Introduction
A common refrain among sports commentators is that professional sports teams avoid trading players with other teams from within their own division (Wong 2017; Simmons 2020). Some offered conjectures on why this should be avoided include: not wanting to improve the competitiveness of a team’s direct rivals (Ley 2017), especially in leagues where games within the division are more common than other match-ups (Bates 2015), or wanting to avoid fans being reminded of players they gave up (Fox Sports 2015), especially if they turn out to play better for their new teams (Ley 2017). Despite the frequency of this speculation, attempts to quantify the actual frequency of such a prohibition has been rare (for one limited exception, see Ahr 2018).
Beyond the popular perception of the avoidance of intra-division trades, there is also ample scholarly literature that would lead us to the same expectation, particularly in professional sports (Stewart and Smith 1999). For example, within a field where the actors are (or are perceived to be) competitors with one another, they may avoid cooperating with one another because it is perceived to be a competitive disadvantage (Hoffmann et al. 2018), though that assumption has been questioned (Bengtsson and Kock 1999; Peng et al. 2012). Instead of cooperating directly with one’s competitors, actors may therefore seek out means of collaboration with those outside the competitive field (Soda and Furlotti 2017). For example, in the case of NBA player development, this could lead to a variety of strategic patterns. Teams may prioritize trading with teams they do not perceive to be direct competitors (Barman 2002), such as those outside their own division. Alternatively, general managers may develop strong stable relationships as resources (Elfenbein and Zenger 2014), for example with particular player agents to provide comparative advantages in access to players on the free agent market (thus reducing their reliance on trades). Teams also could carve out recruitment niches that avoid direct competition (Soltis et al. 2010; Barman 2002). For example, recent shifts in player “apprentice” opportunities through the developmental league and foreign partnerships has opened opportunities for teams to gain contractual rights to players outside of the trade system (Keiper and Barnes 2020). Each of these possibilities could bolster teams’ ability to avoid cooperating through trading with their direct competitors.
In sum, we therefore investigate how strongly division shapes trade partners among National Basketball Association (NBA) franchises, focused especially on whether teams tend to avoid trading with other teams within their own division.
Data
Here, we draw on a database of all player transactions in the NBA from the beginning of the 1976 season—when the NBA merged with the ABA uniting major professional basketball in the US under one league—through the completion of the 2018-19 season (Richardson 2020). We compile this list of 1,977 trades into 43 annual trade networks, with the nodes representing teams, and each edge representing a unique trade between those teams. Each trade is assigned to the “trade season” which runs from the end of the previous season through the corresponding season’s trade deadline.1 The number of teams within each annual slice changes over time as the league gradually expanded from 22 to 30 teams.2. The number of trades observed within each slice varies from year to year (range 22-86 or 0.8-2.9 if expressed in per-team averages; see Figure 1), and teams exhibit differing rates of trading (see Figure 2).
Figure 1: Number of Trades by Year. (normalized to a per team rate)
Figure 1 presents the trend of number of trades per season. Since, the number of teams varies across the window, we standardize these values to represent the number of trades per team in each season. This pattern shows a roughly U-shaped trend, with the highest rates of trades (nearing 2 per team) occurring early and late in the period and the lowest rates occurring in the early 1990s. Notably, this low-point came after the introduction of unrestricted free-agency (in 1988), allowing players to sign with any team after their contract expires. As a corollary, Figure 2 presents the frequency distribution of trades per team across the full observed window. There are some teams (e.g., the Spurs) who consistently trade less frequently, and others who are much more trade active (e.g., the Mavericks).
Figure 2: Number of Trades by Team. (normalized to a per season rate)
To address our research question, we needed to supplement these trade data by compiling a list of each team’s division membership. From the 1976-77 season through 2003-04, there were four divisions (Atlantic, Central, Midwest, and Pacific). From the 2004-05 season onward, there were six divisions (Atlantic, Central, Southeast, Northwest, Pacific, and Southwest).3 While those division/conference names otherwise remained stable, teams changed divisions at various times, whether for geographic reasons or as new expansion teams were added to the league. Accordingly, each team’s division is assigned as current to a particular trade season (for example, the Buffalo Braves were in the Atlantic division in 1977-78, but became the San Diego Clippers and moved to the Pacific division for the 1978-79 season). Combined, these data allow us to investigate the tendency for teams to avoid trading with other teams in their own division.
