Game Theoretic Approaches to Weight Assignments in Data Envelopment Analysis Problems

This paper deals with the problem of fairly allocating a certain amount of divisible goods or burdens among individuals or organizations in the multicriteria environment. It is analyzed within the framework of data envelopment analysis (DEA). We improve the game proposed by Nakabayashi and Tone (2006) and develop an alternative scheme by reassigning the total weight or power for the coalition members. The solutions and equilibria of the new DEA game proposed in this paper are also studied.


Introduction
Recently game theoretic approaches to DEA problems have been often observed.These include, in part, Nakabayashi and Tone [1], Liang et al. [2], Wu et al. [3], Wu et al. [4], and Zhu [5].Among them, Nakabayashi and Tone [1] studied the problem of allocating a fixed amount of reward to players who are evaluated by multiple criteria.They proposed a new scheme for allocating the reward to the players based on cooperative game theory and DEA.Later, Wu et al. [3] and Wu et al. [4] applied the game model by Nakabayashi and Tone [1] to evaluate the cross efficiency of players (decisionmaking units (DMU) in DEA terms) by using solutions in cooperative games, the nucleolus, and the Shapley value.
The game proposed by Nakabayashi and Tone [1] was, however, subadditive.Namely, players lose their power when they cooperate.The reason is clear.Before forming a coalition, each player has a weight of one and puts it on his/her most preferable criterion.To increase their bargaining power, some of the players choose to form a coalition, but, in Nakabayashi and Tone's [1] assumption, the coalition is only given a weight of one in total though each of the members' precoalition weight was one, which causes the subadditivity.To make the game superadditive, they took the dual of the game, called the DEA min game, and studied solutions such as the core, the Shapley value, and the nucleolus.In the DEA min game, each player and each coalition pick up the weight that minimizes their evaluation.No reasonable justification was given in their paper for picking up the minimizing weight under the assumption that players are egoistic and want to maximize their own evaluation.The purpose of this paper is to propose an alternative, and more natural, cooperative game scheme that fits for the problem.We will start with a strategic form game describing the problem posed by Nakabayashi and Tone [1] and then construct a cooperative game from the strategic form game based on the procedures by von Neumann and Morgenstern [6].
The rest of the paper will be organized as follows.In Sections 2 and 3, we review the DEA problem and the DEA game proposed by Nakabayashi and Tone [1].After giving a definition of the strategic form game in Section 4, we will present a cooperative TU game representation in Section 5 and then study its properties and solutions in Sections 6 and 7.The Nash equilibrium, strong Nash equilibrium, and coalition-proof Nash equilibrium for the strategic form game are analyzed in Section 8. Section 9 will close the paper with remarks.

The Model
Let  (>0) denote the fixed amount of benefit to be allocated to players 1, . . ., .Players' contributions are evaluated by multiple criteria and summarized as the score matrix  = (  ) =1,...,, =1,..., , where   is player 's contribution 2 Mathematical Problems in Engineering measured by criterion , called the evaluation index.The problem is to find a weight vector on the criteria determined endogenously by players themselves.Following the DEA analysis, each player  chooses a nonnegative weight vector   = (  1 , . . .,    ) such that ∑  =1    = 1,    ≥ 0 ∀ = 1, . . ., , where    is the weight given to criterion  by player .Then the contribution of player  relative to the total contribution of all players measured by the weight vector   is given by Player  chooses the weight vector that maximizes this ratio.
The weight vector is found by solving the following fractional program: max Each of the other players similarly maximizes the ratio produced by his/her own weight vector.This maximization problem can be reformulated as the following much simpler form.First for each row ( 1 , . . .,   ),  = 1, . . ., , divide each element by the row-sum ∑  =1   .By Charnes-Cooper transformation [7], the maximization problem above is not affected by this operation.Let The matrix   = (   ) =1,...,, =1,..., is called the normalized score matrix and ∑  =1    = 1 is satisfied.Then Due to ∑      = 1, the fractional maximization program above can be expressed as the following linear maximization program: max Let () be the maximal value of the program.Apparently the maximal value is attained by letting   () = 1 for the criterion () such that   () = max =1,...,    and letting    = 0 for all other criteria  ̸ = ().Thus () is the highest relative contribution of player .Namely, Nakabayashi and Tone [1] showed that if each player  claims the portion () of , the sum of the claims generally exceeds the total benefit .Then the following problem arises: how to allocate  reasonably to players?To find a fair allocation of , they proposed to apply cooperative game theory.Let us review their cooperative game model that they call a DEA game.

