1. Introduction
A common refrain among sports commentators is that professional sports teams avoid trading players with other teams from within their own division (Wong, Reference Wong2017; Simmons, Reference Simmons2020). Some conjectures on why this should be avoided include: not wanting to improve the competitiveness of a team’s direct rivals (Ley, Reference Ley2017), especially in leagues where games within the division are more common than other match-ups (Bates, Reference Bates2015), or wanting to avoid fans being reminded of players they gave up (Sports, Reference Sports2015), especially if they turn out to play better for their new teams (Ley, Reference Ley2017). 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, Reference Ahr2018).
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 & Smith, Reference Stewart and Smith1999). These expectations have theoretical roots in literature on organizational competition broadly, and management strategy more specifically.
A primary basis for such an expectation arises from organizations avoiding collaborating with their (perceived) competitors. 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., Reference Hoffmann, Lavie, Reuer and Shipilov2018), though that assumption has been questioned (Bengtsson & Kock, Reference Bengtsson and Kock1999; Peng et al., Reference Peng, Pike, Yang and Roos2012). Instead of cooperating directly with one’s competitors, actors may therefore seek out means of collaboration with those outside the competitive field (Soda & Furlotti, Reference Soda and Furlotti2017). Or they may develop more narrow definitions of who their competitors are (Barman, Reference Barman2002). In combination, these could facilitate the avoidance of intra-division trades through prioritizing trades with teams who are not perceived as their direct competitors—i.e., from outside their own division.
Additionally, the avoidance of competitor collaborations could be facilitated by the availability or pursuit of alternative strategies for developing competitive advantages. In this particular case, such alternatives could focus on strategies to enhance player development (vs. player “acquisition” through trades). This could take the form of enhancing other means of recruitment such as stable pipelines for talent acquisition (Brymer et al., Reference Brymer, Molloy and Gilbert2014; Elfenbein & Zenger, Reference Elfenbein and Zenger2014), wherein general managers could develop stable relationships with particular representatives to provide comparative advantages in access to players on the free agent market. By shifting the focus to alternative strategies of player acquisition and/or development in these ways, teams could further avoid necessitating collaboration (through trades) within their division.
Organizations also carve out recruitment niches that avoid direct competition (Soltis et al., Reference Soltis, Sterling, Borgatti and Ferrier2010; Barman, Reference Barman2002). For example, National Basketball Association (NBA) teams have recently developed improved “apprentice” opportunities through the “Developmental League,” while some teams have focused on foreign partnerships to open opportunities to gain contractual rights to players outside of the trade system (Keiper & Barnes, Reference Keiper and Barnes2020). While we do not empirically test the development of these alternative mechanisms here, each should facilitate teams’ ability to avoid engaging in cooperating with their direct competitors—i.e., by avoiding intra-division trades—directly paralleling the expectation from the popular perception as reported above.
In sum, we therefore investigate how strongly division shapes trade partners among NBA franchises. Drawing both on the popular refrain and supporting scholarly literature, this investigation begins with the expectation that teams may tend to avoid trading with other teams within their own division.
2. Data
To address this question, 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 2018–2019 season (Richardson, Reference Richardson2020). We compile this list of 1977 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.Footnote 1 The number of teams within each annual slice changes over time as the league gradually expanded from 22 to 30 teams.Footnote 2 The number of trades observed within each slice varies from season to season (range 22–86 or 0.8–2.9 when stated as per-team averages; see Fig. 1), and teams exhibit differing rates of trading (see Fig. 2).
Number of trades by season. (standardized to a per team rate).
Average trades by team. (standardized to a per season rate).
Fig. 1 presents the trend of the average 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), which allowed players to sign with any team after their contract expires. As a corollary, Fig. 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 76ers).
To address our main research question, we needed to supplement these trade data by compiling a list of each team’s division membership. From the 1976–1977 season through 2003–2004, there were four divisions (Atlantic, Central, Midwest, and Pacific). From the 2004–2005 season onward, there were six divisions (Atlantic, Central, Southeast, Northwest, Pacific, and Southwest).Footnote 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 (e.g., the Buffalo Braves were in the Atlantic division in 1977–1978, but became the San Diego Clippers and moved to the Pacific division for the 1978–1979 season). Combined, these data allow us to investigate the tendency for teams to avoid trading with other teams in their own division.
