boston Celtics history:


Fullscreen Version

Several eras stand out in the graphic above. The Mikan era was eventually replaced by the Baylor/West dynasty which became the Chamberlain/West years. Notice, incidentally, how West becomes “greener” over the course of his career, indicating a shift away from focusing on scoring, and toward a concentration on other perimeter contributions. A Kareem era follows, though his best years are at this point behind him, and massive team success comes only with the addition of Magic Johnson. Johnson leads the Lakers for ten consecutive years and his retirement marks the end of an era of dominance. LA returns to form in the late 1990s in the hands of O’Neal and Bryant, who turn in some incredible performances–interestingly, there is an obvious breakpoint between 2001 and 2002 on the graphic indicating the switch from the Lakers being “Shaq’s team” to being “Kobe’s team.” The 2006 version of Bryant was forced to carry the scoring load to a massive degree, but the 2008 version (as is evidenced by a much less red color), has been freed up to focus less on point production, and more on doing other things to help his team win. Perhaps the 08-09 season will see Gasol and Bynum float to the top of the column, turning in full, healthy seasons for a very successful LA team.


Fullscreen Version

Boston history is marked even more clearly by the careers of its greatest players. The Celtics of the 1960s are consistently topped by Bill Russell’s defensive-interior blue, and bolstered by some great scorers, like Havlicek and Jones. The 1980s saw a parallel to the Lakers above, in which a perimeter player (here Bird) lead a team supported by strong interior scorers. The 1986 and 1987 Celtics offer an interesting starting lineup of all greens and blues–no single player was responsible for most of the scoring, while other types of contributions were made by all. After years of seasons with little success and narrow columns, the Celtics finally turned it around last season with the addition of Garnett and Allen, almost tripling 06-07′s win total.

The final graphic below offers an alternative take on presenting BoxScores, by tracing the careers of each of 50 NBA greats. Following the peaks and valleys of each player’s tenure, we can also see the years in which many stars were shining brightly. 1972 was a great year for the sport, as was 1990–many of the NBA’s greatest players had good seasons in these years, and the league may be seen to peak at these points. The graphic makes it possible to see the beginning of new eras–witness the start of Bird’s and Johnson’s careers, followed shortly by the rookie seasons of Thomas, Drexler, Jordan, Olajuwon, Stockton, and Barkley. We can see a big dip in strike-shortened 1999, and then another in 2004. This second dip may be troubling–has the quality of the league declined so sharply? Worry not–many modern greats retired in the years just before 2004, meaning that their layers drop out of the picture, setting the stage for the new era of NBA stars we are witnessing today.


Fullscreen Version

Conclusion

As always, I’d very much like to hear your opinions of BoxScores as a measure of value, as well as whether or not you think it gets things right. Was Magic really at his peak in 1987? Were Garnett and Pierce in 2008 in the same league as Bird and McHale in their prime? Should Chris Paul have been the MVP this past season? Were the Rockets really the most balanced of the good teams last year, and will the addition of Ron Artest make them that much more indomitable? Please feel free to leave a comment and take part in the now-customary brief survey below. Next week, I’ll go into much more detail on a new way to describe players, without having to use all those pesky words.

PostScript: Discussion of the “50-Win Standard”
I hope you took the time to read Josh Tucker’s excellent discussion on the established precedent of giving the MVP award only to players on teams with fifty or more wins. I have a few thoughts on how the 50-win minimum precedent fits in with the BoxScores methodology I’ve established here.

The first is that I essentially agree with the implied criteria of such a cutoff (or the implementation of the “Bryant-Nash Rule”). That is, I think that value should determine the MVP, and value is measured in wins, not strictly in individual statistics.

However, as an Arbitrarian, I would tend to shy away from establishing an arbitrary (though precedented) line of demarcation between those who should and should not be under consideration. I think that if you put Wilt Chamberlain on a team with 11 kindergartners, and that team won 41 games, I’d want to consider Chamberlain for MVP. That is, there should be a sliding scale, in the sense that each of the Detroit Pistons individually are less valuable than a single LeBron James, though the Pistons collectively tend to do better than do the Cavaliers collectively.

This is built right into the estimation of BoxScores: A player contributes X% of the production for a team with Y wins, and so he is credited with X•Y of those wins. Fortunately, from the standpoint of the 50-win precedent, as team wins decrease, it gets harder and harder for any player to outproduce a player on a 50+ win team.

For example, imagine a season in which player A contributes 20% of the production for a 50-win team. (20% is on the low end for MVP-candidate PVC, and 50 wins is at the low end for a contender as well, so this is a conservative estimate for an MVP frontrunner.) Such a player accumulated 0.2*50 = 10 BXS. Player B, let’s say, is on a 41-win team. In order to be more valuable than A, B would have to be responsible for 10/41 = 24.4% of his team’s production, which is very high, indeed. In 07-08, only two players had more than a quarter of their team’s valuable contributions, and one of those was the BoxScores MVP, Chris Paul (whose team had 56 wins). LeBron James was the other, and despite contributing more than 2/7 of his team’s production, Cleveland’s win total of 45 reflected James’ second-most-valuable status.

In sum, even if we use BoxScores as our measure of value, it is highly unlikely (although not impossible) that the MVP will come from a sub-50-win team. The precedent will likely remain intact.