Using a Two-Person Zero-Sum Game to Solve a Decision-Making Problem

This study proposes a game-theoretic approach to solve a multiobjective decision-making problem. The essence of the method is that a normalized decision matrix can be considered as a payoff matrix for some zero-sum matrix game, in which the first player chooses an alternative and the second player chooses a criterion. Herein, the solution in mixed strategies of this game is used to construct a weighted sum of the primary criteria that leads to a solution of the primary multiobjective decision-making problem. The proposed method leads to a notionally objective weighting method for multiobjective decisionmaking and provides “true weights” even in the absence of preliminary subjective evaluations. The proposed new method has a simple application. It can be applied to decision-making problems with any number of alternatives/criteria, and its practical realization is limited only by the capabilities of the solver of the linear programming problem formulated to solve the corresponding zero-sum game. Moreover, as observed from the solutions of the illustrative examples, the results obtained with the proposed method are quite appropriate and competitive.


Introduction
A particular type of multiobjective decision-making (MODM) problem, namely the simplest case with a finite number of decision alternatives and criteria, is considered herein. This study aims to propose a mathematical model that is useful when the decision criteria are conflicting and there is no decision-making authority or no evaluation of the importance of the criteria.
In general, a multiobjective formulation is the typical starting point for theoretical and practical analyses of decision-making problems. Thus, various versions of Pareto optimality and a vast arsenal of different methods can be used for Pareto optimization. However, unlike singleobjective optimization, a characteristic feature of Pareto optimality is that the set of Pareto-optimal alternatives is large and all Pareto-optimal alternatives must be considered mathematically equal.
Because the decision made must be usually unique, additional factors are considered for selecting specific or more appropriate (in some sense) alternatives from the set of Pareto-optimal alternatives. Herein, a special game-theoretic approach is proposed for selecting such appropriate alternatives. The essence of the method is that solving a special two-person zero-sum game leads to a notionally objective weighting method for the MODM problem. This game is constructed as follows. Let A and C denote (finite) sets of alternatives and criteria, respectively. The initial data for decision-making is assumed as a decision matrix whose elements exhibit the performance of different alternatives with respect to various criteria. A normalized decision matrix can be considered a payoff matrix for some zero-sum matrix game in which the A-player chooses one of the alternatives from set A and the C-player chooses one of the criteria from set C. In this paper, the solution in mixed strategies of this game is used to construct a weighted sum of the primary criteria that leads to a solution of the MODM problem.
The rest of this paper is organized as follows. Section 2 describes the proposed approach. Section 3 presents two illustrative examples. Finally, Section 4 concludes the study.

Preliminaries
The necessary notation and definitions are first considered. The initial data for decision-making are assumed as a decision matrix X whose elements exhibit the performance of different alternatives with respect to various criteria: 11 12 1 where ij x is the performance measure of alternative i on criterion j and , m n are the numbers of alternatives and criteria, respectively. Furthermore, each criterion is assumed to be classified as either beneficial (for which higher values are desirable) or nonbeneficial (for which lower values are desirable). Moreover, criteria with indexes 1,..., (1 ) B B n n n ≤ ≤ are assumed to be beneficial and, correspondingly, that the other criteria are nonbeneficial. Clearly, the decision matrix must be normalized to ensure that its elements are comparable. Because normalization can be defined variously, the so-called upper-lower-bound approach will be used (for details regarding the problems associated with normalization see [1] The above normalization procedure yields the matrix The elements of ( ) U X are dimensionless numbers on the interval [0,1] and represent the normalized performance of alternative i on criterion . j Note also that, by construction, in the matrix ( ) U X it is predetermined that a lower value is preferable for each criterion (column); in other words, all criteria are nonbeneficial. Therefore, after normalization, the goal of the decision-making procedure is to minimize all criteria (in the sense of matrix ( ) U X ) simultaneously; in other words, a typical multiobjective optimization problem is obtained.
Next, the basic concepts of multiobjective optimization theory are recalled. To this end, the following notation is introduced. Alternatives are denoted by The set of all (weakly) efficient alternatives is denoted by and called the (weakly) Pareto set.
Pareto optimality is an appropriate concept for MODM. However, it must be stressed that, unlike single-objective optimization, a characteristic of Pareto optimality is that the set ( ) we e A A of (weakly) Pareto-optimal alternatives is generally large and all alternatives from ( ) we e A A must be considered mathematically equal (i.e., equally "good"). Because the decision that is made usually must be unique, additional factors are considered for selecting specific or more appropriate (in some sense) alternatives from the set ( ) we e A A . In the following subsection, one possible approach in that direction is considered.

