A Class of Two-Person Zero-Sum Matrix Games with Rough Payoffs

Abstract

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 𝑟-trust maximin equilibrium strategy are defined. Five cases whether the game exists 𝑟-trust maximin equilibrium strategy are discussed, and the technique of genetic algorithm is applied to find the equilibrium strategies. Finally, a numerical example is provided to illustrate the practicality and effectiveness of the proposed technique.

1. Introduction

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 [1]. Peski [2] presented the simple necessary and sufficient conditions for the comparison of information structures in zero-sum games and solved an problem, which is how to find the value of information in zero-sum games. Owena and McCormick [3] analyzed a “manhunting” game involving a mobile hider and consider a deductive search game involving a fugitive, then developed a model based on a base-line model. The traditional game theory assumes that all data of a game are known exactly by players. However, there are some games in which players are not able to evaluate exactly some data in our realistic situations. In these games, the imprecision is due to inaccuracy of information and vague comprehension of situations by players. For these uncertain games, many scholars have made contribution and got some techniques to find the equilibrium strategies of these games. Some scholars, such as, Berg and Engel [4], Ein-Dor and Kanter [5], Takahashi [6], discussed a two-person zero-sum matrix game with random payoffs. Xu [7] made use of linear programming method to discuss two-person zero-sum game with grey number payoff matrix. Harsanyi [8] made a great contribution in treating the imprecision of probabilistic nature in games by developing the theory of Bayesian games. Dhingra et al. [9] combined the cooperative game theory with fuzzy set theory to yield a new optimization method to herein as cooperative fuzzy games and proposed a computational technique to solve the multiple objective optimization problems. Then Espin et al. [10] proposed an innovative fuzzy logic approach to analyze 𝑛-person cooperative games and theoretically and experimentally examined the results by analyzing three-case studies.

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 [11, 12], it is drastic to be applied into many fields, such as, data mining and neural network. Nurmi et al. [13] introduced three uncertainty events in social choice such as the impreciseness of a probabilistic, fuzzy, and rough type, further explored difficult issues of how diverse types of impreciseness can be combined, and in particular the combination of roughness with randomness and fuzziness in voting games. Liu [14] proposed a new concept of rough variable which is a measurable function from rough space to 𝑅. Based on the concept of rough variable, a game with rough payoffs is studied in this paper.

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 [15] has proposed several methods to solve fuzzy matrix games based on linear programming but has not defined explicit concepts of equilibrium strategies. As the extension of the idea of Campos [15], Nishizaki and Sakawa [16] discussed multiobjective matrix game with fuzzy payoffs. Maeda [17] has defined Nash equilibrium strategies based on possibility and necessity measures and investigated its properties.

In this paper, based on the concept of rough variable proposed by Liu [14], we discuss a simplest game, namely, the game in which the number of players is two and rough payoffs which one player receives are equal to rough payoffs which the other player loses. We defined two kinds of concepts of maximin equilibrium strategies and investigate their properties. The rest of this paper is organized as follows. In Section 2, we recalls some definitions and properties about two-person zero-sum game and the rough variable. Then two concepts of equilibrium strategies of two-person zero-sum game with rough payoffs are introduced and then their properties are deduced in Section 3. In Section 4, we proposed the technique of GA to solve some complicated game problems with rough payoffs which can be converted into crisp programming problem. Then a numerical example is discussed to show the effectiveness of the prosed theory and algorithm in Section 5. Finally, the conclusion has been made in Section 6.

2. Basic Concepts of Two-Person Zero-Sum Game and Rough Variable

In this section, let us recall the basic definitions of the two-person zero-sum game in [18]. The concept and properties of rough variable proposed by Liu [14] is also reviewed.

2.1. Two-Person Zero-Sum Game

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 [19]), such a game has been well resolved. But in our realistic world, there are also some noncooperative cases though more cooperation may exist in games. In reality, the non-cooperation between players may be vague. This paper mainly deals with the kind of games with rough payoffs.

