Assessing Overall Performance of Sports Clubs and Decomposing into Their On-Field and Off-Field Efficiency

Abstract

Generally, playing group management performance and financial management performance of sports clubs are assessed separately. We adopt a non-parametric methodology to assess overall performance, first conceptualising overall management as a production process comprising two serially linked subprocesses, namely, playing group management and financial management. Thereafter, we decompose overall performance to obtain estimates of performance at the subprocess level. Through this procedure, it is possible to determine whether a sports club’s on-field performance or off-field performance or both may contribute towards its inefficiency, if any, in overall management. Further, a model is developed to determine targets for inefficient clubs to become overall efficient. The method is applied to 18 clubs in the Australian rules football league. In the 2021 season, the results reveal that on-field performance, on average, is better than off-field performance, and variability in off-field performance is higher than that of on-field performance. The observed overall management inefficiency is mainly due to inefficiency in financial management. Results are robust to the weighting scheme adopted in the overall efficiency configuration.

1. Introduction

Australian rules football (ARF) is a contact sport played by two teams each comprising 18 players. Australian Football League (AFL) is the highest level senior men’s ARF competition in Australia. ARF is a highly competitive team sport played predominantly in Australia. It is gaining popularity in Canada, Unites States and Europe. ARF has its own rules and, in many ways, may be considered a unique sport.

The aim of this paper is to determine how efficiently AFL clubs are managed and suggest pathways for performance improvement for inefficiently managed clubs. A sporting club’s overall performance to a great extent depends on the skill of coaches, players and administration. They are responsible for on-field and off-field performance. Where we differ from previous studies of sports club performance is that we assess overall club performance incorporating both these aspects under a unified framework. In the literature, it is the on-field performance that gets much of the attention. How a club performs overall considering both on-field and off-field performance as a collective receives less attention in the literature. In this paper, we attempt to fill this void.

The methodology that we adopt to assess performance is data envelopment analysis (DEA). In DEA, the management process of an entity (in our case, an AFL club) is conceptualised as a production process. Application of DEA for sports team performance appraisal is not new. Such studies are found in different sporting codes. Among the recent studies of DEA-based appraisal of sports team performance are [1,2,3,4]. For a review of DEA application in sports, see [5]. Football (soccer) is a popular empirical setting for sport management research [6]. Among the other sports codes that have received attention in the literature are basketball, baseball, American football and cricket. Our study adds to sports club performance appraisal literature with evidence from a different sporting code—ARF. An extension of DEA is network DEA. In network DEA, a production process is conceptualised as comprising multiple subprocesses. In the case of sports club management, we conceptualise overall management as a production process comprising two subprocesses, namely, playing group management (PGM) and financial management (FM). In DEA literature, subprocesses are also referred to as stages. Here, we deal with a two-stage process. The two stages are assumed to be linked serially, with PGM as the first stage and FM as the second stage. The idea is to develop a DEA model to assess club overall performance and decompose at the stage level. Thereby, we are able to determine how a club performs overall and from two different management perspectives. How well a sporting team manages its resources to generate on-field performance is useful information to coaching staff, and how well a club manages off-field operations to generate revenue is useful information to club administration. In addition, we develop a DEA model to determine a pathway for a club that performs inefficiently to become efficient in overall management. Through the empirical investigation, we attempt to answer the questions of how efficiently a sports club manages its resources overall in PGM (on-field activity management) and FM (off-field activity management). Some AFL clubs have limited resources due to their small membership (fanbase) and their inability to attract lucrative sponsorship deals. Established clubs generate revenue in many ways, such as through merchandise sales, sponsorship deals and ticket sales. Profitability also varies across AFL clubs. AFL has introduced measures to balance out financial disparity across clubs through equalisation measures such as revenue sharing. AFL has taken measures also to even out on-field competition through salary caps and draft rules. These measures are intended to assist relatively poor-performing clubs. When sports clubs operate under adverse financial and economic conditions, resource management becomes even more challenging. This study intends to provide sports club management teams with information that they may find useful in PGM and FM.

Using DEA for performance appraisal has advantages. Briefly, DEA is a non-parametric frontier technique. Being non-parametric, there is no requirement to prespecify a functional form for the efficient frontier. DEA establishes frontiers from known levels of attainment. Performance of a club is assessed with respect to the established frontier of best performance. Hence, DEA assesses relative performance. For inefficient units, DEA provides a reference set of efficient units that may serve as benchmarks. Further, DEA can assess performance in a multidimensional framework by accommodating multiple inputs and multiple outputs. Network DEA can also uncover which of on-field and off-field management may require more attention in overall management.

We assess performance using output-oriented DEA models under the variable returns to scale (VRS) technology assumption. VRS implies that all clubs may not have access to the same level of technology. Given the significant differences in total assets of clubs, it is plausible that some clubs have access to advanced technology, such as training facilities, that may enhance their chances to excel on-field performance (Differences in training facilities may be attributed to the availability of resources and budget. Rich clubs often have state-of-the-art facilities). When assessing sports team performance using DEA, revenue and points earned are invariably used as output. An objective of PGM is increasing points earned, and an objective of FM is increasing revenue. Hence, it is reasonable to assume that output augmentation is more important than input conservation.

The rest of the paper is organised as follows. In the literature review section (Section 2), we examine relevant studies on performance appraisal of sports teams, with a particular focus on DEA, and offer a concise overview of ARF. Section 3 discusses the development of a network DEA model under the proposed two-stage framework, and Section 4 presents the empirical framework. Section 5 demonstrates application of the proposed model, and Section 6 discusses robustness of the results. The paper concludes in Section 7.

2. Literature Review

2.1. Performance Appraisal of Sports Teams

The literature on performance appraisal of sporting teams is dominated by soccer. Here, we focus on the studies that have paid attention to assessment of team performance. Methods used in these studies may be classified broadly under two categories. Those that use frontier techniques and those that use other techniques. Two frontier techniques are stochastic frontier and deterministic frontier. Application of DEA methodology falls under the deterministic frontier category. DEA is used to evaluate the overall efficiency of sports teams by comparing their inputs (e.g., player wages and coaching staff expenses) to outputs (e.g., number of wins and points earned). Such an analysis helps identify which teams are utilising their resources most effectively and can provide insights improving performance. Examples of early studies that use stochastic frontier methodology for sporting team-related performance appraisal are [7,8,9,10]. Examples of early studies that use deterministic frontier technology are [11,12,13]. Both methods have their strengths, but SFA’s ability to manage noise and allow statistical inference often makes it more suitable when dealing with data containing significant random variations.

2.2. Australian Rules Football

The first ARF game is reported to have been played in 1858 between two school teams (An account of the origin of Australian rules football and its early development is available at https://www.nma.gov.au/defining-moments/resources/australian-rules-football (accessed on 7 June 2023)). An ARF game is played by two teams with each team comprising 18 players. Points are scored by kicking an oval-shaped ball through the opponent’s goalposts. Goalposts are located at either end of an oval-shaped field. At each end, there are four goalposts. The two central posts are taller than the other two posts on either side of them. When the ball is kicked between the two tall posts, a goal is scored worth six points. If the kicked ball goes between a tall post and the shorter post adjacent to it, the score is one point, and that score is called a behind. The ball is moved within the playing field by either running with it, bouncing, passing (punching with a closed fist) or kicking. Tackling and bumping are the actions used to gain the ball from an opposition player. Tackling is allowed to be performed below the shoulders and above the knees.

In the regular season, each team plays 22 games. In this paper, we focus on the performance of these 18 clubs in the 2021 regular season. At the end of the regular season, there is an incentive for being in the top-eight position in the points table. Top-eight teams play in the final series to win the premiership. However, unlike in competitions in Europe, AFL has no relegation. Arguably, when a team realises that they cannot play in the finals, complacency may set in. This is an area that needs further investigation. We do not consider this aspect in performance appraisal.

2.3. Studies of AFL Team Performance

Several studies investigate the performance of AFL teams. Some studies, considering goals scored, accuracy of kicks at goal, effective tackles, contested possessions and entries inside opponents’ goal area (inside 50 m) as performance indicators, investigate which factors may lead to good performance. For example, Ref. [14] consider detailed information on the play, such as kicks, handballs, disposals, hit-outs, tackles, free-kicks and contested marks, to explain match performance. They explain match outcomes using statistical techniques (logistics regression and decision tree analysis). Another area of study is assessing the effectiveness of playing strategies or tactics adopted by the teams. Tactics include game plans, team structures, strategies implemented and adaptation to playing conditions. Ref. [15] investigate the relationship between player actions and match outcome. They consider a wide range of performance indicators and model the relationship between performance indicators and match outcomes, adopting decision tree analysis and generalised linear models. Ref. [16] investigate the association between a team’s numerical advantage during structured phases of play and match event outcomes. They report that increased numerical advantage is likely the advantage gained from clearances or scoring from inside 50 m. A few studies focus on teamwork. Teamwork includes communication and on-field leadership that contribute to team dynamics. Ref. [17] examine the structural relationship between technical, tactical and physical characteristics of match play and report that scoring opportunity and ball movement have direct association with quarter-margin. Ref. [18] study the outcomes of entries inside 50 m of the goal to determine whether there is an association between team formation and team defensive performance after a turnover. The contribution of individual player performance to the team’s success is another area discussed in the literature. Predicting team performance also receives attention in the literature. Predictive models rely on historical data. Such analysis falls under techniques of statistical modelling. More recently, there is evidence of using various algorithms and machine learning techniques for performance prediction. These studies investigate key areas that reveal information useful in player selection and strategising game plans. All of these studies focus on on-field performance. Research in this area is continuing. Our aim is different. We investigate AFL club overall performance incorporating on-field activity and off-field activity as two different management functions, which are linked serially. The methodology we adopt is DEA.

