Game Perspectives of DEA Models and Their Duals

Abstract

We present a series of two-person games, which lead to various DEA models. The relationship between the DEA models and the games is explicit in our setting, although the Nash equilibrium solutions do not generally exist. Besides the classic DEA models, we also establish an explicit relationship between the games and some extended DEA models, such as free disposable hull DEA models and “negative” or “inverted” DEA models.

1. Introduction

DEA has widely been used in performance evaluation or productivity evaluation. The DEA models have been derived from different perspectives like econometrics and operations research; see for example, Charnes et al. [1], Banker et al. [2], and Liu et al. [3] for the details. Also, several researchers found that there are close connections between DEA and the game theory [4]. As a pioneer, Banker et al. [5, 6] explored the connections between DEA and the game theory. In their work, they first built the following payoff matrix: 𝑒_𝑖𝑗=𝑦₀/𝑥_𝑖0𝑦_𝑗/𝑥_𝑖𝑗. (1)

In their formulation, the DMU₀ being evaluated acts as player 1, who wishes to maximize its payoff and has pure strategies defined by the selection of some inputs{𝑖}. On the other hand, player 2 acts as a “central evaluator,” who wishes to minimize the payoff of player 1 and has pure strategies defined by the selection of a competitive DMU{𝑗}. Then, the authors used a mixed strategy formed from the pure strategies and computed an expected payoff function constructed from the finite payoff matrix above. They formulated the game model as the pair of linear programming problems shown in Table 1.

Table 1 

After several transformations of the problem (*), the authors obtained the “CCR ratio form,” which is also the maximal expected gains that DMU₀ can obtain from the corresponding payoff matrix. However, the game model developed by Banker [5] requires that the output in the model is a number. Banker et al. [6] extended their initial work to include multiple outputs, but they had to replace the original CCR model with the BCC model, which makes it possible to write the linear programming problem for player 2 directly.

To establish a rigorous connection between the CCR model and the game theory, Rousseau and Semple [7] developed a class of two-person ratio efficiency games, which have ratio payoff functions solved by two equivalent primal and dual linear programming problems. The authors argued that “the richest information is available from the primal-dual linear programs: one side emphasizes “envelopment” multipliers, the other emphasizes input-output multipliers.” And the most general form of the games is described in Table 2.

Table 2 

The DMU₀, the unit being evaluated, acts as player 1. And the intention of player 1 is to choose some optimal weights(𝑢^∗,𝑣^∗)in𝑃 and minimize the fractional payoff of player 2. Meanwhile, Player 2 acts as a regulator who will select some𝜆in𝑄to construct a virtual “aggregate competitor” and maximize its worst case payoff under all the possible weights chosen by player 2. The authors proved that(𝑢^∗,𝑣^∗,𝜆^∗)constitutes a saddle point for the ratio payoff function in (#) and (##). However, the games (#) and (##) cannot derive CCR model directly and only can yield the information necessary to conduct the CCR efficiency analysis.

To incorporate value judgment, Semple [8] extended the “two-person ratio efficiency game” by including polyhedral cone constraints in the pure strategies played by each player. And the connection between the game theory and DEA has been extended to include the cone ratio models (see [9, 10]).

In another way, Hao et al. [11] followed the line of Banker’s work, which needs to use the payoff matrices and mixed strategies, but extended the two-person zero-sum game to the closed convex cone constraints. Then, the authors proved the connection between the proposed convex cone constrained game model and the generalized DEA model.

In this paper, we will present a game perspective of DEA model using a different payoff function. We will show that many DEA models, including the CCR, BCC, and other models, can be derived from a series of min-max problems. We also show that some max-min problems will derive new DEA models.