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 𝑚×𝑛 matrix: 𝑎𝑃=¹¹⋯𝑎_1𝑛𝑎⋮⋱⋮_𝑛1⋯𝑎_𝑛𝑛, (2.1) where 𝑃 denotes the payoff matrix of player I, 𝑥_𝑖𝑗 is the payoff of player I when player I proposes the strategy 𝑖, and player II proposes the strategy 𝑗. Then, the payoff matrix of player II is −𝑃.

Definition 2.1. A vector 𝑥 in ℜ^𝑚 is said to be a mixed strategy of player I if it satisfies the following condition: 𝑥^𝑇𝑒_𝑚=1, (2.2) where the components of 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑚)^𝑇 are greater than or equal to 0; 𝑒_𝑚 is an 𝑚×1 vector, whose every component is equal to 1. The mixed strategy of the player II is defined similarly. Particularly, a strategy 𝑠_𝑘=(0,0,…,1,…,0) is called a pure strategy of player I. Thereinto, the 𝑘th component of 𝑠_𝑘 is only equal to 1, the other components are equal to 0.

Definition 2.2. If the mixed strategies 𝑥 and 𝑦 are proposed by players I and II, respectively, then the expected payoff of player I is defined by 𝑥^𝑇𝑃𝑦=^𝑛𝑚_𝑗=1𝑖=1𝑎_𝑖𝑗𝑥_𝑖𝑦_𝑗. (2.3) According to Definition 2.2, we have the definition of optimal strategies of players.

Definition 2.3. In one two-person zero-sum game, player I’s mixed strategy 𝑥^∗ and player II’s mixed strategy 𝑦^∗ are said to be optimal strategies if 𝑥^𝑇𝑃𝑦^∗≤𝑥^∗𝑇𝑃𝑦^∗ and 𝑥^∗𝑇𝑃𝑦^∗≤𝑥^∗𝑇𝑃𝑦 for any mixed strategies 𝑥 and 𝑦.

2.2. Rough Variable

Since Pawlak [11] initialized the rough set theory, it has been well developed and applied in a wide variety of uncertainty surrounding real problems.

Definition 2.4 (Slowinski et al. [20]). Let 𝑈 be a universe and 𝑋 a set representing a concept. Then its lower approximation is defined by 𝑋={𝑥∈𝑈∣𝑅⁻¹(𝑥)⊂𝑋}, (2.4) and the upper approximation is defined by 𝑋=_𝑥∈𝑋𝑅(𝑥), (2.5) where 𝑅 is the similarity relationship on 𝑈. Obviously, we have 𝑋⊆𝑋⊆𝑋.

Definition 2.5 (Pawlak [11]). The collection of all sets having the same lower and upper approximations is called a rough set, denoted by (𝑋,𝑋). Its boundary is defined as follows: 𝐵𝑛_𝑅(𝑋)=𝑋−𝑋. (2.6)

Liu [14] also gave a new concept about rough variable. This paper mainly refers to this book. The following results will be used extensively.

Definition 2.6. Let Λ be a nonempty set, 𝒜 a 𝜎-algebra of subsets of Λ,Δ an element in 𝒜, and 𝜋 a nonnegative, real-valued, additive set function. Then (Λ,Δ,𝒜,𝜋) is called a rough space.

Definition 2.7. A rough variable 𝜉 on the rough space (Λ,Δ,𝒜,𝜋) is a function from Λ to the real line ℜ such that for every Borel set 𝑂 of ℜ, we have {𝜆∈Λ∣𝜉(𝜆)∈𝑂}∈𝒜. (2.7) The lower and the upper approximations of the rough variable 𝜉 are then defined as follows: 𝜉={𝜉(𝜆)∣𝜆∈Δ},𝜉={𝜉(𝜆)∣𝜆∈Λ}. (2.8) Liu [14] also defined the trust measure of event 𝐴 by Tr{𝐴}=(1/2)(Tr{𝐴}+Tr{𝐴}), where Tr{𝐴} denotes the upper trust measure of event 𝐴, defined by Tr{𝐴}=𝜋{𝐴}/𝜋{Λ}, and Tr{𝐴} denotes the lower trust measure of event 𝐴, defined by Tr{𝐴})=𝜋{𝐴∩Δ}/𝜋{Δ}.