2.4. Application of DEA for Performance Appraisal in Sports

Ref. [11] assess the performance of Spanish First-Division soccer teams using an output-oriented DEA model. They assess team performance on the field of play considering the number of players used during the season, attacking moves performed, time of ball possession and shots at goal as inputs and the number of points earned as the output. They argue that team performance should not be judged solely on points earned but also on their potential to earn points. Ref. [12] assess English Premier League football team performance adopting an input-oriented DEA model with staff costs and general and other expenses as inputs and points won in the season and total revenue as outputs. Ref. [12] use the Malmquist index to measure change in efficiency and productivity across cross-sections and conclude that there is limited technological change in performance. Both of these studies assume that all clubs have access to the same technology and use constant returns to scale (CRS) DEA models.

Some studies assume VRS technology as well. For example, Ref. [19] consider CRS and VRS models to check the robustness of the results. They use the number of players used, wages, net assets and stadium facility expenses as inputs and points obtained, and attendance and turnover as outputs to assess the performance of 11 clubs in the English Premier Football League. Since their aim is to determine whether wealth affects performance, they consider an input-oriented DEA model. They report that six of the eleven clubs are scale-efficient. That is, they operate in the most productive scale size.

Ref. [13] assess Italian and Spanish football club performance from two different playing strategies, namely, offensive and defensive. They conceptualise offensive strategy as a production process where shots at goal, attacking play (balls kicked to the opposing team’s centre area), and ball possession transform into goals. They consider reciprocal of the same set of inputs of the opposition team as inputs of the defensive strategy process. The output of the defensive strategy process is reciprocal of the number of goals conceded. Ref. [13] report that the Spanish league is more competitive than the Italian league and recommend that the Italian league improve defensive strategy as a better option than improving offensive strategy. Ref. [20] use a weight-restricted input-oriented VRS DEA model to evaluate the performance of Iranian football teams. They use coaches, players and staff wages and fixed assets as inputs and points gained, total revenue and rate of attraction of spectators to the stadium as outputs. They report that there is no association between team ranking based on efficiency and its ranking in the Championship League. They attribute poor performance to high wages paid.

Ref. [21] use two DEA models to assess the performance of the English Premier League football club from the following two different management perspectives: (i) using financial resources (wages and net transfer activity) to maximise the team’s market value, and (ii) using the team’s market value to maximise commercial (total revenue), sporting (athletic output) and social (attendance) output. In case (i), which they refer to as on-field management, they use an input-oriented DEA model, and in case (ii), which they refer to as off-field management, they use an output-oriented DEA model. Their findings suggest off-field operations as generally efficient, and it is the on-field operations that render clubs inefficient. They assess on-field and off-field performance without linking them. In other words, Ref. [21] assess performance from two different management perspectives separately. They do not discuss overall performance.

Ref. [22] use attempts on target, ball recoveries, ball possession and passes as input and sports results as output to assess the performance of teams that participated in the Union of European Football Associations Champions League over a period of 10 seasons. In an input-oriented DEA model, they find two sources of inefficiency, namely, waste of sports resources and selection of poor sports tactics. All studies cited in this section assess performance using single-stage DEA models.

In DEA, single-stage processes are referred to as black box operations. The label black box indicates that when modelling a single-stage production process for performance appraisal using DEA, formulation is not able to accommodate what happens within the production process. In a single-stage process, what is known at best is that a set of inputs are transformed into a set of outputs through an unknown production process. Network DEA models may accommodate what happens inside the black box. For example, Ref. [23] evaluate NBA teams’ performance using DEA, conceptualising the inner structure of club operation as comprising two types of activities, namely, regular season and playoff. In this paper, we conceptualise the AFL club management process as comprising two subprocesses that are linked. Subprocesses are deemed to have different objectives. They are considered working jointly towards overall management. We assess overall performance using a network DEA model and then decompose overall performance into subprocess performance. This way we are able to determine how efficiently an AFL club performs overall and from two different management perspectives.

3. Model Development

We conceptualise the AFL club overall management process as a serially linked two-stage production process, as depicted in Figure 1. Stage A is the PGM process, and stage B is the FM process. Generally, the PGM process includes activities associated with coaching staff managing the playing group with available resources, and the FM process includes activities associated with income generation and fanbase expansion given an outcome of the PGM process. Our aim is to assess overall and stage-level performance, assuming the two-stage process operates under VRS technology. As shown in Figure 1, PGM and FM processes are linked through variables that play a dual role. That is, certain stage A output variables $\left(Z^{AB}\right)$ serve as input at stage B. In DEA, such variables are referred to as intermediate variables. In this paper, we conceptualise the overall AFL club management process as a production process comprising two functionally different subprocesses (PGM—stage A and FM—stage B) linked through intermediate variables. Individual stages are also conceptualised as production processes. When formulating DEA models at the stage level, we consider output-oriented models. This is driven by the nature of variables selected for stages A and B operations. In both stages, we consider that output augmentation is more appropriate than input conservation. When developing a DEA model to assess the relative performance of an overall management process comprising subprocesses, an assumption has to be made on how individual stage-level performance contributes towards overall performance. Following [24], we express overall efficiency as a harmonic average of stage-level efficiencies. In DEA, appraised entities are referred to as decision-making units (DMUs).

Let n = the number of DMUs appraised, $i_A$ = the number of independent inputs at stage A, $\left(x_{1j}^A,x_{2j}^A,\dots ,x_{i_{A}j}^A\right)$ denote stage A inputs of ${\mathrm{D}\mathrm{M}\mathrm{U}}_j$, $i_B$ = the number of independent inputs at stage B, $\left(x_{1j}^B,x_{2j}^B,\dots ,x_{i_{B}j}^B\right)$ denote independent inputs of ${\mathrm{D}\mathrm{M}\mathrm{U}}_j$ at stage B, $D$ = the number of intermediate variables that link stage A to stage B, $\left(z_{1j}^{AB},z_{2j}^{AB},\dots ,z_{Dj}^{AB}\right)$ denote intermediate variables of ${\mathrm{D}\mathrm{M}\mathrm{U}}_j$, $r_A$ = the number of independent outputs at stage A, $\left(y_{1j}^A,y_{2j}^A,\dots ,y_{r_{A}j}^A\right)$ denote outputs of ${\mathrm{D}\mathrm{M}\mathrm{U}}_j$ at stage A, $r_B$ = the number of independent outputs at stage B and $\left(y_{1j}^B,y_{2j}^B,\dots ,y_{r_{B}j}^B\right)$ denote outputs of ${\mathrm{D}\mathrm{M}\mathrm{U}}_j$ at stage B.

When stage-level operations are assumed independent, stage-level relative efficiencies of ${\mathrm{D}\mathrm{M}\mathrm{U}}_0$, $E_0^{\left(A\right)}$ and $E_0^{\left(B\right)}$, under output-orientation and VRS assumption can be obtained using models (1) and (2) (CRS counterparts of models (1) and (2) can be obtained by substituting ${\delta }^A=0$ and ${\delta }^B=0$).

$${\displaystyle \frac{1}{E_0^{\left(A\right)}}}=\mathrm{Min}{\displaystyle \frac{{\sum }_{i=1}^{i_A}{v}_i^A{x}_{i0}^A-{\delta }^A}{{\sum }_{d=1}^D{\omega }_d^{AB\mathrm{’}}{z}_{d0}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{r0}^A}}$$

(1)

$$\mathrm{Subject}\mathrm{to}{\displaystyle \frac{{\sum }_{i=1}^{i_A}{v}_i^A{x}_{ij}^A-{\delta }^A}{{\sum }_{d=1}^D{\omega }_d^{AB\mathrm{’}}{z}_{dj}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{rj}^A}}\ge 1j=\mathrm{1,2},\dots ,n$$

$$v_i^A,u_r^A,{\omega }_d^{AB’}\ge 0;{\delta }^A\mathrm{unrestricted}$$

$${\displaystyle \frac{1}{E_0^{\left(B\right)}}}=\mathrm{Min}{\displaystyle \frac{{\sum }_{i=1}^{i_B}{v}_i^B{x}_{i0}^B+{\sum }_{d=1}^D{\omega }_d^{AB\mathrm{’}\mathrm{’}}{z}_{d0}^{AB}-{\delta }^B}{{\sum }_{r=1}^{r_B}{u}_r^B{y}_{r0}^B}}$$

