Matrix Games with Interval-Valued 2-Tuple Linguistic Information

In this paper, a two-player constant-sum interval-valued 2-tuple linguistic matrix game is construed. The value of a linguistic matrix game is proven as a non-decreasing function of the linguistic values in the payoffs, and, hence, a pair of auxiliary linguistic linear programming (LLP) problems is formulated to obtain the linguistic lower bound and the linguistic upper bound of the interval-valued linguistic value of such class of games. The duality theorem of LLP is also adopted to establish the equality of values of the interval linguistic matrix game for players I and II. A flowchart to summarize the proposed algorithm is also given. The methodology is then illustrated via a hypothetical example to demonstrate the applicability of the proposed theory in the real world. The designed algorithm demonstrates acceptable results in the two-player constant-sum game problems with interval-valued 2-tuple linguistic payoffs.


Introduction
Classical non-cooperative game theory, introduced by Von Neumann and Morgenstern [1], speculates that each player is exposed to the precisely known information of the game.The precise knowledge allows the players to provide exact evaluations of their utility functions in terms of their strategies.The presumption of exact payoffs emerges as a stringent ideology in the era of real problems with uncertainty and ambiguity.The elements of vagueness and imprecision have been integrated into games using various frameworks-to mention but a few, stochastic and fuzzy.Numerous researchers have contributed remarkable theories and outlooks to enrich the literature of fuzzy game [2][3][4] and stochastic game [5,6] theories.However, in this uncertain world, the determination of membership functions of a fuzzy value or probability distributions in case of a stochastic game is not always feasible for players.In some uncertain scenarios, the payoffs may vary within a certain range resulting in matrix-games with interval-valued payoffs [7][8][9][10][11].Collins and Hu [7,8,12] have significantly contributed to developing different perspectives and techniques in order to solve interval-valued matrix-game problems.Li et al. [13] developed auxiliary interval programming models to obtain a pair of bi-objective linear programming models to solve interval-valued matrix games.Li [14] proved the value of the interval-valued matrix-game to be a non-decreasing function of the interval payoffs of the matrix.He formulated a pair of auxiliary linear programming models to derive the upper bound and lower bound of the value of the interval-valued matrix-game.Annexing a new viewpoint to the game theory under uncertainty, Arfi [15,16] developed a linguistic fuzzy logic based game theoretic approach involving the idea of linguistic fuzzy domination and linguistic Nash equilibrium.To enhance the purview of ambiguity and vagueness in game theory, Singh et al. [17] presented a non-cooperative two-player constant-sum matrix game problem in the 2-tuple linguistic framework.The 2-tuple linguistic model is first formalized by Herrera and Martinez [18].Their competence to express any counting of information in the universe of discourse brings the edge over the other existing linguistic model.The veracious and interpretable 2-tuple linguistic model is widely used in decision sciences [18][19][20].However, intermittently, the granularity of vagueness and ambiguity to assess some criteria cannot be catered entirely by the subjectivity of linguistic terms in the predefined linguistic term set.Indeed, the argument to enlarge the term set can always be suggested for enunciating better information; it results in increased cumbersome calculations for aggregating large sets of variables [21].Zhang [21] presented an extended version of the linguistic model involving interval-valued 2-tuple linguistic variables.The model comprises of two 2-tuple linguistic variables in an interval from the predefined linguistic term set, to express the assessments of decision-makers.Various practitioners have adopted the interval-valued 2-tuple linguistic model to solve different real-world problems including but not limited to material selection [22] and supplier selection problems [23].
In this contribution, we have extended the matrix games to the interval-valued 2-tuple linguistic framework.Firstly, a two-player constant-sum interval-valued 2-tuple linguistic matrix game is defined, henceforth called an interval linguistic matrix game.Since the logic claims that the value of an interval linguistic matrix game must be an interval-valued 2-tuple linguistic information, we have verified the value of a linguistic matrix game as a non-decreasing function of the interval-valued payoffs in the matrix, as suggested by Li [14].A pair of auxiliary linguistic linear programming (LLP) problems is then constructed to acquire the linguistic upper bound and linguistic lower bound of the interval-valued linguistic value of the game.The duality principle of LLP is also adopted to prove the equality of interval-valued linguistic lower value and interval-valued linguistic upper value of the game for players I and II, respectively.The designed methodology has a wide variety of applications in the domains where the two-players have the knowledge of their payoffs in terms of interval-valued 2-tuple linguistic variables.
The remainder of this paper is organized as follows.Section 2 presents the rudiments of 2-tuple linguistic model and two-player constant-sum 2-tuple linguistic matrix game.In Section 3, the fundamental structure of a two-player constant-sum interval-valued 2-tuple linguistic matrix game is discussed.Section 4 presents the LLP formulation for solving such a game with a flowchart in Figure 1 to summarize the method.Section 5 illustrates the hypothetical examples to demonstrate the methodology with result analysis.Section 6 concludes the paper.

