DEA EFFICIENCY MEASUREMENTS OF OLYMPIC GAMES WINNING COUNTRIES AND THE SIGNIFICANCE OF COVARIATES

Transcription

1 XV INTERNATIONAL CONFERENCE ON INDUSTRIAL ENGINEERING AND OPERATIONS MANAGEMENT The Industrial Engineering and the Sustainable Development: Integrating Technology and Management. Salvador, BA, Brazil, 06 to 09 October DEA EFFICIENCY MEASUREMENTS OF OLYMPIC GAMES WINNING COUNTRIES AND THE SIGNIFICANCE OF COVARIATES Geraldo da Silva e Souza (Embrapa) geraldo.souza@embrapa.br Eliane Goncalves Gomes (Embrapa) eliane.gomes@embrapa.br Using two stage statistical inference we assess the significance of per capita income and investment in sports on the Olympic efficiency scores of countries. The number of participating athletes is the proxy for investment in sports. These are computed using unit inputs and weights restrictions on the medals distributions. We compare the fits of the distributions gamma and truncated normal to non efficient countries. These include as particular cases the half normal and the exponential, which are of standard use in efficiency analysis. The estimation method is maximum likelihood. We use decision criteria to arrive at a best model. Only medal achieving countries are considered in the sample. Palavras-chaves: DEA; Contextual Variables; Olympics.

2 1. Introduction The Olympic Games were born in ancient Greece, in 776 AC. They were designed for individual contests, trying to maintain the initial spirit of individual competition. Lins et al. (2003) noted that this purpose clearly failed and the games have become later a national competition. In 1896, in Athens, from an initiative of Baron de Coubertin, took place the first edition of the modern games. Wallechinsky (2004) presents a complete history of the modern games, concerning their summer edition. The last Games occurred in Beijing in August, A total of 958 medals were distributed. The number of participating athletes was 11,028, competing in 28 distinct sport modalities, totaling 302 events (International Olympic Committee, 2008). In the 26 editions of the Modern era Olympic Games, the United States was by far the country accumulating the largest number of winnings. Other important winners are China and Soviet Union (while unified), and some countries in West Europe. We believe that the countries development level and their investment in sports are potentially the key factors, overall, for good results in the Olympics. Despite their national character, the Olympic Committee has never issued an official ranking to pick an overall Olympic winner country, i.e., the best performer. The International Olympic Committee and the mass media, however, did so, as they present the medals data in a table that suggests a ranking. This ranking has become a quasi-official one, and it is based on the Lexicographic Multicriteria method (Lins et al., 2003). However, this ranking does not deal properly with the possible existence of competing countries that have won a large number of silver and/or bronze medals but no gold medal, as pointed out by Soares de Mello et al. (2008). In the literature there are a number of alternative ways to assess the countries performance in the Olympics. Some are based on Data Envelopment Analysis that has shown usefulness in performance classification in sports and elsewhere. In this article we seek to generate a performance index to classify countries and to identify contextual variables potentially affecting performance. This is achieved by means of a two stage process. Firstly we compute the measure of efficiency. In a second stage we perform statistical inference to assess the significance of per capita income and investment in sports on the Olympic efficiency scores of countries. The first stage is carried out computing DEA efficiencies with unit inputs and weights restrictions on the medals distribution, similar to the approaches proposed by Soares de Mello et al. (2008, 2009). The data analyzed refers to the Beijing Games. Our discussion proceeds as follows. In Section 2 we present a literature review on performance evaluation on the Olympic Games. In Section 3 we present the DEA models with weights restriction that we use in the first stage of our analysis. Section 4 deals with statistical models adequate to study efficient measurements. Section 5 is on statistical results, and Section 6 summarizes our findings. 2. Evaluating the Performance in Olympics: A Brief Literature Review Beyond the lexicographic method, Olympic rankings can be made, for example, by adding up bronze, silver and gold medals earned by each country. The ranking is then determined by this total, but it under-evaluates the gold medal. Arbitrary evaluations of each medal are also possible. 2