(2)

$$\mathrm{Subject}\mathrm{to}{\displaystyle \frac{{\sum }_{i=1}^{i_B}{v}_i^B{x}_{ij}^B+{\sum }_{d=1}^D{\omega }_d^{AB\mathrm{’}\mathrm{’}}{z}_{dj}^{AB}-{\delta }^B}{{\sum }_{r=1}^{r_B}{u}_r^B{y}_{rj}^B}}\ge 1j=\mathrm{1,2},\dots ,n$$

$$v_i^B,u_r^B,{\omega }_d^{AB’’}\ge 0;{\delta }^B\mathrm{unrestricted}$$

As shown in Figure 1, stage A and stage B are linked. Therefore, when determining the overall efficiency of the two-stage process, individual stages cannot be considered independent. Hence, some form of aggregation of stage-level efficiencies is required. In this case, following [24,25], we express the overall efficiency of the two-stage process, labelled $E_0$, as a weighted harmonic average of stage A efficiency $E_0^{\left(A\right)}$ and stage B efficiency $E_0^{\left(B\right)}$. That is $E_0={\displaystyle \frac{w_A+w_B}{w_A\frac{1}{E_0^{\left(A\right)}}+w_B\frac{1}{E_0^{\left(B\right)}}}}$ where $w_A$ and $w_B$ are user-specified weights and they add to 1 [24] use harmonic average under the CRS assumption. Ref. [25] use harmonic average under the VRS assumption. Both these studies assess stage-level efficiency using output-oriented DEA models). Harmonic weighted mean is a valid aggregation procedure because DEA efficiency scores may be interpreted as ratios. A common approach in weight selection is to define weights so that aggregated efficiency would be fractional linear. We define the weights as a proportion of the implied value of stage-level output to the sum of the implied values of outputs of both stages. (Definition of weights as ratio of the implied value of the outputs of the stage to the total implied value of the outputs of both stages is a reasonable assumption in the output-oriented case because the objective of such models is to achieve the efficient frontier through output augmentation [26]) Further, we assume that the multipliers associated with intermediate variables, ${\omega }_d^{AB’}$ and ${\omega }_d^{AB’’}$ are the same (say ${\omega }_d^{AB}$) when they are used as input and as output. This is common in DEA-based performance appraisal of serially linked multistage processes (e.g., see [27,28,29]). An implication of this assumption is that the implied value of an intermediate variable as output at stage A is equal to its implied value as an input at stage B. Then, for DMU₀, stage A weight would be $w_A={\displaystyle \frac{{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{d0}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{r0}^A}{{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{d0}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{r0}^A+{\sum }_{r=1}^{r_B}{u}_r^B{y}_{r0}^B}}$ and stage B weight would be $w_B={\displaystyle \frac{{\sum }_{r=1}^{r_B}{u}_r^B{y}_{r0}^B}{{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{d0}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{r0}^A+{\sum }_{r=1}^{r_B}{u}_r^B{y}_{r0}^B}}$, and the harmonic weighted average of the two stage-level efficiencies is $1/\left(w_A\left({\displaystyle \frac{{\sum }_{i=1}^{i_A}{v}_i^A{x}_{i0}^A-{\delta }^A}{{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{d0}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{r0}^A}}\right)+w_B\left({\displaystyle \frac{{\sum }_{i=1}^{i_B}{v}_i^B{x}_{i0}^B+{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{d0}^{AB}-{\delta }^B}{{\sum }_{r=1}^{r_B}{u}_r^B{y}_{r0}^B}}\right)\right)$. This simplifies to $E_0={\displaystyle \frac{{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{d0}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{r0}^A+{\sum }_{r=1}^{r_B}{u}_r^B{y}_{r0}^B}{{\sum }_{i=1}^{i_A}{v}_i^A{x}_{i0}^A-{\delta }^A+{\sum }_{i=1}^{i_B}{v}_i^B{x}_{i0}^B+{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{d0}^{AB}-{\delta }^B}}$.

We estimate $E_0$ in model (3).

$${\displaystyle \frac{1}{E_0}}=\mathrm{Min}{\displaystyle \frac{{\sum }_{i=1}^{i_A}{v}_i^A{x}_{i0}^A-{\delta }^A+{\sum }_{i=1}^{i_B}{v}_i^B{x}_{i0}^B+{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{d0}^{AB}-{\delta }^B}{{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{d0}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{r0}^A+{\sum }_{r=1}^{r_B}{u}_r^B{y}_{r0}^B}}$$

(3)

$$\mathrm{Subject}\mathrm{to}{\displaystyle \frac{{\sum }_{i=1}^{i_A}{v}_i^A{x}_{ij}^A-{\delta }^A}{{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{dj}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{rj}^A}}\ge 1j=\mathrm{1,2},\dots ,n$$

$${\displaystyle \frac{{\sum }_{i=1}^{i_B}{v}_i^B{x}_{ij}^B+{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{dj}^{AB}-{\delta }^B}{{\sum }_{r=1}^{r_B}{u}_r^B{y}_{rj}^B}}\ge 1j=\mathrm{1,2},\dots ,n$$

$${LB}_0^A\le {\displaystyle \frac{{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{d0}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{r0}^A}{{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{d0}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{r0}^A+{\sum }_{r=1}^{r_B}{u}_r^B{y}_{r0}^B}}\le 1$$

$${LB}_0^B\le {\displaystyle \frac{{\sum }_{r=1}^{r_B}{u}_r^B{y}_{r0}^B}{{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{d0}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{r0}^A+{\sum }_{r=1}^{r_B}{u}_r^B{y}_{r0}^B}}\le 1$$

$$v_i^A,u_r^A,{\omega }_d^{AB},v_i^B,u_r^B\ge 0;{\delta }^A,{\delta }^B\mathrm{unrestricted}$$

${LB}_0^A$ and ${LB}_0^B$ are lower bounds on the weights $w_A$ and $w_B$, respectively. For example, when the weights are unrestricted, ${LB}_0^A,{LB}_0^B\ge 0$, and when the weights are assumed equal, ${LB}_0^A={LB}_0^B=0.5$. Model (3) is a fractional linear programming model and may be linearised by adopting the Charnes–Cooper transformation [30] (see Appendix A). Let $v_i^{A*},u_r^{A*}$, $v_i^{B*}$, $u_r^{B*}$, ${\omega }_d^{AB*}$, ${\delta }^{A*}$, ${\delta }^{B*}$ denote optimal decision variable values and $1/E_0^{*}$ denote optimal value of a solution to model (3). Then, the overall efficiency of the two-stage process is given by $E_0^{*}$. Efficiencies of stages A and B denoted by $E_0^{\left(1A\right)}$ and $E_0^{\left(1B\right)}$ may be obtained as $E_0^{\left(A\right)}={\displaystyle \frac{{\sum }_{d=1}^D{\omega }_d^{AB*}{z}_{d0}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^{A*}{y}_{r0}^A}{{\sum }_{i=1}^{i_A}{v}_i^{A*}{x}_{i0}^A-{\delta }^{A*}}}$ and $E_0^{\left(B\right)}={\displaystyle \frac{{\sum }_{r=1}^{r_B}{u}_r^{B*}{y}_{r0}^B}{{\sum }_{d=1}^D{\omega }_d^{AB*}{z}_{d0}^{AB}+{\sum }_{i=1}^{i_B}{v}_i^{B*}{x}_{i0}^B-{\delta }^{B*}}}$, respectively. $E_0^{\left(A\right)}$ and $E_0^{\left(B\right)}$ obtained this way may not be unique. One way of resolving this issue is to treat one of the two stages as the preferred stage, as described in [28] (see Appendix B).

Proposition 1.

DMU₀ is overall efficient if and only if stage A and stage B are also efficient.

