Allocation Efﬁciency of Public Sports Resources Based on the DEA Model in the Top 100 Economic Counties of China in Zhejiang Province

: Background: The county is the basic unit of national economic and social development, and is also the foothold and starting point of public sports services. Purpose: Taking the top 100 economic counties of China in Zhejiang Province as the research object, this study explores the allocation efﬁciency and inﬂuencing factors of public sports resources in the period of 2016 to 2020. Methods: The output-oriented Super-SBM model, which is used to measure the static efﬁciency of its public sports resource allocation, is combined with the DEA–Malmquist model to measure the total factor productivity from the perspectives of overall characteristics, regional heterogeneity, and individual differences. Moreover, we objectively evaluate the dynamic evolution and spatiotemporal characteristics of resource quality growth, ﬁnancial management technology, and allocation efﬁciency from the horizontal cross-section and vertical time series. Results: (1) The efﬁciency of allocation of public sports resources in the top 100 economic counties in Zhejiang Province is relatively high, but it presents the characteristics of “extensive” allocation, and the allocation structure is unreasonable. (2) The super-efﬁciency gradient division of public sports resources shows that Yuhuan City ranks ﬁrst with a state of super-efﬁciency allocation; Ruian, Linhai, Wenling, Yiwu, and Haining have a state of high-efﬁciency allocation; and other regions are characterized by a state of medium-or low-efﬁciency allocation. (3) The improvement of total factor productivity depends on the catching-up feature of technological efﬁciency on the production frontier, but it has not yet compensated for the negative effect of the decline of technological progress, resulting in a decline in total factor productivity with an average annual trend of 0.3%. (4) The level of county economic development has a highly signiﬁcant positive effect on the allocation efﬁciency of public sports resources, while the per capita sports ground area has a highly signiﬁcant negative effect on efﬁciency. The county population density has a highly signiﬁcant impact, and regional factors have no signiﬁcant effect on efﬁciency. Conclusions: The results of this study provide useful insights for the development of sound public sports service improvement policies.