The DEA Game by Nakabayashi and Tone
Nakabayashi and Tone [1] construct a characteristic function form game (, ) in the following manner. = {1, . . ., } is the set of players and  is the characteristic function that gives each coalition a value it obtains.For each single player coalition {}, ({}) is given by ().Similarly for each coalition  ⊆ , () is given by the maximum value of the linear maximization program: where   = (  1 , . . .,    ) is the weight vector chosen by .Here they assume that players' evaluation is transferable and take a total of players' evaluation in coalition .Hereafter we call this characteristic function form game NT (Nakabayashi and Tone) DEA game.
Nakabayashi and Tone [1] show a counterintuitive fact that the NT DEA game is subadditive.Namely, players lose their value by forming a coalition.The reason is quite simple.Consider the score matrix in Table 1.
The reason is intuitively clear.Either of players 1 and 2 has a weight of one and puts it on his/her most preferable criterion, respectively, before forming a coalition.To increase their bargaining power, they choose to form a coalition, but after forming a coalition they are only given a total weight of one and put it on their most preferable criterion that maximizes their total evaluation.This is the primary reason for the subadditivity.The coalition of players 1 and 2 should be given weight of two since their total weights are two before forming a coalition.For example, in voting decisions, it is considered to be fair if the coalition is given two votes in total, in the case such that two voters each having one vote form a coalition.
By using this framework, we propose an alternative characteristic function form game. First we construct a strategic form game.Each player's strategy is a weight vector, and the payoff is his/her weighted average contribution calculated by using the simple average of all players' weight vectors.From this strategic form game, we derive a characteristic function form game following the procedure by von Neumann and Morgenstern [6], which will be explained in detail in the following sections.

A Strategic Form DEA Game
Let  = {1, . . ., } be the set of players and let  = {1, . . ., } be the set of criteria.The basic DEA model stated in Section 2 is as follows.Each player  ∈  chooses a weight vector ≥ 0 ∀ ∈ , on the criteria so as to maximize the weighted sum of his/her relative evaluation indices, ∑  =1      .The fixed amount of reward  is shared by the players according to their weighted sums of the evaluation indices.
Therefore the strategic form game naturally reflecting the DEA model will be where  = {1, . . ., } is the set of players, is the strategy set of player  ∈ , and   :  1 × ⋅ ⋅ ⋅ ×   → R is the payoff function of player  ∈ , which is given by Namely, the reward  is shared by players in proportion to the weighted sum of their evaluation indices where the weights are the average weights over all players (another possible definition of )) ; that is, the reward  is shared by players in proportion to their weighted evaluation indices with their own weights.Both payoff functions lead to the same conclusions with respect to the solution concepts).Hereafter we call this game the strategic form DEA game.The equilibria in the strategic form game will be studied in Section 8.

