# Nylon Calculus: In search of the ideal talent distribution for an NBA roster

In discussing the potential impact of new-addition Gordon Hayward on the Celtics’ ability to contend for a title next season, FiveThirtyEight’s Nate Silver introduced an intuitive formula for calculating championship odds based on something he calls “star points.” Silver defined statistical cutoffs for three distinct star levels — “alphas” (Top-6 or so in the league), “betas” (Nos. 7-18) and “gammas” (Nos. 19-35) — and suggested that any roster including a player from each of these star levels could be a championship contender.

Likewise, any other roster combination offering an equivalent number of star points could also work:

"A team with an Alpha and a Beta — say, this year’s Houston Rockets — could probably skip the Gamma if they had a deep rotation. A team with no Alphas but three Betas — say, Jimmy Butler, [Giannis] Antetokounmpo and Kyle Lowry — would more than likely be good enough to contend for a title. A team with a very strong Alpha could go without a Beta and make up for it with two or more Gammas instead — that’s sort of how the current Cavaliers are constructed.-Nate Silver, FiveThirtyEight"

Silver’s quick guide to roster construction intrigued me and it had me thinking: Is there an ideal way to distribute star power up and down a roster? Would three Betas really be just as good as an Alpha-Beta-Gamma model? Or is Danny Ainge barking up the wrong tree by assembling a group of above-average players when none of them seems likely to become a super-duper star? To find out, I examined the history of NBA teams since 1980 in search of the ideal talent distribution for a championship roster.

## Finding the Best Rosters

Higher-quality teams have a greater chance of winning the championship and, generally speaking, the more star players a team can collect, the better. But there’s a more nuanced question to consider:

Given a certain level of overall team quality, what is the most effective way to distribute talent within a roster?

To answer this question, we first need to quantify team quality so that we can account for its effects on championship odds in our calculations. One useful measure of team quality is average margin of victory. M.o.V. is key component used by computer ranking systems like Jeff Saragin’s Predictor rankings, ESPN‘s Basketball Power Index and FiveThirtyEight’s Elo rankings.

I’m including any team with an average margin of victory of +2 points per game or higher since the 1979-80 NBA season. That leaves us with 375 rosters to examine — about 10 per season. Of those 375 teams, 369 (98 percent) made the playoffs (the six exceptions were generally above-.500 teams competing in a loaded Western Conference, most recently the Thunder in 2014-15). This cutoff was selected deliberately to include each of the last 38 NBA champions, even the least-dominant champ of the era, the 1995 Rockets (+2.1 M.o.V.).

The summary measure VORP (value over replacement player) can be thought of as an attempt to parse a team’s margin of victory among the players on the roster using box-score stats. By definition, the sum of all the individual VORP statistics for the players on a team is related to the team’s margin of victory. Here, I’ll use the sum of VORP to quantify a team’s quality and individual VORP to quantify the distribution of talent within a team.

Given the high degree of correlation between team VORP and average margin of victory during the regular season, it makes sense to find that team VORP is also a useful predictor of championship success. For example, since the 1979-80 season, the league-leaders in regular-season team VORP won the title 17 of 38 seasons (45 percent), including the 2016-17 Warriors, who led the league with a collective VORP of +23.6.

We can formalize this observation a bit by creating a logistic model that predicts the probability of winning a championship based on regular-season team VORP. Among the 375 “good” rosters since 1980, there was an obvious positive relationship between team VORP and championship odds, with an infinitesimally-small p-value (1E-9) indicating statistical significance. We’ll use the estimated title probabilities from this model a bit later on.

## Grouping Similar Rosters

Having satisfied myself (and hopefully you, too) that team VORP predicts championship odds, we can turn our attention back to our main question: does the distribution of individual VORP among its players impact a team’s title chances?

For each of the 375 teams considered, I calculated the proportion of team VORP contributed by each individual player. Because roster sizes have increased since 1980, I only considered the Top-12 VORP contributors on each team. I also set any negative VORP scores to zero — so that I’m estimating a quantity that could be described as “the fraction of the team’s star power contributed by each player on the roster.” With these percent-of-team VORP data in hand for the Top-12 players on each team, I used hierarchical clustering (hclust function in R) to group rosters with similar talent distributions.

