We concentrate on discussing a class of two-person zero-sum
games with rough payoffs. Based on the expected value operator and the trust measure of
rough variables, the expected equilibrium strategy and

Game theory is widely applied in many fields, such as, economic and management problems, social policy, and international and national politics since it is proposed by von Neumann and Morgenstern [

Although many cooperative and noncooperative games with uncertain payoffs are researched much by many scholars, there is still a kind of games with uncertain payoffs to be discussed little, that is, games with rough payoffs. Since rough set theory is proposed and studied by Pawlak [

In game theory, it is an important task to define the concepts of equilibrium strategies and investigate their properties. However, in these games with uncertain payoffs, there are no concepts of equilibrium strategies to be accepted widely. Campos [

In this paper, based on the concept of rough variable proposed by Liu [

In this section, let us recall the basic definitions of the two-person zero-sum game in [

In the game theory, the decision makers realize sufficiently the affection of their actions to others. The two-person zero-sum game is the simplest case of game theory in which how much one player receives is equal to how much the other loses. When we assume that both players give pure, mixed strategies (see Parthasarathy and Raghavan [

In the two-person zero-sum game, what one player receives is equal to how much the other loses which could be illustrated by the following

A vector

If the mixed strategies

In one two-person zero-sum game, player I’s mixed strategy

Since Pawlak [

Let

The collection of all sets having the same lower and upper approximations is called a rough set, denoted by

Liu [

Let

A rough variable

When we do not have information enough to determine the measure

Let

Let

Assume that

Let consider the following example before defining the two-person zero-sum game with rough payoffs. When playing a Chinese poker, there are two teams which are constructed by two persons. Without loss of generality, we assume that Team A is the dealer, then its rule is as follows.

If the score Team B gets is less than 40, Team A goes on being a dealer and rises of one grade, denoted as +1.

If the score Team B gets is between 40 and 80, Team B becomes the dealer, denoted as 0.

If the score Team B gets is more than 80, Team B becomes the dealer and rises of one grade, denoted as −1.

From the description, we know that the rule has determined a kind of classification which is regard as an equivalent relation by Pawlak [

Let the rough variable

When player I and player II, respectively, choose the mixed strategies

Because of the vagueness of rough payoffs, it is difficult for players to choose the optimal strategy. Naturally, we consider how to maximize players’ or minimize the opponent’s rough expected payoffs. Based on this idea, we propose the following maximin equilibrium strategy.

Let rough variable

Since the rough variables

According to the definition of trust measure of rough variable, we can get another way to convert the rough variable into a crisp number. Then we propose another definition of Nash equilibrium to this game.

Let rough variable

In the following part, we will introduce the equilibrium strategy under the expected operator and the trust measure, respectively.

When the players’ payoffs are crisp numbers, we know that the game surely has a mixed Nash equilibrium point. Then we will discuss if there is an expected maximin equilibrium strategy when the payoffs

Let

The necessity is apparent. Now we only consider the sufficiency. According to (

Suppose that

Similarly, we can prove

Thus, the strategy

In a two-person zero-sum game, rough variables

Suppose that

Now, we will prove

For the strategy

Similarly, we can prove

Through the proof of Theorem

Let rough variable

According to [

Because of

If

Suppose

Because

If

The proof is similar with that of Theorem

After discussing two particular cases, let us consider the usual case if there exists

In a two-person zero-sum game, rough variables

Let us only discuss one of five cases, the others are considered similarly. Suppose

Since

It is apparent that

Similarly, if

Since

It is apparent that

The other cases can be proved in the same way. This completes the proof.

According to Theorem

For all strategies

For all strategies

For any fixed

For any fixed

We know that only if the two problems have optimal solution, the game exists an equilibrium strategy. Because problems (

For problems (

For all strategies

It can be proved similarly as Theorem

We have discussed many simple cases; there is still a more complicated case that

For many complex problems such as problem (

(

(

(

(

(

Above all, it can be simply summarized in Procedure

initialization( ) by checking the feasiblity;

evaluation(0);

selection( );

crossover( );

mutation( );

evaluation(

Game theory is widely applied in many fields, such as, economic and management problems, social policy, and international and national politics; sometimes players should consider the state of uncertainty. A kind of games are usually characterized by rough payoffs. In this section, we give an example of two-person zero-sum game with rough payoffs to illustrate the effectiveness of the algorithm introduced above. There is a game between player I and player II. When player I gives strategy

Firstly, let us consider the expected maximin equilibrium strategy of this game. According to Remarks

Then, we can get the equilibrium strategy that when player I gives the mixed strategy

Next, let us consider if this game has the

Then we give five predetermined numbers

Apparently,

Similarly,

Because

Apparently,

Then we can make use of simplex method to get the optimal solutions of problems (

Apparently,

For any fixed

For any fixed

The results by GA.

In this paper, we have considered a class of two-person zero-sum matrix games with rough payoffs. Firstly, we have given the definition of the game with rough payoffs and then proposed two kinds of equilibrium strategy. Secondly, we have discussed wether the two-person zero-sum matrix games with rough payoffs exist the equilibrium strategy. Thirdly, we proposed the genetic algorithm to solve the most complicated case. It is an available and efficient way to search the equilibrium of this kind of games with rough payoffs. Lastly, the numerical example illustrated well our research methods. We have only considered one kind of games with uncertain payoffs. Of course, there are many other games with uncertain payoffs which need to be researched.

This research has been supported by the Key Program of NSFC (Grant no. 70831005) and the National Science Foundation for Distinguished Young Scholars, China (Grant no. 70425005).