Nash Bargaining Game Enhanced Global Malmquist Productivity Index for Cross-Productivity Index
[ad_1]
1. Introduction
Data Envelopment Analysis (DEA) has been introduced as a tool for evaluating the performance of homogeneous and similar Decision-Making Units (DMUs). Since then, numerous theoretical and practical developments have been presented for this mathematical programming-based technique, and various topics have been discussed in its scope. One of these topics in the context of evaluating the performance of DMUs is the development of performance appraisal indexes over time, such as the Malmquist, Laspeyres, Paasche, Fisher, Tornqvist, and Hicks-Moorsteen [1]. Among these indexes, the Malmquist Productivity Index (MPI) has received considerable attention and has been the subject of numerous studies in this area.
The MPI, proposed by Caves et al. [2], is based on the first-period technology for performance appraisal. By using the fixed period as the fundamental technology, the MPI maintains certain desirable properties such as consistency in calculating the MPI due to the same basis, transitivity, and circularity. However, the choice of the base time period is arbitrary, and other time periods are not taken into consideration. To address this issue, various suggestions have been proposed, such as using the geometric mean of the MPIs obtained by considering both time periods as the basis [3], and using global technology.
The concept of global technology as the benchmark technology for evaluating the performance of DMUs over time was first added to the MPI concepts at DEA by [4]. The use of this technology as a set of all time versions of all DMUs led to the introduction of the Global Malmquist Productivity Index (GMPI). This index has features such as feasibility, independence of results from different technologies, circularity, and consistency. Consistency means that the basis for calculating the index is the same for different DMUs, which makes the DMUs comparable based on the scores. But the consistency of [4]’s GMPI was only linked to the technology used. The authors of [5] pointed out a kind of inconsistency in this index, which is the possibility of using different weights to calculate the GMPIs of different DMUs. Geometrically, not only different DMUs, but even different time versions of a DMU, can use a different hyperplane of the global technology as the benchmark. To address such inconsistencies, ref. [5] proposed a method involving a Common Weight (CW) for all DMUs across all times, aiming to mitigate disparities arising from varied technologies and support hyperplanes. However, this method showed limitations, lacking peer evaluation and being heavily reliant on weight selection.
Cross-evaluation can also be chosen as an intermediary solution for dynamic evaluation of DMUs: between full weight freedom in [4], and full weight limitation of all DMUs to a CW in [5]. Ding et al. [6] used the concept of cross-evaluation in calculating MPI. However, this applies exclusively to global technology, and cross-evaluation is utilized solely for identifying efficiencies, not productivity indexes. In the conventional MPI formula, only Cross-Efficiency (CE) scores replace the efficiency scores of conventional DEA models. Additionally, the weights used in evaluations are not common, introducing a challenge of inconsistency. In addition, the function used to aggregate performance is of an arithmetic type that is not compatible with MPI studies in the use of geometric functions. Homayoni et al. [7] proposed a cross-productivity index to address the issues identified in [6], where the weights from self-evaluation and peer evaluation were used directly in the productivity index calculation. However, the problem of inconsistency still exists, as the efficiencies in their MPI ratio are calculated based on different weights.
In this paper, we introduce a methodology for calculating GMPIs that addresses the limitations of previous approaches. We consider the global technology set, comprised of all DMUs in two different time periods, as the benchmark. To ensure fair evaluation and account for DMUs’ preferred weights, we suggest using several CWs for self and peer evaluation, instead of relying on just one CW. For both time versions of a DMU, we obtain a CW and use it to calculate the GMPI of that DMU and others.
Due to the competitive nature of the DEA evaluation process, we employ the Nash bargaining game model with suitable breakdown points to establish fair CWs. This cooperative model is formulated to simultaneously maximize the utility of all participants. Additionally, it results in a Pareto-optimal solution, which motivates players to accept it. We create a cross-evaluation matrix containing values of self and peer productivity and aggregate the GMPIs using the geometric mean to maintain the multiplicative structure at the aggregate level. This approach overcomes the inconsistency issue of previous methods and provides a more acceptable evaluation of DMUs. Furthermore, our method accounts for DMUs’ preferences and ensures fairness in the evaluation process.
