Abstract

2×𝑛 or 𝑚×2 interval matrix games are considered, and a graphical method for solving such games is given. Interval matrix game is the interval generation of classical matrix games. Because of uncertainty in real-world applications, payoffs of a matrix game may not be a fixed number. Since the payoffs may vary within a range for fixed strategies, an interval-valued matrix can be used to model such uncertainties. In the literature, there are different approaches for the comparison of fuzzy numbers and interval numbers. In this work, the idea of acceptability index is used which is suggested by Sengupta et al. (2001) and Sengupta and Pal (2009), and in view of acceptability index, well-known graphical method for matrix games is adapted to interval matrix games.

1. Introduction

The simplest game is finite, two-person, zero-sum game. There are only two players, player I and player II, and it can be denoted by a matrix. Thus, such a game is called matrix game. Matrix games have many useful applications, especially in decision-making systems. In usual matrix game theory, all the entries of the payoff matrix are assumed to be exactly given. However, in real-world applications, we often encounter the case where the information on the required data includes imprecision or uncertainty because of uncertain environment. Hence, outcomes of a matrix game may not be a fixed number even though the players do not change their strategies. Therefore, interval-valued matrix, whose entries are closed intervals, is proposed by many researchers to model this kind of uncertainty (see [1–6]).

The solution methods of interval matrix games are studied by many authors. Most of solution techniques are based on linear programming methods for interval numbers (see [1, 2, 5, 6]). We present, in this paper, a simplistic graphical method for solving 2×𝑛 or 𝑚×2 interval matrix games and provide a numerical example to exemplify the obtained algorithm.

1.1. Interval Numbers

An extensive research and wide coverage on interval arithmetic and its applications can be found in [7]. Here, we only define interval numbers and some necessary operations on interval numbers.

An interval number is a subset of real line of the form î€ºğ‘Žğš=−,ğ‘Ž+=î€½ğ‘¥âˆˆâ„âˆ¶ğ‘Žâˆ’â‰¤ğ‘¥â‰¤ğ‘Ž+,(1.1) where ğ‘Žâˆ’,ğ‘Ž+∈ℝ and ğ‘Žâˆ’â‰¤ğ‘Ž+. If ğ‘Žâˆ’=ğ‘Ž+, then 𝐚=[ğ‘Ž,ğ‘Ž] is a real number.

Mid-point m(𝐚) and radius r(𝐚) of interval number 𝐚 is defined as ğ‘Žm(𝐚)=−+ğ‘Ž+2ğ‘Ž,r(𝐚)=+âˆ’ğ‘Žâˆ’2,(1.2) respectively.

Elementary arithmetic operation ⋆∈{+,−,×,÷} between two interval numbers 𝐚 and 𝐛 is given by 𝐚⋆𝐛={ğ‘Žâ‹†ğ‘âˆ¶ğ‘Žâˆˆğš,𝑏∈𝐛}.(1.3) For division operator, it is assumed that 0∉𝐛. For any scalar number 𝛼, 𝛼𝐚 is defined as 𝛼𝐚=ğ›¼ğ‘Žâˆ’,ğ›¼ğ‘Ž+,for𝛼≥0,ğ›¼ğ‘Ž+,ğ›¼ğ‘Žâˆ’î€»,for𝛼<0.(1.4)

1.2. Comparison of Interval Numbers

To compare strategies and payoffs for an interval matrix game, we need a notion of an interval ordering relation that corresponds to the intuitive notion of a better possible outcome or payoff.

A brief comparison on different interval orders is given in [8, 9] on the basis of decision maker’s opinion.

1.2.1. Disjoint Intervals

Let 𝐚=[ğ‘Žâˆ’,ğ‘Ž+] and 𝐛=[𝑏−,𝑏+] be two disjoint interval numbers. Then, it is not difficult to define transitive order relation over these intervals, as 𝐚 is strictly less than 𝐛 if and only if ğ‘Ž+<𝑏−. This is denoted by 𝐚<𝐛, and it is an extension of “<” on the real line.

1.2.2. Nested Intervals

Let 𝐚=[ğ‘Žâˆ’,ğ‘Ž+] and 𝐛=[𝑏−,𝑏+] be two interval numbers such that ğ‘Žâˆ’â‰¤ğ‘âˆ’<𝑏+â‰¤ğ‘Ž+. Then, 𝐛⊆𝐚. Here, the set inclusion only describes the condition that 𝐛 is nested in 𝐚, but it cannot order 𝐚 and 𝐛 in terms of value.