Preliminaries
In this section, the 2-tuple linguistic variable with its extension to interval-valued 2-tuple linguistic variables are reviewed.In addition, the fundamentals of a two-player constant-sum 2-tuple linguistic matrix game are described.

2-Tuple Linguistic Model
The following definitions are taken from the work of Herrera and Martinez [18].The conventional linguistic model is comprised of i , where i depicts the linguistic term from a predefined linguistic terms set LT = { i : i = 0, 1, . . ., g} with cardinality g + 1. Usually, the index g of the granularity of vagueness in information is assumed an even positive integer.The value in {0, 1, . . ., g} represents the index of the closest linguistic label in LT.
The set LT is totally ordered with the following characteristics: The above algorithm to solve the interval-valued 2-tuple linguistic matrix game can be summarized using the following flowchart.
Version August 15, 2018 submitted to Games 14 of 21 The above algorithm to solve the interval-valued 2-tuple linguistic matrix game can be summarized 301 using the following flowchart.

Illustrations and Result Analysis
The game theory has a wide variety of applications in finances, management, and decision sciences.It is also engaged in the real world competing and strategic interactions among players.A 2-tuple linguistic variable is represented by ( i , α), where i is a linguistic term and α is the symbolic translation.

2.
If i = j, i.e., i and j are the same linguistics, then Definition 3. Let β ∈ [0, g] be the value representing the result of a symbolic aggregation.An equivalent 2-tuple linguistic variable can be obtained using the function ∆ : [0, g] → LT × [−0.5, 0.5), defined by The function ∆ is a bijection, and its inverse is given by The literature describing diverse operators on the set of 2-tuple linguistic variables is vast and rich.Here, we cite the weighted average operator from [24] as it is used in the following: Definition 4. Let {( r i , α r i ), r i ∈ {0, 1, . . ., g}, i = 1, . . ., q} be a set of 2-tuple linguistic variables and w = (w 1 , . . ., w q ) T be the weight vector satisfying 0 ≤ w i ≤ 1, i = 1, . . ., q, ∑ q i=1 w i = 1.Then, the weighted average operator is defined as Here, ⊕ depicts the weighted addition of 2-tuple linguistic variables.Since, the usual addition of two 2-tuple linguistic variables is not well-defined, the weighted addition operator, ⊕, is used in place of usual addition.
The negation operator for a 2-tuple linguistic variable v = (l i , α i ) is taken to be Note that when v is a linguistic variable only (that is, v = ( i , 0)), then neg( v) = g−i , which is the same as the standard negation neg( i ) of the linguistic variable.
Remark 1.Here, we consider that the symbolic translations α (L) , α (U) take values in the interval [−0.5, 0.5) in view of a definition given by Herrera and Martinez [18].However, the interval-valued 2-tuple linguistic model defined by Zhang [21] is widely adopted in practice as it does not engage the granularity of the linguistic term set.As we do not require the computations involving interval-valued linguistics, here we are not discussing its operations.One can see [21] for comprehensive knowledge of computation with interval-valued 2-tuple linguistic variables.Now, we review the basics of two-player constant-sum 2-tuple linguistic matrix game from [17].