When we do not have information enough to determine the measure 𝜋 for a real-life problem, we can assumes that all elements in Λ are equally likely to occur. For this case, the measure 𝜋 may be viewed as the Lebesgue measure. In this paper, we only consider the rough variable 𝜉=([𝑎,𝑏],[𝑐,𝑑]) such 𝜉(𝑥)=𝑥 for all 𝑥∈Λ, where 𝑐≤𝑎<𝑏≤𝑑.

Definition 2.8. Let 𝜉 be a rough variable on the rough space (Λ,Δ,𝒜,𝜋). The expected value of 𝜉 is defined by 𝐸[𝜉]=₀^+∞Tr{𝜉≥𝑟}𝑑𝑟−⁰_−∞Tr{𝜉≤𝑟}𝑑𝑟. (2.9)

Remark 2.9. Let 𝜉=([𝑎,𝑏],[𝑐,𝑑]) be a rough variable with 𝑐≤𝑎≤𝑏≤𝑑. Then we have 1𝐸[𝜉]=4(𝑎+𝑏+𝑐+𝑑). (2.10)

Remark 2.10. Assume that 𝜉 and 𝜂 are both variables with finite expected values. Then for any real numbers 𝑎 and 𝑏, we have 𝐸[𝑎𝜉+𝑏𝜂]=𝑎𝐸[𝜉]+𝑏𝐸[𝜂]. (2.11)

3. Two Kinds of Equilibrium Strategies of Two-Person Zero-Sum Game with Rough Payoffs

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.

(1)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.(2)If the score Team B gets is between 40 and 80, Team B becomes the dealer, denoted as 0.(3)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 [12] on the universe [0,100]. This means that obtaining 45 or 75 expresses the same meaning, and they are equivalent or indiscernible. Thus, the rough variable 𝜉=([40,80],[0,100]) is applied to describe the above process and its trust measure expresses the probability that Team A obtains +1, or 0, or −1 in every game. In the following part, we will only consider the rough variable which is combined by the payoff.

Let the rough variable 𝜉_𝑖𝑗 represent the payoff that the player I receives or player II loses, then a rough payoff matrix is presented as follows to denote a two-person zero-sum game: 𝜉𝑃=¹¹𝜉₁₂⋯𝜉_1𝑛𝜉₂₁𝜉₂₂⋯𝜉_2𝑛𝜉⋮⋮⋱⋮_𝑚1𝜉_𝑚2⋯𝜉_𝑚𝑛. (3.1)

When player I and player II, respectively, choose the mixed strategies 𝑥 and 𝑦, the rough payoffs of player I are 𝑥^𝑇𝑃𝑦=^𝑛𝑚_𝑗=1𝑖=1𝜉_𝑖𝑗𝑥_𝑖𝑦_𝑗. (3.2)

3.1. Basic Definition of Two Kinds of Equilibrium 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.

Definition 3.1. Let rough variable 𝜉_𝑖𝑗 (𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛) represent the payoffs that the player receives or player II loses when player I gives the strategy 𝑖 and player II gives the strategy 𝑗. Then (𝑥^∗,𝑦^∗) is called a rough expected maximin equilibrium strategy if 𝐸[𝑥^𝑇𝑃𝑦^∗]≤𝐸[𝑥^∗𝑇𝑃𝑦^∗]≤𝐸[𝑥^∗𝑇𝑃𝑦], (3.3) where 𝑃 is defined by (3.1).

Remark 3.2. Since the rough variables 𝜉 are independent, then for any mixed strategies 𝑥 and 𝑦, according to Remark 2.10, we have that 𝐸[𝑥^𝑇𝑃𝑦]=𝐸[^𝑛𝑚_𝑗=1𝑖=1𝜉_𝑖𝑗𝑥_𝑖𝑦_𝑗]=^𝑛𝑚_𝑗=1𝑖=1𝐸[𝜉_𝑖𝑗]𝑥_𝑖𝑦_𝑗. (3.4)

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.