3 Balmer et al. (2001) obtained indices of home advantage, based on the medals won by competing nations, for each event held at the Winter Olympics from 1908 to They aimed at to assess overall home advantage in the Winter Olympics and to determine the influence of factors thought to have a bearing on home advantage. The same authors proposed an enhanced approach (Balmer et al., 2003), in which they compared home advantage between groups of events with varying styles of officiating. They selected five event groups from the Summer Olympic Games between 1896 and 1996, and put home advantage in team games in context with other sports. Statistical approaches were also proposed. Morton (2002), for instance, examined the least squares fitting of a log-linear allometric model to the data related to the 80 countries which won at least one bronze medal in Sydney Games. Bernard and Busse (2004) considered the role of population and economic resources in determining medal totals. Olympic rankings can be established using DEA models. Lozano et al. (2002) used the DEA BCC model to measure the performance of the nations participating at five Summer Olympic Games. They considered two inputs (GDP and population) and three outputs (number of gold, silver and bronze medals won, with weight restrictions). Lins et al. (2003) built a new DEA model, the so-called Zero Sum Gains DEA model (ZSG- DEA, taking into account one more constraint: the total amount of medals is a constant. A methodology combining data mining techniques and DEA to analyze the real achievements of various Olympics participants was presented by Churilov and Flitman (2006). Instead of using as outputs the number of gold, silver and bronze medal, they used four linear combinations of these figures. Hai (2007) used vote-ranking and cross-evaluation to evaluate the results at the Sydney 2000 Olympics. The author considered that the weight differences with gold medal and silver medal is no less than the differences with silver medal and bronze medal. The same statements were used by Soares de Mello et al. (2008, 2009). Soares de Mello et al. (2008) proposed a method based on DEA, in which the outputs were the number of all three medals that each country won at both the Salt Lake City and Sydney games; a constant input was considered for all countries. They defined restrictions based on the importance of each medal. A weight average for each output, based on a modified Cross Evaluation model, was developed and used as the coefficient in the weighted sum that established the final ranking. In Soares de Mello et al. (2009), in order to take into account the difference in winning values for different sports, competitions were aggregated into clusters. Then a modified crossevaluation DEA model with weighted restrictions was used for each cluster, considering the results of the Athens Olympic Games. The outputs are the number of gold, silver and bronze medals and the input is a unitary constant for all countries. The authors also took into account that medals won in different competitions do not have the same value. Li et al. (2008) took into consideration the variability in the economic status of participating nations. They extended previous DEA studies by incorporating multiple sets of nationspecific assurance regions (Thanassoulis et al., 2004) into the DEA, for six summer Olympic Games. They believe that they established fair models for measuring and benchmarking the performance of nations. A modified DEA game cross-efficiency model was presented by Wu et al. (2009). In their model each DMU is viewed as a competitor via non-cooperative game. The authors provided 3

4 a modified variable returns to scale model that yields non-negative cross-efficiency scores. The proposed approach was applied to six Summer Olympic Games. 3. DEA Models with Weights Restrictions From an Economics point of view, DEA has for objective the assessment of efficiency of a set of production units, the so-called Decision-Making Units (DMUs). DEA uses linear programming problems to identify DMUs that represent the best production practices in the sense that they make the best use of the inputs available to produce outputs. These units are considered efficient and act as benchmarks to the other DMUs. The efficiency of production of a DMU, in a typical DEA analysis, is defined by the ratio of a weighted average of the outputs by a weighted average of the inputs. These weights are shadow prices. They are distinct for outputs and inputs, and are determined by linear programming. For each DMU, the linear programming problem is set in such a way to maximize the efficiency ratio. Subjective or preconceived perceptions of the investigator about the relative importance of outputs and inputs may be included as restrictions in the linear programming problem. There are two classical DEA models (cooper et al., 2004): the CCR, also known as CRS or constant returns to scale, and the BCC, or VRS variable returns to scale. The two models differ according to the convexity restriction imposed in the BCC DEA model. Two orientations are possible in any DEA model. One may look for maximal outputs given input levels, or one may look for minimal input utilization, given output levels. In DEA modeling one has to define the DMUs, the production variables (inputs and outputs), the orientation and the model. In our case the DMUs are the countries that one at least one medal. The gold, silver and bronze medals won define the output vector. To measure the efficiency of each winning country we assume unitary inputs for all units. This idea is not strange in the DEA literature (see Thompson et al., 1986; Lovell and Pastor, 1999; De Koeijer et al., 2002; Leta et al., 2005; Gomes et al., 2008) and, notably, in the studies about DEA efficiency measurements in Olympics, as brought forward in Section 2. The model we consider is DEA CCR, with output orientation. We note that there are no issues of scale here, since the BCC solution for unit inputs coincide with the CCR solution, as stated and proved by Lovell and Pastor (1999). The assumption of unit inputs is adequate for our case, since it puts all countries on the same basis for comparisons, and real differences in the response vector are therefore due to the influence of contextual variables. In this context, using DEA scores as the response variable, the effects of contextual variables are to be assessed in a second stage, as follows. We now define the general DEA CCR model. Assume that each DMU k, k = 1...n, is a production unit that uses m inputs (non-negatives, not all zeros) x ik, i =1 r, to produce s outputs (non-negatives, not all zeros) y jk, j =1 s. The efficiency h o of DMU o, with production values ( x, y ), is defined by the solution of the linear programming r o o problem Min h = vx, subject to the restrictions s j jk i ik j= 1 i= 1 r o i io i= 1 u y vx 0, k= 1,..., n, u j, v i 0, j,i. s uy j jo = 1, j= 1 4