After examining the literature on CW and MPI in DEA, it is apparent that no attempt has been made to obtain a CW for the time versions of a DMU. Furthermore, no study has yet calculated a GMPI matrix. A review of the literature also suggests that game theory has not been used to generate productivity indexes in DEA.
The proposed integrated framework of GMPI, CW, Nash bargaining, and cross-evaluation allows the constructed GMPI to possess certain properties. The use of a CW in calculating the efficiencies of time versions of a DMU ensures that the GMPI fraction is well defined and consistent across all DMUs. The use of global technology ensures that the models used to estimate efficiencies are always feasible [4]. Additionally, both self-evaluation and peer evaluation are considered, resulting in an aggregated GMPI that inherits the desired cross-evaluation features. This includes reducing the dependence of results on a particular weight, increasing stability and reality, avoiding overestimation, being fair due to the use of desired weights of each unit, and reflecting reality more accurately.
The paper is structured as follows. The second section offers an overview of essential background information. In the third section, we introduce the proposed method along with its properties, and the fourth section presents an examination of the method through a numerical example. Finally, the last section presents our conclusions and suggestions for future research.
4. Application of Proposed Method to the Example
In this section, we delve into a comprehensive numerical example using the dataset previously introduced in the motivation section. Our aim is to showcase the functionality of our proposed integrated approach in computing the GMPI in detail. The primary focus here is to elucidate the methodology introduced in this study, outlining its intricacies, steps, and distinctive features. Employing the dataset, we provide a detailed, step-by-step breakdown of how our method is applied and its functionalities, emphasizing its unique attributes and advantages. Towards the end, we conduct a brief comparison between our method and existing methodologies, specifically the GMPI and CWGMPI, to highlight the distinct advantages observed in our proposed method through the numerical illustration.
Describing the proposed method involves revisiting the numerical example outlined in Section 3, and focusing on DMU C. The maximum efficiency of C, using model (5), is 0.571429. With this efficiency held constant for DMU C, we use model (7) to obtain the minimum efficiency for C’, which is the bargaining breakdown point for C’, at 0.666667. Similarly, considering as the leader DMU, and the unit as the follower DMU, and solving model (8), the breakdown value for the C unit will be 0.55. By employing these values in bargaining model (10), the optimal CW between DMUs C and C’ is (v1*, v2*, u*) = (0.34476397, 0.24142703, 0.56035676), and the value of the game is 0.0003337. Using this common weight, the efficiencies of DMUs C and C’ are 0.560357 and 0.571429, respectively. The SGMPI for C is equal to (0.698888)/(0.560357) =1.247219, which indicates progress in the performance of unit C from time t to t + 1. Similarly, we can obtain the CW and SGMPI values for all DMUs (A,A’), (B,B’), …, (H,H’). Table 7 shows the minimum and maximum efficiency values of the models (7)/(8) and (5).
Table 8 displays the outcomes of the implementation of the bargaining model (10). The optimal values of CW are presented in columns 2 to 4, while the fifth column shows the value of the game.
We utilized the weights obtained earlier to evaluate the productivity of other DMUs and determine their corresponding productivity ratios. Table 9 presents the Cross- GMPI matrix. As mentioned before, each row exhibits the rating values of the other DMUs according to the weights linked with the DMUs listed in that respective row. For instance, in the second row, the corresponding line of A and A’, the values for the other DMUs are derived from the CW of the bargaining game between A and A’. The three numbers in each cell denote the efficiencies in time 1, time 2, and the performance change index (efficiency of time 2/efficiency of time 1) from top to bottom, respectively. The CGMPI values are listed in the last row of Table 9.
Based on the CGMPI values in the last row of Table 9, DMUs B, G, and H have experienced regression in productivity, while other units show improvement, as can be seen in Figure 1 and Figure 2.
Upon careful examination of the game values, it has been observed that for DMUs whose lower efficiency is 1, the value of the game will always be zero. In such cases, we can avoid the computation of the game model (10) by considering the optimal weight of the other unit as a common weight. For instance, let us consider DMUs A and A’. Since A’ is located on the global frontier, its optimal weight is (0.61538462, 0.30769231, 0.84615385) (line segment parallel to the vertical axis), and the optimal weight of A is (0.61538462, 0.30769231, 0.84615385) (line segment A’B). As the optimal value of A’ does not change, we can use the weight vector related to A as the common weight for both DMUs. This reduces the computational burden significantly.
