Australian rules football (ARF) is a contact sport played by two teams each comprising of eighteen 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 as 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 receive 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, management process of an entity (in our case, an AFL club) is conceptualized as a production process. Application of DEA for sports team performance appraisal is not new. Such studies are found in different sporting codes. However, most studies focus on soccer. Examples of studies that use DEA for sports team performance are [1,2,3,4,5,6,7,8,9,10,11,12]. For a review of DEA application in sports, see [13]. Our study adds to sports club performance appraisal using DEA literature with evidence from a different sporting code- ARF.

An extension of DEA is network DEA. In network DEA, a production process is conceptualized as comprising of multiple subprocesses. In the case of sports club management, we conceptualize overall management as a production process comprising of two subprocesses; 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 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-filed 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; how efficiently a sports club manages its resources overall, in PGM (on-field activity management) and FM (off-field activity management). Established clubs generate revenue in many ways such as through merchandise sales, sponsorship deals and ticket sales. Some AFL clubs have limited resources due to small membership (fanbase) and their inability to attract lucrative sponsorship deals. Profitability also varies across AFL clubs. AFL has introduced measures to balance out financial-disparity across clubs through equalization measures such as revenue sharing. AFL has taken measures also to even out on-filed 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 information that they may find useful in PGM and FM. Some studies consider detailed information on the play to explain match performance [14]. Another area of study is assessing effectiveness of playing strategies or tactics adopted by the teams [15,16,17,18]. Further, there is evidence of using various algorithms and machine learning techniques for performance prediction. See for example, [19]. These studies investigate key areas that reveal information useful in player selection and strategizing game plans. All of these studies focus on on-field performance.

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 frontier 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. 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 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 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 organized as follows. In the next section, we provide a brief description of Australian rules football and review the literature on performance appraisal of sporting teams with special reference to DEA. Section 3 discusses 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.

We conceptualise 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, PGM process includes activities associated with coaching staff managing the playing group with available resources and 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 plays a dual role. That is, certain stage A output variables $\left(Z^{AB}\right)$ serves as input at stage B. In DEA, such variables are referred to as intermediate variables. In this paper, we conceptualize overall AFL club management process as a production process comprising of two functionally different subprocesses (PGM- stage A and FM- stage B) linked through intermediate variables. Individual stages are also conceptualized 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 stage B operations. In both stages, we consider that output augmentation is more appropriate than input conservation. When developing a DEA model to assess relative performance of overall management process comprising of subprocesses, an assumption has to be made on how individual stage-level performance contribute towards overall performance. Following Wang and Chin (2010), we express overall efficiency as 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 links 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).¹

Subject to ${\displaystyle \frac{{\sum }_{i=1}^{i_A}{v}_i^A{x}_{ij}^A-{\delta }^A}{{\sum }_{d=1}^D{\omega }_d^{AB\text{'}}{z}_{dj}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{rj}^A}}\ge 1$ $j=\mathrm{1,2},\dots ,n$ $v_i^A,$ $u_r^A$, ${\omega }_d^{AB\text{'}}\ge 0$; ${\delta }^A$ unrestricted

Subject to ${\displaystyle \frac{{\sum }_{i=1}^{i_B}{v}_i^B{x}_{ij}^B+{\sum }_{d=1}^D{\omega }_d^{AB\text{'}\text{'}}{z}_{dj}^{AB}-{\delta }^B}{{\sum }_{r=1}^{r_B}{u}_r^B{y}_{rj}^B}}\ge 1$ $j=\mathrm{1,2},\dots ,n$ $v_i^B,$ $u_r^B,{\omega }_d^{AB\text{'}\text{'}}\ge 0$; ${\delta }^B$ unrestricted

As shown in Figure 1, stage A and stage B are linked. Therefore, when determining overall-efficiency of the two-stage process, individual stages cannot not be considered as independent. Hence, some form of aggregation of stage-level efficiencies is required. In this case, following [20] and [21], we express 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 add to 1. [20] use harmonic average under the CRS assumption. [21] 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. Common approach in weight selection is to define weights so that aggregated efficiency would be fractional linear. We define the weights as proportion of implied value of stage-level output to 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 [22]. Further, we assume that the multipliers associated with intermediate variables, ${\omega }_d^{AB\text{'}}$ and ${\omega }_d^{AB\text{'}\text{'}}$ 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 multi-stage processes. See for example, [23,24,25]. An implication of this assumption is that 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).