Similarly, we assume that there are two players in our game. One player is the DMU₀ being evaluated and the other player is a central evaluator. In this game, DMU₀ wishes to maximize (or minimize) its gain (or loss). On the contrary, player 2 wants to minimize (or maximize) its loss (or gain). Corresponding to the players, there are two pure strategy spaces in the game. One is the weight space𝑊in the multiplier formulation of DEA and the other one is the index{𝑗}, which selects a competitiveDMU_𝑗. Clearly, such two strategy spaces are not “symmetric” in the sense that the second is smaller and thus has a weaker influence on the game results. In order to have a fairer game, one remedy is to enlarge it. For example, one can expand it into selections of all virtual DMUs. That is precisely the strategy space in Semple [8]. However, the connections between DEA models and the min-max or max-min programming become less clear and more complicated. In the following, we explain another idea to handle this issue.

For any game,max_𝑦∈𝑌min_𝑥∈𝑋𝜙(𝑥,𝑦)≤min_𝑥∈𝑋max_𝑦∈𝑌𝜙(𝑥,𝑦)always holds, where𝜙(𝑥,𝑦)is the loss function of player 1 and𝑋,𝑌are the pure strategy spaces of player 1 and player 2, respectively. Whenmax_𝑦∈𝑌min_𝑥∈𝑋𝜙(𝑥,𝑦)=min_𝑥∈𝑋max_𝑦∈𝑌𝜙(𝑥,𝑦), we say that in this game there exists at least one Nash equilibrium. The existence of the Nash equilibriums shows the min-max/max-min that is an optimal strategy for both players, and the game may have no solution if the Nash equilibriums do not exist. However, the min-max strategy is the safest strategy for player 1 if player 2 has the right to terminate the game (so it has the right to select the strategies for the final step of the game), and the max-min strategy is the safest strategy for player 2 if player 1 has the right. In our case, Nash equilibriums generally do not exist since the second strategy space is too small. We compensate this by assuming that the player whose strategy space is the second has the right to terminate the game. Thus, in the present situation, the min-max strategies should be the safest for player 1. The advantage of this approach seems is that many different DEA models can be directly connected to the safest strategies of the player.

The paper is organized as follows. To illustrate the main ideas clearly, Section 2 presents the min-max and max-min formulations for DEA models with index data at first. In Section 3, we discuss the general case of the min-max and max-min formulations for the CCR, BCC, and other DEA models, and the conclusions and discussions are given in Section 4.

2. A Game Perspective of Index DEA Models

We start our investigation from a class of simpler DEA models, where the inputs are assumed to be standardized, or DEA models without inputs. This class of DEA models in fact uses index data and therefore is referred to as index DEA models; see Lovell and Pastor [12], Halkos and Salamouris [13], and Liu et al. [3] for more details.

2.1. Min-Max Problems of Index DEA Models

In this paper, a pure strategy of weights is denoted by a vector 𝐮=(𝑢₁,…,𝑢_𝑠)^𝑡in a weight space𝑊. Now, let us consider the following case: the pure strategy of DMU₀ is the weight space. This means that DMU₀ can freely select weights from the weight space to account its weighed output score∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0. Due to the conflicting interests, the evaluator may not be happy to use these weights for accounting their scores. Here, we assume that they can select aDMU_𝑗from competitive DMUs’ space𝐽={1,…,𝑛}to increase the relative loss function of∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗/∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0to force DMU₀ to change its strategies. Obviously, the larger is∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗/∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0, the worse is the relative score of DMU₀.

If the DMU₀ is risk aversion, then the DMU₀ will consider the possible worst case; that is, player 2 will choose best practiceDMU_𝑗 to maximize the performance gap with DMU₀. Then the DMU₀ will choose a weight that will minimize the gap with all the possible competitiveDMU_𝑗. Under this circumstance, DMU₀ will choose the “min-max” strategy. In the following mathematical formulation, we will use the normalization{𝑢∣∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0=1,𝑢_𝑟≥0}, thus,∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗/∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0can be simplified as∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗, and then the min-max strategy illustrated above can be written as follows: min_𝑢∈𝑊max^𝑠_{𝑗∈𝐽𝑟=1}𝑢_𝑟𝑦_𝑟𝑗, (2) where𝑊={𝑢∣∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0=1,𝑢_𝑟≥0}and𝐽={1,…,𝑛}.