Introduction
With the rapid development of China's economy and the improvement of quality of life, physical exercise has become one of the most important activities in people's daily life. Regular exercise can not only improve physical health, but also alleviate negative psychological conditions, such as anxiety and depression, and promote self-expression in social interactions [1]. The Chinese government is increasingly emphasizing the importance of public participation in sports, and mass sports have become a vital component of the public sports service system. Mass sports is where people participate in physical exercise and competitive activities whose purpose is to strengthen physique and serve the learning and labor production directly [2]. In the "13th Five-Year Plan" to Promote the Equalization of Basic Public Services, the comprehensive promotion of the equalization of basic public services was proposed [3]. "The Outline of Building a Sports Powerful Country" takes the construction of the public sports service system as a major step to implement the strategy of national fitness and assist with creating a healthy China [4]. Meanwhile, the development of mass sports cannot be achieved without government financial support. Certain issues are of great concern to the government and the entire society, such as the current performance of financial investment in mass sports in China, as to whether the input reaches the maximum output. The systematic and scientific evaluation of the efficiency of public sports resource allocation is the prerequisite for optimizing public sports services in China. Therefore, evaluating the efficiency of public sports resource allocation to maximize the effectiveness of limited resource investment can better promote the development of mass sports.
At this stage, driven by the dual background of "national fitness" and "healthy China", the level of public sports resources allocation has been greatly improved, but the unbalanced allocation of public sports resources has also gradually come to the fore [5,6]. Yu [7] conducted an evaluation on the efficiency of financial investment in sports in China and reported a decreasing trend from 2003 to 2008. Shao [8] analyzed the efficiency of mass sports financial investment in 30 Chinese provinces in 2011 and found it to be generally low. The allocation of public sports resources refers to the process of allocating human, material, and financial resources provided by the government, society, and enterprises for public sports services in different regions under the guidance of public sports service policies, so as to meet the needs of groups for public sports [9]. The equalization of the level of development of public sports resources will certainly be an important element in enhancing the equalization of public sports services and will be an important step in advancing China from a large sports country to a strong sports country.
Such a state of disparity in the allocation of public sports resources is not unique to China; it is a worldwide problem. By investigating sports facilities in different regions of the United States, Gordon et al. found that areas with better economic and social development were significantly higher than those with less development [10]. Giles et al. reached similar conclusions by investigating the allocation of public sports resources in different cities in Australia [11]. The issue of equitable allocation of public sport resources is very important and prevalent in a country and, indeed, globally.
In response to the problems of insufficient allocation level, unbalanced regional development and unreasonable structure in the allocation of public sports resources, scholars have conducted a series of studies related to the allocation of public sports resources. The current methods for evaluating efficiency include parametric and non-parametric analysis methods; the commonly used methods are stochastic frontier analysis (SFA) and data envelopment analysis (DEA) [12]. The model assumptions of SFA are more complex and require more input and output data. If the input and output data do not conform to the basic assumptions of the model, the skewness problem of the above analysis will easily occur and eventually lead to the failure of the calculation. DEA, as a non-parametric estimation method, circumvents various limitations of parametric methods, does not need to set the parameters and specific form of the frontier production function, does not need to consider the contours between indicators, and can use linear programming to carry out the model weights calculation, thus circumventing the interference of artificial subjective weights. Due to the advantages of the DEA method, it has been gradually applied to the evaluation of sports financial input performance in recent years. Ren (2011) [13] used a DEA model to evaluate the efficiency of public state sports services in 31 Chinese provinces in 2016. Iversen (2018) [14] discussed the consequences of the UK government's response to the abatement of sports facility subsidies after the 2008 economic and financial crisis, and derived the results based on the implementation of the New Public Governance through a qualitative case study of non-profit private stadiums in the UK. Bernardino (2012) [15] implemented a DEA model to estimate the determinants of efficiency in terms of local physical public sports facilities in Spain.
Although DEA methods can be used in the performance evaluation of resource allocation, they have limitations. These are relative efficiency evaluation methods that cannot replace the traditional ratio analysis of the absolute efficiency assessment. They are only applicable for the evaluation of efficiency, but the specific factors causing DEA to be invalid need to be further analyzed. In addition, if only a single DEA-Charnes-Cooper-Rhodes (CCR) model [16] or DEA-Banker-Charnes-Cooper (BCC) model [17] is used, the combined efficiency cannot be resolved into pure technical efficiency and scale efficiency, and will be subject to the shape of the production frontier surface, which cannot further distinguish the degree of difference in allocation efficiency among technically efficient decision-making units (DMUs). The traditional DEA-CCR and DEA-BCC models do not take into account the influence of external environmental factors and stochastic factors on redundancy, i.e., the efficiency value of the DMU will fluctuate within a certain range due to the interference of stochastic factors when the observed data are insufficient; therefore, the calculation results of the traditional DEA model lack robustness.
The Super Slacks Based Measure (SBM) analysis model can compensate for the inability of the BCC and CCR models to rank the resource allocation efficiency of effective DMU [18]. The Super-SBM model is a non-parametric static efficiency evaluation method that determines the production frontier surface by measuring the relative efficiency among DMUs with the same attributes in the same period. The Super-SBM model is based on the premise of expanding outputs with relatively stable existing inputs. It has the advantage of incorporating slack variables into the objective function, which makes the efficiency measurement more accurate and puts more emphasis on maximizing the output of public sports resources.
In order to present a rigorous and accurate evaluation of the research object in the actual research process, we need to comprehensively compare the changes of production factors of this production unit in different periods, including the factors of technological progress and efficiency changes. Since the static efficiency of traditional DEA evaluation is based on the actual production frontier constructed by the input and output of the DMU in the same period, the technical management level and resource input and output changes of the production unit in different periods are different, and the obtained efficiency is also different. Therefore, the time longitudinal comparison of efficiency factors measured by many previous scholars by using short panel data lacks scientific and comparable validity. In order to be able to objectively evaluate the changes in total factor productivity of DMU at the time longitudinal level, the Malmquist index model is introduced. The Malmquist productivity index was first proposed by Malmquist (1953) [19]. Subsequently, Fare (1994) [20] combined the nonparametric linear programming method developed by Farrell (1957) [21] and the productivity measure proposed by Caves (1982) [22] to construct the Malmquist productivity index based on the DEA method, in order to measure the change of DMU productivity over time.
Therefore, the present study will examine the space-time characteristics of resource allocation efficiency in three dimensions: the overall characteristics of public sports resource allocation, regional differences, and individual development, in order to reveal the differences in the efficiency of public sports resource allocation and the problems in the market-based allocation process in China. Firstly, the DEA-Super-SBM model is constructed in the static dimension to analyze the overall efficiency, pure technical efficiency, scale efficiency, scale payoff, super-efficiency, input redundancy, and output deficiency values of public sports resource allocation. Secondly, the sources of total factor productivity growth are dissected in the dynamic dimension with the help of the premise assumption of constant returns to scale, decomposed into information on the indices of technical progress and technical efficiency change. Third, in the spatial dimension, technical efficiency is decomposed into an index of pure technical efficiency change and an index of scale efficiency change with the help of the variable payoffs of scale premise assumption.

Materials and Methods
Four main processes are included in this study to achieve the above task, namely, the selection of the decision-making unit (DMU) to be evaluated, the construction of the evaluation index system of the model, the establishment of the mathematical econometric model, and the empirical analysis of the model (as shown in Figure 1). In this work, static efficiency will be evaluated in terms of technical efficiency, pure technical efficiency, scale efficiency, return to scale efficiency, super efficiency, and slack variable improvement, whereas dynamic efficiency will be evaluated in terms of total factor productivity index, technical change index, and technical efficiency change index.
progress and technical efficiency change. Third, in the spatial dimension, technical efficiency is decomposed into an index of pure technical efficiency change and an index of scale efficiency change with the help of the variable payoffs of scale premise assumption.

