A Graphical Method for Solving Interval Matrix Games

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 𝐚=𝑎⁻,𝑎⁺=𝑥∈ℝ∶𝑎⁻≤𝑥≤𝑎⁺,(1.1) where 𝑎⁻,𝑎⁺∈ℝ and 𝑎⁻≤𝑎⁺. If 𝑎⁻=𝑎⁺, then 𝐚=[𝑎,𝑎] is a real number.

Mid-point m(𝐚) and radius r(𝐚) of interval number 𝐚 is defined as m(𝐚)=𝑎⁻+𝑎⁺2,r(𝐚)=𝑎⁺−𝑎⁻2,(1.2) respectively.

Elementary arithmetic operation ⋆∈{+,−,×,÷} between two interval numbers 𝐚 and 𝐛 is given by 𝐚⋆𝐛={𝑎⋆𝑏∶𝑎∈𝐚,𝑏∈𝐛}.(1.3) For division operator, it is assumed that 0∉𝐛. For any scalar number 𝛼, 𝛼𝐚 is defined as 𝛼𝐚=𝛼𝑎⁻𝛼𝑎₊,𝛼𝑎⁺,for𝛼≥0,,𝛼𝑎⁻,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=[𝑎⁻,𝑎⁺] 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 𝑎⁺<𝑏⁻. This is denoted by 𝐚<𝐛, and it is an extension of “<” on the real line.

1.2.2. Nested Intervals

Let 𝐚=[𝑎⁻,𝑎⁺] and 𝐛=[𝑏⁻,𝑏⁺] be two interval numbers such that 𝑎⁻≤𝑏⁻<𝑏⁺≤𝑎⁺. 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=𝐚𝐛,,iifftthheeDDMMiissoppetsismimisitsitci,c.(1.5) Here, the notation “⋁” denotes the maximum among the interval numbers 𝐚 and 𝐛. Similarly, we can write 𝐚𝐛=𝐛𝐚,,iifftthheeDDMMiisspoepstsimimisitsitcic,.(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 𝑎⁺≤𝑏⁻,(ii)𝒜(𝐚≺𝐛)∈(0,1) for m(𝐛)>m(𝐚) and 𝑏⁻≤𝑎⁺, (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)1+2=38>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 𝐚𝐛=⎧⎪⎪⎨⎪⎪⎩𝐛𝐚𝐛,,,iiifff𝒜𝒜𝒜(((𝐚𝐚𝐚≺≺≺𝐛𝐛𝐛)))>==000,,,rr((𝐚𝐚))<<rr((𝐛𝐛)),,DDMMiisspoepstismimisitsitci,c,(1.10) where, 𝐚⋁𝐛 denotes the maximum of the interval numbers 𝐚 and 𝐛. Similarly, 𝐚𝐛=⎧⎪⎪⎨⎪⎪⎩𝐛𝐚𝐛,,,iiifff𝒜𝒜𝒜(((𝐛𝐛𝐛≺≺≺𝐚𝐚𝐚)))>==000,,,rr((𝐚𝐚))>>rr((𝐛𝐛)),,DDMMiissoppestismimisitsitci,c.(1.11) Here, the notation 𝐚⋀𝐛 represents the minimum of the interval numbers 𝐚 and 𝐛.

Table 1 

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)=mr((𝐜𝐜))+−rm(𝐛(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], then𝒜(𝐚≺𝐛)=1121,𝒜(𝐛≺𝐜)=323,(1.15) but𝒜(𝐚≺𝐜)=121<323.(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. If𝒜(𝐚≺𝐛)≥0,𝒜𝒜𝒜(((𝐜𝐚𝐚≺≺≺𝐝𝐛𝐛)))≥≥≥000⟹⟹⟹𝒜𝒜𝒜(((𝐚−𝐚+1+⋅𝐜𝐜𝐛≺≺≺𝐛𝐛−++1𝐝𝐜))⋅≥𝐚≥)00≥,,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 𝑋_𝑚=𝑥=𝑥₁,…,𝑥_𝑚∈ℝ^𝑚∶𝑥_𝑖≥0,^𝑚_𝑖=1𝑥_𝑖=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 𝑌_𝑛=𝑦=𝑦₁,…,𝑦_𝑛∈ℝ^𝑛∶𝑦_𝑖≥0,^𝑛_𝑖=1𝑦_𝑖=1.(1.19)

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

If player I plays row 𝑖 (i.e., pure strategy 𝑥=(0,0,…,𝑥_𝑖=1,0,…,0) and player II plays column 𝑗 (i.e., pure strategy 𝑦=(0,0,…,𝑦_𝑗=1,0,…,0), then player I receives payoff 𝑔_𝑖𝑗, and player II pays 𝑔_𝑖𝑗, where 𝑔_𝑖𝑗 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)=𝑥^𝑇𝐺𝑦,(1.20) where 𝑥^𝑇 denotes the transpose of 𝑥.