Let 𝐚 and 𝐛 be two cost intervals, and the minimum cost interval is to be chosen. (i)If the decision maker (DM) is optimistic, then he/she will prefer the interval with maximum width along with the risk of more uncertainty giving less importance. (ii)If DM is pessimistic, then he/she will pay more attention on more uncertainty. That is, on the right hand points of the intervals, and he/she will choose the interval with minimum width.

The case will be reverse when 𝐚 and 𝐛 represent profit intervals. Therefore, we can define the ranking order of 𝐚 and 𝐛 as 𝐚𝐛=𝐚,iftheDMisoptimistic,𝐛,iftheDMispessimistic.(1.5) Here, the notation “⋁” denotes the maximum among the interval numbers 𝐚 and 𝐛. Similarly, we can write 𝐚𝐛=𝐛,iftheDMisoptimistic,𝐚,iftheDMispessimistic.(1.6) Likewise, the notation “⋀” denotes the minimum among the interval numbers 𝐚 and 𝐛.

1.2.3. Partially Overlapping Intervals

The above-mentioned order relations cannot explain ranking between two overlapping closed intervals.

Here, we use the acceptability index idea suggested by [8, 10]. This comparison method is mainly based on the mid-points of the intervals.

Let 𝕀ℝ be the set of all interval numbers. The function 𝒜∶𝕀ℝ×𝕀ℝ⟶ℝ,𝒜(𝐚≺𝐛)=m(𝐛)−m(𝐚)r(𝐚)+r(𝐛),(1.7)r(𝐚)+r(𝐛)≠0 is called acceptability function.

For m(𝐛)≥m(𝐚), the number 𝒜(𝐚≺𝐛) is called the grade of acceptability of the interval number 𝐚 to be inferior to the second interval number 𝐛.

By the definition of 𝒜, for any interval numbers 𝐚 and 𝐛, we have (i)𝒜(𝐚≺𝐛)≥1 for m(𝐛)>m(𝐚) and ğ‘Ž+≤𝑏−,(ii)𝒜(𝐚≺𝐛)∈(0,1) for m(𝐛)>m(𝐚) and ğ‘âˆ’â‰¤ğ‘Ž+, (iii)𝒜(𝐚≺𝐛)=0 for m(𝐛)=m(𝐚). In this case, if r(𝐚)=r(𝐛), then 𝐚 is identical with 𝐛. If r(𝐚)≠r(𝐛), then the intervals 𝐚 and 𝐛 are noninferior to each other. In this case, the acceptability index becomes insignificant, so DM has to negotiate with the widths of 𝐚 and 𝐛.Let 𝐚 and 𝐛 be two cost intervals, and minimum cost interval is to be chosen. If the DM is optimistic, then he/she will prefer the interval with maximum width along with the risk of more uncertainty giving less importance. Similarly, if the DM is pessimistic, then he/she will pay more attention on more uncertainty, that is, on the right end points of the intervals, and will choose the interval with minimum width. The case will be reverse when 𝐚 and 𝐛 represent profit intervals.

Example 1.1. Let 𝐚=[−5,−3] and 𝐛=[2,6] be two intervals. Then, 𝒜(𝐚≺𝐛)=4−(−4)=81+23>1.(1.8) Thus, the DM accept the decision that 𝐚 is less than 𝐛 with full satisfaction.

Example 1.2. Let 𝐚=[2,8] and 𝐛=[1,15] be two intervals. Then, 𝒜(𝐚≺𝐛)=8−53+7=0.3∈(0,1).(1.9) Hence, the DM accept the decision that 𝐚 is less than 𝐛 with grade of satisfaction 0.3.

Example 1.3. Let 𝐚=[2,18] and 𝐛=[5,15] be two intervals. Then, 𝒜(𝐚≺𝐛)=0. Here, both of the intervals are noninferior to each other. In this case, the DM has to negotiate with the widths of 𝐚 and 𝐛 as listed inTable 1.
These can be written explicitly as ğšî˜âŽ§âŽªâŽªâŽ¨âŽªâŽªâŽ©ğ›=𝐛,if𝒜(𝐚≺𝐛)>0,𝐚,if𝒜(𝐚≺𝐛)=0,r(𝐚)<r(𝐛),DMispessimistic,𝐛,if𝒜(𝐚≺𝐛)=0,r(𝐚)<r(𝐛),DMisoptimistic,(1.10) where, 𝐚⋁𝐛 denotes the maximum of the interval numbers 𝐚 and 𝐛. Similarly, ğšî—âŽ§âŽªâŽªâŽ¨âŽªâŽªâŽ©ğ›=𝐛,if𝒜(𝐛≺𝐚)>0,𝐚,if𝒜(𝐛≺𝐚)=0,r(𝐚)>r(𝐛),DMisoptimistic,𝐛,if𝒜(𝐛≺𝐚)=0,r(𝐚)>r(𝐛),DMispessimistic.(1.11) Here, the notation 𝐚⋀𝐛 represents the minimum of the interval numbers 𝐚 and 𝐛.

