A Novel Multiperson Game Approach for Linguistic Multicriteria Decision Making Problems

1 Department of Industrial Education and Technology, National Changhua University of Education, 2 Shi-Da Road, Changhua County, Changhua City 500, Taiwan 2Department of Information Management, National United University, 1 Lienda Road, Miaoli County, Miaoli City 36003, Taiwan 3 Institute of Mechatronoptic Systems and Department of Automation Engineering, Chienkuo Technology University, No. 1 Chiehshou North Road, Changhua City 500, Taiwan 4Department of International Business, National Chi Nan University, 1 University Road, Nantou County, Puli City 54561, Taiwan


Introduction
In decision science field, game theory provides an effective way for handing the interactive optimization problems.Game theory started with the publication of "the theory of games and economics behavior" in 1944 by von Neumann and Morgenstern [1].It is a special tool to analyze the interaction result among players and has been broadly applied in business, financial, politics, education, sports, and so forth [2][3][4][5].
There are three basic components in a game such as players, players' strategies, and the performances (payoffs) with respect to the strategies of players.Strategy form and extensive form are two main ways to describe the interaction between players [6].In strategy form, each player executes his/her strategy simultaneously and the payoff is decided based on the strategy combination of each player.In extensive form, each player executes his/her strategy sequentially and the payoff is the final result based on the decision of each player.In general, the extensive form of a game can be transferred to the strategy form.The different classification types of game included cooperation or noncooperation games, zero-sum or non-zero-sum games, one round or multiround games, two-person or multiperson games.
Recently, the development trend of game theory is to integrate multicriteria decision making (MCDM) method to deal with the decision-making problems in real situations.There are some literatures that have been proposed by combining multicriteria decision method (MCDM) with game theory for coping with the decision making problems.
Campos [7] proposed a two-person zero sum fuzzy matrix game and applied fuzzy linear programming to calculate the mixed strategy probability of each player.Sakawa and Nishizaki [8] applied the max-min concept to integrate fuzzy goals and fuzzy payoffs in a two-person zero sum game.Song and Kandel [6] applied fuzzy set to formulate the goals of players and the strategy probability of their competitors.The mixed strategy probability is calculated based on considering their goals and the probability of each strategy of competitors simultaneously.The drawback of Song and Kandel's model is that it is difficult to compute the game matrix with multiple persons.Wang [9] built a fuzzy linear programming model to deal with -person multiattribute noncooperative game.Liao [10] applied fuzzy linear goal programming to solve a two-person zero sum game for analyzing the wireless market in Taiwan.Angelou and Economides [11] integrated analytic hierarchy process, game theory, and real options to analyze the business alternatives of information and communication technology (ICT).Reneke [12] used the vector function to evaluate long term investment alternatives for predicting oil prices and environmental degradation under the conditions of risk and uncertainty.Madani and Lund [13] used Monte-Carlo game theory (MCGT) technology to handle the uncertainty problem with the deterministic strategic games.Monroy and Fernández [14] extended the Shapley-Shubik index from conventional simple games to deal with the multi-criteria game problem.Barough et al. [15] used the traditional game approach to deal with two types of project construction conflict problem.Li and Hong [16] developed an effective methodology for handling the constrained matrix game with fuzzy payoffs.Monroy and Fernández [17] used voting systems to handle multi-criteria simple games in social-choice situation.The stable set and the core are the solution concept in the multi-criteria simple games.Different kinds of aggregation operators such as union, intersection, marginalization, and composition can be applied in their method.Lozan and Ungureanu [18] used graphs intersection method of the best response mappings to deal with the two-criterion games.Pusillo and Stef Tijs [19] used the improvement sets for developing the equilibrium condition for noncooperative multi-criteria games.Kawamura et al. [20] extended neutrally and evolutionarily stable strategies from single-criterion game to multi-criteria games.In entropy environment, Roy and Das [21] handled the multi-criteria bimatrix goal game problem by determining Ggoal security strategies.They applied the real coded genetic algorithm to acquire the bounds of the objectives of the proposed game.The fuzzy programming technique is used to solve the formulated model.
Although many literatures have been proposed by applying game theory to make a decision, few of them can integrate MCDM and game theory to handle the multiperson multicriteria game in a fuzzy environment.In real environment, each player will compete with other players.A good player not only should consider his/her strategy for approaching his/her goal but also need to forecast the behaviors of competitors.There is usually more than one influenced factor that should be considered by each player in a game model.In addition, the uncertainty and fuzziness will happen in the real competitive environment because it is not easy to collect the decision information completely and the decision time is limited for making a decision.A good model must provide a mechanism such as linguistic value or fuzzy number for experts to express their opinions flexibly.
However, the original game model usually considers one dimension or criterion to make a decision and the crisp values are used to evaluate the performance with respect to the dimension by single decision maker.In order to overcome the drawbacks of original game model, the main purpose of this study is to develop a new decision making method, linguistic multiperson multi-criteria game (LMPMCG) model, for dealing with the game problems under multiperson and multi-criteria environment.According to the linguistic variable, the decision makers can easily express their opinions with respect to each criterion for each alternative (strategy combination).By using the linear programming method, we can find the optimal solution of a game matrix effectively.
The reminder of this study is organized as follows.In Section 2, the definitions of linguistic variables and fuzzy numbers will be introduced.After that, the linguistic multiperson multi-criteria game model is presented at Section 3.And then, an example is implemented for the new mobile phone development project selection problem.Finally, conclusion and future research are discussed at the end of this paper.