For the purpose of analysis, we introduce an auxiliary variable𝑣such that𝑣=Max_{𝑗=1,…,𝑛}{∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗}. Then, model (2) may be rewritten as Min_𝑢𝑣subjectto∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗≤𝑣,_𝑠𝑟∑₌₁𝑢_𝑟𝑦_𝑟0𝑢=1,_𝑟≥0,𝑗=1,…,𝑛. (3) And the optimal value𝑣^∗is the worst possible relative score of DMU₀. Now, we consider its dual model, which will be shown to be the following index DEA model: Max𝜃subjectto∑^𝑛_𝑗=1𝑌_𝑗𝜆_𝑗≥𝜃𝑦₀,_𝑛𝑗∑₌₁𝜆_𝑗𝜆=1,_𝑗≥0,𝑗=1,…,𝑛. (4) Model (4) is first introduced by Lovell and Pastor [12].

Proposition 1. Model (4) is the dual model of model (3).

Proof. Model (3) can be rewritten as follows: Min𝑣subjectto𝑌^𝑇𝑢−𝑣𝐼≤0,−𝑌^𝑇₀𝑢+1=0,𝑢≥0, (5) where𝑌₀=(𝑦₁₀,…,𝑦_𝑠0)^𝑇,𝐘=(𝑌₁,…,𝑌_𝑛),𝐮=(𝑢₁,…,𝑢_𝑠)^𝑇, and𝐼=(1,…,1)^𝑇. The Lagrange function of model (5) is𝐿(𝑢,𝑣,𝜃,𝜆)=𝑣+𝜆^𝑇(𝑌^𝑇𝑢−𝑣𝐼)+𝜃(−𝑌^𝑇₀𝑢+1), and its dual function is 𝑞(𝜃,𝜆)=inf_{𝜇≥0,𝑣}𝐿𝜇,𝜆=inf_{𝜇≥0,𝑣}𝜃+𝜆^𝑇𝑌^𝑇−𝜃𝑌^𝑇₀𝑢+−𝜆^𝑇𝑣=𝐼+1𝜃,if−𝜃𝑌^𝑇₀+𝜆^𝑇𝑌^𝑇≥0,−𝜆^𝑇𝐼+1=0,−∞,if−𝜃𝑌^𝑇₀+𝜆^𝑇𝑌^𝑇≤0,−𝜆^𝑇𝐼+1≠0. (6) Thus, the dual problem is Max𝑞(𝜃,𝜆)Subjectto𝜆≥0, (7) which reads Max𝜃subjectto∑^𝑛_𝑗=1𝑌_𝑗𝜆_𝑗≥𝜃𝑌₀,_𝑛𝑗∑₌₁𝜆_𝑗𝜆=1,_𝑗≥0,𝑗=1,…,𝑛. (8)
Hence, model (4) is the dual of model (3).
Let us investigate model (3) further. Let𝜇_𝑟=𝑢_𝑟/𝑣, and then model (3) can be easily changed to the following model: Max∑^𝑠_𝑟=1𝜇_𝑟𝑦_𝑟0subjectto∑^𝑠_𝑟=1𝜇_𝑟𝑦_𝑟𝑗𝜇≤1,_𝑟≥0,𝑗=1,…,𝑛. (9) Now, we can understand it by intuition: model (9) (therefore (4)) is simply an index DEA model, which selects weights to maximize its weighted score like a classical DEA model. Thus, the min-max strategy just leads to index DEA models.

2.2. Max-Min Problem of Index DEA

As discussed before, the min-max strategy leads to index DEA models. Now, we consider the “max-min” strategy as follows: Max_𝑗∈𝐽Min^𝑠_{𝑢∈𝑊𝑟=1}𝑢_𝑟𝑦_𝑟𝑗, (10) where𝑊={𝑢∣∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0=1,𝑢_𝑟≥0}and𝐽={1,…,𝑛}. We can calculate this model through the following two-stage algorithm.