Subject to ${\displaystyle \frac{{\sum }_{i=1}^{i_A}{v}_i^A{x}_{ij}^A-{\delta }^A}{{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{dj}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{rj}^A}}\ge 1$ $j=\mathrm{1,2},\dots ,n$ ${\displaystyle \frac{{\sum }_{i=1}^{i_B}{v}_i^B{x}_{ij}^B+{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{dj}^{AB}-{\delta }^B}{{\sum }_{r=1}^{r_B}{u}_r^B{y}_{rj}^B}}\ge 1$ $j=\mathrm{1,2},\dots ,n$ ${LB}_0^A\le {\displaystyle \frac{{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{d0}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{r0}^A}{{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{d0}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{r0}^A+{\sum }_{r=1}^{r_B}{u}_r^B{y}_{r0}^B}}\le 1$ ${LB}_0^B\le {\displaystyle \frac{{\sum }_{r=1}^{r_B}{u}_r^B{y}_{r0}^B}{{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{d0}^{AB}+{\sum }_{r=1}^{r_A}{u}_r^A{y}_{r0}^A+{\sum }_{r=1}^{r_B}{u}_r^B{y}_{r0}^B}}\le 1$ $v_i^A,$ $u_r^A$, ${\omega }_d^{AB},v_i^B,$ $u_r^B\ge 0$; ${\delta }^A,{\delta }^B$ unrestricted

${LB}_0^A$ and ${LB}_0^B$ are lower bounds on the weights, $w_A$ and $w_B$. 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 linearized by adopting Charnes-Cooper transformation [26]. 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 stage 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 $E_0^{\left(B\right)}={\displaystyle \frac{{\sum }_{r=1}^{r_B}{u}_r^{B*}{y}_{r0}^B}{{\sum }_{d=1}^D{\omega }_d^{AB*}{z}_{d0}^{AB}+{\sum }_{i=1}^{i_B}{v}_i^{B*}{x}_{i0}^B-{\delta }^{B*}}}$. $E_0^{\left(A\right)}$ 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 in [24]. See Appendix B.

Proposition 1.DMU₀ is overall efficient if and only if stage A and stage B are efficient.

Proof of Proposition 1. 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 stage 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 ${\sum }_{i=1}^{i_B}{v}_i^B{x}_{ij}^B+{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{dj}^{AB}-{\delta }^B={\sum }_{r=1}^{r_B}{u}_r^B{y}_{rj}^B$. Hence, follows ${\sum }_{i=1}^{i_A}{v}_i^A{x}_{ij}^A-{\delta }^A+{\sum }_{i=1}^{i_B}{v}_i^B{x}_{ij}^B+{\sum }_{d=1}^D{\omega }_d^{AB}{z}_{dj}^{AB}-{\delta }^B=1$. Now we have numerator and denominator of the objection function of model (3) is 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 the second constraints of model (3), we have that each component in the round brackets are non-negative. Hence their sum is zero only when each component is zero suggesting stage A and stage B are efficient. □

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

We conceptualize the overall management process of a sports club as comprising two subprocesses: on-field management and off-field management. The survival of sports clubs hinges on the efficient collective management of these subprocesses. This study aims to highlight how the performance in on-field and off-field management can impact the club's overall performance.

To achieve 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 prioritized over off-field performance, and another where off-field performance is prioritized 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 performance of clubs in other team sports.