Materials and Methods
Four main processes are included in this study to achieve the above task, namely, the selection of the decision-making unit (DMU) to be evaluated, the construction of the evaluation index system of the model, the establishment of the mathematical econometric model, and the empirical analysis of the model (as shown in Figure 1). In this work, static efficiency will be evaluated in terms of technical efficiency, pure technical efficiency, scale efficiency, return to scale efficiency, super efficiency, and slack variable improvement, whereas dynamic efficiency will be evaluated in terms of total factor productivity index, technical change index, and technical efficiency change index.
where represents the efficiency evaluation indicator; denotes the i-th input vector of the j-th DMU; denotes the r-th output vector of the j-th DMU; indicates the amount of input slack adjustment; indicates the amount of output slack adjustment; and are the weights of the DMU j.
where ρ represents the efficiency evaluation indicator; x ij denotes the i-th input vector of the j-th DMU; y rj denotes the r-th output vector of the j-th DMU; s − indicates the amount of input slack adjustment; s + indicates the amount of output slack adjustment; and λ j are the weights of the DMU j.

Malmquist Index Model
This study will use the DEA-Malmquist model to analyze the total factor productivity of public sports resource allocation in Zhejiang Province in three dimensions: overall characteristics, regional heterogeneity, and individual disparities. The Malmquist productivity index measures the change in total factor productivity. It is constructed by solving several distance functions based on the efficiency values calculated by the DEA method, and then combining the distance functions. Under the assumption of constant returns to scale, the Malmquist index based on time period t can be expressed as: where (x t , y t ) and (x t+1 , y t+1 ) denote the input and output vectors for periods t and t + 1, respectively; D t c (x t , y t ) represents the distance function of the input-output vector in period t with reference to the technology in period t; D t c (x t+1 , y t+1 ) represents the distance function of the input-output vector in period t + 1 with reference to the technology in period t + 1.
The results of Malmquist indices under different technology levels can be inconsistent, thus creating a problem for evaluating total factor productivity. To solve this problem, Fare et al. referred to the construction method of Fisher's ideal index and took the geometric mean of the Malmquist productivity index with reference to the technology in period t and period t + 1 as the Malmquist productivity index. Thus, the Malmquist index can be expressed as: A Malmquist index value greater than 1 implies an improvement in the effectiveness of production in period t compared to period t + 1, i.e., an increase in total factor productivity; a value less than 1 implies a regression in total factor productivity, and a value equal to 1 implies no change in productivity. The Malmquist index can be decomposed into two components, technical change (TC) and technical efficiency change (TEC), which represent the jumping and catching-up characteristics of DMU relative to the production frontier, respectively. TEC can be further decomposed into pure technical efficiency change (PTEC) and scale efficiency change (SEC), reflecting the efficiency changes driven by management and scale factors, respectively, using the following equations: The first part of the above equation represents the PTEC, the second part represents the SEC, and the third part represents the TC.

Data and Variables
The county is the basic unit of national economic and social development, and it is also the foundation and starting point of public sports services. Furthermore, the county as the main playing field of high-quality development, precise poverty alleviation, and the overall building of a well-off society is directly related to the development of grassroots public sports services. For many years, based on the socio-economic statistics of more than 2000 counties across the country, China's National Bureau of Statistics has measured the comprehensive socio-economic development of counties in terms of development level, development vitality, and development potential. Zhejiang Province has been among the top three within the top 100 economic counties in China and is also the main site of China's reform and opening-up. By taking the top 100 economic counties in Zhejiang Province as the research object, the regional, exploratory, and exemplary characteristics and roles can be better reflected; in addition, empirical research on the static and dynamic efficiency of the input and output of human, material, and financial resources of public sports in Zhejiang Province can have far-reaching influence on the future reform of grassroots sports in China.

