A Graphical Method for Solving Interval Matrix Games
or 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.
The simplest game is finite, two-person, zero-sum game. There are only two players, player and player , 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 or 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 . 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 where and . If , then is a real number.
Mid-point and radius of interval number is defined as respectively.
Elementary arithmetic operation between two interval numbers and is given by For division operator, it is assumed that . For any scalar number , is defined as
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.
1.2.1. Disjoint Intervals
Let and be two disjoint interval numbers. Then, it is not difficult to define transitive order relation over these intervals, as is strictly less than if and only if . This is denoted by , and it is an extension of “<” on the real line.
1.2.2. Nested Intervals
Let and be two interval numbers such that . Then, . Here, the set inclusion only describes the condition that is nested in , but it cannot order and in terms of value.
Let and 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 and represent profit intervals. Therefore, we can define the ranking order of and as Here, the notation “” denotes the maximum among the interval numbers and . Similarly, we can write Likewise, the notation “” denotes the minimum among the interval numbers and .
1.2.3. Partially Overlapping Intervals
The above-mentioned order relations cannot explain ranking between two overlapping closed intervals.
Let be the set of all interval numbers. The function is called acceptability function.
For , the number is called the grade of acceptability of the interval number to be inferior to the second interval number .
By the definition of , for any interval numbers and , we have (i) for and ,(ii) for and , (iii) for . In this case, if , then is identical with . If , then the intervals and are noninferior to each other. In this case, the acceptability index becomes insignificant, so DM has to negotiate with the widths of and .Let and 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 and represent profit intervals.
Example 1.1. Let and be two intervals. Then, Thus, the DM accept the decision that is less than with full satisfaction.
Example 1.2. Let and be two intervals. Then, Hence, the DM accept the decision that is less than with grade of satisfaction 0.3.
Example 1.3. Let and be two intervals. Then, . Here, both of the intervals are noninferior to each other. In this case, the DM has to negotiate with the widths of and as listed inTable 1.
These can be written explicitly as where, denotes the maximum of the interval numbers and . Similarly, Here, the notation represents the minimum of the interval numbers and .
Proposition 1.4. The interval ordering by the acceptability index defines a partial order relation on .
Proof. (i) If and , then , and we say that and are noninferior to each other. Hence, it is reflexive.
(ii) For any interval number and , , and implies . Therefore, is antisymmetric.
(iii) For any interval numbers if then . Hence, Thus, is transitive.
On the other hand, acceptability index must not interpreted as difference operator of real analysis. Indeed, while and , the inequality may not hold.
Actually, let , then but
For any two interval numbers and from , either , or , or . Also, can be interpreted as the interval number is inferior to the interval number , since .
Additionally, for any interval numbers , , , and , the following properties of acceptability index is obvious.
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 ), in which he proved the famous minimax theorem for zero-sum games. It was also basis for .
The simplest game is finite, two-person, zero-sum game. There are only two players, player and player , and it can be denoted by a matrix. Thus, such a game is called matrix game. More formally, a matrix game is an matrix of real numbers.
A (mixed) strategy of player is a probability distribution over the rows of , that is, an element of the set Similarly, a strategy of player is a probability distribution over the columns of , that is, an element of the set
A strategy of player is called pure if it does not involve probability, that is, for some , and it is denoted by . Similarly, pure strategies of player is denoted by for .
If player plays row (i.e., pure strategy ) and player plays column (i.e., pure strategy ), then player receives payoff , and player pays , where is the entry in row and column of matrix .
If player plays strategy and player plays strategy , then player receives the expected payoff where denotes the transpose of .
A strategy is called maximin strategy of player in matrix game if for all and a strategy is called minimax strategy of player in matrix game if for all .
Therefore, a maximin strategy of player maximizes the minimal payoff of player , and a minimax strategy of player minimizes the maximum that player has to pay to player .