I considered a few possibilities for describing the clusters formed by the dendrogram and settled on using six groups that I found meaningful. The top-level split in the dendrogram separates the relatively evenly-distributed rosters — “Team Efforts,” “Big 4s” and “Dynamic Duos” in pink, green and orange — from the relatively imbalanced rosters — “Alphas et al.,” “Big 3s” and “Solo Acts” in the blues and purple. You can see the champions who represent each cluster as well as some of the best regular-season teams who did not win a championship from each group.

Teams have a tendency to evolve from year-to-year with contiguous rosters from a single franchise sometimes jumping from cluster to cluster. Consider Michael Jordan’s best-ever career — he had eight distinct seasons included in the analysis that ended up stretched over four different clusters. During Jordan’s early “Raging Bull” period, Chicago put together two of the few Solo Acts (light blue) good enough to achieve a +2 margin of victory, in 1988 and 1990. Subsequently, the Bulls’ first three-peat (1991-1993) was facilitated by a Big-3 model of Jordan, Scottie Pippen and Horace Grant, which accounted for as much as 94 percent of the team’s VORP (in 1992). Finally, during the Bulls’ second three-peat (1996-1998), the talent distribution was much more egalitarian, with role players like Toni Kukoc, Dennis Rodman, Steve Kerr and Ron Harper each pitching in at least 5 percent of the team’s VORP. These three rosters straddled the line between dynamic duos (1997) and team efforts (1996 and 1998).

The chart below represents the variety of ways that talent was distributed among teams from the six clusters:

The “Team Efforts” are headlined by three Piston champions — from 1989, 1990 and 2004 — who Nate Silver noted as rare examples of starless rosters that relied on a collection of “above-average players at nearly every position, a deep bench and a cohesive rotation” for a successful title run. But, by my measure, the quintessential team effort was turned in by the 2014 Spurs who had seven different players contributing at least 9 percent of team VORP and nobody accounting for more than Kawhi Leonard at 18 percent.

Like the 1996 and 1998 Bulls, some of the rosters found in the Team-Effort cluster had a definitive alpha dog leading the way. The 2015 Warriors and the 1987 Lakers — two other teams who also featured a league-MVP — were grouped in the pink cluster, too. In each of these four instances, the deep collection of talent found on the roster insured an individual VORP contribution of no more than 37 percent of the team sum.

The dynamic-duo cluster included a slew of Laker tandems: Magic Johnson and Kareem Abdul-Jabbar (28/25 percent, 1985), Johnson and Byron Scott (29/26 percent, 1988), Shaquille O’Neal and Kobe Bryant (40/26 percent, 2001 || 28/25 percent, 2002) and, most recently, Pau Gasol and Bryant (26/24 percent, 2009).

The subsequent addition of Metta World Peace (13 percent of team VORP) and the improved play of Lamar Odom (19 percent) pushed the 2010 Lakers into the Big-4 cluster. That version of the Lakers shared their group with the Julius Erving-Moses Malone-Mo Cheeks-Bobby Jones quartet (82 percent of team VORP, collectively) of the 1982-83 76ers.

The two blue clusters — “Alphas et al.” and “Solo Acts” — are filled with teams that are dominated by a single player, but teams in the former group had more support for that team leader. No modern superstar champion accounted for more of his team’s quality than Hakeem Olajuwon on the 1994 Rockets (47 percent). Rosters with even less balance — such as the LeBron James-led Cavaliers of 2006 and 2010 — have yet to yield any modern champs.

## Evaluating Which Type of Roster is Most Likely to Win a Title

Now that we’ve grouped the teams by VORP distribution, we can evaluate which type of roster has had the most success in winning NBA championships.

For five of the six roster clusters, the average team VORP was right around +16, give or take a few tenths of a point. It follows that our simplest-possible championship probability model (using team VORP as the only predictor) expects to see a similar fraction of champions from each of these five groups (~10 percent). The one exception was the Solo-Acts cluster, which contained lower quality teams (average team VORP = +14.6) and, therefore, was expected to produce a smaller fraction of champions (5 percent).