Consider model (3). Model (3) is solved considering its linear programming counterpart, where ${\sum }_{d=1}^D{\omega }_d^{AB}{z}_{d0}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{r0}^A+{\sum }_{r=1}^{r_B}{u}_r^B{y}_{r0}^B=1$ is a constraint. If stages A and B are efficient, ${\sum }_{i=1}^{i_A}{v}_i^A{x}_{ij}^A-{\delta }^A={\sum }_{d=1}^D{\omega }_d^{AB}{z}_{dj}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{rj}^A$ and ${\sum }_{i=1}^{i_B}{v}_i^B{x}_{ij}^B+{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{dj}^{AB}-{\delta }^B={\sum }_{r=1}^{r_B}{u}_r^B{y}_{rj}^B$, respectively. Hence, they follow the following: ${\sum }_{i=1}^{i_A}{v}_i^A{x}_{ij}^A-{\delta }^A+{\sum }_{i=1}^{i_B}{v}_i^B{x}_{ij}^B+{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{dj}^{AB}-{\delta }^B=1$. Now we have the numerator and denominator of the objection function of model (3) as 1. Now suppose that DMU₀ is overall efficient. Then, as ${\sum }_{d=1}^D{\omega }_d^{AB}{z}_{d0}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{r0}^A+{\sum }_{r=1}^{r_B}{u}_r^B{y}_{r0}^B=1$, we have ${\sum }_{i=1}^{i_A}{v}_i^A{x}_{ij}^A-{\delta }^A+{\sum }_{i=1}^{i_B}{v}_i^B{x}_{ij}^B+{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{dj}^{AB}-{\delta }^B=1$, leading to ${\sum }_{d=1}^D{\omega }_d^{AB}{z}_{d0}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{r0}^A+{\sum }_{r=1}^{r_B}{u}_r^B{y}_{r0}^B={\sum }_{i=1}^{i_A}{v}_i^A{x}_{ij}^A-{\delta }^A+{\sum }_{i=1}^{i_B}{v}_i^B{x}_{ij}^B+{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{dj}^{AB}-{\delta }^B$. Re-arranging the terms we obtain, $\left({\sum }_{i=1}^{i_B}{v}_i^B{x}_{ij}^B+{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{dj}^{AB}-{\delta }^B-{\sum }_{r=1}^{r_B}{u}_r^B{y}_{r0}^B\right)+\left({\sum }_{i=1}^{i_A}{v}_i^A{x}_{ij}^A-{\delta }^A-{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{d0}^{AB}-{\sum }_{r=1}^{r_A}{u}_r^A{y}_{r0}^A\right)=0$. From the first and second constraints of model (3), we have that each component in the round brackets is non-negative. Hence their sum is zero only when each component is zero, suggesting that stage A and stage B are efficient.

This means, under our modelling framework, a club may be overall efficient if and only if it manages on-field and off-field operations efficiently. It is possible that a club that is not overall efficient is efficient in either on-field or off-field operations and inefficient in either on-field or off-field operations or inefficient in both operations. Improving efficiency in different types of operations requires different sets of management skills. In the proposed analytical framework, it is possible to determine which of the two operations (on-field and off-field) render a club overall inefficient.

4. Empirical Framework

4.1. Variable Selection

When formulating DEA-type models, management processes are regarded as production processes where inputs are transformed to generate outputs. We conceptualise the AFL-club overall management process as a serially linked two-stage process comprising PGM and FM processes. In total, we selected three input variables, two output variables and one intermediate variable. Stage A is a process comprising two inputs and two outputs, and stage B is a process comprising two inputs and one output, as shown in Figure 2. One output of stage A serves as an input of stage B, and hence it serves as an intermediate variable that links stage A and stage B. The two output variables proxy athletic output and commercial output. Ref. [31] alludes to three outputs that a football club may aspire to satisfy. They are athletic output, commercial output and social output. In a DEA application, the number of variables used is constrained by sample size. A norm is that three times the number of input–output variables used should not exceed sample size. If not, the discriminatory power of DEA may diminish [32]. In our empirical analysis, we investigate the performance of 18 clubs. Hence, to be in line with this norm, we restrict the number of variables to six. As a consequence, factors such as social output, team statistics, individual player performance and quality and stability of coaching staff are not accommodated in the overall performance appraisal [32] uses stadium utilisation and [21] use total number of spectators attracted within the season as a proxy for social output. However, social activities should also target fostering connections between sports organisations and the communities they serve. Among the commonly adopted community engagement programmes are health and well-being, youth development and community outreach. Obtaining data on social outputs for sports clubs involves tapping into a mix of internal records, public sources and partnership data. Some studies check robustness of the results by estimating efficiency with different input–output sets (e.g., see [33]). We do not do this).

We regard stage A as a process that utilises financial resources available for football operations to train the players for on-field success. Stage A inputs are football operation expenses and AFL distribution. AFL distribution is funding that a club receives from the AFL (Sourced from https://resources.afl.com.au/afl/document/2022/03/10/76a16be1-6439-4020-af33-1cac86639f7e/2021-AFL-Annual-Report.pdf, accessed on 19 May 2023). These funds generally vary from one club to another. They are given to clubs by the AFL based on their financial need to level out the competition. Invariably, new clubs receive the most funding. Football operation expenses include wages paid to coaching staff and players, travel expenses, administration expenses and expenditure on facilities used in playing group management. We investigate performance in the cross-section, and hence we do not model time lag between playing field achievement and availability of financial resources. Moreover, as money spent on construction and renovation of stadiums incurs at different points in time, we do not consider such expenses in the analysis. We postulate that football operation expenses and AFL distribution reflect funds available within the year for football operations in that year.

Playing group return is captured through playing group rating and points earned in the regular national competition. At the end of the season, all players who took part in the competition are ranked by Stats Insider (Stats Insider ranks every player in the AFL based on multiple factors such as media votes, coaches’ votes, match reports and from other sources where the player has received mention. Individual player rankings are available in https://www.statsinsider.com.au/sport-hub/afl/player-ratings, accessed on 19 June 2023). We compute for each club a composite score labelled ‘playing group rating’ based on their individual player ratings. To compute playing group rating, we consider the players who are ranked between 1 (top rank) and 560. One club had only 28 players ranked between 1 and 560, and therefore, we compute the playing group rating of each club by averaging the ratings of its highest-ranked 28 players. Playing group rating may be viewed as a proxy for the technical ability of the playing group. Playing group rating, however, does not qualify as an output measure because a club that has a playing group with relatively good technical skills will have a low average. Therefore, to maintain the higher the better norm output characteristic in DEA application, we take the inverse of player group rating. Points earned are the other output. This is a commonly used measure of on-field performance. Inefficiency in stage A operations implies that finances available for football operations are not utilised to their full potential for on-field success and playing group technical ability improvement.

Stage B is a process that generates commercial output. We proxy commercial output with revenue. Revenue includes membership fees, corporate marketing, sponsorships, sale of merchandise, gate receipts, gaming revenue and food and beverage sales. Points earned in the competition and leverage risk are the inputs at stage B. Leverage risk is computed as the ratio of liabilities to total assets. While leverage can be a useful tool for financing growth and investments, it also exposes sports clubs to additional financial risks. In the context of sports clubs, leverage risk can manifest in several ways. We focus on the aspect that if a club fails to generate sufficient revenue to service its debt, it may face financial distress or even bankruptcy. We conceptualise that while stage A aims to earn as many points as possible, stage B aims to generate as much revenue as possible through good use of on-field performance. By virtue of playing a dual role, points earned (an output at stage A and an input at stage B) are internal to the overall management process and hence become an intermediate variable. The rationale for using points earned as an input at stage B is that on-field success may have considerable influence on revenue. On-field success may increase attendance at matches (high ticket sales) and increase enthusiasm of fans (increase merchandise sales). Advertisers and sponsors also tend to associate with clubs that perform well in the field. Inefficiency in stage B operations implies that on-field success is not utilised to its full potential in revenue generation while undertaking financial risk.

4.2. Data

We demonstrate application of the methodology using cross-sectional data of the AFL clubs that played in the 2021 season. In 2021, eighteen clubs took part in the AFL national competition (In the 2021 season, each club played 22 home-and-away games. The season was played during the second year of the COVID-19 pandemic. There were some disruptions—relocation of games outside their originally fixtured venues and re-scheduling some match dates). We obtained data from multiple sources. (AFL club annual reports are available at https://www.footyindustry.com/?page_id=90122, accessed on 19 May 2023) Some summary statistics of the input, intermediate and output variables considered in performance appraisal and of some other club-specific characteristics (total assets, membership, Brownlow votes (The Brownlow Medal is awarded annually to the best and fairest player in the AFL during the season, as judged by the umpires. After each match, the three field umpires confer and award three votes to the player they deem best on ground, two votes to the second-best player, and one vote to the third-best player. These votes are tallied at the end of the season, and the player with the most votes wins the medal. Brownlow votes is the sum of the Brownlow votes received by all players who played in the season) and home and away attendance (Home and away attendance in sports, particularly in the context of leagues like the Australian Football League (AFL), refers to the number of spectators attending games played by a team at their home venue (home games) and the number of spectators attending games when the team is playing at their opponent’s venue (away games)) are reported in

Table 1. Summary statistics of input, intermediate and output measures and some characteristics.