Neumann proved that for every matrix game , there is a real number with the following properties.(1)A strategy of player guarantees a payoff of at least to player (i.e., for all strategies of player ) if and only if is a maximin strategy. (2)A strategy of player guarantees a payment of at most by player to player (i.e., for all strategies of player ) if and only if is a minimax strategy.
Hence, player can obtain a payoff at least by playing maximin strategy, and player can guarantee to pay not more than by playing minimax strategy. For these reasons, the number is also called the value of the game .
A position is called saddle point if for all and for all , that is, if is maximal in its column and minimal in its row . Evidently, if is a saddle point, then 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 is maximizing or optimistic player and player is minimizing or pessimistic player. We assume that player will try to make maximum profit and player will try to minimize the loss.
The two person interval matrix game is defined by matrix whose entries are interval numbers.
Let be an interval matrix game
and , ; that is, and are strategies for players and . Then, the expected payoff for player is defined by Here, denotes the Minkowski’s sum of intervals.
Example 1.5. Let
be interval matrix game.
For this game, if player plays second row () and player plays third column (), then player receives, and correspondingly, player pays a payoff .
On the other hand, for pair of strategies and , the expected payoff for player belongs to .
Now, define to be lower and upper value of interval matrix game.
It is naturally clear that .
In addition, if , then is called the value of interval matrix game.
Theorem 1.6 (Fundamental Theorem). Let be an interval matrix game. Then, and both exist and are equal.
Let be the value of game. Then, the is called the solution of interval matrix game if Furthermore, the strategies and 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 is . On the other hand, if we consider the left and right endpoints of the entries of interval matrix as two different matrix games, that is, 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 be an interval matrix game.
Let be value of interval matrix game . In general, the width of can vary. To normalize the width of in order to investigate a method for solving such games, from now on, we will assume that all entries of are of same length; that is, for some and , .
A useful idea in solving 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 is the solution of interval matrix game (2.1) and the pure strategies and are worthwhile strategies, then the equalities hold.
Proof. We assume without loss of generality that player has an optimal strategy . Here, , and . We also assume that player has an optimal strategy and value of the game is . Hence, .
Furthermore, for any pure strategy () of player , we have since player playing optimal strategy .
We set . Then, using following equality: and (2.4) we find Thus, acceptability index must be 0, and so for all . Therefore, playing the worthwhile strategy against optimal strategy gives a payoff that is the value of the game.
The proof of (2.3) is similar.
Theorem 2.2. If is value of interval matrix game (2.1), then the equalities 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 . Then, we have
Since for all , then using (2.10), we get
for all . Therefore, it follows that
Using (2.13), we find
Now, let us show that
Suppose, contrary to what we wish to show, that
For notational simplicity, we assume that
Then, there exists such that
Since and acceptability index is transitive, then we obtain
for all .
Additionally, for all we can write Therefore, we have for all . Since , we get from (2.11) that for all .
Using (1.28), (2.21), and (2.22), we conclude that 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 interval matrix game.
We denote the value of by the symbol .
Since player has only two strategies, for any , we can write , . Then, we obtain for . Setting we get Using Theorem 2.2, we have Therefore, the following algorithm can be used to find solution of interval matrix games.
Algorithm 2.3. Step 1. Plot for all on the plane.Step 2. Draw the function by setting (see Figure 1).Step 3. By means of definition of calculate the on ; that is, find Step 4. Using definition of , calculate the , where attains its maximum. Thus, , and the optimal strategy of player is obtained.Step 5. Using the property of worthwhile strategies, find the optimal strategy of player .
To see how this works, let us solve a game.
3. Numerical Example
Example 3.1. Consider the interval matrix game 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 has no saddle point.
Now, we can calculate the on ; that is, we can find the value of the game
Since, the function attains its maximum at , we obtain
Furthermore, the optimal strategy of player is .
On the other hand,
Therefore, using the property of worthwhile strategies, we obtain that only pure strategies , , and are worthwhile strategies of player .
If is optimal strategy of player , then the systems are valid. Thus, is optimal strategy of player .
As a consequence, we obtain as the solution of the game (3.1).
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 or 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.
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