Proposition 1.4. The interval ordering by the acceptability index defines a partial order relation on 𝕀ℝ.

Proof. (i) If 𝒜(𝐚≺𝐛)=0 and r(𝐚)=r(𝐛), then 𝐚≡𝐛, and we say that 𝐚 and 𝐛 are noninferior to each other. Hence, it is reflexive.
(ii) For any interval number 𝐚 and 𝐛, 𝒜(𝐚≺𝐛)>0, and 𝒜(𝐛≺𝐚)>0 implies 𝐚=𝐛. Therefore, 𝒜 is antisymmetric.
(iii) For any interval numbers 𝐚,𝐛,𝐜∈𝕀ℝ if 𝒜(𝐚≺𝐛)=m(𝐛)−m(𝐚)r(𝐛)+r(𝐚)≥0,𝒜(𝐛≺𝐜)=m(𝐜)−m(𝐛)r(𝐜)+r(𝐛)≥0,(1.12) then m(𝐜)−m(𝐚)=[m(𝐜)−m(𝐛)]+[m(𝐛)−m(𝐚)]≥0. Hence, 𝒜(𝐚≺𝐜)=m(𝐜)−m(𝐚)r(𝐜)+r(𝐚)≥0.(1.13) Thus, 𝒜 is transitive.

On the other hand, acceptability index must not interpreted as difference operator of real analysis. Indeed, while 𝒜(𝐚≺𝐛)≥0 and 𝒜(𝐛≺𝐜)≥0, the inequality 𝒜(𝐚≺𝐜)≥max{𝒜(𝐚≺𝐛),𝒜(𝐛≺𝐜)}(1.14) may not hold.

Actually, let 𝐚=[−2,2],𝐛=[−1,21/20]and𝐜=[−1/2,3/4], then1𝒜(𝐚≺𝐛)=2121,𝒜(𝐛≺𝐜)=33,(1.15) but1𝒜(𝐚≺𝐜)=<22133.(1.16)

For any two interval numbers 𝐚 and 𝐛 from 𝕀ℝ, either 𝒜(𝐚≺𝐛)>0, or 𝒜(𝐚≺𝐛)=𝒜(𝐛≺𝐚)=0, or 𝒜(𝐛≺𝐚)>0. Also, 𝒜(𝐚≺𝐛)<0 can be interpreted as the interval number 𝐛 is inferior to the interval number 𝐚, since 𝒜(𝐛≺𝐚)>0.

Additionally, for any interval numbers 𝐚, 𝐛, 𝐜, and 𝐝, the following properties of acceptability index is obvious. 𝒜If𝒜(𝐚≺𝐛)≥0,𝒜(𝐜≺𝐝)≥0⟹𝒜(𝐚+𝐜≺𝐛+𝐝)≥0,(𝐚≺𝐛)≥0⟹𝒜(𝐚+𝐜≺𝐛+𝐜)≥0,𝒜(𝐚≺𝐛)≥0⟹𝒜(−1⋅𝐛≺−1⋅𝐚)≥0.(1.17)

1.3. Matrix Games

Game theory is a mathematical discipline which studies situations of competition and cooperation between several involved parties, and it has many applications in broad areas such as strategic warfares, economic or social problems, animal behaviours, political voting systems. It is accepted that game theory starts with the von Neumann’s study on zero-sum games (see [11]), in which he proved the famous minimax theorem for zero-sum games. It was also basis for [12].

The simplest game is finite, two-person, zero-sum game. There are only two players, player I and player II, and it can be denoted by a matrix. Thus, such a game is called matrix game. More formally, a matrix game is an 𝑚×𝑛 matrix 𝐺 of real numbers.

A (mixed) strategy of player I is a probability distribution 𝑥 over the rows of 𝐺, that is, an element of the set 𝑋𝑚=𝑥𝑥=1,…,𝑥𝑚∈ℝ𝑚∶𝑥𝑖≥0,𝑚𝑖=1𝑥𝑖=1.(1.18) Similarly, a strategy of player II is a probability distribution 𝑦 over the columns of 𝐺, that is, an element of the set 𝑌𝑛=𝑦𝑦=1,…,𝑦𝑛∈ℝ𝑛∶𝑦𝑖≥0,𝑛𝑖=1𝑦𝑖=1.(1.19)

A strategy 𝑥 of player I is called pure if it does not involve probability, that is, 𝑥𝑖=1 for some 𝑖=1,…,𝑚, and it is denoted by I𝑖. Similarly, pure strategies of player II is denoted by II𝑗 for 𝑗=1,…,𝑛.