Definition 3.3. Let rough variable 𝜉_𝑖𝑗 (𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛) represent the payoffs that the player receives or player II loses when player I gives the pure strategy 𝑖 and player II gives the pure strategy 𝑗.𝑟 is the predetermined level of the payoffs, 𝑟∈𝑅. Then (𝑥^∗,𝑦^∗) is called the 𝑟-trust maximin equilibrium strategy if Tr{𝑥^𝑇𝑃𝑦^∗≥𝑟}≤Tr{𝑥^∗𝑇𝑃𝑦^∗≥𝑟}≤Tr{𝑥^∗𝑇𝑃𝑦≥𝑟}, (3.5) where 𝑃 is defined by (3.1).

3.2. The Existence of Two Kinds of Equilibrium Strategies

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

3.2.1. The Existence of Expected Maximin Equilibrium Strategies

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 𝜉_𝑖𝑗 are characterized as rough variables.

Lemma 3.4. Let 𝜉_𝑖𝑗 (𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛) be rough variables with finite expected values. Then strategy (𝑥^∗,𝑦^∗) is an expected maximin equilibrium strategy to the game if and only if for every pure strategy 𝑠_𝑘 (𝑘=1,2,…,𝑚) of player I and 𝑠_𝑡 (𝑡=1,2,…,𝑛) of player II, one has 𝐸[𝑠^𝑇_𝑘𝑃𝑦^∗]≤𝐸[𝑥^∗𝑇𝑃𝑦^∗]≤𝐸[𝑥^∗𝑇𝑃𝑠_𝑡], (3.6) where 𝑃 is defined by (3.1).

Proof. The necessity is apparent. Now we only consider the sufficiency. According to (3.6), for every 𝑘=1,2,…,𝑚, 𝐸[𝑠^𝑇_𝑘𝑃𝑦^∗]≤𝐸[𝑥^∗𝑇𝑃𝑦^∗]. (3.7)
Suppose that 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑚) is any one mixed strategy of player I. In (3.7), for every 𝑘, we multiply 𝑥_𝑘 to every inequality. Then 𝐸[𝑠^𝑇_𝑘𝑃𝑦^∗]𝑥_𝑘≤𝐸[𝑥^∗𝑇𝑃𝑦^∗]𝑥_𝑘⇒^𝑚_𝑘=1𝐸[𝑠^𝑇_𝑘𝑃𝑦^∗]𝑥_𝑘≤^𝑚_𝑘=1𝐸[𝑥^∗𝑇𝑃𝑦^∗]𝑥_𝑘⇒𝐸[𝑥^𝑇𝑃𝑦^∗]≤𝐸[𝑥^∗𝑇𝑃𝑦^∗]^𝑚_𝑘=1𝑥_𝑘⇒𝐸[𝑥^𝑇𝑃𝑦^∗]≤𝐸[𝑥^∗𝑇𝑃𝑦^∗]. (3.8)
Similarly, we can prove 𝐸[𝑥^∗𝑇𝑃𝑦^∗]≤𝐸[𝑥^∗𝑇𝑃𝑦]. (3.9)
Thus, the strategy (𝑥^∗,𝑦^∗) is an expected maximin equilibrium strategy to the game. This completes the proof.

Theorem 3.5. In a two-person zero-sum game, rough variables 𝜉_𝑖𝑗 (𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛) represent the payoffs player I receives or player II loses, and the payoff matrix 𝑃 is defined by (3.1). Then there at least exists an expected maximin equilibrium strategy to the game.