Analysis and Findings
Analytically, we proceed in three steps, which allow us to address two simultaneous aims. Primarily, these steps allow us to build an appropriate test of our research question. Secondarily, these steps also allow us to illustrate the proper way to statistically condition a question such as this, and to explain the need for doing so. This combination motivates our usage of a relatively recent development for statistically modeling weighted network data.
Video 1. Dynamic Visualization of Annual Trade Networks Note: Intra-division trades are highlighted in red. Node color and position refers to division.
Our first step is to tabulate the number of trades observed both within and across divisions. This tabulation is presented in the form of what is referred to as a mixing matrix, which shows—at the level of divisions—how many trades occur between teams of the same division and between teams of differing divisions. We present these mixing matrices in two aggregations. The first represents all years prior to division expansion, when there were four divisions, a period which spans from 1976-2004. The second window began in 2005 and runs through the end of our analytic period at the end of the 2018-19 season, during which the teams were separated into six divisions. To provide a visual representation of the changes in these division-specific trade patterns across the examined period, Video 1 presents a dynamic visualization of each year’s trade network.4
| 1976 - 2004 | Atlantic | Central | Midwest | Pacific |
|---|---|---|---|---|
| Atlantic | 49 | 135 | 140 | 124 |
| Central | 62 | 156 | 175 | |
| Midwest | 69 | 114 | ||
| Pacific | 53 |
| 2005 - 2019 | Atlantic | Central | Northwest | Pacific | Southeast | Southwest |
|---|---|---|---|---|---|---|
| Atlantic | 23 | 51 | 62 | 56 | 42 | 71 |
| Central | 18 | 52 | 42 | 45 | 41 | |
| Northwest | 33 | 41 | 45 | 55 | ||
| Pacific | 9 | 52 | 54 | |||
| Southeast | 16 | 61 | ||||
| Southwest | 31 |
As can be seen in Tables 1 and 2, these tabulations would appear to suggest that there is an avoidance of trading with members of one’s own division. For example in the pre-expansion era only 22% (231/1062) of trades were within division, a ratio of nearly 3.6 trades across divisions for every one trade within division.
While these mixing matrices may appear to support the prohibition against teams trading within their division, there are two caveats for interpreting these patterns which we address in each of our subsequent analytic steps. In order to test for whether teams avoid trading within their division we must first account for each team’s tendency to trade at all, before we can assess with whom they conduct those trades. For example, in the 1976-77 trade season, there were 35 trades, of which 7 were within division and 28 across divisions. If we account for how many trades each team made, yet assumed that they made those trades without respect to division of their trade partner, random expectation would be that on average 7.6 trades (sd 2.2) would be within division and 27.4 (sd=2.2) would be across divisions. That is, this observed mixing ratio of 4.0 trades across division for every one within is actually consistent with the random expectations, and thus does not conform to a within division avoidance pattern.
Our second analytic step therefore addresses these conditional distribution assumptions that are not apparent from the mixing matrices alone. To accomplish these properly conditioned versions of the test we use what is known as the \(\alpha\)-index for computing segregation in networks (Moody 2001). Essentially, what the \(\alpha\)-index does is allow the estimates of frequency of ties between teams from the same division to be estimated as an odds ratio, which is properly conditioned to account for the number of trade partners one would expect to be within and across division, given the observed base rates of trading by each team, if they chose with whom to trade without respect to division (Bojanowski and Corten 2014). Since this is a homophily statistic, testing for nodes within the same division, a significantly negative \(\alpha\)-index would be consistent with teams following the proposed prohibition. Figure 3 presents this \(\alpha\)-index for each trade season individually. What can be seen here is that in none of the observed trade years does the \(\alpha\)-index differ significantly from random expectation (OR=1). In other words this improved conditional form of the test suggests that there is no observed prohibition against teams trading within division.
Figure 3: Divisional Homophily in NBA Trades by Season. (as computed by Moody’s \(\alpha\)-index).