Fuzzy Set and Linguistic Variable
Fuzzy set theory is first introduced by Zadeh in 1965 [22].Fuzzy set theory is a very feasible method to handle the imprecise and uncertain information in a real world [23,24].Especially, it is more suitable for experts to express their subjective judgments and qualitative assessments in the decision making processes [25][26][27].
A positive triangular fuzzy number (PTFN) T can be defined as T = (, , ), where  ≤  ≤  and  > 0 (shown in Figure 1).The membership function  T() of positive triangular fuzzy number (PTFN) T is defined as [28,29] A linguistic variable is a variable whose values are expressed in linguistic terms, in other words, variable whose values are not numbers but words or sentences in a nature or artificial language [30][31][32].For example, "weight" is a linguistic variable whose values can be very low, low, medium, high, very high, and so forth.These linguistic values can also be represented by fuzzy numbers.There are two advantages for using triangular fuzzy number to express linguistic variable [33,34].First, it is a rational and simple method to use triangular fuzzy number to express the opinions of experts.Second, it is easy to make the arithmetic operations between fuzzy numbers when using triangular fuzzy numbers to express the linguistic variables.It is suitable to represent the degree of subjective judgment in qualitative aspect than crisp value.Some linguistic variables and their membership functions can be illustrated as Table 1 and Figure 2.
Let T1 = ( 1 ,  1 ,  1 ) and T2 = ( 2 ,  2 ,  2 ) be two PTFNs.The additive operation of PTFNs can be calculated as [28,33] Many ranking methods have been developed to transform fuzzy number into crisp value.Lee and Li [35] presented the generalized mean value method to rank fuzzy numbers.It is very easy to compare fuzzy numbers by this method.This method has been applied in decision science, personnel selection, weapon selection and supplier selection fields, and so forth [36][37][38][39].Suppose that T = (, , ) is a PTFN, the defuzzied value is easily computed as [35,36] If ( T1 ) > ( T2 ), then T1 > T2 .

Linguistic Multiperson Multicriteria Game (LMPMCG) Model
In real environment, each player must compete with other players to determine the actions.A good player not only must consider the own strategies for approaching the goals but also need to forecast the behaviors of their competitors to select the best reaction.Under this situation, many influenced criteria should be considered by each player to make a strategy decision in a game system.Because the different strategies of players will influence the performance of each other, we need to consider the strategies of each player based on a strategy combination.A strategy combination means combination of strategies of players in a specific situation.From this viewpoint, a new game model with linguistic variables for the multiperson multi-criteria problem is proposed in this study.