	Football $\mathbf{Operation}\mathbf{Expenses}(\mathbf{AUD}{10}^6$)	AFL $\mathbf{Distribution}(\mathbf{AUD}{10}^6$)	Playing Group Rating	Points Earned	Leverage Risk	Total $\mathbf{Revenue}(\mathbf{AUD}{10}^6$)	$\mathbf{Total}\mathbf{Assets}(\mathbf{AUD}{10}^6$)	$\mathbf{Membership}({10}^3$)	$\mathbf{Attendance}({10}^3$)	Total Brownlow Votes
Mean	24.95	16.72	60.18	44	0.56	55.04	55.86	60.01	423.25	66
Median	24.28	14.95	60.66	40	0.47	52.46	50.72	57.19	430.28	64.50
SD	2.79	4.16	4.95	16.06	0.28	14.03	30.05	22.80	103.73	18.09
Range	11.911	14.04	18.21	52	0.91	42.55	109.75	84.63	362.92	57
Minimum	21.772	12.27	50.51	18	0.16	39.17	19.29	18.82	243.76	39
Maximum	33.683	26.32	68.72	70	1.07	81.72	129.03	103.45	606.68	96
CV	0.11	0.25	0.08	0.36	0.51	0.25	0.54	0.38	0.25	0.27

. According to coefficient of variation (CV), total assets have relatively the highest variation (0.54) closely followed by leverage risk with CV at 0.51. The CV of other variables varies between 0.38 and 0.08. This is an indication that there is considerable variation in the financial assets and financial risk of AFL clubs.

5. Results

5.1. Stage-Level Performance When Assumed Independent

First, we discuss stage-level performance, assuming they are independent processes.

Table 2. Stage-level efficiency assuming stages operate independently.

Club	Playing Group Management (PGM) Performance		Financial Management (FM) Performance
	Efficiency	Ranking	Efficiency	Ranking
Adelaide	0.9924	8	0.7092	10
Brisbane	1.0000	1	1.0000	1
Carlton	0.8865	14	1.0000	1
Collingwood	0.9594	11	1.0000	1
Essendon	1.0000	1	0.7577	7
Fremantle	0.9518	13	0.6664	11
Geelong	1.0000	1	0.6500	12
Gold Coast	0.8111	17	0.5773	15
GWS	0.8497	16	0.5222	18
Hawthorn	1.0000	1	0.7326	8
Melbourne	1.0000	1	0.6056	14
North Melbourne	0.7574	18	1.0000	1
Port Adelaide	1.0000	1	0.6137	13
Richmond	0.9578	12	1.0000	1
St Kilda	0.8844	15	0.7154	9
Sydney	0.9817	9	0.5448	17
West Coast	1.0000	1	1.0000	1
Western Bulldogs	0.9702	10	0.5505	16
Average	0.9446		0.7581
Standard deviation	0.0748		0.1874
CV	0.0792		0.2472

Note: Stage A efficiency is estimated in model (1), and stage B efficiency is estimated in model (2), with the restriction that all multipliers $v_i^A$, $u_r^A$, ${\omega }_d^{AB’},{\omega }_d^{AB’’},v_i^B$, $u_r^B$ are $\ge {10}^{-15}.$ Restricting the multipliers to $\ge {10}^{-15}$ ensures that no variable is left out in performance evaluation. The underlined clubs played in the finals. CV is the coefficient of variation.

presents the relative efficiency scores and the corresponding rankings. PGM and FM performance are assessed using models (1) and (2), respectively. Table 2 reveals that seven clubs are efficient in PGM and six clubs are efficient in FM. Two clubs manage both stages efficiently. They are Brisbane and West Coast. Five out of the seven PGM-efficient clubs played in the finals. On the other hand, only one club (Brisbane), out of the six FM efficient clubs, played in the finals. Melbourne deemed efficient in stage A management topped the points table and went on to win the premiership.

The coefficient of variation (CV) reported in the last row in Table 2 reveals that variation in FM performance is much higher than the variation in PGM performance. The CV of PGM efficiency scores is 0.0792, and that of the FM efficiency scores is 0.2472. In this preliminary investigation, it appears that off-field management performance may induce variability in overall management performance more than on-field performance.

The Spearman rank correlation between PGM and FM efficiency scores is 0.036 and is not statistically significant at the 10 per cent level. The evidence here reveals that there is no association between PGM performance and FM performance. This is not surprising given that they are functionally different processes. However, as discussed later in Section 5.3, these two subprocesses have a strong association with some club-specific characteristics not considered in the overall performance appraisal. This we advance as empirical evidence to justify linking the two subprocesses for overall performance appraisal and dismiss the notion of their independence. Therefore, independently assessed stage-level performance may not be aggregated to determine overall performance.

5.2. Overall Performance

Here, we discuss the overall performance of the 18 AFL clubs in the 2021 regular season.

Table 3. Overall efficiency.

Club	Weighting Scheme				Difference in the UW and EW Ranking
	Unrestricted Weights (UW)		Equal Weight (EW)
	Efficiency	Ranking	Efficiency	Ranking
Adelaide	0.9924	9	0.7434	12	−3
Brisbane	1.0000	1	1.0000	1	0
Carlton	1.0000	1	0.9398	4	−3
Collingwood	0.9594	12	0.7952	7	5
Essendon	1.0000	1	0.8622	5	−4
Fremantle	0.9518	14	0.7839	10	4
Geelong	1.0000	1	0.7879	9	−8
Gold Coast	0.8111	17	0.6052	17	0
GWS	0.8497	16	0.6428	16	0
Hawthorn	1.0000	1	0.8409	6	−5
Melbourne	0.9541	13	0.7409	13	0
North Melbourne	0.7574	18	0.5982	18	0
Port Adelaide	0.9950	8	0.7592	11	−3
Richmond	1.0000	1	0.9785	3	−2
St Kilda	0.8844	15	0.7910	8	7
Sydney	0.9808	10	0.7007	15	−5
West Coast	1.0000	1	1.0000	1	0
Western Bulldogs	0.9686	11	0.7020	14	−3
Average	0.9503		0.7929
Standard deviation	0.0742		0.1255
CV	0.0780		0.1583

Note: Overall performance comprises PGM performance and FM performance. Overall efficiency is estimated in model (3), with the restriction that all multipliers $v_i^A$, $u_r^A$, ${\omega }_d^{AB},v_i^B$, $u_r^B$ are $\ge {10}^{-15}.$ Restricting multipliers to $\ge {10}^{-15}$ ensures that no variable is left out in performance evaluation. Unrestricted weight is the case where ${LB}_0^A={LB}_0^B=0$ and equal weight is the case where ${LB}_0^A={LB}_0^B=0.5$ in model (3). Underlined clubs played in the finals. CV is the coefficient of variation. Introduction of any lower bound on multipliers may not always produce feasible solutions to DEA problems. A discussion on this issue is available at https://doi.org/10.1287/opre.48.2.344.12381.

reports the VRS relative efficiency scores and the corresponding rankings of clubs determined under two weighting schemes of stage-level efficiency aggregation. In one scheme, there is no weight restriction $\left({0\le w}_A,w_B\le 1,w_A+w_B=1\right)$, and in the other, both stages are given equal weight $\left(w_A=w_B=0.5\right)$ in aggregation. Later, we check robustness of the results to two more sets of weighting schemes as follows: $w_A=$0.75, $w_B=0.25$ and $w_A=0.25$, $w_B=0.75$.

Table 3 reveals that average overall efficiency under no weight restriction is higher than when PGM performance and FM performance are considered equally important. This is an expected result. When weights are unrestricted, there is more freedom to choose values for multipliers and show overall performance in the best possible manner than when the weights are restricted. Hence, the overall efficiency scores obtained under the unrestricted weights case are higher than or equal to the overall efficiency scores obtained under weight restriction. When weights are unrestricted, seven clubs are overall efficient and under the restricted case, only two clubs (Brisbane and West Coast) are overall efficient. In fact, they are overall efficient under both weighting schemes. Moreover, the difference in the average and the variability in overall efficiency scores obtained under the two weighting schemes are also considerable. When the weights are unrestricted, the average overall efficiency score is 0.9503, and CV is 0.0780, and when the weights are assumed equal, the average overall efficiency is 0.7929 and CV is 0.1583. A higher average overall efficiency score under the unrestricted weighting scheme implies, generally, that the clubs operate closer to the frontier of best performance than when the weights are restricted.

In DEA analysis, rankings are more informative than efficiency scores, as efficiency scores are relative. Therefore, we refer to the rankings of the clubs when discussing performance at the individual club level. The correlation between the rankings obtained under the two weighting schemes is 0.820 (statistically significant at the 1 per cent level). Under the weight restriction, Geelong suffers the most. Geelong is relatively efficient when weights are unrestricted and loses 8 positions in ranking when weights are restricted. The rankings of six clubs do not change. Two out of these six clubs (Brisbane and West Coast) are efficient under both weighting schemes and thereby retain their ranking. The other four clubs that retain their ranking under both weighting schemes are inefficient performers. They are Melbourne (ranked 13), GWS (ranked 16), Gold Coast (ranked 17) and North Melbourne (ranked 18). A change in the weighting scheme favours only two clubs. They are Collingwood and St Kilda. It appears that, for clubs that perform poorly overall, choice of weighting scheme does not make a notable difference in their rankings based on overall efficiency scores.