3. 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 further motivates our usage of a relatively recent development for statistically modeling weighted network data.
Our first step is to tabulate the number of trades observed both within and across divisions. This tabulation is presented in the form of a mixing matrix, which shows—at the level of divisions—how many trades occur between teams of the same division (diagonal) and between teams of differing divisions (off-diagonal). We present these mixing matrices in two aggregations. The first represents when there were four divisions, i.e., all seasons prior to division expansion, a period which spans from 1976 to 2004. The second window began in 2005 and runs through the end of our analytic period at the end of the 2018–2019 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 season’s trade network.Footnote 4
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 (1976–2004) only 22% (233/1,077) of trades were within division, a ratio of approximately 3.6 trades across divisions for every one trade within division.
Mixing matrix for the period 1976–2004
Mixing matrix for the period 2005–2019
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. First, 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 1976–1977 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 properly condition a test to account for baseline expectations, we use what is known as the
$\alpha$
-index for computing segregation in networks (Moody, Reference Moody2001). 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 & Corten, Reference Bojanowski and Corten2014). Since this is a homophily statistic, testing for ties between teams within the same division, an
$\alpha$
-index significantly less than one would be consistent with teams following the proposed prohibition. Fig. 3 presents this
$\alpha$
-index (with 95% confidence interval) for each trade season individually. What can be seen here is that in none of the observed trade seasons 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.
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 (ERGMs, Strauss & Ikeda, Reference Strauss and Ikeda1990; Lusher et al., Reference Lusher, Koskinen and Robins2013; Handcock et al., Reference Handcock, Hunter, Butts, Goodreau, Krivitsky and Morris2019), that allow for estimation with weighted networks (Krivitsky, Reference Krivitsky2012, Reference Krivitsky2019). 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, and transitivity. The magnitude of effect corresponding with each term is measured by an associated coefficient. In the player trade networks we are interested in verifying whether there is a (negative) divisional homophily effect, i.e., analogously to the above presented
$\alpha$
. This would indicate if (multiple) trades are more or less likely in pairs of teams belonging to the same division compared to team 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).
Appropriate modeling for a weighted network via an ERGM also requires adapting model specification to the distribution of trade counts across dyads by choosing an appropriate reference measure (Krivitsky, Reference Krivitsky2012, sec. 5.2). While the typical approach involves assuming that the counts are Poisson-distributed, in practice empirical data distributions often exhibit an over-abundance of 0s (zero-inflation) and/or greater variation than the Poisson distribution assumes (overdispersion). Table 3 summarizes that trading is relatively rare in general and it happens even more rarely that a pair of teams trades more than once in the same season. Formal tests of this distribution do indicate overdispersion (Cameron & Trivedi, Reference Cameron and Trivedi1990; Kleiber & Zeileis, Reference Kleiber and Zeileis2008).
Distribution of dyadic trade counts. Frequency of number of trades in a season in pairs of teams
At this point we could either choose a non-Poissonian reference measure or simplify data by dichotomizing the counts (any number of trades vs no trades at all) and proceeding with a binary ERGM. We ran both versions, reaching the same conclusion. The homophily coefficient in the binary ERGM fit to all seasonal trade networks simultaneously, including season- and division-specific effects, is equal to 0.012 (standard error of 0.065, z-statistic of 0.188) again showing no avoidance patterns. The complete model is presented in Table B1 in the Appendix.
Valued ERGM fitted to pooled seasonal data. The table excludes seasonal constants; complete table is presented in Table C1 the Appendix
Applying the valued ERGM, to account for the caveats raised above, and with the Conway–Maxwell–Poisson reference measure, we again find that in none of the seasons do teams exhibit a tendency to avoid trading with other teams from within their division. This is summarized with the “funnel plot” presented in Fig. 4 that shows each of homophily coefficients from season-specific weighted ERGMs fall within the white cone of statistical insignificance.
Homophily coefficients and standard errors from weighted exponential random graph models. (including base-rate effects).