If player I plays row 𝑖 (i.e., pure strategy 𝑥=(0,0,…,𝑥𝑖=1,0,…,0) and player II plays column 𝑗 (i.e., pure strategy 𝑦=(0,0,…,𝑦𝑗=1,0,…,0), then player I receives payoff 𝑔𝑖𝑗, and player II pays 𝑔𝑖𝑗, where 𝑔𝑖𝑗 is the entry in row 𝑖 and column 𝑗 of matrix 𝐺.

If player I plays strategy 𝑥 and player II plays strategy 𝑦, then player I receives the expected payoff 𝑔(𝑥,𝑦)=𝑥𝑇𝐺𝑦,(1.20) where 𝑥𝑇 denotes the transpose of 𝑥.

A strategy 𝑥∗ is called maximin strategy of player I in matrix game 𝐺 if 𝑥min∗𝑇𝐺𝑦,𝑦∈𝑌𝑛𝑥≥min𝑇𝐺𝑦,𝑦∈𝑌𝑛,(1.21) for all 𝑥∈𝑋𝑚 and a strategy 𝑦∗ is called minimax strategy of player II in matrix game 𝐺 if 𝑥max𝑇𝐺𝑦∗,𝑥∈𝑋𝑚𝑥≤max𝑇𝐺𝑦,𝑥∈𝑋𝑚,(1.22) for all 𝑦∈𝑌𝑛.

Therefore, a maximin strategy of player I maximizes the minimal payoff of player I, and a minimax strategy of player II minimizes the maximum that player II has to pay to player I.

Neumann proved that for every matrix game 𝐺, there is a real number 𝜈 with the following properties.(1)A strategy 𝑥 of player I guarantees a payoff of at least 𝜈 to player I (i.e., 𝑥𝑇𝐺𝑦≥𝜈 for all strategies 𝑦 of player II) if and only if 𝑥 is a maximin strategy. (2)A strategy 𝑦 of player II guarantees a payment of at most 𝜈 by player II to player I (i.e., 𝑥𝑇𝐺𝑦≤𝜈 for all strategies 𝑥 of player I) if and only if 𝑦 is a minimax strategy.

Hence, player I can obtain a payoff at least 𝜈 by playing maximin strategy, and player II can guarantee to pay not more than 𝜈 by playing minimax strategy. For these reasons, the number 𝜈 is also called the value of the game 𝐺.

A position (𝑖,𝑗) is called saddle point if 𝑔𝑖𝑗≥𝑔𝑘𝑗 for all 𝑘=1,…,𝑚 and 𝑔𝑖𝑗≤𝑔𝑖𝑙 for all 𝑙=1,…,𝑛, that is, if 𝑔𝑖𝑗 is maximal in its column 𝑗 and minimal in its row 𝑖. Evidently, if (𝑖,𝑗) is a saddle point, then 𝑔𝑖𝑗 must be the value of the game.

1.4. Interval Matrix Games

Interval matrix game is the interval generation of classical matrix games, and it is the special case of fuzzy games.

Here, we consider a nonzero sum interval matrix game with two players, and we assume that player I is maximizing or optimistic player and player II is minimizing or pessimistic player. We assume that player I will try to make maximum profit and player II will try to minimize the loss.

The two person interval matrix game is defined by 𝑚×𝑛 matrix 𝐆 whose entries are interval numbers.

Let 𝐆 be an interval matrix game 𝐆=II1II2⋯II𝑛I1I2⋮Iğ‘šâŽ›âŽœâŽœâŽœâŽœâŽœâŽğ 11𝐠21⋮𝐠𝑚1𝐠12𝐠22⋮𝐠𝑚2⋯⋯⋱⋯𝐠1𝑛𝐠2ğ‘›â‹®ğ ğ‘šğ‘›âŽžâŽŸâŽŸâŽŸâŽŸâŽŸâŽ ,(1.23)

and 𝑥∈𝑋𝑚, 𝑦∈𝑌𝑛; that is, 𝑥 and 𝑦 are strategies for players I and II. Then, the expected payoff for player I is defined by 𝐠(𝑥,𝑦)=𝑥𝑇𝐆𝑦=𝑖𝑗𝑥𝑖𝑦𝑗𝐠𝑖𝑗.(1.24) Here,∑ denotes the Minkowski’s sum of intervals.