First Stage. We can divide model (10) into𝑛separate models as follows: ℎ^∗_𝑗=Min∑_𝑢^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗subjectto∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0𝑢=1,_𝑟≥0. (11) Letℎ^∗_𝑗be the optimal value of (11). Then, we obtain all the infimum{ℎ^∗₁,ℎ^∗₂,…,ℎ^∗_𝑛}.

Second Stage.  Let ℎ^∗=max_{𝑗=1,…,𝑛}{ℎ^∗_𝑗}.

From the min-max theory, it always holds that Max_{𝑗=1,…,𝑛}Min_𝑢∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗≤Min_𝑢Max_{𝑗=1,…,𝑛}∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗, where∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0=1and𝑢≥0. And ifMax_{𝑗=1,…,𝑛}Min_𝑢∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗=Min_𝑢Max_{𝑗=1,…,𝑛}∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗, we can conclude that this game has a Nash equilibrium solution. Unfortunately, Nash equilibrium solution does not always exist actually unless DMU₀ only has one reference peer in model (4). Let us consider the dual model of (11), which can be written as Max𝜃_𝑗subjectto𝑌_𝑗≥𝜃_𝑗𝑌₀. (12) And considering𝜃^∗=max_{𝑗=1,…,𝑛}{𝜃^∗_𝑗}, we can write its dual as Max𝜃subjectto∑^𝑛_𝑗=1𝑌_𝑗𝜆_𝑗≥𝜃𝑌₀,_𝑛𝑗∑₌₁𝜆_𝑗𝜆=1,_𝑗=1or0,𝑗=1,…,𝑛, (13) which is a free disposable hull DEA model without inputs. Thus, in general, there is no Nash equilibrium solution for this game formulation. If the DMU₀ had the right to terminate the game, assumed to be risk aversion, the central evaluator would adopt the max-min strategy in our situation, which derives a FDH DEA model.

2.3. Max-Min Problem of a Different Game

Now, we consider a different game, where DMU₀ and the central evaluator exchange their pure strategy spaces. That is DMU₀ can control the variable𝑗, and the evaluator can control the weight𝑢. This implies that DMU₀ itself can select the worst practiceDMU_𝑗as its benchmark freely. On the contrary, the central evaluator would choose the best weights. As explained in Section 1, the safest strategy for the evaluator is the max-min strategy, which can be modeled as follows: Max_𝑢∈𝑊Min^𝑠_{𝑗∈𝐽𝑟=1}𝑢_𝑟𝑦_𝑟𝑗, (14) where𝑊={𝑢∣∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0=1,𝑢_𝑟≥0}and𝐽={1,…,𝑛}. By introducing an auxiliary variable𝑣=Min_{𝑗=1,…,𝑛}{∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗}, the model above will be transformed to Max_𝑢𝑟𝑣subjectto∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗≥𝑣,_𝑠𝑟∑₌₁𝑢_𝑟𝑦_𝑟0𝑢=1,_𝑟≥0,𝑗=1,…,𝑛. (15) And its dual model is as follows: Min𝜃subjectto∑^𝑛_𝑗=1𝑌_𝑗𝜆_𝑗≤𝜃𝑌₀,_𝑛𝑗∑₌₁𝜆_𝑗𝜆=1,_𝑗≥0,𝑗=1,…,𝑛. (16) Then, its meaning is clearer. Model (16) compares DMU₀ with the worst DMU, instead of comparing it with the best one as in the classic DEA. This idea was first used by Takamura and Tone [14] and Yamada et al. [15]. Using transformations𝜇_𝑟=𝑢_𝑟/𝑣, then model (15) is changed to Min∑_𝑢𝑟^𝑠_𝑟=1𝜇_𝑟𝑦_𝑟0subjectto∑^𝑠_𝑟=1𝜇_𝑟𝑦_𝑟𝑗𝑢=1,_𝑟≥0,𝑗=1,…,𝑛. (17)

Then, this idea is more clearly illustrated. For the purpose of relocating several government agencies out of Tokyo, Takamura and Tone [14] used both models (9) and (17) with weight restrictions to deal with the problem of site selection for that project. They argued that “Each site is compared with these worst performers and is gauged by its efficiency “negatives” as the ratio of distances from the “worst” frontiers in the same way as in ordinary DEA.” Yamada et al. [15] named this worst side approach “Inverted DEA.”