Our third analytic step addresses one additional caveat necessary for appropriate interpretation. The \(\alpha\)-index used above only allows for ties in the network to be dichotomous. That is, within each trade season the test only asks whether each pair of teams trades with one another or not. However in these data it is possible (and observed) that some pairs of teams trade with each other multiple times within the same trade season. So we further need a version of the test that allows for the weighting of the edges in the network, to properly allow for these multiple trades to be incorporated into the estimate. We rely on a recently developed advance in Exponential-family Random Graph Models (ERGM, Strauss and Ikeda 1990; Lusher, Koskinen, and Robins 2013; Handcock et al. 2019), that allow for estimation with weighted networks (Krivitsky 2012, 2019). ERGM is a statistical model for explaining the structure of a network by means of various local tendencies for ties to be present or absent quantified by model terms such as density, degree, homophily, transitivity and so on. The magnitude of effect of each term is measured by an associated coefficient. In the player trade networks we are interested in verifying whether there is any divisional homophily effect, i.e. analogously to the above presented \(\alpha\). This would indicated if (multiple) trades are more or less likely in pairs of teams belonging to the same division compared to to pairs in different divisions. A positive value of the coefficient would suggest homophily while negative values would indicate heterophily (the hypothesized avoidance of within-division trading).
Once properly conditioning the test for each of these necessary caveats, we find that in none of the seasons are teams more likely to avoid trading with other teams from within their division. As can be verified in the “funnel plot” presented in Figure 4, each the seasonal homophily coefficients fall within the white cone of statistical insignificance.
Figure 4: Homophily Coefficients and Standard Errors from Weighted Exponential Random Graph Models. (including base-rate effects)
To summarize the results from Figure 4 in a single model, we also fit a pooled ERGM across all seasons, including a base-rate effect for each year’s trade volume. These results are presented in Table 3.
| Effect | Estimate | SE | 95% CI | p-value |
|---|---|---|---|---|
| Homophily | 0.0099434 | 0.0587577 | (-0.11; 0.13) | 0.866 |
We find an aggregate same-division homophily effect of 0.0099 (standard error of 0.059, z-statistic of 0.169), again showing no avoidance pattern. Fitting similar models to the periods 1976-2004 and 2004-2019 separately leads to almost identical results.5
Discussion
In short, the empirical patterns do not support the popular claim that NBA teams are likely to avoid trading with teams from within their own division. However, the lack of support for this prohibition is only observed once the strategy for testing the question first accounts for the differential rates at which teams trade and therefore have the opportunity to trade with teams from within/outside their division. The lingering weak associations are even further reduced once including the weighting that is possible from multiple trades between partners in a single season.
It is worth mentioning that the models we have estimated in the results for Figure 4 are actually no different from what would have been possible through a generalized linear model predicting the (weighted) number of ties observed between pairs of teams. However, the value of introducing the weighted ERGM approach with these data is that if subsequent researchers want to take the next step to provide an explanatory account that further examines which teams are likely to trade with which others, this modeling framework would allow for the simultaneous estimation of predictive factors that variously operate at the node level (e.g., is a team that finished lower in the previous year’s standings more likely to trade), the dyad level (e.g., are teams with higher total salaries more likely to trade with other teams who have lower total salaries), or structural features above the dyad (e.g., the tendency to form trade-triples or larger configurations through multi-team deals). The GLM framework would not be able to estimate each of these types of features in such a model without violating model assumptions (e.g., some of these are explicitly modeling the dependency between multiple trades). As such, we hope that the illustration if how the weighted ERGM could be beneficial serves as a useful introduction to this modeling framework, which could be extended by future researchers using the data we provide.