Proposed Approach
The considerations based on game theory, [2], and, correspondingly, the game-theoretic approach to solving MODM problems are presented herein. The proposed method considers the matrix ( ) as a payoff matrix for some zero-sum matrix game. This game can be interpreted as follows. The row player (A-player) chooses one of the alternatives , a A ∈ and the column player (C-player) chooses one of the criteria represents the sum paid to the A-player by the C-player when the former chooses alternatives i a A ∈ and the latter chooses criteria . j c C ∈ A mixed strategy for the Aplayer is vector m ξ ∈ ∆ and a mixed strategy for the C-player represents the probabilities of the A-player (C-player) choosing alternative/criterion k. Therefore, for mixed strategies the expected payoff for the A-player is 1 1 is considered as a "true" aggregation of the performance criteria. Moreover, it is well known that any solution of the minimization problem is always Pareto optimal, [3]. Therefore, the proposed approach allows selecting some Pareto-optimal alternative that can be considered as "appropriate." Clearly, a relevant interpretation of the aforementioned procedure is required to determine in what sense this obtained Pareto-optimal alternative is appropriate. The A-player and the C-player are represented by populations named the A-population and the C-population, respectively. It is assumed that each alternative (criterion) corresponds to the subpopulation of individuals that dispose of this and only this alternative (criterion) in the A-population (C-population) and that such subpopulations cover all A-populations (C-populations). Furthermore, component ( )

Examples
This section focuses on three illustrative examples of using the proposed method, which is applied by solving the corresponding zero-sum game. To this end, the standard approach of reducing a game-theoretic problem to a linear programming problem is used. All the necessary calculations are performed using MATLAB. Note that the obtained results are quite appropriate and competitive and are found with no prior estimates of criterion importance.

Material Selection
Consider an example that involves selecting material for the mast of a sailing boat. The component in question is a hollow cylinder that is subjected to axial compression (the parameters are a length of 1,000 mm, outer diameter ≤ 100 mm, inner diameter ≥ 84 mm, mass ≤ 3 kg, and a total compressive axial force of 153 kN; see [4]). This problem has been faced by several researchers using various methods such as weightedproperties method (WPM), VIKOR (multicriteria optimization through the concept of a compromise solution), CVIKOR (comprehensive VIKOR), fuzzy-logic approach (FLA), multiobjective optimization based on ratio analysis (MOORA), MULTIMOORA (a multiplicative form of MOORA), and the reference-point approach (RPA) etc., [4][5][6][7][8]. Note also that the material selection problem is an important application of MODM until today, [9,10].
The following criteria are defined for the problem in hand: specific strength (SS), specific modulus (SM), corrosion resistance (CR), and cost category (CC), [4]. The choice must be made from 15 alternative materials. The corresponding decision-making data are given in Table 1, the considered sample materials ranked in descending order as obtained by different methods are given in Table 2 and the normalization  results are given in Table 3. Therefore, in accordance with the proposed approach, the "proper" performance of the alternatives can be obtained, aggregating their (normalized) performances with weights * ζ . Notes: CR scale: 1 = poor; 2 = fair; 3 = good; 4 = very good; 5 = excellent; CC scale: 1 = very high; 2 = high; 3 = moderate; 4 = low; 5 = very low. Source: [8] The results of the corresponding calculations are presented in Table 4. Figure 1 shows the solution obtained using the proposed method (dark gray line) in comparison with those obtained using other methods (see Table 3). Note that for the proposed method with the considered decision matrix and the method for its normalization, the following materials have special status: AISI 1020, epoxy-70% glass fabric, epoxy-63% carbon fabric, and epoxy-62% aramid fabric. Vertical axis: rank; horizontal axis: material (see the main text for an explanation)   11  5  8  6  6  6  5  6  12  8  5  2  5  3  7  8  13  2  2  3  3  12  2  2  14  3  3  1  2  1  1  1  15  1  1  5  1  5  3  3 Sources: * [8]; ** [4]; *** [7].