Similarly, the min-max strategy will lead to a free-disposal hull inverted DEA model.

3. Min-Max/Max-Min Problem of DEA Models

3.1. Min-Max/Max-Min Problem of CCR

After exploring the game perspective of DEA models with index data, we turn to the classical input-output DEA formulation. We first consider the same game situation set in Section 2.1 and use a similar relative loss function as before. Then, the min-max problem reads Min_{𝑢,𝑣∈𝑊}Max_𝑗∈𝐽∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗/∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0/∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖0, (18) where𝑊={𝑢∣∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0=1,𝑢≥0}∪{𝑣≥0}and𝐽={1,…,𝑛}.

Let𝑡=max_𝑗{∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗/∑^𝑠_𝑟=1𝑣_𝑖𝑥_𝑖𝑗}. Then, we can transform the model above to the following model: Min_𝑢,𝑣𝑡∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖0∑subjectto^𝑠𝑟=1𝑢_𝑟𝑦_𝑟𝑗∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗≤𝑡,_𝑠𝑟∑₌₁𝑢_𝑟𝑦_𝑟0=1,𝑢≥0,𝑣≥0,𝑗=1,…,𝑛. (19) Further, let𝑣_𝑖=𝑣_𝑖𝑡, and then the model is further transformed to Min^𝑚_𝑢,𝑣∑_𝑖=1𝑣_𝑖𝑥_𝑖0subjectto∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗−∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗≤0,_𝑠𝑟∑₌₁𝑢_𝑟𝑦_𝑟0=1,𝑢≥0,𝑣≥0,𝑗=1,…,𝑛. (20) This is the standard output-oriented CCR DEA model. If we replace the normalization∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0=1with∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖0=1, then after several similar transformations as above, we will derive the input-oriented CCR model.

Similarly, we can examine the max-min strategy as follows: Max_𝑗∈𝐽Min_{𝑢,𝑣∈𝑊}∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗/∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0/∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖0, (21) where𝑊={𝑢∣∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0=1,𝑢≥0}∪{𝑣≥0} and 𝐽={1,…,𝑛}.We again calculate the model through the following two-stage algorithm.

First Stage. We divide model (21) into𝑛separate models as follows: ℎ^∗_𝑗=Min^𝑚_𝑢,𝑣∑_𝑖=1𝑣_𝑖𝑥_𝑖0subjectto∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗−∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗≤0,_𝑠𝑟∑₌₁𝑢_𝑟𝑦_𝑟0=1,𝑢≥0,𝑣≥0. (22) And letℎ^∗_𝑗be the optimal value of model (22). Then, we obtain all the infimum{ℎ^∗₁,ℎ^∗₂,…,ℎ^∗_𝑛}.

Second Stage. Letℎ^∗=max_{𝑗=1,…,𝑛}{ℎ^∗_𝑗}.

Then, we consider the dual model of (22) as before, which can be written as Max𝜃_𝑗subjectto𝑋_𝑗𝜆_𝑗≤𝑋₀,𝑌_𝑗𝜆_𝑗≥𝜃_𝑗𝑌₀,𝜆_𝑗≥0. (23) And we know𝜃^∗=max_{𝑗=1,…,𝑛}{𝜃^∗_𝑗}. Then, the programming problem above can be transformed further toMax𝜃subjectto∑^𝑛_𝑗=1𝑋_𝑗𝜆_𝑗≤𝑋₀,_𝑛𝑗∑₌₁𝑌_𝑗𝜆_𝑗≥𝜃𝑌₀,𝜆_𝑗=0oranyotherpositiverealnumber.Andfor𝑗=1,…,𝑛,thereisonlyone𝜆_𝑗notzero. (24)

This can be viewed as the FDH CCR DEA model.