TU DEA Game
In the characteristic function form game theory, it is commonly considered that the characteristic function gives each coalition the value that the coalition members can surely win by themselves.Here the term "by themselves" means that they can win the value even if outsiders' behavior is least favorable to them.In our game, every player wishes to maximize his/her evaluation so as to maximize the share of .Hence the least favorable behavior of outsiders is to minimize the coalition's evaluation, that is, the weighted sum of the players' evaluation indices inside the coalition.This behavior is also most favorable to the players outside the coalition since in our game players share a fixed amount of reward .Namely, minimizing the coalition's evaluation leads to the maximization of the outsiders' evaluation.
Therefore following von Neumann and Morgenstern [6], the characteristic function V gives the following value V() to each coalition  ⊆ : where  is the number of players in .From the perspective of the bargaining story we mentioned in the previous section, the worst situation for  is when outsiders put their whole weights on the criterion with the lowest evaluation for the coalition.For example, in Table 2, coalition {1, 2}'s maximum and minimum evaluation indices are 0.9 and 0.2.Therefore the value for coalition {1, 2} is properly measured by the weighted average of 0.9 and 0.2 with weights of 2/3 and 1/3, respectively.Thus we would propose (2/3)×0.9+(1/3)×0.2 = 2/3 as the value of the characteristic function for coalition {1, 2}.Similarly for the single player coalition {3}, the value would be (2/3) × 0.1 + (1/3) × 0.8 = 1/3 since player 3's maximum and minimum evaluation indices are 0.8 and 0.2.Note that V() =  since ∑ ∈   = 1 ∀ = 1, . . ., .We call the characteristic function form game (, V) the TU DEA game since we allow for side payments among players assuming transferable utility.Nontransferable utility case will be studied in the future extension work.

Properties of the TU DEA Game
TU DEA games satisfy interesting properties.First TU DEA games are superadditive.
Furthermore TU DEA games are constant-sum.
holds for all  ⊆ . ( Then it is quite simple to show that

Solutions to the TU DEA Game
7.1.Core.In characteristic function form games, solutions are considered within the concept of imputations.
The set of imputations of (, V) is denoted by (V).
Definition 6 (core).The set is called the core of (, V).

Definition 7 (inessentiality). A characteristic function form game
holds for all  ⊆ .
If a characteristic function form game (, V) is inessential, then its imputation set is a singleton (V({})) ∈ .Each player can receive the "safety" amount, guaranteeing the individual value being not worse off compared with the precoalition amount.Hence it is essential games that are of interest to us.The following theorem is well known; for the proof see Owen [8].
Theorem 8. Suppose that a characteristic function form game (, V) is constant-sum.Then if it is essential, its core (V) = 0.
As proved in the previous section, the TU DEA game (, V) is superadditive and constant-sum, and thus the core is nonempty only when the game is inessential, which is equal to the unique imputation set.The following theorem characterizes the inessential TU DEA game.Theorem 9. A TU DEA game is inessential if and only if, for all  ∈ ,   =     holds for all ,   = 1, . . ., .
Therefore in the TU DEA game, the core is nonempty if and only if the evaluation indices for all the criteria are identical for each player.It is usually not the case in reality and thus the core is generally empty.We can see that the allocation by the core concept is not of significant meaning for the TU DEA game.

Shapley Value.
If the concept of the core is to give a set of stable imputations without distinguishing the most preferable payoff vector, although it might be empty, then the Shapley value is trying to assign the game a specific payoff vector.It is defined below.
Definition 10 (Shapley value).In a characteristic function form game (, V), for each  ∈ , is called the Shapley value for player .The vector (V) = (  (V)) ∈ is called the Shapley value.
By applying the constant-sum property of the TU DEA game, the Shapley value can be represented in a much simpler form, which is also an exercise problem in Owen [8].