Selection of Input and Output Indicators
The reasonable determination of input and output indicators is a prerequisite for evaluating the efficiency of public sports services through the DEA model. The study follows the empirical pre-selection (initial indicators)-Delphi method (round 1)-indicator screening-Delphi method (round 2)-Hierarchical analysis-Indicator assignment-Final indicator logical sequence and framework to select indicators. First, based on the studies of Chen (2016) [23], Yu (2019) [24], Li (2017) [25],  [6], and other researchers, an evaluation index system for the efficiency of county public sports resources allocation was initially constructed based on the principles of rationality, representativeness, measurability, and data accessibility in the construction of efficiency indicators. Second, after two rounds of the Delphi method with reference to 14 experts' suggestions, useless indicators were excluded and the Friedman and Kendall coefficient tests were conducted. The results showed that: p < 0.05 in the Friedman test, rejecting the original hypothesis, indicating that there is a significant difference in the experts' assignment of the six evaluation indicators; the original hypothesis of the Kendall coefficient is that there is no significant consistency among the experts' evaluation indicators; p < 0.05 in the Kendall test, therefore rejecting the original hypothesis, indicating that there is significant consistency in the experts' overall evaluation of the six evaluation indicators; according to the value of ω (0.45), this indicates that the coordination of expert evaluations is good. Finally, the DEA efficiency indicators of public sports resource allocation in the top 100 economic counties were derived.
In this study, the number of public sports services and managers, the amount of investment, and the total area of sports venues in the county were selected as input indicators, and the number of people who regularly participate in physical exercise in the county (i.e., the total number of people who participate in physical exercise more than three times a week and exercise for more than 30 min each time) and the number of social organizations were selected as output indicators. To examine the efficiency of public sports resource allocation is to measure how limited human, financial, and organizational resources are invested in the development, construction, and allocation of public sports resources in order to produce what social and economic benefits. Considering the shortcomings of previous studies that focused only on economic efficiency but not on social justice efficiency, the output indicators of both social and economic benefits were selected to be examined in this study's output indicators. The output indicator of social efficiency is expressed in terms of the number of people who regularly participate in sports and exercise in the county, while the indicator of economic efficiency focuses on examining the level of public finance management and the degree of organization of mass sports activities, which is mainly measured from the tangible material level; here, the number of county sports social organizations is measured.
Environmental variables refer to factors that can influence the efficiency of public sport services and are subjectively uncontrollable. Considering the factors affecting the efficiency of public sports resources, this study uses county GDP per capita and population density as environmental variables. Regional GDP is a core national economic accounting indicator that reflects the economic status and development of a region, while the affluence of a place positively influences the efficiency of its government to provide better and more efficient public sport services. In addition, any rapid change in the size of the population will have a significant impact on the size of the society in that region, both in terms of social resources and environmental development.

Data Sources
This study analyzes 14 of the top 100 economic counties in Zhejiang Province as the research objects, specifically Ruian, Yueqing, Linhai, Yuhuan, Wenling, Yiwu, Zhuji, Dongyang, Ninghai, Cixi, Yuyao, Haining, Jiashan, and Tongxiang. The data on public sports service and management personnel, the amount of financial investment in public sports, and the number of social sports organizations were obtained from the field research of the sports bureaus of the "Top 100 counties" in Zhejiang Province. County sports social organizations refers to the county government departments approved by law in the Civil Affairs Bureau or registered or filed in the Sports Bureau of the non-profit nature of sports social organizations, including specifically mass sports associations, mass sports foundations, and sports private non-enterprise units. Therefore, the number of social organizations were obtained directly from the statistical database of each county and city sports bureau. The number of people who regularly participate in physical activity was obtained from the annual report statistics of each county and city government or sports bureau. Data on the total area of sports venues and per capita area of sports venues were extracted from the report of Zhejiang Institute of Sports Science. The population density data were obtained from World Pop and the statistical yearbook of each city. Gross Domestic Product (GDP) per capita was obtained from the China County Statistical Yearbook (2016-2020). Some missing data were acquired from the Zhejiang Sports Yearbook, government sports work reports, and the official website of the Sports Bureau. Please refer to Table 1 for detailed data.

Static Efficiency Analysis of Public Sports Resource Allocation Based on the Super-SBM Model
Three efficiency values, namely, technical efficiency (TE), pure technical efficiency (PTE), and scale efficiency (SE), were measured using the Super-SBM model under output orientation for the public sports resources input-output data of the top 100 economic counties in Zhejiang Province from 2016 to 2020.

Technical Efficiency (TE) Analysis
At the overall level, the mean value of TE of 14 economically strong counties in Zhejiang Province was 0.833, and the value of coefficient of variation was 0.012. These data indicate that the overall allocation efficiency of economically strong counties in Zhejiang Province was relatively stable from 2016 to 2020, and the degree of variation in the mean value of annual efficiency among counties was not significant. The highest TE occurred in 2016 (0.849), followed by 2019 (0.838). The overall trend was decreasing and then increasing, but it was still in a weakly effective DEA state at the end of the study period. The mean value of TE in 2020 was 0.81, with 57% of the counties and cities having a mean value of overall efficiency below the mean, indicating that the overall allocation efficiency of the economically strong counties in 2020 was at a moderate to high level.
At the individual level, the only three economically strong counties in Zhejiang Province with an effective DEA allocation of public sports resources in 2016 were Yiwu, Cixi, and Jiashan, accounting for only 21%. The two counties with an effective DEA allocation in 2017 were Yuhuan and Cixi, accounting for only 14%. All counties were in a state of ineffective configuration in 2018. The number of DEA effective configurations by 2019 was back to three counties again, namely, Linhai, Yiwu, and Jiashan. In 2020, the impact of Coronavirus Disease 2019 (COVID-19) and the closure of stadiums and public sports facilities to ensure the safety of people's lives caused the output indicator of the number of regular physical activity participants to drop, and the number of valid DEAs returned to one again. Detailed TE data results can be found in Table A1 and Figure A1.