3.2. Min-Max/Max-Min Problem of BCC

To derive the min-max/max-min problem of BCC model, we can change the relative loss function as follows: Min_{𝑢,𝑣,𝑢0}Max_{𝑗=1,…,𝑛}∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗/∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗+𝑢₀∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0/∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖0+𝑢₀subjectto∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0=1,𝑢≥0,𝑣≥0,𝑢₀isfree. (25) Again let𝑡=max_𝑗{∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗/(∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗+𝑢₀)}. Then, we transform the model above to the following: Min_{𝑢,𝑣,𝑢0}𝑡∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖0+𝑢₀∑subjectto^𝑠𝑟=1𝑢_𝑟𝑦_𝑟𝑗∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗+𝑢₀≤𝑡,_𝑠𝑟∑₌₁𝑢_𝑟𝑦_𝑟0=1,𝑗=1,…,𝑛,𝑢≥0,𝑣≥0,𝑢₀isfree. (26) Then, further let𝑣_𝑖=𝑡𝑣_𝑖and𝑢₀=𝑡𝑢₀. Then, the model is transformed to Min^𝑚_𝑢,𝑣∑_𝑖=1𝑣_𝑖𝑥_𝑖0+𝑢₀subjectto∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗−∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗−𝑢₀≤0,_𝑠𝑟∑₌₁𝑢_𝑟𝑦_𝑟0=1,𝑗=1,…,𝑛,𝑢≥0,𝑣≥0,𝑢₀isfree. (27) This is the standard output-oriented BCC DEA model.

Similarly the “max-min” strategy leads to the following formulation: Max_{𝑗=1,…,𝑛}Min_{𝑢,𝑣,𝑢0}∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗/∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗+𝑢₀∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0/∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖0+𝑢₀Subjectto∶∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0=1,𝑢≥0,𝑣≥0,𝑢₀isfree. (28) Again, we can show that its dual reads Max𝜃subjectto∑^𝑛_𝑗=1𝑋_𝑗𝜆_𝑗≤𝑋₀,_𝑛𝑗∑₌₁𝑌_𝑗𝜆_𝑗≥𝜃𝑌₀,_𝑛𝑗∑₌₁𝜆_𝑗=1,𝜆_𝑗=1or0. (29) This can be viewed as the FDH BBC DEA model.

3.3. Max-Min Problem When Exchanging Strategy Spaces

Similar to Section 2.3, we exchange the pure strategies of DMU₀ and central evaluator, and then the max-min formulation can be written as follows: Max_𝑢,𝑣Min_{𝑗=1,…,𝑛}∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗/∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0/∑^𝑚_𝑟=1𝑣_𝑖𝑥_𝑖0subjectto∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖0=1,𝑢≥0,𝑣≥0. (30) Letting𝑡=min_𝑗{∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗/∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗}, we transform the model above to the following model: Max_𝑢,𝑣𝑡∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0∑subjectto^𝑠𝑟=1𝑢_𝑟𝑦_𝑟𝑗∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗≥𝑡,_𝑚𝑖∑₌₁𝑣_𝑖𝑥_𝑖0=1,𝑗=1,…,𝑛,𝑢≥0,𝑣≥0. (31) Further, let𝑢_𝑟=𝑢_𝑟/𝑡, and then we have Max_𝑢,𝑣1∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0subjectto∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗−∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗≥0,_𝑚𝑖∑₌₁𝑣_𝑖𝑥_𝑖0=1,𝑗=1,…,𝑛,𝑢≥0,𝑣≥0. (32) Or Min^𝑠_𝑢,𝑣∑_𝑟=1𝑢_𝑟𝑦_𝑟0subjectto∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗−∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗≥0,_𝑚𝑖∑₌₁𝑣_𝑖𝑥_𝑖0=1,𝑗=1,…,𝑛,𝑢≥0,𝑣≥0. (33)