Proof. Suppose that 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑚) is any one mixed strategy of player I. For every pure strategy 𝑠_𝑘 (𝑘=1,2,…,𝑚) and any strategy 𝑦, we define 𝜌_𝑘(𝑥)=max{0,𝐸[𝑠^𝑇_𝑘𝑃𝑦]−𝐸[𝑥^𝑇𝑃𝑦]}. (3.10) For every 𝑥_𝑘 (𝑘=1,2,…,𝑚), we define 𝑓_𝑘=𝑥_𝑘+𝜌_𝑘(𝑥)1+∑^𝑚_𝑘=1𝜌_𝑘.(𝑥) (3.11) Obviously, 𝑓_𝑘≥0 (𝑘=1,2,…,𝑚), ∑^𝑚_𝑘=1𝑓_𝑘=1. Thus, 𝑓=(𝑓₁,𝑓₂,…,𝑓_𝑚) is a mixed strategy of player I and is a continous function about 𝑥. According to Brouwer’s fixed-point theorem, there exists a point 𝑥^∗=(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑚) satisfying 𝑥^∗_𝑘=𝑥^∗_𝑘+𝜌_𝑘(𝑥^∗)1+∑^𝑚_𝑘=1𝜌_𝑘(𝑥^∗),𝑘=1,2,…,𝑚. (3.12)
Now, we will prove 𝐸[𝑠^𝑇_𝑘𝑃𝑦]≤𝐸[𝑥^∗𝑇𝑃𝑦]. For any mixed strategy 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑚), let 𝐸[𝑠^𝑇_ℎ𝑃𝑦]=min_𝑘∈𝐽𝐸[𝑠^𝑇_𝑘𝑃𝑦],(𝐽={1,2,…,𝑚}). Then 𝐸[𝑠^𝑇_ℎ𝑃𝑦]=𝐸[𝑠^𝑇_ℎ𝑃𝑦]^𝑚_𝑘=1𝑥_𝑘≤^𝑚_𝑘=1𝑥_𝑘𝐸[𝑠^𝑇_𝑘𝑃𝑦]=𝐸[𝑥^𝑇𝑃𝑦]. (3.13) Namely, there exists a pure strategy 𝑠_ℎ satisfying 𝐸[𝑠^𝑇_ℎ𝑃𝑦]≤𝐸[𝑥^𝑇𝑃𝑦]. Thus, for 𝑥^∗=(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑚), there exists a pure strategy 𝑠_ℎ such that 𝐸[𝑠^𝑇_ℎ𝑃𝑦]≤𝐸[𝑥^∗𝑇𝑃𝑦]. According to (3.10), we have 𝜌_ℎ(𝑥^∗)=0.
For the strategy 𝑥^∗_ℎ (𝑥^∗_ℎ>0) of player 𝐼, according to (3.12), we have 𝑥^∗_ℎ=𝑥^∗_ℎ+𝜌_ℎ(𝑥^∗)1+∑^𝑚_𝑘=1𝜌_𝑘(𝑥^∗)=𝑥^∗_ℎ1+∑^𝑚_𝑘=1𝜌_𝑘(𝑥^∗)⇒^𝑚_𝑘=1𝜌_𝑘(𝑥^∗)=0. (3.14) By the definition of 𝜌_𝑘(𝑥^∗), we know that 𝜌_𝑘(𝑥^∗) is nonegative. Therefore, for every 𝑘=1,2,…,𝑚,𝜌_𝑘(𝑥^∗)=0. Then 𝐸[𝑠^𝑇_𝑘𝑃𝑦]−𝐸[𝑥^∗𝑇𝑃𝑦]≤0,⇒𝐸[𝑠^𝑇_𝑘𝑃𝑦]≤𝐸[𝑥^∗𝑇𝑃𝑦],𝑘=1,2,…,𝑚. (3.15)
Similarly, we can prove 𝐸[𝑥^∗𝑇𝑃𝑦^∗]≤𝐸[𝑥^∗𝑇𝑃𝑠_𝑡] for every pure strategy 𝑠_𝑡(𝑡=1,2,…,𝑛) of player II. By Lemma 3.4, we know that (𝑥^∗,𝑦^∗) is an expected maximin equilibrium strategy to the game. This completes the proof.

3.3. The Existence of 𝑟-Trust Maximin Equilibrium Strategies

Through the proof of Theorem 3.5, we know that there at least exists an expected maximin equilibrium strategy to any two-person zero-sum game with rough payoffs. Now we will discuss the existence of 𝑟-trust maximin equilibrium strategy to this kind of game.