Example 1.5. Let 𝐆=II1II2II3II4I1I2I3⎛⎜⎜⎜⎝[][][][][][][][][][][][]⎞⎟⎟⎟⎠−10,1−9,8−3,−2−8,7−1,7−1,5−8,−73,8−2,2−6,78,9−3,7(1.25) be interval matrix game.
For this game, if player I plays second row (𝑥=(0,1,0)) and player II plays third column (𝑦=(0,0,1,0)), then player I receives, and correspondingly, player II pays a payoff 𝐠(I2,II3)∈[3,8].
On the other hand, for pair of strategies 𝑥=(1/2,1/2,0) and 𝑦=(0,1/3,1/3,1/3), the expected payoff for player I belongs to 𝐠(𝑥,𝑦)=[−2,31/6].

Now, define 𝝂𝐿=𝑥∈𝑋𝑚𝑦∈𝑌𝑛𝝂𝐠(𝑥,𝑦),𝑈=𝑦∈𝑌𝑛𝑥∈𝑋𝑚𝐠(𝑥,𝑦),(1.26) to be lower and upper value of interval matrix game.

It is naturally clear that 𝒜(𝝂𝐿≺𝝂𝑈)≥0.

In addition, if 𝝂𝐿=𝝂𝑈=𝝂, then 𝝂 is called the value of interval matrix game.

Theorem 1.6 (Fundamental Theorem). Let 𝐆 be an interval matrix game. Then, ⋁𝑥∈𝑋𝑚⋀𝑦∈𝑌𝑛𝐠(𝑥,𝑦) and ⋀𝑦∈𝑌𝑛⋁𝑥∈𝑋𝑚𝐠(𝑥,𝑦) both exist and are equal.

Let 𝝂 be the value of game. Then, the (𝑥∗,𝑦∗,𝝂)∈𝑋𝑚×𝑌𝑛×𝕀ℝ is called the solution of interval matrix game if 𝑦∈𝑌𝑛𝐠𝑥∗=,𝑦𝑥∈𝑋𝑚𝑦∈𝑌𝑛𝐠(𝑥,𝑦)=𝝂,(1.27)𝑥∈𝑋𝑚𝐠𝑥,𝑦∗=𝑦∈𝑌𝑛𝑥∈𝑋𝑚𝐠(𝑥,𝑦)=𝝂.(1.28) Furthermore, the strategies 𝑥∗ and 𝑦∗ are called optimal strategies for players.

In view of Theorem 1.6, we can conclude that every two person interval matrix game has a solution.

The value of the interval matrix game 𝐆=II1II2II3II4I1I2I3⎛⎜⎜⎜⎝[][][][][][][][][][][][]⎞⎟⎟⎟⎠−10,1−9,8−3,−2−8,7−1,7−1,5−8,−73,8−2,2−6,78,9−3,7(1.29) is [−9,8]. On the other hand, if we consider the left and right endpoints of the entries of interval matrix 𝐆 as two different matrix games, that is, 𝐺−=II1II2II3II4I1I2I3⎛⎜⎜⎝38⎞⎟⎟⎠,𝐺−10−9−3−8−1−1−8−2−6−3+=II1II2II3II4I1I2I3⎛⎜⎜⎝1877582797⎞⎟⎟⎠,−2−7(1.30) then value of these matrix games are −3 and 7, respectively. Hence, there is no trivial relation between the value of interval matrix game and the value of endpoint matrix games.

2. A Graphical Method for Solving Interval Matrix Games

Let 𝐆=II1II2⋯II𝑛I1I2⋮Iğ‘šâŽ›âŽœâŽœâŽœâŽœâŽœâŽğ 11𝐠21⋮𝐠𝑚1𝐠12𝐠22⋮𝐠𝑚2⋯⋯⋱⋯𝐠1𝑛𝐠2ğ‘›â‹®ğ ğ‘šğ‘›âŽžâŽŸâŽŸâŽŸâŽŸâŽŸâŽ (2.1) be an interval matrix game.

Let 𝝂 be value of 𝑚×𝑛 interval matrix game 𝐆. In general, the width of 𝝂 can vary. To normalize the width of 𝝂 in order to investigate a method for solving such games, from now on, we will assume that all entries of 𝐆 are of same length; that is, 𝑔+𝑖𝑗=𝑔−𝑖𝑗+𝜆 for some 𝜆>0 and 𝑖=1,2,…,𝑚, 𝑗=1,2,…,𝑛.

The solution methods of interval matrix games are studied by many authors. Most of solution techniques are based on linear programming methods for interval numbers (see [1–4, 6]).

A useful idea in solving 2×𝑛 interval matrix games is that of worthwhile strategies. A worthwhile strategy is a pure strategy which appears with positive probability in an optimal strategy.

The following proposition is the important property about such strategies.