A strategy 𝑥^∗ is called maximin strategy of player I in matrix game 𝐺 if min𝑥^∗^𝑇𝐺𝑦,𝑦∈𝑌_𝑛≥min𝑥^𝑇𝐺𝑦,𝑦∈𝑌_𝑛,(1.21) for all 𝑥∈𝑋_𝑚 and a strategy 𝑦^∗ is called minimax strategy of player II in matrix game 𝐺 if max𝑥^𝑇𝐺𝑦^∗,𝑥∈𝑋_𝑚≤max𝑥^𝑇𝐺𝑦,𝑥∈𝑋_𝑚,(1.22) for all 𝑦∈𝑌_𝑛.

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., 𝑥^𝑇𝐺𝑦≥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., 𝑥^𝑇𝐺𝑦≤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 𝑔_𝑖𝑗≥𝑔_𝑘𝑗 for all 𝑘=1,…,𝑚 and 𝑔_𝑖𝑗≤𝑔_𝑖𝑙 for all 𝑙=1,…,𝑛, that is, if 𝑔_𝑖𝑗 is maximal in its column 𝑗 and minimal in its row 𝑖. Evidently, if (𝑖,𝑗) is a saddle point, then 𝑔_𝑖𝑗 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⋯II_𝑛I₁I₂I⋮_𝑚⎛⎜⎜⎜⎜⎜⎝𝐠¹¹𝐠₂₁𝐠⋮_𝑚1𝐠₁₂𝐠₂₂𝐠⋮_𝑚2⋱⋯⋯⋯𝐠_1𝑛𝐠_2𝑛𝐠⋮_𝑚𝑛⎞⎟⎟⎟⎟⎟⎠,(1.23)

and 𝑥∈𝑋_𝑚, 𝑦∈𝑌_𝑛; 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)=𝑥^𝑇𝐆𝑦=_𝑖_𝑗𝑥_𝑖𝑦_𝑗𝐠_𝑖𝑗.(1.24) Here,∑ denotes the Minkowski’s sum of intervals.

Example 1.5. Let 𝐆=II¹II₂II₃II₄I₁I₂I₃⎛⎜⎜⎜⎝[[−[−−319,0,−,812]]][[[−−−811,,,775]]][−[−[832,,,−82]7]][[−−[863,,,977]]]⎞⎟⎟⎟⎠(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 𝝂_𝐿=_{𝑥∈𝑋𝑚𝑦}_∈𝑌𝑛𝝂𝐠(𝑥,𝑦),_𝑈=_{𝑦∈𝑌𝑛𝑥}_∈𝑋𝑚𝐠(𝑥,𝑦),(1.26) to be lower and upper value of interval matrix game.

It is naturally clear that 𝒜(𝝂_𝐿≺𝝂_𝑈)≥0.

In addition, if 𝝂_𝐿=𝝂_𝑈=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∈𝑌𝑛}𝐠(𝑥,𝑦) and ⋀_{𝑦∈𝑌𝑛}⋁_{𝑥∈𝑋𝑚}𝐠(𝑥,𝑦) both exist and are equal.

Let 𝝂 be the value of game. Then, the (𝑥^∗,𝑦^∗,𝝂)∈𝑋_𝑚×𝑌_𝑛×𝕀ℝ is called the solution of interval matrix game if _𝑦_∈𝑌𝑛𝐠𝑥^∗,𝑦=_{𝑥∈𝑋𝑚𝑦}_∈𝑌𝑛𝐠(𝑥,𝑦)=𝝂,(1.27)_{𝑥∈𝑋𝑚}𝐠𝑥,𝑦^∗=_{𝑦∈𝑌𝑛𝑥}_∈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⎛⎜⎜⎜⎝[[−[−−319,0,−,812]]][[[−−−811,,,775]]][−[−[832,,,−82]7]][[−−[863,,,977]]]⎞⎟⎟⎟⎠(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⁻=II¹II₂II₃II₄I₁I₂I₃𝐺⎛⎜⎜⎝−−−1930−−−811−−382−−863₊⎞⎟⎟⎠,=II¹II₂II₃II₄I₁I₂I₃⎛⎜⎜⎝−182775−827797⎞⎟⎟⎠,(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⋯II_𝑛I₁I₂I⋮_𝑚⎛⎜⎜⎜⎜⎜⎝𝐠¹¹𝐠₂₁𝐠⋮_𝑚1𝐠₁₂𝐠₂₂𝐠⋮_𝑚2⋱⋯⋯⋯𝐠_1𝑛𝐠_2𝑛𝐠⋮_𝑚𝑛⎞⎟⎟⎟⎟⎟⎠(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)∈𝑋_𝑚×𝑌_𝑛×𝕀ℝ is the solution of interval matrix game (2.1) and the pure strategies I_𝑖∗ and II_𝑗∗ are worthwhile strategies, then the equalities 𝐠𝐠𝑥𝑥^∗,𝑦^∗=𝐠I_𝑖∗,𝑦^∗=𝝂,(2.2)_∗,𝑦^∗=𝐠𝑥^∗,II_𝑗∗=𝝂(2.3) hold.

