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In the case of the proportional output reduction strategy with a single output, the Variable-Returns-to-Scale (VRS) Zero-Sum Gains Data Envelopment Analysis (ZSG-DEA) efficiency scores can be obtained from the VRS conventional DEA efficiency scores by means of the Target's Assessment Theorem (TAT). Using TAT as a departure point, two relations for computing the ZSG-DEA efficiency scores appear in the literature. Our objective in this note is to compare, contrast and challenge them on both theoretical and empirical grounds. For the latter, three different data sets are used.

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In this paper, we present for the first time the on-field production process of soccer teams as a mixed (serial and parallel) structure two-stage network system. According to this, the first stage consists of two distinct sub-processes (offense and defense) that operate in parallel. These respectively use players' offensive and defensive actions as inputs to produce two different intermediate measures, namely, goals scored and prevention of goals conceded. These, in turn, are the inputs of the second stage (points' accumulation sub-process) that produces accumulated points. Furthermore, based on a two-stage network Data Envelopment Analysis (DEA) model, we estimate the offensive, defensive, and athletic efficiency of soccer teams during a league season.According to our proposed framework, these three different efficiency scores are provided (for each soccer team under evaluation) by a single linear programming problem. For this purpose, aggregate-over-games statistics from the 2013-14 Greek premier soccer league are used. IntroductionSince the study of Carmichael et al. (2001), on-field performance evaluation in soccer seeks to answer whether athletic efficiency leads teams to sporting success. To this end, Espitia-Escuer and García-Cebrián (2004, 2006 proposed to estimate the athletic efficiency of soccer teams during a league season based on a single-stage Data Envelopment Analysis (DEA) model. According to this model, the on-field actions of soccer teams are considered as inputs and their accumulated points as a single output. This variable is preferred over the number of teams' victories (or their winning percentage) during a league season (Dawson et al., 2000, p. 407). This is due to the fact that (in each soccer game) there are three probable results, each of which is rewarded differently. In particular, victories (losses) receive three (zero) points and ties one point for each soccer team. Some years later, the studies of Boscá et al. (2009) andSala-Garrido et al. (2009) took a step further in the on-field performance evaluation of soccer teams. Specifically, these studies proposed to separately estimate the offensive and defensive efficiency of soccer teams during a league season based on two different single-stage DEA models.According to these models, teams' goals scored and the inverse of teams' goals conceded are independently used as a single output. In this way, the above studies