Proposition 2.1. When a worthwhile strategy plays an optimal strategy, the payoff is the value of the game.

It means that if (𝑥∗,𝑦∗,𝝂)∈𝑋𝑚×𝑌𝑛×𝕀ℝ is the solution of interval matrix game (2.1) and the pure strategies I𝑖∗ and II𝑗∗ are worthwhile strategies, then the equalities 𝐠𝑥∗,𝑦∗I=𝐠𝑖∗,𝑦∗𝐠𝑥=𝝂,(2.2)∗,𝑦∗𝑥=𝐠∗,II𝑗∗=𝝂(2.3) hold.

Proof. We assume without loss of generality that player I has an optimal strategy 𝑥∗=(𝑥1,𝑥2,…,𝑥𝑘,0,…,0). Here, 𝑥𝑖>0, 𝑖=1,2,…,𝑘 and ∑𝑘𝑖=1𝑥𝑖=1. We also assume that player II has an optimal strategy 𝑦∗ and value of the game is 𝝂. Hence, 𝐠(𝑥∗,𝑦∗)=𝝂.
Furthermore, for any pure strategy I𝑖 (1≤𝑖≤𝑘) of player I, we have 𝒜𝐠I𝑖,𝑦∗≺𝝂≥0,(2.4) since player II playing optimal strategy 𝑦∗.
We set 𝐠(I𝑖,𝑦∗)=𝝂𝑖. Then, using following equality: 𝑥𝝂=𝐠∗,𝑦∗=𝑘𝑖=1𝑥𝑖𝐠I𝑖,𝑦∗=𝑘𝑖=1𝑥𝑖𝝂𝑖,(2.5) and (2.4) we find 𝒜𝝂=𝑘𝑖=1𝑥𝑖𝝂𝑖≺𝑘𝑖=1𝑥𝑖𝝂=𝝂≥0.(2.6) Thus, acceptability index must be 0, and so 𝝂𝑖=𝝂 for all 𝑖=1,2,…,𝑘. Therefore, playing the worthwhile strategy I𝑖 against optimal strategy 𝑦∗ gives a payoff that is the value of the game.
The proof of (2.3) is similar.

Theorem 2.2. If 𝝂 is value of 𝑚×𝑛 interval matrix game (2.1), then the equalities 𝑥∈𝑋𝑚𝑦∈𝑌𝑛𝐠(𝑥,𝑦)=𝑥∈𝑋𝑚𝑗=1,2,…,𝑛𝐠𝑥,II𝑗=𝝂,(2.7)𝑦∈𝑌𝑛𝑥∈𝑋𝑚𝐠(𝑥,𝑦)=𝑦∈𝑌𝑛𝑖=1,2,…,𝑚𝐠𝐼𝑖,𝑦=𝝂(2.8) hold.