Basic Notation of LMPMCG Model.
Generally speaking, the contents of linguistic multiperson multi-criteria game (LMPMCG) model can be illustrated as follows.
. A strategy combination means a combination of the strategy of each player in a specific situation.
(4) A set of criteria with respect to each strategy combination of each player is  = { 1 ,  2 , . . .,  V }, where The  1 ,  2 , . . .,  V is the number of the criteria for player 1, player 2, . .., and player V.The    represents the th strategy criteria of player .
For a strategy combination ( 1  ,  2  , . . .,    , . . .,  V  ), the aggregated performance evaluation of decision makers with respect to criterion    can be computed as where x( The aggregated performance evaluation of decision makers with respect to criterion    under the strategy combination ( 1  ,  2  , . . .,    , . . .,  V  ) can be defuzzied as where ( x(  1).
The aggregated weight of criterion    can be computed as where w   represents the aggregated weight with respect to criterion    based on the opinions of decision makers.The aggregated weight of criterion    can be defuzzified as where (w

Decision Process.
For the strategy combination ( ), the integrated performance of player  with respect to all criteria can be calculated as where Ψ ( The optimal probability of each strategy for player  can be computed by considering all strategy combinations of all players.Under this situation, player  will maximize the expected performance by calculating the optimal probability of each strategy.Therefore, the problem of optimal probability Strategy combination of each strategy for player  can be formulated as a linear programming model as follows:

Strategy combination
where V  represents the expected performance of player  and Φ   is the probability of the strategy    .

Numerical Example
Suppose that there are three enterprises (players)  1 ,  2 , and  3 who can produce high technology mobile phone in the market.Each enterprise possesses its strong point and weakness.The product development strategy and manufacture strategy of each enterprise will influence the competitive performance of other enterprises in the different dimensions.
Based on R&D ability and the brand impressions of each enterprise, the strategy of each enterprise will be limited to develop a new product.Enterprise  1 invites three experts to analyze and select a suitable strategy for developing a new mobile phone.The first expert is a marketing director who comes from the marketing department in the enterprise.The second expert is an R&D manager who is the project leader of the research and development department in the enterprise.The third expert is a strategy professor who is invited from the university.Enterprise  1 has three development strategies and also knows that competitor  2 has three strategies and competitor  3 has two strategies for developing a new mobile phone.Enterprise  1 considers four criteria to make a decision.Enterprise  1 also knows that competitor  2 considers three criteria for making decision and competitor  3 considers four criteria for making decision.The strategies and evaluation criteria of enterprises are shown in Table 2.
According to the computational process of linguistic multiperson multi-criteria game (LMPMCG) model, the problem can be solved as follows.

Aggregating the Evaluations of Experts
Step 1.Each expert uses the linguistic variables to evaluate the performance of each enterprise with respect to each criterion based on different strategy combinations as Tables 3, 4, and 5.
Step 2. Transform the linguistic evaluation of the performance of each enterprise with respect to each criterion into PTFN as Tables 6, 7, and 8.And then, the performance of each enterprise with respect to each criterion can be aggregated as Table 9.
Step 3. The aggregated performance of each enterprise with respect to each criterion can be defuzzied as Table 10.
Step 4. Each expert uses the linguistic variables to evaluate the weight of each criterion as Table 11.
Step 5. Transform the linguistic evaluation of the weight of each criterion into PTFN as Table 12.And then, the aggregated weight of each criterion can be computed as Table 12.
Step 6.The aggregated weight of each criterion can be defuzzied as Table 12.

Analysis of Mixed Strategy.
After aggregating the evaluations of experts, the optimal strategy of each enterprise can be determined by considering all strategy combinations.At first, the weighted performance of each strategy combination based on the opinions of experts must be calculated.And then, the optimal probability of each strategy for each Table 5: The linguistic performance of enterprise  3 with respect to criteria for each strategy combination.