5 The quantities v i e u j are the shadow prices (weights) of inputs and outputs, respectively. These weights assigned to each variable are chosen in complete freedom in classical DEA models. In our case this means that in order to achieve their efficiency scores some DMUs may overvalue the silver or the bronze medals, and may ignore the gold ones. Such choices are against the common sense. However, it is possible that some outputs should be regarded as more important than others. This is such the case, as the medals won have different value. As discussed in Thanassoulis et al. (2004), it is possible to define DEA models incorporating these perceptions (value judgments), and reducing the flexibility of DMUs in choosing their value system. One of those methods is the so-called Weight Restrictions (Allen et al., 1997; Thanassoulis et al., 2004), for which some different approaches are available. Here we chose the Cone Ratio method, also called Assurance Region of Type I (Thompson et al., 1990), proposed by Charnes et al. (1990). The Cone Ratio method needs less more than ordinal considerations on the importance of each variable. In our case, this means that a gold medal is more important than a silver one, and this one is more important than a bronze medal. The additional restrictions are ugold usilver and usilver ubronze. 4. Statistical Models Following the literature on DEA models in the presence of contextual variables, we now consider statistical models flexible enough to assess the significance of covariates. The goal is to investigate the covariates of interest likely to affect the restricted weights DEA performance on the Olympics. We mimic the results of Banker (1993), Souza and Staub (2007), Simar and Wilson (2007) on DEA, and Daraio and Simar (2007) and Souza et al. (2009) on the FDH to fit a class of flexible density functions by maximum likelihood methods. Our concern is to model Olympics efficiency as dependent of two variables: size of delegation and gross domestic product per capita. Let z 1 and z 2 represent these variables. First we consider the specification ϕi = δ0 + δ1zi 1+ δ2zi2 + wi, where ϕ i is the efficiency response, the δ j are parameters to be estimated, the z ij are realizations of the z j, and 2 the truncation of the normal N(0, σ ) at µ i( = δ0 + δ1zi 1+ δ2zi2). w i is 2 These assumptions are consistent with non-negative truncations of the N ( µ i, σ ). This model is also considered in stochastic frontier analysis to describe inefficiency errors. See Khumbhakar and Lovell (2000) and Coelli et al (2005). It may be fitted by maximum. One 2 notices that the mean of the positive truncation of the N ( µ i, σ ), µ i + σλi and the variance, σ 2 [ 1 λ i( µ i/ σ + λi) ], are both monotonic functions of µ i, where λi = φ ( µ i/ σ ) Φ ( µ i/ σ), φ (). and Φ. () being the density and the distribution function of the standard normal, respectively. The formulation therefore allows heteroscedasticity and for unequal distributions for the efficiencies. The same ideas above may be applied to the exponential family of densities λ j exp{ λ t j }, imposing λ j = exp{ µ j}. These two families, i.e., the exponential and the truncated normal, as in the stochastic frontier analysis, seem to provide enough flexibility in applications but, for the sake of generality and added flexibility, we extend the exponential distribution to the p p gamma family. For this purpose one considers the density λ 1 jt exp{ λjt} /Γ( p), imposing as in the exponential case λ = exp{ µ }. j j 5

6 In any case the maximum likelihood estimates are to be computed ignoring efficient units, like in Simar and Wilson (2007). It should be said that our approach here is exploratory, in the sense that no actual production model is being modeled for the efficiency measurements as in Souza and Staub (2007) or Simar and Wilson (2007). 5. Statistical Results Annex I shows the data we used in our analysis. Our perception motivating the analysis is that investment in sports, measured by the delegation size, and development, measured by the gross domestic product, are the key factors to explain performance in the Olympic Games The impression one has from the box-plot and the histogram shown in Figures 1 and 2 is of a skewed distribution, with potential and definite outliers. The later are the cases of China and the US. From this observation one sees that is a god practice to eliminate efficient units from the maximum likelihood analysis efficiency Figure 1 - Box-Plot of efficiency distribution. 6