Proof. By Theorem 1.6, we know that interval matrix game (2.1) has a solution. Let us denote the solution of (2.1) by (𝑥∗,𝑦∗,𝝂). Then, we have 𝑥∈𝑋𝑚𝑦∈𝑌𝑛𝐠(𝑥,𝑦)=𝑦∈𝑌𝑛𝑥∈𝑋𝑚𝒜𝑥𝐠(𝑥,𝑦)=𝝂,(2.9)𝝂≺𝐠∗,𝑦≥0,∀𝑦∈𝑌𝑛𝒜𝐠,(2.10)𝑥,𝑦∗≺𝝂≥0,∀𝑥∈𝑋𝑚.(2.11) Since II𝑗∈𝑌𝑛 for all 𝑗=1,2,…,𝑛, then using (2.10), we get 𝒜𝑥𝝂≺𝐠∗,II𝑗≥0,(2.12) for all 𝑗=1,2,…,𝑛. Therefore, it follows that 𝒜𝝂≺𝑗=1,2,…,𝑛𝐠𝑥∗,II𝑗≥0.(2.13) Using (2.13), we find ğ’œâŽ›âŽœâŽœâŽî˜ğ‚â‰ºğ‘¥âˆˆğ‘‹ğ‘šî—ğ‘—=1,2,…,𝑛𝐠𝑥∗,IIğ‘—âŽžâŽŸâŽŸâŽ î€¸î€¾â‰¥0.(2.14) Now, let us show that ğ’œâŽ›âŽœâŽœâŽî˜ğ‚â‰ºğ‘¥âˆˆğ‘‹ğ‘šî—ğ‘—=1,2,…,𝑛𝐠𝑥∗,IIğ‘—âŽžâŽŸâŽŸâŽ î€¸î€¾=0.(2.15) Suppose, contrary to what we wish to show, that ğ’œâŽ›âŽœâŽœâŽî˜ğ‚â‰ºğ‘¥âˆˆğ‘‹ğ‘šî—ğ‘—=1,2,…,𝑛𝐠𝑥∗,IIğ‘—âŽžâŽŸâŽŸâŽ î€¸î€¾>0.(2.16) For notational simplicity, we assume that 𝝂∗=𝑥∈𝑋𝑚𝑗=1,2,…,𝑛𝐠𝑥∗,II𝑗.(2.17) Then, there exists ̃𝑥=(̃𝑥1,…,̃𝑥𝑚)∈𝑋𝑚 such that 𝝂∗=𝑗=1,2,…,𝑛𝐠̃𝑥,II𝑗.(2.18) Since 𝒜(𝝂∗≺𝐠(̃𝑥,II𝑗))≥0 and acceptability index is transitive, then we obtain 𝒜𝝂≺𝐠̃𝑥,II𝑗>0,(2.19) for all 𝑗=1,2,…,𝑛.
Additionally, for all 𝑦=(𝑦1,…,𝑦𝑛)∈𝑌𝑛 we can write 𝐠(̃𝑥,𝑦)=𝑛𝑚𝑗=1𝑖=1̃𝑥𝐠𝑖𝑗𝑦𝑗=𝑛𝑗=1𝑚𝑖=1̃𝑥𝐠𝑖𝑗𝑦𝑗=𝑛𝑗=1𝐠̃𝑥,II𝑗𝑦𝑗.(2.20) Therefore, we have 𝒜𝝂∗≺𝐠(̃𝑥,𝑦)≥0,(2.21) for all 𝑦∈𝑌𝑛. Since 𝒜(𝝂≺𝝂∗)>0, we get from (2.11) that 𝒜𝐠𝑥,𝑦∗≺𝝂∗≥0,(2.22) for all 𝑥∈𝑋𝑚.
Using (1.28), (2.21), and (2.22), we conclude that (̃𝑥,𝑦∗,𝝂∗) is also a solution of (2.1), contradicting the fact that value of the game is unique. This concludes that (2.15) holds. Hence, from (2.9) and (2.15), we obtain the validity of (2.7).
By completely analogous arguments, we can also obtain (2.8).

Now, using Theorem 2.2, we can show how to solve interval matrix games, where at least one of the players has two pure strategies.

Let us consider 𝐆=II1II2⋯II𝑛I1I2𝐠11𝐠21𝐠12𝐠22⋯⋯𝐠1𝑛𝐠2𝑛,(2.23)2×𝑛 interval matrix game.

We denote the value of 𝐆 by the symbol 𝝂∗.

Since player I has only two strategies, for any 𝑥=(𝑥1,𝑥2)∈𝑋2, we can write 𝑥=(𝑡,1−𝑡), 𝑡∈[0,1]. Then, we obtain 𝐠𝑥,II𝑗=𝑡𝐠1𝑗+(1−𝑡)𝐠2𝑗,(2.24) for 𝑡∈[0,1]. Setting 𝝓𝑗(𝑡)=𝐠𝑥,II𝑗,𝑗=1,…,𝑛,(2.25) we get 𝝓𝑗(𝑡)=𝑡𝐠1𝑗+(1−𝑡)𝐠2𝑗,𝑗=1,…,𝑛.(2.26) Using Theorem 2.2, we have 𝝂∗=𝑥∈𝑋2𝑗=1,…,𝑛𝐠𝑥,II𝑗=[]𝑡∈0,1𝑗=1,…,𝑛𝝓𝑗(𝑡).(2.27) Therefore, the following algorithm can be used to find solution of 2×𝑛 interval matrix games.

Algorithm 2.3. Step 1. Plot 𝝓𝑗∶[0,1]→𝕀ℝ for all 𝑗=1,…,𝑛 on the plane.Step 2. Draw the function 𝝍∶[0,1]→𝕀ℝ by setting 𝝍(𝑡)=𝑗=1,…,𝑛𝝓𝑗(𝑡),(2.28) (see Figure 1).Step 3. By means of definition of 𝝍 calculate the ⋁{𝝍(𝑡)} on [0,1]; that is, find 𝝂∗=[]𝑡∈0,1{𝝍(𝑡)}.(2.29)Step 4. Using definition of 𝝍, calculate the 𝑡∗∈[0,1], where 𝝍 attains its maximum. Thus, 𝑥∗=(𝑡∗,1−𝑡∗), and the optimal strategy of player I is obtained.Step 5. Using the property of worthwhile strategies, find the optimal strategy of player II.

To see how this works, let us solve a game.