The top eight teams on points earned during the regular season play in the finals. When weights are unrestricted, seven clubs are deemed overall-efficient. The senior team of three of the seven clubs deemed overall efficient under the unrestricted weight case played in the finals. They are Brisbane, Essendon and Geelong. The relative efficiency score of Port Adelaide under this weighting scheme is 0.9950 ($\cong 1$). Hence, four out of the eight clubs that played in the finals may be considered among the best overall performers that DEA analysis identified. Under the unrestricted-weights case, four teams deemed overall inefficient also did play in the finals. They are ranked 10 (Sydney), 11 (Western Bulldogs), 13 (Melbourne) and 16 (GWS). Their average overall efficiency score is 0.9383. These four clubs are deemed overall inefficient under the equal weighting scheme as well. In that case, their average overall efficiency is 0.6966, and the rankings of two of these four clubs get worse.

Melbourne is an exception. Melbourne, which topped the points table and won the premiership in 2021, performs poorly in overall management (Melbourne is one of the oldest clubs in the league and won their last premiership in 1964. Given this drought, Melbourne was the sentimental favourite in 2021. AFL faced challenges and disruptions in 2021. Some games were played in empty stadiums or with limited attendance to avoid spread of the COVID-19 virus. This resulted in revenue losses. Disruptions had an impact on players’ mental health too. Given these, 2021 may not be considered a typical AFL season). Melbourne is PGM efficient when stages are analysed independently and is ranked 13 on overall performance under both weighting schemes. Overall performance is assessed combining both on-field and off-field performance. Therefore, overall efficiency score masks stage-level performance in aggregation. We discuss overall efficiency decomposition next.

Decomposition of Overall Performance

In overall efficiency decomposition, one stage is given priority (preference) over the other. Giving priority to a particular stage implies that the performance of that stage is considered more important than the performance of the other stage. Then, the stage with priority will be able to obtain the highest efficiency score possible, conditional on maintaining the overall efficiency determined in model (3). For the discussion, we consider the overall efficiency computed in model (3), assuming equal weights (Then, the overall efficiency may be expressed as $E_0=2/\left({\displaystyle \frac{1}{E_0^{\left(PGM\right)}}}+{\displaystyle \frac{1}{E_0^{\left(FM\right)}}}\right)$, where $E_0^{\left(PGM\right)}$ and $E_0^{\left(FM\right)}$ are PGM and FM efficiency scores, respectively).

We decompose the overall efficiency obtained under the equal-weighted scenario under two priority schemes, namely, giving priority to PGM and giving priority to FM.

Table 4. Stage-level performance when weights are equal.

Club	Priority Assigned to Stage A		Priority Assigned to Stage B
	Stage A Efficiency	Stage B Efficiency	Stage A Efficiency	Stage B Efficiency
Adelaide	0.9791	0.5992	0.8434	0.6646
Brisbane	1.0000	1.0000	1.0000	1.0000
Carlton	0.8694	0.6455	0.8865	1.0000
Collingwood	0.6941	0.9308	0.6802	0.9570
Essendon	1.0000	0.7577	1.0000	0.7577
Fremantle	0.9518	0.6664	0.9518	0.6664
Geelong	1.0000	0.6500	1.0000	0.6500
Gold Coast	0.5948	0.6159	0.7025	0.5315
GWS	0.7625	0.5555	0.8358	0.5222
Hawthorn	1.0000	0.7255	1.0000	0.7255
Melbourne	0.6745	0.8218	0.9541	0.6056
North Melbourne	0.6912	0.5272	0.4756	0.8057
Port Adelaide	1.0000	0.6118	0.9950	0.6137
Richmond	0.9578	1.0000	0.9578	1.0000
St Kilda	0.7863	0.7957	0.8844	0.7154
Sydney	0.9358	0.5600	0.9817	0.5448
West Coast	1.0000	1.0000	1.0000	1.0000
Western Bulldogs	0.9314	0.5633	0.9686	0.5505
Average	0.8794	0.7237	0.8954	0.7395
SD	0.1392	0.1648	0.1443	0.1784
CV	0.1583	0.2277	0.1612	0.2413

Note: Stage A is PGM, and stage B is FM. Overall efficiency is decomposed assuming priority to stage A (stage A performance is preferred to stage B performance) and priority to stage B (stage B performance is preferred to stage A performance). Overall efficiency scores obtained under the equal-weighted scheme are given in Table 2. Underlined clubs played in the finals. SD is standard deviation, and CV is coefficient of variation.

reports the results. The second column in Table 4 reveals that, when PGM is given priority over FM, six clubs seem to manage their playing groups efficiently. They are Brisbane, Essendon, Geelong, Hawthorn, Port Adelaide and the West Coast. Four of them (Port Adelaide, Geelong, Brisbane and Essendon) played in the finals. A noteworthy absentee is the club that won the premiership in Melbourne. The other three inefficient clubs that played in the finals are Sydney, Western Bulldogs and GWS. Their PGM efficiency scores are 0.9358, 0.9314 and 0.7625. The PGM efficiency score of Melbourne is 0.6745. Under the equal-weighted scheme, Adelaide, Richmond and Freemantle also record high overall efficiency scores of 0.9791, 0.9578 and 0.9518, respectively. They did not play in the finals. Gold Coast, with a relative efficiency score of 0.5948, is at the bottom of the list (ranked 18). Overall, when PGM is given priority over FM in overall efficiency decomposition, PGM efficiency is positively associated with points earned during the regular season. The Spearman rank correlation between PGM performance and points earned during the regular season is statistically significant at the 1 per cent level, with the Spearman rank correlation at 0.604. Consistency in the rankings based on PGM efficiency and points earned is more pronounced among the top (best) five and bottom (worst) five teams. The exception is with Melbourne. Melbourne earned 70 out of the maximum possible 88 points during the regular season and went on to win the premiership. However, Melbourne is ranked 17 based on PGM performance.

The last column in Table 4 gives FM efficiency scores when FM is given priority over PGM. In this case, four clubs are FM efficient. They are Brisbane, Carlton, Richmond and West Coast. The next best performer is Collingwood with an FM efficiency score of 0.9570. FM efficiency scores of the remaining 13 clubs vary between 0.8057 and 0.5222. The two worst performers in terms of FM under the equal weighting scheme are GWS (ranked 18) and Gold Coast (ranked 17). Their FM efficiency scores are 0.5222 and 0.5315. These two clubs have a relatively low asset base, low membership and operate under high financial risk. Membership of Gold Coast is 18,823 and that of GWS is 25,000. These are low compared to the other clubs. The average club membership is approximately 60,000. These two clubs have a low asset base as well. GWS reports AUD 19.25 million and Gold Coast reports AUD 20.12 million. Average total asset base of clubs is AUD 55.86 million. This average is more than double the amount reported for GWS and for Gold Coast. The leverage ratios of these two clubs are close to 1. The leverage ratios of GWS and Gold Coast are 0.986 and 1.075, respectively. There is only one other club that has leverage ratio close to 1. That is Sydney with leverage ratio at 1.074. Leverage ratio is an input of the FM process. The other input at the FM process is points earned. Gold Coast earned 28 points, and GWS earned 40 points in the regular home and away seasons. The clubs that played in the finals obtained 44 or higher numbers of points in the regular season.

When PGM is given priority over FM, the average PGM efficiency score (0.8794) is higher than the average FM efficiency score (0.7237). This pattern is unchanged when priority is reversed. The average FM efficiency score then is 0.7395, and the average PGM efficiency score is 0.8954. However, overall efficiency decomposition at the individual club level may not always follow this pattern. For example, when FM is given priority over PGM, the PGM efficiency scores of ten clubs are higher than the corresponding FM efficiency scores. When PGM is given priority over FM, the FM efficiency scores of four clubs exceed their corresponding PGM efficiency scores. As expected, PGM and FM efficiency scores are equal to 1 when overall efficiency score is 1 (see Proposition 1 in Section 3).

Table 4 reveals that irrespective of priority assignment, the average PGM efficiency score is higher than the average FM efficiency score. According to the expression for overall efficiency given in footnote 14, an approximate estimate of average overall efficiency under priority given to PGM over FM is $\widehat{E}=2/\left({\displaystyle \frac{1}{\overline{E_0^{\left(PGM\right)}}}}+{\displaystyle \frac{1}{\overline{E_0^{\left(FM\right)}}}}\right)=2/\left({\displaystyle \frac{1}{0.8794}}+{\displaystyle \frac{1}{0.7237}}\right)$ = 2/(1.1371 + 1.3818) = 0.7940. When FM is assigned priority PGM, an estimate of average overall efficiency may be obtained as $\widehat{E}=2/\left({\displaystyle \frac{1}{0.8954}}+{\displaystyle \frac{1}{0.7395}}\right)$ = 2/(1.1168 + 1.3523) = 0.8100. These two overall efficiency estimates are comparable with the average overall efficiency obtained using model (3), which is 0.7929. Since differences in priority assignment do not make a significant difference on average stage-level performance under an equal-weighted scenario, next turn attention to variability in the efficiency scores.