Constant-Sum Linguistic Matrix Game
Definition 6.A two-player constant-sum linguistic game G is expressed as a quadruplet (S n , S m , LT, A), where 1.
S n is a set of n strategies for player I defined as, 2. S m is a set of m strategies for player II given as, S m = (y 1 , . . ., y m ) : y j ≥ 0, j = 1, . . ., m, ∑ m j=1 y j = 1 ; 3. LT = { 0 , 1 , . . ., g } is a predefined linguistic term set with cardinality g + 1 for both the players, and 4.
A = [ a ij ] n×m is the linguistic payoff matrix for player I where a ij ∈ LT, i = 1, . . ., n, j = 1, . . ., m, while neg( A) = [neg( a ij )] n×m represents the payoff matrix for player II.Here, a ij , i = 1, . . ., n, j = 1, . . ., m can depict a linguistic variable or a 2-tuple linguistic variable.The linguistic payoff determines how beneficial and favourable it is for a player to play a given strategy.
Using the lexicographic ordering of 2-tuple linguistic variables, the value of the linguistic matrix game G is defined as follows.The lower value v − describes the minimum linguistic benefit that player I is assured to receive, whereas the upper value v + is player II's linguistic loss-ceiling.The value v of game G is defined when v The strategies i * and j * with the payoff a i * j * , such that a i * j * = v − = v + , are optimal pure strategies for player I and player II, respectively.In this case, the strategy pair (i * , j * ) is termed as the saddle point of game G.
Since the existence of saddle point in each linguistic game G is hardly possible, it is essential to define the linguistic expected payoff of players as follows.
Definition 8. Given a pair of mixed strategies x = (x 1 , . . ., x n ) ∈ S n and y = (y 1 , . . ., y m ) ∈ S m for player I and player II, respectively, the linguistic expected payoff of player I is defined by ( The linguistic expected payoff of player II is neg( E A (x, y)).
It is noteworthy that the lower value and upper value of the linguistic game G given in Definition 7 are described in light of pure strategies.The structures of the two values change to some degree in the case of mixed strategies.
Suppose player I chooses any strategy x ∈ S n .Then, player I's linguistic expected gain-floor is defined as follows: Here, E A (x, y) can be considered as the weighted average of player I's payoffs if the player I chooses mixed strategy x ∈ S n against pure strategy of player II.Hence, the minimum is achieved by some pure strategy of player II: where e j ∈ S m is the unit vector with 1 at the jth position.Thus, player I should select x * ∈ S n in the interest of maximising v − (x) to attain lower value of the game G as follows: The strategy x * ∈ S n is called optimal strategy for player I and v − = v − (x * ) is called the lower value of the game G.
The computation of optimal strategy x * ∈ S n to achieve v − (x * ) is equivalent to solving the following linguistic linear programming (LLP) problem: x 1 , x 2 , . . . ,x n ≥ 0.
Games 2018, 9, 62 7 of 19 Furthermore, correspondingly, the upper value of the game G can be defined in view of minimizing player II's linguistic expected loss-ceiling.Suppose that player II plays strategy y ∈ S m , then his/her linguistic expected loss-ceiling is given as where e i ∈ S n is the unit vector with 1 at ith position.Hence, player II should select strategy y * ∈ S m such that the minimum of v + (y) is obtained.Hence, Here, v + is the upper value of the game G with optimal strategy y * ∈ S m of player II.
The equivalent LLP problem to evaluate player II's optimal strategy y * ∈ S m to obtain the upper value of the game is as follows: subject to Remark 2. Models (M1) and (M2) form a pair of primal-dual linguistic linear programs.
The models (M1) and (M2) are transformed to conventional (crisp) primal-dual linear programming problems as suggested in [17] using the definitions of ∆ and ∆ −1 .The models provide the optimal mixed strategy vectors x * and y * of players I and II, respectively, with a 2-tuple linguistic value of the game.We will stick to the same methodology for the conversion of linguistic linear programming models to classical linear programming models as discussed in [17].

Constant-Sum Interval-Valued Linguistic Matrix Game Definition 9.
A two-player constant-sum interval-valued linguistic matrix game G Int is characterized by a quadruplet (S n , S m , LT, A Int ), where S n , S m and LT are the same as discussed in Definition 6.A Int is the interval-valued linguistic payoff matrix of player I in defiance of player II.

The payoff matrix
for player I consists of linguistic intervals where a indicates the range of linguistic payoff that player I may achieve if he/she decides to play strategy i, i = 1, . . ., n in response to strategy j, j = 1, . . ., m of player II while the payoff of player II varies within [neg( a ij )] with payoff matrix neg( A Int ) such that the payoffs of two players sum up to a constant linguistic value ( g , 0).