Proof. We assume without loss of generality that player I has an optimal strategy 𝑥^∗=(𝑥₁,𝑥₂,…,𝑥_𝑘,0,…,0). Here, 𝑥_𝑖>0, 𝑖=1,2,…,𝑘 and ∑^𝑘𝑖=1𝑥_𝑖=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 I_𝑖 (1≤𝑖≤𝑘) of player I, we have 𝒜𝐠I_𝑖,𝑦^∗≺𝝂≥0,(2.4) since player II playing optimal strategy 𝑦^∗.
We set 𝐠(I_𝑖,𝑦^∗)=𝝂_𝑖. Then, using following equality: 𝝂=𝐠𝑥^∗,𝑦^∗=^𝑘_𝑖=1𝑥_𝑖𝐠I_𝑖,𝑦^∗=^𝑘_𝑖=1𝑥_𝑖𝝂_𝑖,(2.5) and (2.4) we find 𝒜𝝂=^𝑘_𝑖=1𝑥_𝑖𝝂_𝑖≺^𝑘_𝑖=1𝑥_𝑖𝝂=𝝂≥0.(2.6) Thus, acceptability index must be 0, and so 𝝂_𝑖=𝝂 for all 𝑖=1,2,…,𝑘. Therefore, playing the worthwhile strategy I_𝑖 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∈𝑋𝑚𝑗=1}_,2,…,𝑛𝐠𝑥,II_𝑗=𝝂,(2.7)_{𝑦∈𝑌𝑛𝑥}_∈𝑋𝑚𝐠(𝑥,𝑦)=_{𝑦∈𝑌𝑛𝑖=1}_,2,…,𝑚𝐠𝐼_𝑖,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∈𝑌𝑛𝑥}_∈𝑋𝑚𝒜𝝂≺𝐠𝑥_∗𝐠(𝑥,𝑦)=𝝂,(2.9)𝒜𝐠𝑥,𝑦_∗,𝑦≥0,∀𝑦∈𝑌_𝑛,(2.10)≺𝝂≥0,∀𝑥∈𝑋_𝑚.(2.11) Since II_𝑗∈𝑌_𝑛 for all 𝑗=1,2,…,𝑛, then using (2.10), we get 𝒜𝝂≺𝐠𝑥^∗,II_𝑗≥0,(2.12) for all 𝑗=1,2,…,𝑛. Therefore, it follows that 𝒜𝝂≺_{𝑗=1,2,…,𝑛}𝐠𝑥^∗,II_𝑗≥0.(2.13) Using (2.13), we find 𝒜⎛⎜⎜⎝𝝂≺_{𝑥∈𝑋𝑚𝑗=}_1,2,…,𝑛𝐠𝑥^∗,II_𝑗⎞⎟⎟⎠≥0.(2.14) Now, let us show that 𝒜⎛⎜⎜⎝𝝂≺_{𝑥∈𝑋𝑚𝑗=1}_,2,…,𝑛𝐠𝑥^∗,II_𝑗⎞⎟⎟⎠=0.(2.15) Suppose, contrary to what we wish to show, that 𝒜⎛⎜⎜⎝𝝂≺_{𝑥∈𝑋𝑚𝑗=1}_,2,…,𝑛𝐠𝑥^∗,II_𝑗⎞⎟⎟⎠>0.(2.16) For notational simplicity, we assume that 𝝂^∗=_{𝑥∈𝑋𝑚𝑗=1}_,2,…,𝑛𝐠𝑥^∗,II_𝑗.(2.17) Then, there exists 𝑥̃=(𝑥̃₁,…,𝑥̃_𝑚)∈𝑋_𝑚 such that 𝝂^∗=_{𝑗=1,2,…,𝑛}𝐠𝑥̃,II_𝑗.(2.18) Since 𝒜(𝝂^∗≺𝐠(𝑥̃,II_𝑗))≥0 and acceptability index is transitive, then we obtain 𝒜𝝂≺𝐠𝑥̃,II_𝑗>0,(2.19) for all 𝑗=1,2,…,𝑛.
Additionally, for all 𝑦=(𝑦₁,…,𝑦_𝑛)∈𝑌_𝑛 we can write 𝐠(𝑥̃,𝑦)=^𝑛_𝑚𝑗=1_𝑖=1𝑥̃𝐠_𝑖𝑗𝑦_𝑗=^𝑛_𝑗=1^𝑚_𝑖=1𝑥̃𝐠_𝑖𝑗𝑦^𝑗=^𝑛_𝑗=1𝐠𝑥̃,II_𝑗𝑦_𝑗.(2.20) Therefore, we have 𝒜𝝂^∗≺𝐠(𝑥̃,𝑦)≥0,(2.21) for all 𝑦∈𝑌_𝑛. Since 𝒜(𝝂≺𝝂^∗)>0, we get from (2.11) that 𝒜𝐠𝑥,𝑦^∗≺𝝂^∗≥0,(2.22) for all 𝑥∈𝑋_𝑚.
Using (1.28), (2.21), and (2.22), we conclude that (𝑥̃,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⋯II_𝑛I₁I₂𝐠𝐠¹¹₂₁𝐠₁₂𝐠₂₂⋯⋯𝐠_1𝑛𝐠_2𝑛,(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,II_𝑗=𝑡𝐠_1𝑗+(1−𝑡)𝐠_2𝑗,(2.24) for 𝑡∈[0,1]. Setting 𝝓_𝑗(𝑡)=𝐠𝑥,II_𝑗,𝑗=1,…,𝑛,(2.25) we get 𝝓_𝑗(𝑡)=𝑡𝐠_1𝑗+(1−𝑡)𝐠_2𝑗,𝑗=1,…,𝑛.(2.26) Using Theorem 2.2, we have 𝝂^∗=_{𝑥∈𝑋2𝑗=}_1,…,𝑛𝐠𝑥,II_𝑗=_{[]𝑡∈0,1𝑗=}_1,…,𝑛𝝓_𝑗(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 𝝓_𝑗∶[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}𝝓_𝑗(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.