The most evenly-distributed teams (pink and green) had smaller-than-expected championship returns relative to model predictions based on team VORP in the regular season. Aside from the Solo Acts group, the least evenly-distributed teams (dark blue, purple) were the most likely to exceed the model’s predictions, producing more championships than expected from regular-season VORP. More formally, including the standard deviation of individual VORP scores across the roster as an additional predictor of title odds in the model indicated a statistically significant positive relationship between roster imbalance and title odds. In other words, for a given level of overall team quality, an unbalanced roster was more likely to produce a championship than a balanced one.

Intuitively, this finding makes sense. In the playoffs, benches tend to get “shorter” as fewer players are given an opportunity to play in the most-important moments of the season. Thus, sacrificing depth in favor of star power seems like a logical strategy for championship success. Still, there was obviously an upper limit to this type of beneficial imbalance, with no more than a 50:50 split between an alpha dog and his pack yielding a title.

## Looking at Roster Types for Next Season

Eight 2016-17 rosters were included in the clustering analysis across five different groups. And — if they had been good enough to make the cut (+0.8-point margin of victory) — the Thunder would have inhabited the sixth cluster (Solo Acts, light blue) because Russell Westbrook accounted for an overwhelming 78 percent of his team’s VORP (whoa).

The league-champion Warriors employed a Big-4 model with Curry, Kevin Durant, Draymond Green and Andre Iguodala accounting for 78 percent of the team’s VORP. A more conventional choice for fourth might have been Klay Thompson, but VORP tends to underestimate his defensive contribution relative to a stat like Regularized Adjusted Plus-Minus. In any event, all five of the Warriors key players will be returning next season, so a similar distribution of responsibilities can be expected.

The runner-up Cavaliers were grouped in the Alpha et al. cluster, with James contributing nearly half of the team’s VORP (49 percent). Given their lack of activity during a rudderless offseason so far, the Cavs seem likely to employ a similar model next season.

The Spurs and the Raptors had mirror-image talent distributions last season (although the Spurs had a much higher-quality roster — +19.0 team VORP vs. +15.2). Leonard paced the Spurs with 31 percent of team VORP followed by nine players with 4-12 percent each, whereas Kyle Lowry led the Raptors with 33 percent and he was trailed by eleven players with up to 13 percent of team VORP each. Despite the departures of some important contributors from both rosters, team approaches will likely be emphasized again next year.

Several other teams may be adopting new talent distributions next season. The Clippers — formerly a Big-3 of Chris Paul (32 percent of team VORP), DeAndre Jordan (23 percent) and Blake Griffin (20 percent) in 2017 — will be looking to shift more focus towards Griffin next season with the departures of Paul and J.J. Redick. On the other side of the coin, James Harden was an undisputed Alpha last season in Houston (51 percent of team VORP, dark blue cluster); but the arrival of Paul may change the Rockets’ formula.

Finally, the Jazz paired a talented duo of Rudy Gobert (33 percent) and Gordon Hayward (25 percent) last year. With Hayward leaving Utah for Boston in free agency, the Jazz are left with a defense-first leader and they seem positioned for a team effort next season. The 2017 Celtics fit into the “team effort with a definitive leader” mold as Isaiah Thomas headlined the show (34 percent of team VORP) followed by his co-stars Al Horford (20 percent) and Jae Crowder (15 percent) and a supporting cast of Amir Johnson, Marcus Smart, Kelly Olynyk and Avery Bradley (5-12 percent each). Johnson, Olynyk and Bradley will all be playing elsewhere next season, which may shift Boston towards a Big-4 talent distribution.

Obviously, the Big-4 model can be an effective roster construction — it worked just last year, after all. But Big-4s have produced only three champions out of 70 good rosters since 1980, a rate lower than expected based on team VORP. Moreover, those three champions that succeeded with a Big-4 each had higher team VORP during the regular season (23.6 for 2017 Warriors, 18.5 for 1983 Sixers and 16.1 for 2010 Lakers) than the Celtics did last season (12.5).

Next: Charting the first week of NBA Free Agency

In the end, my assessment for the Celtics is no different than Nate Silver’s: Unless Hayward takes a huge leap in production next season and assumes the role of a true alpha player, his addition will not be sufficient to put Boston in contention for the title in 2018.