7 Density efficiency Figure 2 - Histogram of efficiency distribution. Table 1 shows goodness of fit statistics of the three families of models we postulated for this analysis, i.e., exponential, gamma, and truncated normal. We see that by far the truncated normal density fits better. The gamma model shows an increase in performance over the exponential model as expected. Model -2 ll AIC BIC Exponential Gamma Truncated Normal Table 1 - Goodness of fit statistics for the exponential, gamma and truncated normal distributions. -2ll is the negative of two times the likelihood, AIC is Akaike information criterion (smaller the better) and BIC is Shwarz Bayesian information criterion (smaller the better). Table 2 shows maximum likelihood estimates. This analysis is performed with SAS procedure NLMIXED. We see that the dominant effect is delegation, which positively causes efficiency. The variable income (GDP per capita) is non significant. It is interesting to notice that the negative sign in income is detected for all models. The p-values for the corresponding test statistics are 0.08 (standard regression analysis), 0.35 (exponential model) and 0.13 (gamma model). We interpret this as negative marginal significance of the GDP per capita on efficiency. Parameter Estimate Standard Error t-value p-value δ (constant) δ (Delegation) < δ (GDP per capita) σ <

8 6. Conclusions Table 2 - Maximum likelihood estimates. Truncated normal distribution. We compute DEA efficiency measurements with weight restrictions to data on country medal s performance during the 2006 Olympic Games. The medal distribution is right skewed with outliers and potential outliers. We model the data on non efficient units as dependent on max normalized covariates size of delegation and per capita income by means of three families of statistical distributions, in such way so that the means and variances are monotonic functions of a linear construct in these variables. We conclude that the dominating variable in the relation is size of delegation with a strong positive effect, and that income has a marginal negative effect. References ALLEN, R.; ATHANASSOPOULOS, A.; DYSON, R.G. & THANASSOULIS, E. Weights restrictions and value judgements in data envelopment analysis: evolution. development and future directions. Annals of Operations Research Vol 73, p , BALMER, N.J.; NEVILL, A.N. & WILLIAMS, A.M. Home advantage in the Winter Olympics ( ). Journal of Sports Sciences Vol. 19, p , BALMER, N.J.; NEVILL, A.N. & WILLIAMS, A.M. Modelling home advantage in the Summer Olympic Game. Journal of Sports Sciences Vol. 21, p , BANKER, R.D. Maximum likelihood, consistency and DEA: a statistical foundation. Management Science Vol. 39, n. 10, p , BERNARD, A.B.; BUSSE, M.R. Who wins the Olympic Games; economic resources and medal totals. Review of Economics and Statistics Vol. 86, n. 1, p , CHARNES, A.; COOPER, W.W.; HUANG, Z.M. & SUN, D.B. Polyhedral cone-ratio DEA models with an illustrative application to large commercial banks. Journal of Econometrics Vol. 46, p , CHURILOV, L. & FLITMAN, A. Towards fair ranking of Olympics achievements: The case of Sydney Computers and Operations Research Vol. 33, p , COELLI, T.J.; PRASADA RAO, D.S.; O'DONNELL, C.J. & BATTESE, G.E. An Introduction to Efficiency and Productivity Analysis. 2nd Edition, Springer, COOPER, W.W.; SEIFORD L.M. & ZHU J. Handbook on Data Envelopment Analysis. Kluwer Academic Publishers, DARAIO, C. & SIMAR, L. Advanced Robust and Nonparametric Methods in Efficiency Analysis. Springer, DE KOEIJER, T.J.; WOSSINK, G.A.A.; STRUIK, P.C. & RENKEMA, J.A. Measuring agricultural sustainability in terms of efficiency: the case of Dutch sugar beet growers. Journal of Environmental Management Vol. 66, p. 9-17, GOMES, E.G.; SOUZA, G.S. & VIVALDI, L.J. Two-stage inference in experimental design using DEA: an application to intercropping and evidence from randomization theory. Pesquisa Operacional Vol. 28, p , HAI, H.L. Using vote-ranking and cross-evaluation methods to assess the performance of nations at the Olympics. WSEAS Transactions on Systems Vol. 6, p , INTERNATIONAL OLYMPIC COMMITTEE Official website of the Olympic movement Retrieved from Accessed in 12 Jan KUMBHAKAR, S. & LOVELL, C.A.K. Stochastic Frontier Analysis, Cambrigde University Press,

