The simplest game is finite, two-person, zero-sum game. There are only two players, player

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 [

An extensive research and wide coverage on interval arithmetic and its applications can be found in [

An interval number is a subset of real line of the form

Mid-point

Elementary arithmetic operation

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 [

Let

Let

Let

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.

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

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

Here, we use the acceptability index idea suggested by [

Let

For

By the definition of

Let

Let

Let

These can be written explicitly as

Optimistic outlook | Pessimistic outlook | |
---|---|---|

Profit intervals | DM will prefer the interval | DM will prefer the interval |

Cost intervals | DM will pay more attention on the minimum cost of 2 unit, that is, the left hand points both of the cost intervals | DM will again prefer the interval |

The interval ordering by the acceptability index defines a partial order relation on

(i) If

(ii) For any interval number

(iii) For any interval numbers

On the other hand, acceptability index must not interpreted as difference operator of real analysis. Indeed, while

Actually, let

For any two interval numbers

Additionally, for any interval numbers

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 [

The simplest game is finite, two-person, zero-sum game. There are only two players, player

A (mixed) strategy of player

A strategy

If player

If player

A strategy

Therefore, a maximin strategy of player

Neumann proved that for every matrix game

A strategy

A strategy

Hence, player

A position

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

The two person interval matrix game is defined by

Let

and

Let

For this game, if player

On the other hand, for pair of strategies

Now, define

It is naturally clear that

In addition, if

Let

Let

In view of Theorem

The value of the interval matrix game

Let

Let

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 [

A useful idea in solving

The following proposition is the important property about such strategies.

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

It means that if

We assume without loss of generality that player

Furthermore, for any pure strategy

We set

The proof of (

If

By Theorem

Additionally, for all

Using (

By completely analogous arguments, we can also obtain (

Now, using Theorem

Let us consider

We denote the value of

Since player

Plot

Draw the function

By means of definition of

Using definition of

Using the property of worthwhile strategies, find the optimal strategy of player

Step

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

Consider the

If player

Graph of

Graph of

Now, we can calculate the

Since, the function

Furthermore, the optimal strategy of player

On the other hand,

Therefore, using the property of worthwhile strategies, we obtain that only pure strategies

If

As a consequence, we obtain

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