In what follows, we compare the proposed Cross-GMPI method with two other methods, namely the GMPI method proposed by [4] and the CWGMPI method proposed by [5]. The reason for this choice is that these two indexes are based on the consideration of global technology.
First, we consider the GMPI model. In this method, each of the two-time versions of the 8 DMUs are compared to the global boundary separately, and as a result, 16 separate weights are obtained. Table 10 shows these weights. In the last column, the value of GMPI is given. Since the output orientation of the CCR model (1) is used in the evaluation of units relative to the frontier, and considering that the output of all units is equal to the constant number one, then the optimal value of the objective function of the CCR model is equal to the optimal weight of U*. So, the numbers in the fourth column of Table 10, in addition to the optimal value for the output weight of the units at any time, indicate the efficiency value of the DMU of that row. For example, the efficiency of unit A in the first time is equal to 0.846154, and in the second time is equal to 1. To calculate GMPI, the division of two efficiencies is used. The optimal weights corresponding to these two efficiency vectors are (0.615385, 0.307692, 0.846154) and (4, 0, 1), respectively. As can be seen, the weight vectors are not the same. As a result, two efficiency values are divided for calculating the GMPI, which are based on different weights. Apart from this, the index calculated for the units is based on self-evaluation strategy, not peer evaluation. In the proposed cross-MPI method, instead of 16 different weight vectors, eight weight vectors were used (Table 8); that is, for both time versions of a DMU, a common weight vector is used, so the basis for calculating the efficiency of two DMUs is the same. Also, these vectors were used to calculate the efficiencies, and as a result the GMPI, of other units in the form of peer evaluation.
Now we consider the CWGMPI model. If we use this model to evaluate the 8 DMUs of the numerical example, only one common weight vector is generated for all units, and all the efficiencies used to calculate the CWGMPI are based on this one vector. In Table 11, GMPI values based on this vector for units are given. As you can see, not only is this index not based on peer evaluation, but it is not even based on self-evaluation, and all calculations and rankings depend on this common weight.
Now let us review the CGMPI values obtained by the proposed method which can be seen in Table 9. For a better visual understanding, we use a radar chart (Figure 3).
In a radar chart, the different levels, or concentric polygons (from inside to outside), represent the values or scales of the dimensions being measured. The distance of a point from the center of the chart to a specific level on each axis corresponds to the value of the variable for that particular entity. The interpretation of levels in the radar chart based on the data provided enables a quick assessment of the relative performance of DMUs (A to H) across multiple weights (Common weights between DMUs A,A’ to H,H’ in Table 7). DMUs A, B, G, and H seem to have nearly identical performance across all weights. They maintain consistent values across the radar chart, indicating similar strengths and weaknesses in the measured weights. DMUs C, D, E, and F exhibit more variation in their performance across different weights. The third numbers in the columns of Table 8 are the vertices of the octagons of Figure 3. When one level is inside another level, it suggests that the DMU corresponding to the outer level performs better compared to the DMU associated with the inner level. Briefly, we notice that DMU B is at the innermost level of the graph, and this means that this unit undoubtedly has the least change in performance compared to other units. After that unit, G and H units are located. On the other hand, units E and C are the largest octagons, i.e., polygons with the greatest distance from the center of the radar. These results are consistent with the CGMPI values obtained by the proposed method in the last row of Table 8. Therefore, it can be inferred that the conclusion regarding the state of change in the performance of the units is based on an average value as represented by an octagon. It is obvious that the inference and analysis of the situation based on such a process is much more reliable than the analysis based on only one of its values.
The proposed integrated framework of GMPI, CW, Nash bargaining, and cross-evaluation allows the constructed GMPI to possess certain properties. The use of a CW in calculating the efficiencies of time versions of a DMU ensures that the GMPI fraction is well defined and consistent across all DMUs. This means that the basis of comparison is the same for performance appraisal. That is, not only is the efficiency used in calculating the Malmquist index based on common boundaries, but also on common weights or common supporting hyperplanes of boundaries [5]. Since a common weight is employed in the computation of productivity indexes for all units each time, the base for comparison remains consistent, ensuring the consistency and comparability of results.