Proposition 2. The dual form of model (33) is as follows: Max𝜃subjectto∑^𝑛_𝑗=1𝑥_𝑖𝑗𝜆_𝑗≥𝜃𝑥_𝑖0,_𝑛𝑗∑₌₁𝑦_𝑟𝑗𝜆_𝑗≤𝑦_𝑟0,𝜆_𝑗≥0,𝑗=1,…,𝑛. (34)

Proof. Model (33) can be written as follows: Min𝑌^𝑇₀𝑢subjectto𝑌^𝑇𝑢−𝑋^𝑇𝑣≥0,−𝑋^𝑇₀𝑣+1=0,𝑢≥0,𝑣≥0, (35) where𝑌₀=(𝑦₁₀,…,𝑦_𝑠0)^𝑇,  𝑋₀=(𝑥₁₀,…,𝑥_𝑚0)^𝑇,𝐘=(𝑌₁,…,𝑌_𝑛),𝐗=(𝑋₁,…,𝑋_𝑛),𝐮=(𝑢₁,…,𝑢_𝑠)^𝑇, and 𝐯=(𝑣₁,…,𝑣_𝑚)^𝑇. The Lagrange function of model (35) reads:𝐿(𝑢,𝑣,𝜃,𝜆)=𝑌^𝑇₀𝑢+𝜆^𝑇(−𝑌^𝑇𝑢+𝑋^𝑇𝑣)+𝜃(−𝑋^𝑇₀𝑣+1), and its dual function is 𝑞(𝜃,𝜆)=inf_{𝑢,𝑣≥0}{𝐿(𝑢,𝑣,𝜃,𝜆)}=inf_{𝑢,𝑣≥0}𝜃+𝑌^𝑇₀−𝜆^𝑇𝑌^𝑇𝑢+𝜆^𝑇𝑋^𝑇−𝜃𝑋^𝑇₀𝑣=𝜃,if𝑌^𝑇₀−𝜆^𝑇𝑌^𝑇≥0,𝜆^𝑇𝑋^𝑇−𝜃𝑋^𝑇₀≥0,−∞,if𝑌^𝑇₀−𝜆^𝑇𝑌^𝑇<0,𝜆^𝑇𝑋^𝑇−𝜃𝑋^𝑇₀<0. (36) Thus, the dual problem is Max𝑞(𝜃,𝜆)subjectto𝜆≥0, (37) which reads Max𝜃subjectto𝑋𝜆≥𝜃𝑋₀,𝑌𝜆≤𝑌₀,𝜆≥0. (38) Hence, model (34) is the dual of model (33).

Similarly, we can induce the output-oriented models: Max_𝑢,𝑣Min_{𝑗=1,…,𝑛}∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗/∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0/∑^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖0subjectto∑^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟0=1,𝑢≥0,𝑣≥0,Min𝜃subjectto∑^𝑛_𝑗=1𝑥_𝑖𝑗𝜆_𝑗≥𝑥_𝑖0,_𝑛𝑗∑₌₁𝜆_𝑟𝑗𝜆_𝑗≤𝜃𝑦_𝑟0,𝜆_𝑗≥0,𝑗=1,…,𝑛. (39)(40)

4. Conclusions

In this paper, we presented a series of two-person games, which further derive various DEA models. Unlike the previous work, the relationship between DEA models and the games is more direct, although the Nash solutions do not generally exist. To sum up, our contribution can be summarized into the following conclusions.(1)Classical DEA models like the CCR can be viewed as the safest solutions of the game processes, where the DMU₀ is one player and the other player is the central evaluator. The DMU₀ has the right to choose the weights to maximize its performance score, and the evaluator will try to choose a competitiveDMU_𝑗to minimize the performance score and has the right to terminate the game.(2)If the two players exchange their strategy spaces, then we can derive nonclassic DEA models. In these models, DMU₀ will choose the worst practice DMU as the reference to compare, instead of the best practice. We can call this kind of models “negative” DEA or “inverted” DEA like Yamada et al. [15] in order to differentiate them from the classical DEA models.