Strategy combination
enterprise can be formulated by a linear programming model.Finally, the occurrence probability of each strategy combination can be computed by solving the linear programming model.The computational steps can be illustrated as follows.
Step 1.For any strategy combination, the weighted performance of each enterprise with respect to each criterion can be calculated as Table 13.
Step 2. Calculate the aggregated performance of each enterprise with respect to all criteria for any strategy combination.
Step 3. The aggregated performance of each enterprise for the strategy combination can be arranged as a multiperson multi-criteria noncooperation game as Table 14.According to Table 14, the performance of enterprise  1 is 0.5254 when enterprise  1 chooses strategy  1 1 , enterprise  2 chooses strategy  2  1 , and enterprise  3 chooses strategy  3 1 .The performance of enterprise  1 is 0.5354 when enterprise  1 chooses strategy  1 2 , enterprise  2 chooses strategy  2 1 , and enterprise  3 chooses strategy  3  1 .The performance of enterprise  1 is 0.4821 when enterprise  1 chooses strategy  1 3 , enterprise  2 chooses strategy  2  1 , and enterprise  3 chooses strategy  3  1 .Based on the strategy combinations in game model, the larger aggregated performance is the better strategy for the enterprise.
Step 4.Under the competitive situation, each enterprise hopes that the overall performance is higher than other enterprises as the probability of each strategy is computed.Therefore, the expected performance of the mixed strategy of each enterprise should be higher than the performance of each enterprise although the strategies are selected by other enterprises.According to the aggregated performances of all strategies, enterprise  1 will maximize the expected performance by formulating a linear programming model as follows: By solving this linear programming model, the optimal probability of strategies  ,  1 2 , and  1 3 are Φ 1 1 = 0.2632, Φ 1 2 = 0.7368, and Φ 1 3 = 0.0000, respectively.According to the aggregated performances of all strategies, enterprise  2 will maximize the expected performance by formulating a linear programming model as follows: Max V 2 subject to 0.7056 * Φ By solving this linear programming model, the optimal probability of strategies  2 1 ,  2 2 ,  2 3 are Φ 2 1 = 0.2136, Φ 2 2 = 0.4629, and Φ 2 3 = 0.3235, respectively.According to the aggregated performances of all strategies, enterprise  3 will maximize the expected performance by formulating a linear programming model as follows:

Strategy combination
Enterprise According to the EPV of each enterprise, the ranking order of competition ability is  1 ≻  3 ≻  2 .

Conclusion and Future Research
Facing the dynamic and competitive environment, a new decision making model is presented to deal with group MCDM problems by combining MCDM with game theory in this paper.Original game model usually considers one dimension to make decision based on crisp values.By using the linguistic variables, the decision makers can express their opinions flexibly and easily in the proposed method.According to the proposed method, each player in a game can develop the best strategy or alternative by considering the performance with respect to multiple criteria and the reactions of competitors simultaneously.A particular strongpoint of this proposed method is that the evaluation criteria can be flexible to determine by players for fitting their objections.Under this situation, the proposed method provides a more reasonable and systematic solution for each player in a competitive environment.
In fact, the number of players of a game model can increase more than two players flexibly in the proposed method.In addition, the most important contribution of proposed method is to present the expected performance value (EPV) to judge the competitive ability of each player in a game model.In the future, multiple types of decision information will be considered in the proposed model such as interval value, crisp value, and type-2 fuzzy set.In order to enhance the computational efficiency, an interactive program will be designed based on the proposed model.
A set of decision makers or experts is  = { 1 ,  2 , . . .,   }, where  represents the number of decision makers.

Table 2 :
The strategies and evaluation criteria of each enterprise.

Table 3 :
The linguistic performance of enterprise  1 with respect to criteria for each strategy combination.

Table 4 :
The linguistic performance of enterprise  2 with respect to criteria for each strategy combination.

Table 6 :
The PTFN performances of enterprise  1 with respect to criteria for each strategy combination.

Table 10 :
The defuzzied performance of each enterprise with respect to criteria under all strategy combinations.

Table 11 :
The linguistic weight of each criterion.

Table 12 :
The PTFN information of each criterion.

Table 13 :
The weighted performance of each enterprise with respect to criteria under all strategy combinations.