Nucleolus.
Instead of applying a general axiomatization of fairness to a value function defined by the set of characteristic functions, the nucleolus looks at a fixed characteristic function, V, and tries to find an imputation  = (  ) ∈ that minimizes the worst inequity.That is, we ask each coalition  ⊆  how dissatisfied it is with the proposed imputation  and we try to minimize the maximum dissatisfaction.
Definition 13 (excess).As a measure of the inequity of an imputation  = (  ) ∈ for a coalition  ⊆ , the excess is defined as which measures the amount by which coalition  falls short of V() by the proposed allocation .
From the definition of the core, we immediately know that an imputation  is in the core if and only if all of its excesses are negative or zero.
The property such that (, ) + (,  \ ) = 0 can reduce the computational complexity of the nucleolus, which can be proved by the constant-sum property of the TU DEA game.Thus we just have to check the excesses for half of the coalitions.
Theorem 16.In the 3-player TU DEA game (, V), the allocations by the Shapley value and the nucleolus coincide.
Proof.Assuming  = 1, the characteristic function for each coalition  ∈ {1, 2, 3} is The Shapley value allocation (V) is decided by Mathematical Problems in Engineering 7 With the allocation above, the excess for each single player coalition is It can be easily verified that ((V), {1}) = ((V), {2}) = ((V), {3}) is satisfied: By the constant-sum property, any deviation from the allocation (V) will increase at least one of the excesses.Hence in the 3-player TU DEA game, the allocations by the Shapley value and the nucleolus always coincide.
For the cases that players are more than 3, here is a counterexample showing that the theorem above does not necessarily hold.
Example 17.The Shapley value allocation is (16/60, 16/60, 15/60, 13/60) with the data set in Table 4.It can be easily verified that the maximum excess is not minimized with the allocation above, and hence the nucleolus allocation does not coincide with the Shapley value allocation in this example.

Equilibria in the Strategic Form DEA Game
In this section we study the equilibria in the strategic form DEA game, which is defined in Section 4.
Next we give a condition under which the coalition-proof Nash equilibrium exists and is uniquely determined.
Assume ∃ ⊆  such that  has a credible deviation from  * , and let   = (  ) ∈ ∈   denote the deviation.Then some {} ⊊  must have a deviation at (  ,  * − ) by Theorem 19.This deviation is a credible one by Definition 23.Therefore   is not a credible deviation at  * .Hence  * is the unique coalition-proof Nash equilibrium.
Theorem 26 shows that if for each player the criteria set giving him/her the maximum evaluation index is a singleton, then the coalition-proof Nash equilibrium exists and is uniquely determined.

Concluding Remarks
In this paper, we have improved the DEA min game proposed by Nakabayashi and Tone [1] by reassigning the total weight for the coalition members and developed a more natural, superadditive cooperative game scheme for this kind of problems.We first introduced a strategic form DEA game with the average weights over all players regarding each criterion in Section 4 and then defined the TU DEA game based on the definition of the strategic form game in Section 5 and studied its properties and solutions in Sections 6 and 7, respectively.By applying the constant-sum property, we found the sufficient and necessary condition for the inessentiality of the TU DEA game, which is also the condition for a nonempty core.On showing that the core is usually empty in the TU DEA game, we next introduced the Shapley value and nucleolus and also proved that the allocations by these two concepts coincide with each other in the 3-player setting.Section 8 analyzed the equilibria for the strategic form DEA game.We gave a condition under which the coalition-proof Nash equilibrium exists and is uniquely determined.
Our future research subjects include the following: (1) the solutions for the NTU DEA game, namely, the existence of -core and -core; (2) the role of other imputations, that is, the kernel, the proportional nucleolus, and so forth; (3) the application of our proposed scheme in the voting game.

Table 1 :
Score matrix for single player coalition case.

Table 4 :
Incoincidence of the Shapley value and nucleolus.

)
(∀ ∈ ) is given; in order to maximize    ( 1 , ...,   ), player   should choose a weight vector maximizing ∑  =1 (      ).It is obvious that only criteria with the highest evaluation indices should be assigned with positive weights.Applying this procedure for all other players, the Nash equilibrium should be  * = ( * 1 , . . .,  *  ),  *   = 0 ∀ ∈  ∀ ∉   .For the necessary condition, assume that player   ∈  deviates from  *   to    such that ∃  ∉    with = 1, there must exist at least one criterion   ∈    to which the weight assigned becomes less

Table 6 :
A counterexample for the existence of CPNE.