Pure Technical Efficiency (PTE) Analysis
TE can be decomposed into PTE and SE under the premise of variable returns to scale, which is used to reflect the utilization rate of public sports resources and financial allocation effectiveness. At the overall level, the PTE mean value of economically strong counties was 0.936, which is close to the effective state of PTE. The percentage of areas with PTE values greater than the mean value was 57%, indicating that the public sports financial management techniques and management capabilities of economically strong county governments were better during 2016-2020. There was an overall trend of folding line fluctuation of rising-declining-rising again-declining again. From the overall standard deviation and coefficient of variation values, the disparity between regions is more obvious. The largest number of effective PTEs reached seven in 2020, which were Ruian, Yueqing, Yuhuan, Wenling, Yiwu, Zhuji, and Cixi. However, the largest standard deviation and coefficient of variation also occurred in 2020, indicating that the individual differences in PTE of the strongest economic counties in 2020 were large, and the counties with high efficiency and low efficiency kept pulling away from each other, gradually showing the polarization effect.
At the individual level, the PTE remained effective for five consecutive years in Yueqing, followed by Wenling, which was in PTE invalid status only in 2018 and remained valid in the rest of the studied years. Ninghai County had the lowest ranking (0.786), indicating that Ninghai's sports public finance management technology and public sports venue management level needed further improvement. Yiwu, Zhuji, and Haining showed a rising trend year-on-year, reaching the effective state of PTE in 2018, 2019, and 2020, respectively, indicating that the layout and implementation of the 13th Five-Year Plan for Sports Development in Zhejiang Province and the implementation of the national fitness strategy in Zhejiang Province played a certain role, making the public management organizations of the grassroots government constantly improve their public management and optimize the resource allocation structure. On the contrary, Dongyang, Jiashan, and Tongxiang showed an overall decreasing trend. Detailed PTE data results can be found in Table A2 and Figure A1.

Scale Efficiency (SE) Analysis
SE stands for the ratio of technical efficiency frontier output to optimal scale output; the larger the SE, the closer the production scale is to the optimal value. At the overall level, the mean value of SE in economically strong counties in Zhejiang Province was 0.892, which is close to the scale effective state, but lower than PTE. Therefore, it can be considered that the main factor leading to the ineffectiveness of PTE in economically strong counties is the low SE, and PTE plays a secondary role. In other words, the current institutional arrangement is not reasonable enough for the rationing of the scale of public sports resource input and output.
The SE of economic power counties in Zhejiang for 2016-2020 showed a decreasing trend. The reason is that the rapid economic development of the main counties drives the increase in financial investment in public sports resources. As the amount of input becomes larger, the output indicators of social sports organizations and the number of regular exercise participants, which reflect the output of resources, do not increase accordingly, because an organization needs time and a good institutional environment to be established. The coefficient of variation and standard deviation overall were 0.099 and 0.088, respectively, indicating that there were large scale differences between the overall DEA and that of individual regions. From the decomposition of TE, it can be seen that the mean change curve of SE is much smaller than that of PTE. Therefore, it can be concluded that the main factor leading to the ineffectiveness of the overall DEA of economically strong counties in Zhejiang Province is the problem of low PTE. To improve the efficiency of resource allocation, the most important development directions are strengthening the introduction of sports management talents in economically strong counties, improving the professional level of resource management of policy makers, and guaranteeing the implementation and acceptance of public sports policies and regulations. Detailed SE data results can be found in Table A3 and Figure A1.

Returns to Scale (RS) Analysis
RS reflects the relationship between changes in output caused by changes in the scale of public sports resource input in the Top 100 Economic Counties of China in Zhejiang Province. By the end of the study period, the Top 100 Economic Counties of China in Zhejiang Province were overall in a state of diminishing RS. If the input of public sports production resources continues to increase under decreasing RS, it will only lead to inefficient resource allocation, negative externalities, and waste. The diminishing RS is mainly caused by various factors such as the expansion of manufacturers' production scale, shortcomings in the availability of production factors and the regulation of public sports resources, the unsoundness of the market information mechanism, and the inefficient management of resources. Between 2016 and 2020, the trend of decreasing RS shows an upward-then downward-then upward trend. The highest number of unchanged RS occurred in 2016 and 2019, both at three; while there was only one increasing RS in all years (Jiashan).
At the specific level, those that had decreasing RS from 2016 to 2020 are Ruian, Yueqing, Yiwu, Zhuji, Dongyang, Ninghai, Cixi, Yuyao, and Tongxiang. The only county that featured increasing RS is Jiashan County, which indicates that the current production specialization and resource intensification in Jiashan County should continue to expand the resource input of human, financial, and material resources in public sports to maximize the production capacity and obtain a higher proportion of resource output. Yuhuan was in the same estimated RS except for 2017, in which it was in the incremental stage, and in 2018, in which it was in the decremental stage. Please refer to Table 2 for specific RS analysis data.