Table 4 reveals that the variation in FM efficiency scores is much higher than the variation in PGM efficiency scores. This disparity is seen under both priority schemes. The coefficients of variation of the FM efficiency scores are 0.2277 and 0.2413, and the coefficients of variation of the PGM efficiency scores are 0.1583 and 0.1612. Further, the average FM efficiency of the six PGM efficient clubs shown in the second column in Table 4 is 0.7646, and the average PGM efficiency of the four FM efficient clubs shown in the fifth column in Table 4 is 0.9158. Given these notable differences in average performance and variation in performance at the stage level, we infer that, generally, FM performance may contribute more towards variation in overall performance than PGM performance. This is consistent with what we observed in Section 5.1, when both stages are assumed to operate independently. More can be learnt on this issue with the correlation between overall and stage-level performance. We discuss this in the next section.

5.3. Determinants of Performance

Here, we investigate the association between overall and stage-level performance and total Brownlow votes received by the playing group and club size (proxied by membership and total assets).

Table 5. Spearman rank correlation.

Spearman Rank Correlation	PGM Efficiency	FM Efficiency	Overall Efficiency	Membership	Brownlow Votes	Points Earned	Total Assets
PGM eff	1.00
FM eff	0.29	1.00
Overall eff	0.68	0.87	1.00
Membership	0.45	0.53	0.63	1.00
Brownlow votes	0.31	−0.02	0.06	−0.14	1.00
Points earned	0.31	0.02	0.08	−0.14	0.98	1.00
Total assets	0.49	0.55	0.64	0.55	0.09	0.04	1.00

Note: The club that receives the highest value for a variable is ranked 1. PGM and FM efficiency rankings are based on the relative efficiency scores obtained when PGM is afforded priority in overall efficiency decomposition. The relative efficiency scores are given in columns 2 and 3 in Table 4.

reports the Spearman rank correlation between overall on-field and off-field performance and total Brownlow votes, membership and total assets.

First, we discuss the association between PGM performance and total Brownlow votes. The results reveal that the association between PGM efficiency and total Brownlow votes is positive. Spearman rank correlation is 0.31 and is statistically significant at the 1 per cent level. On the other hand, there is no evidence to suggest that FM efficiency is associated with total Brownlow votes. Spearman rank correlation is −0.02 and is not statistically significant at the 10 per cent level. FM performance indicates how well a club manages on-field success to generate revenue under risk. Moreover, the association between Brownlow votes and overall performance is not statistically significant. Spearman rank correlation is 0.06. We advance these findings as empirical evidence that conceptualisation of the overall club management process as a two-stage process separating on-field operation from off-field operation as subprocesses may be a valid proposition.

Membership has a strong positive association with overall efficiency and stage-level efficiency. The Spearman rank correlation between overall efficiency and membership is 0.63, FM efficiency is 0.53 and PGM efficiency is 0.45. Membership plays a part in both subprocesses. Attendance of supporters at games is well known to influence player performance. In a study of home advantage in AFL, Ref. [34] highlights crowd effect and ground familiarity as two determinants of home advantage. Membership may play a part in FM as well. Large attendances benefit clubs financially. The association between total assets and performance is similar. Clubs with large financial asset bases generally have modern training equipment and facilities. That can have an impact on player performance. Clubs with a high financial asset base may have modern stadiums that attract crowds, resulting in increased revenue.

The Spearman rank correlation between FM performance and overall management performance (0.87) is higher than that between PGM performance and overall management performance (0.68). However, the association between PGM performance and FM performance is weak. Spearman rank correlation is 0.29. This evidence indirectly suggests that FM performance may influence overall performance more than PGM performance.

Good FM performance is important for the financial health of the club. Clubs that adopt wasteful practises and resort to inefficient practices may end up being cash trapped. The financial status of a club depends on many factors, including membership, sponsorships and on-field performance. As on-field performance may have an impact on revenue, stage A and stage B operations are seemingly linked. Hence, this justifies using a serially linked two-stage process for overall performance appraisal.

5.4. Determination of a Pathway for an Overall Inefficient Club to Become Overall Efficient

We determine overall efficiency using model (A1) given in Appendix A. Model (A1) is the linear programming counterpart of model (3). Model (A1) solution does not provide estimates of intermediate measures to project inefficient units to the frontier and thereby become efficient. Following [35,36], we obtain estimates of input, intermediate and output measures for frontier projection by solving model (A5) given in Appendix C.

For the demonstration, we selected two clubs, namely, Richmond and North Melbourne. Under the equal-weight scenario, the overall efficiency of Richmond is 0.9785 and of North Melbourne is 0.5982. Richmond has the highest overall efficiency score among all overall inefficient clubs, while North Melbourne has the lowest (see column 4 in Table 3).

Table 6. Input, intermediate and output targets for overall efficiency.

	Richmond			North Melbourne
	Observed	Target	% Change	Observed	Target	% Change
Inputs
$\mathrm{Football}\mathrm{operation}\mathrm{expenses}(\mathrm{AUD}{10}^6$)	23.428	23.428	0.00	23.647	23.647	0.00
$\mathrm{AFL}\mathrm{distribution}(\mathrm{AUD}{10}^6$)	14.192	13.661	−3.74	17.441	13.707	−21.41
Leverage risk	0.419	0.419	0.00	0.514	0.513	−0.19
Intermediate measure
Points earned	38	61.340	61.42	18	63.634	253.52
Outputs
Playing group rating	63.029	65.803	4.40	50.507	66.684	32.03
$\mathrm{Total}\mathrm{revenue}(\mathrm{AUD}{10}^6$)	78.376	78.376	0.00	39.268	79.442	102.31

Note: Input, intermediate and output targets for overall efficiency are obtained in model (A5) given in Appendix C.

gives the observed input, intermediate and output measures and a set of input, intermediate and output targets that make these two clubs efficient.

The targets shown in Table 6 reveal that points earned are a major concern for both clubs. Revenue target for North Melbourne suggests doubling. This is a big ask in the short term. According to the targets in Table 6, North Melbourne requires a considerable increase in playing group rating as well. For both clubs, targets suggest a reduction in AFL distribution. In practise, this may be interpreted as funds available for spending to boost playing group performance are not utilised effectively. Model (A5) provides one pathway for performance improvement. Targets, in general, may be used as guidance as they are established relative to how all clubs in the league utilise their resources.

6. Robustness Check

6.1. Change in the Weighting Scheme

Table 3 reports overall efficiency obtained when stage-level efficiencies are unweighted (${LB}_0^A={LB}_0^B=0$) and equally weighted (${LB}_0^A={LB}_0^B=0.5$). Here, we consider two more weighting schemes as follows: ${LB}_0^A=0.75$, ${LB}_0^B=0.25$ (on-field performance is more important than off-field performance) and ${LB}_0^A=0.25$, ${LB}_0^B=0.75$ (off-field performance is more important than on-field performance).

Table 7. Overall efficiency under two other weighting schemes.

Club (Listed in Alphabetical Order)	Weighting Scheme
	${\mathit{L}\mathit{B}}_0^{\mathit{A}}=0.75$$\mathbf{and}{\mathit{L}\mathit{B}}_0^{\mathit{B}}=0.25$		${\mathit{L}\mathit{B}}_0^{\mathit{A}}=0.25$$\mathbf{and}{\mathit{L}\mathit{B}}_0^{\mathit{B}}=0.75$
	Efficiency	Ranking	Efficiency	Ranking
Adelaide	0.8500	11	0.6605	13
Brisbane	1.0000	1	1.0000	1
Carlton	0.9124	6	0.9690	4
Collingwood	0.8696	8	0.7325	8
Essendon	0.9260	4	0.8066	5
Fremantle	0.8597	10	0.7204	9
Geelong	0.8814	7	0.7124	10
Gold Coast	0.6932	17	0.5365	18
GWS	0.7319	16	0.5729	16
Hawthorn	0.9164	5	0.7690	6
Melbourne	0.8341	13	0.6664	12
North Melbourne	0.6684	18	0.5413	17
Port Adelaide	0.8612	9	0.6787	11
Richmond	0.9680	3	0.9891	3
St Kilda	0.8351	12	0.7513	7
Sydney	0.8177	14	0.6130	15
West Coast	1.0000	1	1.0000	1
Western Bulldogs	0.8140	15	0.6171	14
Average	0.8577		0.7409
Standard deviation	0.0933		0.1556
CV	0.1087		0.2101

Note: Overall efficiency is estimated in model (3) with the restriction that all multipliers $v_i^A$, $u_r^A$, ${\omega }_d^{AB},v_i^B,\mathrm{a}\mathrm{n}\mathrm{d}$ $u_r^B$ are $\ge {10}^{-15}.$ Underlined clubs played in the finals.

reports the results. Analysis of the rankings reported in Table 3 and the rankings reported in Table 7 reveals a high positive correlation between them. Spearman rank correlation varies between 0.663 (Spearman rank correlation between equal-weighted and unweighted schemes) and 0.946 (Spearman rank correlation between 0.75–0.25 and 0.25–0.75 weighting schemes). Across all three weighting schemes investigated, the same two clubs (West Coast and Brisbane) consistently emerge as overall efficient. This finding indicates that the ranking of overall efficiency is robust to the choice of weighting scheme used in aggregating stage-level efficiency for overall performance appraisal. This result may be specific to the data set. The selection of a specific weighting scheme depends on the aim of the analysis. Such an approach can be considered decision-making guided by individual rationalisation. In practice, the decision-maker may select lower bounds to ensure that no one stage completely dominates another in aggregation. When a weighting scheme is not pre-specified, the weights would be determined by the model and hence are data-driven.