(a)

(b)

Figure 1 

Step 1 and Step 2 of Algorithm 2.3.

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[−[12,,30]][−[01,,21]][−[01,,21]][−[53,,7−]1][−[44,,6−]2].(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,…,5}𝐠I_𝑖,II_𝑗={[−4,−2],[−3,−1]}=[−3,−1_{𝑗=1,2,…,5}],_𝑖_=1,2𝐠I_𝑖,II_𝑗={[1,3],[0,2],[0,2],[5,7],[4,6]}=[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 𝝓_𝑗(𝑡)=𝐠𝑥,II_𝑗=𝑡𝐠_1𝑗+(1−𝑡)𝐠_2𝑗.(3.3) Hence, 𝝓𝝓₁(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 𝝍∶[0,1]⟶𝕀ℝ,𝝍(𝑡)==⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩𝝓𝝓₁(𝑡),𝝓₂(𝑡),𝝓₃(𝑡),𝝓₄(𝑡),𝝓₅(𝑡)₄(𝑡),𝑡∈0,2𝝓₃7,(𝑡),𝑡∈2𝝓₁7,12,(𝑡),𝑡∈1𝝓₅2,35,(𝑡),𝑡∈35,1.(3.5) see Figure 3.

Figure 2 

Graph of 𝝓𝑗(𝑡), 𝑗=1,…,𝑛.

Figure 3 

Graph of 𝝍(𝑡).

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=−21,32.(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^∗,𝐠𝑥^∗,II₂=−12,32=𝝂^∗,𝐠𝑥^∗,II₃=−12,32=𝝂^∗𝐠𝑥_∗,,II₄=[1,3]≠𝝂^∗,𝐠𝑥^∗,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 ⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑦𝐠₁+I𝑦₂+𝑦₃=1,₁𝐠I₂,𝑦^∗=𝝂^∗,𝑦₁,𝑦^∗=𝝂^∗,>0,𝑦₂>0,𝑦₃>0⟹⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑦¹−2+𝑦𝑦₂₁+𝑦₃=1,−𝑦₂,𝑦₂+2𝑦₃=−1𝑦₁2,32,−𝑦₃,3𝑦₁+2𝑦₂+𝑦₃=−1𝑦₁2,32,>0,𝑦₂>0,𝑦₃>0(3.9) are valid. Thus, 𝑦^∗=−12+𝑠,32−2𝑠,𝑠,0,0,21<𝑠<34(3.10) is optimal strategy of player II.

As a consequence, we obtain 12,12,−21+𝑠,32−2𝑠,𝑠,0,0,−21,32,21<𝑠<34,(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.