Intercompany Comparison
Next, the problem of comparing companies as an MODM problem is considered. In a previous study, seven companies were compared using four criteria { } c and reversal debt ratio 4 c (note that taking the reversal value of debt ratio as the criterion instead of the debt ratio itself makes all criteria beneficial), [11]. Moreover, the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method was considered and different techniques were used for criteria weighting, namely EM (Entropy Measure), CRITIC (CRiteria Importance Through Intercriteria Correlation), SD (Standard Deviation), and MW (Mean Weight), [11]. Note also that the MODM methods are frequently used to develop relevant "composite indicators" for various applications, [12]. The decision matrix for this case study is presented in Table 5. Table 6 lists the companies ranked in descending order obtained using different methods, and Table 7 presents the normalization results.   1  3  1  2  2  2  6  5  6  5  3  7  7  7  7  4  1  3  1  1  5  5  6  5  6  6  4  4  4  4  7  2  3  3  3 Source: [11] The solution (equilibrium) to the corresponding zero-sum game with mixed strategies is Therefore, in accordance with the proposed approach, the "proper" performance of the alternatives can be obtained, aggregating their (normalized) performances with weights * .
ζ The results of the corresponding calculations are presented in Table 8.  Figure 2 shows the solution obtained using the proposed method (dark gray line) in comparison with those obtained using other methods (see Table 6). Note that for the proposed method, with the considered decision matrix and the method for its normalization, the priorities shift in the directions of the market position and reversal debt ratio. In addition, companies 3 and 6 appear to have special status. Vertical axis: rank; horizontal axis: company (see the main text for an explanation)

Employee Selection
Finally, the problem of selecting employees is considered, as investigated in a previous study [13] (note also that application of MODM methods for employee selection problem was also considered recently, e.g. [14] Table 9. The applicants ranked in descending order determined using different methods are presented in Table 10; the decision matrix normalization results are presented in Table 11. Table 9. Decision matrix for the employee selection problem. LT  PT  ST  PS  CS  PI1  II1  PI2  II2  PI3  II3  PI4  II4  1  2  3  4  Therefore, in accordance with the proposed approach, the "proper" performance of the alternatives can be obtained, aggregating their (normalized) performances with weights * .

# Persons
ζ The results of the corresponding calculations are presented in Table 12. Figure 3 shows the solution obtained using the proposed method (dark gray line) in comparison with those obtained using other methods (see Table 10). In this example; only three criteria play decisive roles, namely the test of professional knowledge and the face-to-face interviews with managers 2 and 4. Moreover, the aforementioned interviews have roughly the same significance and are greater than that of the test of professional knowledge. In addition, only applicants P04, P11, and P12 appear to have privileged positions.

Conclusion
As is well known, reaching an agreement about the relative importance of criteria in MODM problems is difficult. Herein, a special game-theoretic approach is proposed to solve this problem. The proposed method leads to a notionally objective weighting method for MODM problems by solving a special two-person zero-sum game. Moreover, the proposed method provides notionally true weights even in the absence of preliminary subjective evaluations.
Further, the proposed method can be applied to decisionmaking problems with any number of alternatives/criteria and its practical realization is limited only by the capabilities of the solver of the linear programming problem formulated to solve the corresponding zero-sum game. As observed from the solutions of the illustrative examples indicate, the results obtained with the proposed method are quite appropriate and competitive.