The Malmquist index measures productivity change by comparing a unit’s efficiency at time t + 1 to time t. In DEA, these efficiencies are relative and reliant on a frontier comparison. Malmquist indexes are categorized based on single or multiple technology use: reference-based (using only one technology, e.g., time t or global technology) or adjacent-based (employing more than one time technology). Adjacent-based indexes may encounter infeasibility issues due to some units’ super-efficiency in one period relative to another. Employing global technology ensures feasibility in efficiency estimation models [4]. By utilizing global technology, our proposed index avoids potential inefficiency concerns in efficiency calculations.
DEA’s notable functionality lies in its ability to assist each DMU in selecting the most advantageous weights or multipliers for inputs and outputs during efficiency calculations. However, traditional DEA models, particularly in measuring MPI, tend to overstate efficiency due to the flexibility in weight selection for inputs and outputs. Consequently, this overestimation affects the rationality of derived MPI values. Various strategies address this issue, notably the common set of weights and the cross-evaluation process. Cross-evaluation, a significant component of DEA theory, surpasses the common set of weight method in applications [111]. It presents clear benefits, including personalized DMU ordering and the avoidance of unrealistic weight schemes without necessitating expert weight restrictions [112].
Utilizing a self-evaluation and peer evaluation methodology akin to cross-evaluation to compute cross GMPI, our approach yields realistic values for judgment. Cross GMPI, derived from the mean of productivity scores, represents an average productivity change, ensuring stability and reliability in assessment.
5. Conclusions
This study presents an innovative, integrated approach to compute the MPI through a combination of global technology, Common Weights, Nash bargaining, and cross-evaluation techniques. Our proposition advocates for obtaining CWs for two-time versions of a DMU based on global technology, facilitating efficiency calculations in tandem with a singular frontier, and ensuring consistency and comparability in results across different evaluations.
By employing the Nash bargaining game model, these CWs satisfy desirable properties, ensuring incentive alignment among DMUs and yielding Pareto-efficient solutions. Additionally, the novelty of incorporating CWs for peer and cross-sectional evaluations stands out as a novel contribution, enhancing the method’s robustness and applicability.
This integrated framework yields a proposed index boasting several desirable properties: feasibility, non-arbitrariness in base time period selection, technological and weight consistency, result stability and reliability, and equitable assessments. The comparison of our Cross-GMPI method with existing models, namely the GMPI and CWGMPI, underscores its superiority in offering a holistic evaluation while maintaining fairness and reliability across evaluations.
The numerical example showcases the effectiveness of our proposed method. Utilizing radar charts for visual representation allows for a swift yet comprehensive assessment of DMU performance across various weights. This visual depiction highlights consistent performances among certain DMUs and variations in others, providing nuanced insights into their relative strengths and weaknesses.
In essence, our integrated approach not only resolves efficiency overestimation concerns prevalent in traditional DEA models, but also ensures a more equitable and reliable assessment. Its distinct advantages, such as avoidance of unrealistic weight schemes, and enhanced stability, underline its superiority in performance evaluation.
The present study could be improved or expanded in several ways. This research discussed a methodology for determining a CW vector of DEA based on the Nash bargaining solution. In general, the resulting Nash bargaining game model is non-linear, given the nature of ratio forms of DEA efficiency. Converting the nonlinear model into a parametric linear programming problem with one parameter whose lower and upper bounds can be determined, and using a heuristic search on the single parameter employing the Kalai-Smorodinsky solution (which is the unique allocation rule for two-player bargaining problems with linear programming), or using Egoist’s dilemma for this purpose, could be considered for future research. Developing the present study for more than two time periods using the Extended Nash bargaining game, using more advanced and newer indexes in the literature to calculate cross-sectional MPI values, and considering the effects of return to scale and of internal structures for DMUs, could also be guidelines for future research. It is recommended that studying the effects of common inputs, merging and decomposition of DMUs, and decision-making in a centralized scenario and in uncertainty conditions, be considered for future research.
[ad_2]