To summarize the results from Fig. 4 in a single model, we also fit a pooled ERGM across all seasons, including a base-rate effect for each season’s trade volume, similarly to the binary ERGM mentioned above.Footnote 5 These results are presented in Table 4. In this final specification, we find an aggregate same-division homophily effect (coefficient 0.006, standard error of 0.059, z-statistic of 0.102), that again shows no avoidance pattern. Fitting similar models to the periods 1976–2004 and 2004–2019 separately leads to almost identical results.Footnote 6
4. 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. We are reticent to suggest that these findings necessarily alter the theoretical claims within organizational literature that competitors may avoid cooperating with one another (Hoffmann et al., Reference Hoffmann, Lavie, Reuer and Shipilov2018; Stewart & Smith, Reference Stewart and Smith1999); however they do suggest that future scholarship may want to examine potential variation between the contexts where such prohibitions exist compared to others like the NBA case, where they seem to be violated.
It is worth mentioning that the models we have estimated in the results for Fig. 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 season’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).Footnote 7 As such, in addition to our primary empirical conclusions–that within-division trades are not avoided by NBA franchises–we hope that the illustration of how the weighted ERGM could be beneficial serves as a useful introduction to this modeling framework, and may be put to further examination by future researchers using the data and code we provide.
Acknowledgments
We appreciate feedback we received on previous iterations of this work from Skye Bender-deMoll, Pavel Krivitsky, and David Schaefer.
Competing interests
None.
Appendix
A. Details of model specification
Basic notation:
-
$N$
—number of unique team-season pairs -
$i, j \in \{1, \ldots, N\}$
—team playing in a season (a team-season pair)
-
$\{s_1, \ldots, s_i, \ldots, s_N\}$
—season of each team-season pair
-
$s_i \in \{1, \ldots, s, \ldots, S\}$
—seasons
-
$\{d_1, \ldots, d_i, \ldots, d_N\}$
—division of each team-season pair
-
$d_i \in \{1, \ldots, d, \ldots, D\}$
—divisions
-
$[y_{ij}]_{N \times N}$
— A block-diagonal adjacency matrix of trades between teams in seasons. For simplicity, but abusing notation slightly,
$y_{ij}$
is the number of trades in a season between teams
$i$
and
$j$
in case of valued ERGMs and, in the case of binary ERGM, trade count dichotomized to
$\{0,1\}$
(no trade vs any number of trades).
In words, each node of the network corresponds to a team playing in the particular season. Nodes are grouped into seasonal blocks such that trades can happen only within seasons (ties within blocks) but cannot happen between seasons (no ties between blocks).
The models use the following sufficient statistics:
-
Season-specific propensity to trade represented by the number of trade participations by teams in season
$s$
:
\begin{equation*}g_1(y, s) = \sum _{i,j:\; s_i=s \,\vee \, s_j=s} y_{ij}\end{equation*}
-
Division-specific propensity to trade represented by the number of trade participations by teams in division
$d$
:
\begin{equation*}g_1(y, d) = \sum _{i,j:\; d_i = d \,\vee \, d_j=d} y_{ij}\end{equation*}
-
Propensity towards within-division trading represented by the number of trades involving teams from the same division:
\begin{equation*}g_2(y) = \sum _{i,j:\; d_i = d_j} y_{ij}\end{equation*}
-
Over-/under-dispersion statistic which modifies the reference measure into a Conway–Maxwell–Poisson reference ERGM (c.f. Krivitsky, Reference Krivitsky2012, sec. 5.2.3):
Pooled binary ERGM is specified as:
\begin{equation*}g_3(y) = \sum _{i,j} \log (y_{ij}!)\end{equation*}
Pooled valued ERGM is specified as:
\begin{equation*}P( Y = y ) = \frac {\exp \left ( \sum _{s=2}^S \theta _s g_1 (y, s) + \sum _{d=1}^D \theta _d g_1 (y, d) + \theta _2 g_2 (y) \right )}{\kappa }\end{equation*}
\begin{equation*}P( Y = y ) = \frac {\exp \left ( \theta _3 g_3 (y) + \sum _{s=1}^S \theta _s g_1 (y, s) + \theta _2 g_2 (y) \right )}{\kappa }\end{equation*}
B. Binary ERGM model fit
Full results for the binary ERG model fit to pooled data
C. Weighted ERGM model fit
Full results for the weighted ERG model (Conway–Maxwell–Poisson reference) fit to pooled data