9 LETA, F.R.; SOARES DE MELLO, J.C.C.B.; GOMES, E.G. & ANGULO MEZA, L. Métodos de melhora de ordenação em DEA aplicados à avaliação estática de tornos mecânicos. Investigação Operacional Vol. 25, n. 2, p , LI, Y.; LIANG, L.; CHEN, Y. & MORITA, H. Models for measuring and benchmarking Olympics achievements. Omega Vol. 36, p , LINS, M.P.E.; GOMES, E.G.; SOARES DE MELLO, J.C.C.B. & SOARES DE MELLO, A.J.R. Olympic ranking based on a zero sum gains DEA model. European Journal of Operational Research Vol. 148, p , LOVELL, C.A.K. & PASTOR, J.T. Radial DEA models without inputs or without outputs. European Journal of Operational Research Vol. 118, n. 1, p , LOZANO, S.; VILLA, G.; GUERRERO, F. & CORTÉS, P. Measuring the performance of nations at the Summer Olympics using data envelopment analysis. Journal of the Operational Research Society Vol. 53, p , MORTON, R.H. Who won the Sydney 2000 Olympics?: an allometric approach. The Statistician Vol. 51, n. 2, p , SIMAR, L. & WILSON, P.W. Estimation and inference in two-stage, semi-parametric models of production processes. Journal of Econometrics Vol. 136, n. 1, p , SOARES DE MELLO, J.C.C.B.; ANGULO MEZA, L. & BRANCO DA SILVA, B.P. A ranking for the Olympic Games with unitary input DEA models. IMA Journal of Management Mathematics,, 2009 (forthcoming). SOARES DE MELLO, J.C.C.B.; GOMES, E.G.; ANGULO MEZA, L. & BIONDI NETO, L. Cross evaluation using weight restrictions in unitary input DEA models: Theoretical aspects and application to Olympic Games ranking. WSEAS Transactions on Systems Vol. 7, p , SOUZA, G.S. & STAUB, R.B. Two-stage inference using data envelopment analysis efficiency measurements in univariate production models. International Transactions in Operational Research Vol. 14, p , SOUZA, G.S.; GOMES, E.G. & STAUB, R.B. Influence of contextual variables: an application to agricultural research evaluation in Brazil. In Proceedings International Data Envelopment Analysis Symposium, THANASSOULIS, E.; PORTELA, M.C.S. & ALLEN, R. Incorporating value judgments in DEA. In Handbook on Data Envelopment Analysis (W.W. Cooper, L.M. Seiford and J. Zhu, Eds.) Kluwer Academic Publishers, p , THOMPSON R.G.; SINGLETON JR. F.D.; THRALL R.M. & SMITH B.A. Comparative evaluation for locating a high-energy physics lab in Texas. Interfaces Vol. 16, p , THOMPSON, R.G.; LANGEMEIER, L.N.; LEE, C.T.; LEE, E. & THRALL, R.M. The role of multiplier bounds in efficiency analysis with application to Kansas farming. Journal of Econometrics Vol. 46, p , WALLECHINSKY, D. The Complete Book of the Summer Olympics. Aurum Press, WU, J.; LIANG, L. & CHEN, Y. DEA game cross-efficiency approach to Olympic ranking. Omega Vol. 37, p ,

10 Annex 1: Number of medals, size of delegation ( z 1 ), normalized (maximum) gross domestic product per capita ( z 2 ), and efficiency with weights restrictions. Country Gold Silver Bronze Delegation ( z 1 ) GDP ( z 2 ) Efficiency Afghanistan Algeria Argentina Armenia Australia Austria Azerbaijan Bahamas Bahrain Belarus Belgium Brazil Bulgaria Cameroon Canada Chile China Colombia Croatia Cuba Czech Republic Denmark Dominican Rep Ecuador Egypt Estonia Ethiopia Finland France Georgia Germany Greece Hungary Iceland India Indonesia Iran Ireland Israel Italy Jamaica

11 Japan Kazakhstan Kenya Kyrgyzstan Latvia Lithuania Malaysia Mauritius Mexico Moldova Mongolia Morocco Netherlands New Zealand Nigeria North Korea Norway Panama Poland Portugal Romania Russia Serbia Singapore Slovakia Slovenia South Africa South Korea Spain Sudan Sweden Switzerland Taiwan Tajikistan Thailand Togo Trinidad and Tobago Tunisia Turkey Ukraine United Kingdom United States Uzbekistan Venezuela Vietnam

12 Zimbabwe