ERGM model fit
| Effect | Estimate | SE | 95% CI | p-value |
|---|---|---|---|---|
| Seasonal base rates | ||||
| 1976-1977 | -0.9026534 | 0.0792530 | (-1.06; -0.75) | 0.000 |
| 1977-1978 | -0.8427925 | 0.0724760 | (-0.98; -0.70) | 0.000 |
| 1978-1979 | -0.9263500 | 0.0796106 | (-1.08; -0.77) | 0.000 |
| 1979-1980 | -0.8762352 | 0.0771426 | (-1.03; -0.73) | 0.000 |
| 1980-1981 | -0.9910653 | 0.0854415 | (-1.16; -0.82) | 0.000 |
| 1981-1982 | -0.9050884 | 0.0754043 | (-1.05; -0.76) | 0.000 |
| 1982-1983 | -0.8069531 | 0.0682278 | (-0.94; -0.67) | 0.000 |
| 1983-1984 | -0.9092292 | 0.0766022 | (-1.06; -0.76) | 0.000 |
| 1984-1985 | -1.0526877 | 0.0872369 | (-1.22; -0.88) | 0.000 |
| 1985-1986 | -1.0782139 | 0.0930195 | (-1.26; -0.90) | 0.000 |
| 1986-1987 | -0.9198185 | 0.0757693 | (-1.07; -0.77) | 0.000 |
| 1987-1988 | -0.9403157 | 0.0781448 | (-1.09; -0.79) | 0.000 |
| 1988-1989 | -1.1192392 | 0.0854069 | (-1.29; -0.95) | 0.000 |
| 1989-1990 | -1.1674130 | 0.0852806 | (-1.33; -1.00) | 0.000 |
| 1990-1991 | -1.1015000 | 0.0789942 | (-1.26; -0.95) | 0.000 |
| 1991-1992 | -1.3314738 | 0.1014767 | (-1.53; -1.13) | 0.000 |
| 1992-1993 | -1.2131200 | 0.0885597 | (-1.39; -1.04) | 0.000 |
| 1993-1994 | -1.2949074 | 0.0953019 | (-1.48; -1.11) | 0.000 |
| 1994-1995 | -1.3952511 | 0.1080338 | (-1.61; -1.18) | 0.000 |
| 1995-1996 | -1.2254082 | 0.0852894 | (-1.39; -1.06) | 0.000 |
| 1996-1997 | -1.2243585 | 0.0837553 | (-1.39; -1.06) | 0.000 |
| 1997-1998 | -1.1148373 | 0.0727527 | (-1.26; -0.97) | 0.000 |
| 1998-1999 | -1.2492779 | 0.0861211 | (-1.42; -1.08) | 0.000 |
| 1999-2000 | -1.2789135 | 0.0904522 | (-1.46; -1.10) | 0.000 |
| 2000-2001 | -1.0430409 | 0.0700097 | (-1.18; -0.91) | 0.000 |
| 2001-2002 | -1.0608807 | 0.0696335 | (-1.20; -0.92) | 0.000 |
| 2002-2003 | -1.2653868 | 0.0888219 | (-1.44; -1.09) | 0.000 |
| 2003-2004 | -1.0915922 | 0.0733866 | (-1.24; -0.95) | 0.000 |
| 2004-2005 | -1.0507651 | 0.0685301 | (-1.19; -0.92) | 0.000 |
| 2005-2006 | -1.0769347 | 0.0703190 | (-1.21; -0.94) | 0.000 |
| 2006-2007 | -1.1925306 | 0.0774736 | (-1.34; -1.04) | 0.000 |
| 2007-2008 | -1.1603188 | 0.0759557 | (-1.31; -1.01) | 0.000 |
| 2008-2009 | -0.9697841 | 0.0623303 | (-1.09; -0.85) | 0.000 |
| 2009-2010 | -0.9605862 | 0.0613118 | (-1.08; -0.84) | 0.000 |
| 2010-2011 | -0.9335487 | 0.0588953 | (-1.05; -0.82) | 0.000 |
| 2011-2012 | -1.1465080 | 0.0721213 | (-1.29; -1.01) | 0.000 |
| 2012-2013 | -0.9791587 | 0.0622563 | (-1.10; -0.86) | 0.000 |
| 2013-2014 | -0.9789426 | 0.0643977 | (-1.11; -0.85) | 0.000 |
| 2014-2015 | -0.8316791 | 0.0542273 | (-0.94; -0.73) | 0.000 |
| 2015-2016 | -1.0257114 | 0.0647017 | (-1.15; -0.90) | 0.000 |
| 2016-2017 | -1.0875715 | 0.0680398 | (-1.22; -0.95) | 0.000 |
| 2017-2018 | -1.0105469 | 0.0640282 | (-1.14; -0.89) | 0.000 |
| 2018-2019 | -0.9172200 | 0.0586872 | (-1.03; -0.80) | 0.000 |
| Homophily | 0.0099434 | 0.0587577 | (-0.11; 0.13) | 0.866 |
Acknowledgements
We appreciate feedback we received on this paper from Skye Bender-deMoll, Pavel Krivitsky and David Schaefer.
The opening of the trade season begins either (1) with the conclusion of the previous season’s playoffs (early in the observed period), or on a specified date each summer (later in the observed window). Using these dates, there were no ambiguous trades that we were unable to attribute to a particular season.↩︎
Unless otherwise specifically noted, all historical accounts in the text were pulled from Wikipedia and confirmed on nba.com.↩︎
The divisions are aggregated into conferences, with the Atlantic, Central, and Southeast being in the Eastern Conference, and the Midwest, Pacific, Northwest, and Southwest being in the Western Conference.↩︎
The appendices also include full replication code for all analyzes conducted and the generation of this visualization.↩︎
Results can be obtained from the authors upon request.↩︎