Super Efficiency Analysis
The efficiency values measured by the SBM model are determined by linear scales to obtain the production frontier surface of the DMU, so the efficiency of the top 100 economic counties can only be evaluated in the [0, 1] interval. Tone (2002) proposed a super-efficiency model based on slack variables based on the SBM model, which can further evaluate and compare regions with efficiency values greater than 1 [26]. For the period of 2016 to 2020, Yuhuan City had the first Super efficiency ranking, which reached the highest in 2020 at 1.227, showing a trend of increasing year by year. Ninghai County had the lowest ranking, which is lower than the provincial average value of super efficiency. Only five areas exceeded the provincial average, namely, Linhai, Yuhuan, Wenling, Haining, and Jiashan. The most unstable performance was in Linhai, with the largest coefficient of variation and standard deviation of 0.160 and 0.145, respectively. The magnitude of the super-efficiency is related to the technical stability, which is fitted to the previous analysis of TE, PTE, and SE in this paper. The larger the value of super-efficiency, the better the stability of technical efficiency. The values of the overall coefficient of variation and standard deviation of super-efficiency from the top 100 economic counties were large, indicating the significant individual differences and polarization characteristics of public sports resource allocation efficiency among the top 100 economic counties.

Input Redundancy and Inadequate Output Analysis
According to the static efficiency of resource allocation of public sports resources in the top 100 Economic Counties of China in Zhejiang Province, previously measured using DEA, more regions exist with resource allocation in the state of invalid envelope data. To further explore the path of resource allocation efficiency improvement, the slack variable values of public sports input and output resources measured in 2020 were used as an example to improve the analysis of counties with invalid DEA to reach an effective production frontier surface. Negative numbers indicate the number of inputs to be reduced and positive numbers indicate the number of inputs to be increased. The measured results are shown in Table 3.
The evaluation results of the SBM model show that 42% of the regions had inadequate input of public sports resources and 92% of the regions had inadequate output of public sports resources in 2020. Among them, the regions with redundant input of public sports service and management personnel accounted for 42.8%, and the regions with redundant input of the number of public sports input amount accounted for 42.8%. The regions with insufficient output of the number of regular sports participants accounted for 71.4%, and the regions with insufficient output of the number of social sports organizations accounted for 71.4%. The above shows that most of the non-DEA effective counties and cities failed to make full use of public sports services and management personnel, and perform public sports financial capital investment, and many input redundancies occurred. Due to the unreasonable structure of input-output factors, which leads to the prominent phenomenon of insufficient output, the top 100 economic counties urgently need to adjust the corresponding input factors of public sports resources to achieve the DEA effective expectations. Specifically, the layoff of 459 people in Yiwu's public sports services and management personnel is needed, and a reduction of 230,000 yuan in public sports investment funds. In addition, to maintain the year-by-year proportional growth of public sports venues area at the same time, the number of regular participants in physical exercise should be 30,469 people. Moreover, the number of social sports has to be increased by 38 to achieve an effective allocation of resources. To achieve the above goals, the government must relax the registration and approval of grassroots social sports organizations and conduct standardized training and management. In addition, they need to actively explore and vigorously carry out various types of folk competition programs with sports characteristics in some qualified townships, as well as increase the scale of construction of sports social organizations. At the same time, they should adopt effective training and incentive measures to improve the business level and guidance rate of social sports instructors and create effective incentives to continuously send volunteers to townships, so as to increase the awareness and enthusiasm of township residents to participate in sports activities and slow down the polarization effect of urban and rural sports atmosphere. More attention should be paid to the allocation structure of urban and rural resources, increasing the construction of mass sports and fitness facilities resources, renovating and using abandoned factories to build them into free or low-charge comprehensive sports venues, and improving the efficiency of resource utilization.

Analysis of the Overall Characteristics of Public Sports Resources in a County
Total factor productivity is an important concept in macroeconomic theory and a useful tool for analyzing economic efficiency and the trends and sources of economic growth. This parameter for public sports resources in Zhejiang Province is able to capture the root causes of changes in their productivity and trace them under the assumption of variable returns to scale. It can identify the main factors that contribute to or hinder the productivity progress or decline, and then address specific problems by constructing a series of effective public management policies or resource allocation mechanisms to promote the high-quality development and efficient allocation of public sports resources.
During the sample observation period, total factor productivity change in Zhejiang Province declined at an average annual rate of 0.3%. Based on the decomposition indicators, technical progress decreased at an average annual rate of 0.4%, while technical efficiency increased at an average annual rate of 0.8%. From the curve of development trend of total factor productivity movement index (MI), it is synchronized with the development trend of technological progress index (TC). Therefore, it can be shown that the regression of technological progress has a significant negative impact on the total factor productivity increase in public sports resources in the top 100 economic counties. The economic implication of technological progress is the increase in output caused by technological progress alone while keeping the total factor inputs of production constant, which includes advanced sports management personnel, scientific techniques of resource management, construction and planning, as well as external economic cycles and institutional changes. Therefore, if we want to encourage the growth of total factor productivity, improving the technical progress of resource management is an indispensable starting point.