Remark 3. The bounds of the linguistic interval
ij ], i = 1, . . ., n, j = 1, . . ., m can be linguistic variables or 2-tuple linguistic variables and the intervals encompass all 2-tuple linguistic variables between the lower and upper bounds.In the latter case, it is called a two-player constant-sum interval-valued 2-tuple linguistic matrix game.Here, we present an example to illustrate the structure of an interval linguistic matrix game for the better understanding of readers.
Example 1.Consider the two contending insurance schemes proposed by two companies in the insurance market.The companies are planning to launch the schemes in the coming few months.The available options for the commencement are one month, two months and three months from now.The payoffs reflect how much market share a company can anticipate in view of the month of introducing the respective schemes.Since the estimation of the payoffs as 2-tuple linguistic variables is not always possible, the payoffs of the companies appear in the form of linguistic intervals from a predefined linguistic term set LT = {Very Low: VL, Low: L, Average: Avg, High: H, Very High: VH}.Let the payoff matrix of company I be as follows: For instance, if company I and company II launch the corresponding insurance schemes in the second and third months, respectively, then company I is expected to receive Low to Very High (i.e., [ a ]) market share.The payoff of company I, in case companies I and II decide to introduce their schemes in the first and third months, respectively, is anticipated to be Very High as it can be easily expressed as linguistic interval [V H, V H].Furthermore, the payoff matrix corresponding to company II is expressed as follows: The matrix neg( A Int ) suggests that on selecting second and third months by companies I and II, respectively, to launch their schemes, company II will receive Very Low to High market share.In particular, if company I gets High (H ∈ [L, V H]) market share on the given selection of months by the two companies, company II will receive neg(H) = L ∈ [VL, H] market shares as it is a constant-sum interval linguistic matrix game.The interval payoffs provide a range of expected benefits to the players.

Linguistic Linear Programming Approach to Solve Two-Player Constant-Sum Interval-Valued 2-Tuple Linguistic Matrix Game
This section proposes the LLP formulation of a constant-sum interval-valued linguistic matrix game G Int .Suppose the predefined linguistic term set with granularity g be LT = { 0 , 1 , . . ., g }.Here, g is considered as a positive even integer.Consider the interval-valued linguistic payoff matrix of player I as follows: For any given linguistic payoffs ij ], i = 1, . . ., n, j = 1, . . ., m, a linguistic payoff matrix, A, is given as It is evident from Equations ( 4) and ( 5) that the lower value of the linguistic game with payoff matrix A is function of the values a ij , i = 1, . . ., n, j = 1, . . ., m.Hence, the value v − can also be written as Additionally, the optimal strategy x * ∈ S n of player I can also be considered as a function of the a ij values, i.e., x * = x * ( A) = x * ( a ij , i = 1, . . ., n, j = 1, . . ., m).

Furthermore, consider the payoff matrices
, the lower value, the upper value and hence the linguistic value of the game is a non-decreasing function of the values ãij ; i = 1, . . ., n, j = 1, . . ., m,.The validation of the inequalities can be easily performed in the light of Equations ( 5) and (7).
In view of the above discussion, it can be concluded that the lower value, upper value and value of the interval game G Int results in being closed intervals of linguistic variables and can be defined as follows.
Definition 10.Given a two-player constant-sum interval-valued linguistic matrix game G Int = (S n , S m , LT, A Int ), the interval-valued linguistic lower value, v − Int , and the interval-valued linguistic upper value, v + Int of the game G Int depicts the ranges of the gain-floor of player I and loss-ceiling of player II, respectively.The values are computed as follows: where A Here, the optimal pure strategies for the interval-valued linguistic payoff matrix A Int is challenging to deduce, whereas the optimal pure strategy for A (L) and A (U) could be easily computed.Now, we elucidate the above discussion via an example.The corresponding matrices of lower terms and upper terms i.e., A (L) and A (U) are given below: 302

Example 2 .
Consider the illustration given in Example 1.The interval-valued linguistic payoff matrix of company I is as follows: the above models, y * (L) and y * (U) can be calculated along withv + Int = [ v +(L) , v +(U) ]. and max V (U) = Y