6.2. Influence of Potential Outliers

West Coast is the only club that is overall efficient under all four weighting schemes (including the unweighted case) considered in the analysis. Because the West Coast overall efficiency score is robust to different weighting schemes, we checked whether West Coast has any influence (is an outlier) on the relative performance of the other clubs. To investigate this issue, we removed West Coast from the sample and repeated the analysis. The results reveal that the correlation between the rankings of clubs based on overall performance with and without West Coast is very high, with Spearman rank correlation at 0.914 (statistically significant at the 1 per cent level). We advance this finding as evidence that our results may not be influenced by outliers.

However, analysing data from a single season limits the generalisability of the findings. Additionally, our study period, 2021, was marked by significant challenges for AFL clubs due to the hardships and disruptions caused by COVID-19, which affected the season’s dynamics. Moreover, a club’s overall efficiency can be influenced by its investment strategies, and therefore, it is essential that an overall efficiency analysis is conducted spanning several seasons [37]. Because the DEA frontier is established with observed data, the choice of variables in the DEA analysis influences the results. Adding or removing variables can significantly alter the efficiency scores. Some studies suggest using principal component analysis to guide variable selection [38]. For an integrated framework for conducting robustness analysis in the application of DEA models, see [39]).

6.3. Change in Returns to Scale

Here, we check whether the change in assumption from VRS to CRS makes a significant difference in the results. The CRS assumption implies that clubs have access to the same technology. For the discussion, we present the results obtained in the unrestricted-weights case. The results under the VRS assumption are reported in Table 3. The results under the CRS assumption are given in

Table 8. Overall efficiency under CRS and scale efficiency.

Club	CRS Efficiency	Ranking	Scale Efficiency	Ranking
Adelaide	0.9618	7	0.9692	10
Brisbane	0.9889	3	0.9889	7
Carlton	0.8419	16	0.8419	18
Collingwood	0.8802	14	0.9174	16
Essendon	0.9627	6	0.9627	11
Fremantle	0.9145	11	0.9608	13
Geelong	1.0000	1	1.0000	2
Gold Coast	0.7794	17	0.9609	12
GWS	0.8486	15	0.9987	4
Hawthorn	0.9145	10	0.9145	17
Melbourne	0.9035	12	0.9469	15
North Melbourne	0.7569	18	0.9994	3
Port Adelaide	0.9724	4	0.9773	9
Richmond	0.9534	9	0.9534	14
St Kilda	0.8817	13	0.9970	5
Sydney	0.9702	5	0.9892	6
West Coast	1.0000	1	1.0000	1
Western Bulldogs	0.9551	8	0.9861	8
Average	0.9159		0.9647

Notes: The undelined clubs played in the finals series.

. Under the CRS assumption, only two clubs are overall efficient. They are West Coast and Geelong. In the VRS case, we observed seven efficient clubs, including West Coast and Geelong (Higher number of clubs deemed DEA-efficient under the VRS assumption than in the CRS case is an expected result). Being efficient under both returns to scale schemes, West Coast and Geelong are scale efficient with scale efficiency equal to 1. Scale efficiency does not imply technical efficiency. For example, St Kilda and North Melbourne also have very high-scale efficiency scores. However, they perform poorly overall. The CRS efficiency of St Kilda and North Melbourne is 0.8817 and 0.7569, and their VRS efficiency is 0.8844 and 0.7574. The inefficiency of St Kilda and North Melbourne may not be attributed to their scale of operation. St Kilda and North Melbourne perform poorly overall due to their inefficient management practices.

The overall efficiency score-based rankings obtained under the two technologies are positively correlated. Spearman rank correlation is 0.644 and is statistically significant at the 0.1 per cent level. Three clubs, namely, Carlton, Hawthorn and Richmond, stand out in terms of the difference in the rankings under the two returns to scale schemes. Overall efficiency-based rankings of Carlton, Hawthorn and Richmond under CRS technology are 8, 9 and 15 placings lower than the overall efficiency-based rankings under VRS technology. These three clubs are overall efficient under VRS. Carlton, Hawthorn and Richmond suffer from scale inefficiency (Chen and Zhu (2019) show that overall scale efficiency in two-stage network DEA is consistent with scale efficiency in conventional DEA. Therefore, as our modelling framework falls under the two-stage network DEA framework, we interpret scale efficiency similar to the conventional case). The scale efficiency of Carlton, Hawthorn and Richmond is 0.8419 (ranked 18th), 0.9145 (ranked 17th) and 0.9534 (ranked 14th), respectively. This is an indication that their size of operations may not be optimal. These clubs have a high asset base. Carlton and Richmond are ranked second and third in terms of assets. The asset base of Hawthorn is about average. Membership in these clubs is also high. Richmond’s membership is the highest at 103,450. Carlton and Hawthorn membership exceeds 81,000 and 70,000, respectively. The average membership of all clubs is approximately 60,000. Carlton and Richmond dominate in revenue as well.

6.4. Using a Single-Stage Model

Thus far, we have been discussing the results obtained under a two-stage modelling framework. Here, we conceptualise the overall club management operation as a single-stage process. In the two-stage case, we consider points earned as an intermediate (linking) variable. Some studies disregard intermediate variables when a two-stage process is conceptualised as a single-stage process (see [21]). We do differently. As points earned are clearly a desirable outcome, we consider points earned as an output variable. Using points earned as an output variable is common practice in sports team performance appraisal using a single-stage DEA model (e.g., see [20]). Our single-stage modelling framework is depicted in Figure 3.

When we assume CRS, the number of clubs deemed overall efficient is eight. The average efficiency under the CRS case is 0.9461. In the results obtained under output-orientation and VRS assumption, eleven clubs are overall efficient, with average efficiency at 0.9614. We find here that, as expected, discriminatory power of relative performance diminishes when the overall management process is conceptualised as a single-stage process. In this case, eight clubs are scale-efficient. They are Brisbane, Essendon, Geelong, Melbourne, Port Adelaide, Richmond, West Coast and Western Bulldogs. Six of these eight clubs played in the finals. Under both technologies, North Melbourne is the worst performer. However, its scale efficiency is 0.9884. This is an indication that North Melbourne operates close to its most productive scale size, and its inefficiency may be attributed to poor management. The average scale efficiency is 0.9841. The evidence here suggests that, generally, AFL clubs’ inefficiency in overall management may be due to managerial inefficiency. In our modelling framework, AFL club operation involves two distinguishable management functions, namely, PGM and FM. These functions require different sets of management skills. The inefficiency of which of these two skill sets may contribute more towards overall inefficiency can be determined by the methodology we propose in this paper.

7. Concluding Remarks

We conceptualise the overall management process of a sports club as comprising two subprocesses, namely, on-field management and off-field management. Sports clubs’ survival hinges on the efficient collective management of these subprocesses. This study aims to highlight how the performance of on-field and off-field management can impact a club’s overall performance.

To investigate this, we employ data envelopment analysis (DEA). Initially, we assess the overall performance and then decompose it into subprocess performance under two priority schemes: one where on-field performance is prioritised over off-field performance, and another where off-field performance is prioritised over on-field performance. Our modelling framework allows for the investigation of the sensitivity of overall performance to the weights assigned to on-field and off-field performance in their aggregation and the sensitivity of subprocess-level performance to the priority scheme. This approach enables clubs to evaluate their overall, on-field and off-field performance relative to other clubs in the league. It also helps identify whether managerial inefficiency, scale inefficiency or both contribute to overall inefficiency.

In our empirical investigation using 2021 data, we assess the overall performance (relative efficiency) of Australian rules football league clubs. The results indicate that, during the investigation period, the average on-field performance surpassed the average off-field performance, and off-field performance exhibited greater variability than on-field performance. Analysis at the individual club level reveals that efficient on-field performance or efficient off-field performance alone does not guarantee overall efficiency. These model-driven insights align with outcomes that may be desired by sports clubs in their overall management.

The proposed methodology offers valuable information to club management for making short-term decisions in line with long-term strategic plans. Applying this approach over a longer period could reveal how resilient clubs’ on-field and off-field performances are to non-discretionary factors, changing economic conditions, and financial positions. Additionally, our method can be easily adapted to assess the overall performance of clubs in other team sports.