Analysis of Regional Heterogeneity of Public Sports Resources
The 14 studied regions were divided according to their geographical location regional attribution, and the differences were evaluated between their regions. As shown in Table 4, the highest MI index was 1.00052 in southern Zhejiang, followed by 0.995 in central Zhejiang and 0.984 in northern Zhejiang, which are close to the total factor constant state. The TC of northern Zhejiang was 0.984 and the EC was 1.029. The growth of technical efficiency drove the growth of MI, of which SE improved at an average annual rate of 3.4% accounting for the main contributing factor. Therefore, perfecting the government's public sports resource allocation mechanism and innovating the resource allocation management capacity are the main directions to be focused on regarding southern Zhejiang.

Total Factor Productivity Index Analysis of County Public Sports Resources
As shown in Table 5, the total factor productivity of the top 100 economic counties changed more evenly in the [−6%, 5%] range. The total factor productivity index grew in 42.8% and declined in 57.2% of the studied regions. Among them, the highest was in Yuhuan, followed by Cixi, and the most serious decline in growth rate occurred in Tongxiang. Only 28% of the regions showed positive growth in the technological progress index and 72% of the regions exhibited growth in technical efficiency. Therefore, improving the technical level of resource allocation should be the trend of future reforms. Among the regions where the total factor productivity index grew, the growth of technical efficiency was the main contributing factor. Taking Yiwu as an example, the rate of technical progress was 0.5% and the growth rate of technical efficiency was 2.2%; together, they promoted the total factor productivity improvement, in which EC growth was the main contribution factor. After further decomposition of the model, it can be seen that the comprehensive efficiency improvement in Yiwu was jointly driven by the scale efficiency and pure technical efficiency, among which the scale efficiency improvement was the main contributing factor. Most of the regions achieved a slow increasing trend in overall technical efficiency during 2016-2018, while Yiwu, Jiashan, and Wenling showed a decreasing trend. After 2018 and until 2019, except Linhai that continued to improve, all counties were in a declining state during the sample period. The possible explanation for this phenomenon is that it takes time for public sports policies to be implemented and the construction of sports venues has a time lag effect. The growth in 2016-2018 was mainly due to the institutional driving effect of the national fitness strategy that led to the construction of public sports venues and arenas in various regions and reached the highest overall efficiency growth trend in 2018. Without considering the role of factors such as the scale of resource allocation, the continued input of resources continues to be redundant, which fits the results of the study of diminishing scale efficiency mentioned above. The economic implication of diminishing scale efficiency is that, as the resource input increases, the resource output declines faster than the input, suggesting that continuing to increase the input at this time will make some resources redundant.

Discussion
The results of the empirical study based on data envelopment analysis show that the comprehensive efficiency values of the top 100 economic counties in Zhejiang province are in a weak DEA effective state of medium to high. The research results indicate that the efficiency of the inputs and outputs of public sports resources is not yet fully effective and still exists to some extent unreasonably. The results of the model analysis based on Malmquist's total factor production index show that the total factor productivity of the top 100 economic counties in Zhejiang Province shows a slight downward trend of 0.3%. From the total factor productivity fluctuation curve, it is consistent with the fluctuation trend of its technological progress, so the regression of technological progress is the main source to restrain the decline of total factor productivity. The trend of total factor productivity fluctuations in the top 100 economic counties is highly compatible with the timing and characteristics of macro public sports policies enacted, thus this study points out that institutional arrangements have a significant positive impact on the efficiency of public sports resource allocation.
Since 1994, China's public sports financial resources have steadily increased, public sports organizations have been strengthened, sports human resources have developed rapidly, and the process of building stadium resources has accelerated [27]. According to Wang, although the supply of public sports services for national fitness in China has been increasing, the problem of insufficient supply of diversified public sports resources is still evident as the number of people working out continues to increase [28]. Zeng et al. (2015) [29] and Xiao et al. (2005) [30] pointed out that, from 1998 to 2013, the overall level of public sports resources allocation in China showed a decreasing development trend, with large differences in the technical efficiency of public sports resources between regions. Moreover, there is a disparity in the allocation of financial resources for public sports between urban and rural areas [31]. According to the findings of this study, until now, China still has problems such as inefficient comprehensive resource allocation, insufficient scale efficiency, unreasonable scale structure, a lacking government resource management model, and limitations of technological upgrading. Based on the data of the top 100 economic counties, it can be found that the efficiency level of public sports resources allocation is not directly proportional to the economic level and urbanization level. This finding is consistent with existing studies, such as Wang et al. (2016), who suggested that the level of public sports resources allocation did not increase with the level of urbanization, but on the contrary, the level of public sports resources allocation was low in areas with high urbanization levels [32].
Although this study explores the efficiency of public sports resource allocation in China's top 100 economic counties through five consecutive years of data and draws some meaningful conclusions, the study still has limitations. This research only evaluates the efficiency level of the top 100 counties, which can be combined in the future from the perspective of the fairness of public sports resource allocation in the top 100 counties. Fairness evaluation mainly adopts the Gini coefficient method to draw the Lorenz curve and compare the fairness between different counties and cities according to the curvature of the curve. Firstly, the above data are standardized to eliminate the influence of the dimension between indicators, and then the entropy weight method is used to calculate the weights by weighting so as to obtain the comprehensive Gini coefficient value. Finally, the size of Gini coefficients of public sports in terms of people, money, and materials between different regions based on population, economy, and geographical area distribution were calculated separately.
The allocation of public sports resources is a systemic, complex, and non-linear dynamic project. This study mostly focuses on static and linear governmental role positioning, market mechanism innovation, collaborative governance, and external structure of supply paths, but does not make an in-depth and specific analysis on the structure of specific internal resource elements. The development of public sports is the result of the coupling of multiple internal and external resource elements. Therefore, system simulation can be conducted on the basis of DEA efficiency decomposition combined with system dynamics modeling methods to draw causal feedback diagrams within the boundary of the system, and finally arrive at a combination strategy for the optimization path of public sports resources structure.
The resources available to human beings are limited and scarce. Because of this, the contradiction between the infinite demand for resources and the scarcity of resources in the development of human society has become an important issue since the beginning, and resource allocation has become an important means to solve this problem. Resource allocation is based on the scarcity of resources, and the scarcity of public sports resources provides the pre-condition for reasonable allocation of public sports resources. The study points out that, in the process of public sports resource allocation, simply by expanding the scale of investment in public sports human and financial resources, it cannot fully and effectively improve the allocation efficiency of public sports resources. In order to effectively promote the overall improvement of total factor production efficiency of public sports resources, scientific and reasonable resource allocation methods and paths should be chosen, and the supervision and management of resource allocation should be improved. This study analyzes the current situation of public sports resources allocation in counties and provides quantifiable decision-making unit improvement strategies, breaking through the traditional "experience-program" research paradigm in the field of public sports services. In addition, the results of this study provide a reference for relevant decision-making departments to improve the efficiency of county public sports resources allocation. This study finds that it is conducive to solving the problems of supply and demand of public sports resources, unbalanced and inadequate allocation of total constraints and structural constraints, and can also give full play to the effectiveness of public sports resources allocation, guarantee citizens' fair enjoyment of public welfare sports rights, and promote the equalization of basic public services.