3. Numerical Example

Example 3.1. Consider the 2×5 interval matrix game 𝐆=II1II2II3II4II5I1I2[][][][][][][][][][].−2,01,3−1,10,20,2−1,15,7−3,−1−4,−24,6(3.1) In this game, no player has a dominant strategy; that is, there is no strategy which is always better than any other strategy, for any profile of other players' actions. Additionally, since 𝑖=1,2𝑗=1,2,…,5𝐠I𝑖,II𝑗={[],[][],−4,−2−3,−1}=−3,−1𝑗=1,2,…,5𝑖=1,2𝐠I𝑖,II𝑗={[],[],[],[],[][],1,30,20,25,74,6}=0,2(3.2)𝐆 has no saddle point.

If player I's optimal strategy is 𝑥=(𝑡,1−𝑡), 𝑡∈[0,1], then for 𝑗=1,2,…,5, we get 𝝓𝑗(𝑡)=𝐠𝑥,II𝑗=𝑡𝐠1𝑗+(1−𝑡)𝐠2𝑗.(3.3) Hence, 𝝓1[],𝝓(𝑡)=1−3𝑡,3−3𝑡2[],𝝓(𝑡)=−𝑡,2−𝑡3[],𝝓(𝑡)=−1+𝑡,1+𝑡4[],𝝓(𝑡)=−3+8𝑡,−1+8𝑡5[].(𝑡)=4−8𝑡,6−8𝑡(3.4) We can show this graphically, plotting each payoff as a function of 𝑡 (see Figure 2). Then, we get []𝝓𝝍∶0,1⟶𝕀ℝ,𝝍(𝑡)=1(𝑡),𝝓2(𝑡),𝝓3(𝑡),𝝓4(𝑡),𝝓5=âŽ§âŽªâŽªâŽªâŽªâŽªâŽ¨âŽªâŽªâŽªâŽªâŽªâŽ©ğ“(𝑡)42(𝑡),𝑡∈0,7,𝝓32(𝑡),𝑡∈7,12,𝝓11(𝑡),𝑡∈2,35,𝝓53(𝑡),𝑡∈5.,1(3.5) see Figure 3.

Now, we can calculate the ⋁{𝝍(𝑡)} on [0,1]; that is, we can find the value of the game 𝝂∗=[]𝑡∈0,1{𝝍(𝑡)}.(3.6)

Since, the function 𝝍(𝑡) attains its maximum at 𝑡∗=1/2, we obtain 𝝂∗=[]𝑡∈0,1𝝍(𝑡)=𝝓112=−12,32.(3.7)

Furthermore, the optimal strategy of player I is 𝑥∗=(𝑡∗,1−𝑡∗)=(1/2,1/2).

On the other hand, 𝐠𝑥∗,II1=−12,32=𝝂∗𝑥,𝐠∗,II2=−12,32=𝝂∗𝑥,𝐠∗,II3=−12,32=𝝂∗,𝐠𝑥∗,II4=[]1,3≠𝝂∗𝑥,𝐠∗,II5=[]0,2≠𝝂∗.(3.8)

Therefore, using the property of worthwhile strategies, we obtain that only pure strategies II1, II2, and II3 are worthwhile strategies of player II.

If 𝑦∗=(𝑦1,𝑦2,𝑦3,0,0)∈𝑌5 is optimal strategy of player II, then the systems âŽ§âŽªâŽªâŽªâŽ¨âŽªâŽªâŽªâŽ©ğ‘¦1+𝑦2+𝑦3𝐠I=1,1,𝑦∗=𝝂∗,𝐠I2,𝑦∗=𝝂∗,𝑦1>0,𝑦2>0,𝑦3âŸ¹âŽ§âŽªâŽªâŽªâŽªâŽ¨âŽªâŽªâŽªâŽªâŽ©ğ‘¦>01+𝑦2+𝑦3=1,−2𝑦1−𝑦2,𝑦2+2𝑦3=−12,32,𝑦1−𝑦3,3𝑦1+2𝑦2+𝑦3=−12,32,𝑦1>0,𝑦2>0,𝑦3>0(3.9) are valid. Thus, 𝑦∗=−123+𝑠,2,1−2𝑠,𝑠,0,023<𝑠<4(3.10) is optimal strategy of player II.

As a consequence, we obtain 12,12,−123+𝑠,2,−2𝑠,𝑠,0,0−12,32,123<𝑠<4,(3.11) as the solution of the game (3.1).

4. Conclusion

In this paper, we have adapted graphical method for matrix games to interval matrix games by means of acceptability index notion. It is showed that by means of this method, solution of 2×𝑛 or 𝑚×2 interval matrix games can be easily calculated. Since the method can be easily programmed, it is very useful for interval matrix games even though players have at most two pure strategies after domination.