Conclusions
This study uses the Super-SBM model and the DEA-Malmquist index model to systematically measure and analyze the efficiency of public sports resource allocation in China's top 100 economic counties (taking Zhejiang Province as an example). The results lead to the following conclusions: (1) The allocation efficiency of public sports resources in the top 100 economic counties of Zhejiang Province is high; however, it shows the characteristics of "rough and loose" allocation, the allocation structure is not reasonable, and many counties and cities have different degrees of redundant resource input or insufficient resource output.
(2) The super-efficiency gradient of public sports resources shows that Yuhuan City ranks first in super-efficient allocation, while Ruian, Linhai, Wenling, Yiwu, and Haining feature high-efficiency allocation, and other regions are in the medium-and low-efficiency allocation.
(3) The improvement of total factor productivity relies on the catch-up feature of technical efficiency on the production frontier, but it still cannot compensate for the negative effect of the decline of technical progress, making the total factor productivity increase annually by 0.1%.
(4) The level of economic development of a county has a highly significant positive effect on the allocation efficiency of public sports resources, while the area of sports venues per capita exerts a highly significant negative effect on the allocation efficiency. The population density of the county has a highly significant effect, but regional factors do not have a significant effect on the allocation efficiency.
From the perspective of increasing returns to scale, although increasing the investment of public sports resources has certain benefits, the unreasonable allocation of resources can cause waste and reduce efficiency. Based on the results of the empirical study and the actual situation of the top 100 economic counties in Zhejiang Province, the following recommendations are made: (1) The public sector should establish the sports development concept of serviceoriented government, clarify the consciousness of democracy, service, and responsibility, and aim to maximize the public interest in the supply of public sports resources.
(2) The total supply and quality of public sports resources in the county should be improved, the allocation structure of factor resources should be adjusted, and the regional spatial layout of resources should be reasonably planned.
(3) New types of management talents and innovative market-based resource supply models should be introduced.
(4) A series of institutional arrangements should be constructed, such as the public interest expression platform, political participation, and the political knowledge platform, to realize intensive allocation of resources, in order to effectively improve the allocation efficiency of public sports resources in the county.
Although this study explored the efficiency of public sports resource allocation in the Top 100 Economic Counties in China through five consecutive years of data and came up with some meaningful findings, future studies will need to construct TOBIT fixed-effects regression models to deeply examine the factors and degree of influence of economic, population, sports venue allocation scale, and regional factors on the efficiency of public sports resource allocation in the Top 100 Economic Counties of China.  Data Availability Statement: Datasets generated and analyzed during this current study are not publicly available due to participant confidentiality but are available from the corresponding author on reasonable request.