Lemma 3.6. Let rough variable 𝜉_𝑖𝑗 (𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛) represent the payoffs that the player receives or player II loses when player I gives the strategy 𝑖 and player II gives the strategy 𝑗. Suppose that 𝑟 is a given number and rough variable 𝜉_𝑖𝑗=([𝑎_𝑖𝑗,𝑏_𝑖𝑗],[𝑐_𝑖𝑗,𝑑_𝑖𝑗]) (𝑐_𝑖𝑗≤𝑏_𝑖𝑗≤𝑎_𝑖𝑗≤𝑑_𝑖𝑗), then one has Tr{𝑥^𝑇𝑃𝑦≥𝑟}=0if𝑥^𝑇𝑥𝐷𝑦≤𝑟,_𝑇𝐷𝑦−𝑟2(𝑥^𝑇𝐷𝑦−𝑥^𝑇𝐶𝑦)if𝑥^𝑇𝐵𝑦≤𝑟≤𝑥^𝑇1𝐷𝑦,2(𝑥^𝑇𝐷𝑦−𝑟𝑥^𝑇𝐷𝑦−𝑥^𝑇+𝑥𝐶𝑦^𝑇𝐵𝑦−𝑟𝑥^𝑇𝐵𝑦−𝑥^𝑇𝐴𝑦)if𝑥^𝑇𝐴𝑦≤𝑟≤𝑥^𝑇1𝐵𝑦,2(𝑥^𝑇𝐷𝑦−𝑟𝑥^𝑇𝐷𝑦−𝑥^𝑇𝐶𝑦+1)if𝑥^𝑇𝐶𝑦≤𝑟≤𝑥^𝑇𝐴𝑦,1if𝑟≤𝑥^𝑇𝐶𝑦, (3.16) where 𝑎𝐴=¹¹𝑎₁₂⋯𝑎_1𝑛𝑎₂₁𝑎₂₂⋯𝑎_2𝑛𝑎⋮⋮⋱⋮_𝑚1𝑎_𝑚2⋯𝑎_𝑚𝑛𝑏,𝐵=¹¹𝑏₁₂⋯𝑏_1𝑛𝑏₂₁𝑏₂₂⋯𝑏_2𝑛𝑏⋮⋮⋱⋮_𝑚1𝑏_𝑚2⋯𝑏_𝑚𝑛,𝑐𝐶=¹¹𝑐₁₂⋯𝑐_1𝑛𝑐₂₁𝑐₂₂⋯𝑐_2𝑛𝑐⋮⋮⋱⋮_𝑚1𝑐_𝑚2⋯𝑐_𝑚𝑛𝑑,𝐷=¹¹𝑑₁₂⋯𝑑_1𝑛𝑑₂₁𝑑₂₂⋯𝑑_2𝑛𝑑⋮⋮⋱⋮_𝑚1𝑑_𝑚2⋯𝑑_𝑚𝑛. (3.17)

Proof. According to [14], we have 𝑥^𝑇𝑃𝑦=^𝑚𝑛_𝑖=1𝑗=1𝑥_𝑖𝜉_𝑖𝑗𝑦_𝑗=^𝑚𝑛_𝑖=1𝑗=1([𝑥_𝑖𝑦_𝑗𝑎_𝑖𝑗,𝑥_𝑖𝑦_𝑗𝑏_𝑖𝑗],[𝑥_𝑖𝑦_𝑗𝑐_𝑖𝑗,𝑥_𝑖𝑦_𝑗𝑑_𝑖𝑗])=([^𝑚𝑛_𝑖=1𝑗=1𝑥_𝑖𝑦_𝑗𝑎_𝑖𝑗,^𝑚𝑛_𝑖=1𝑗=1𝑥_𝑖𝑦_𝑗𝑏_𝑖𝑗],[^𝑚𝑛_𝑖=1𝑗=1𝑥_𝑖𝑦_𝑗𝑐_𝑖𝑗,^𝑚𝑛_𝑖=1𝑗=1𝑥_𝑖𝑦_𝑗𝑑_𝑖𝑗]). (3.18) Suppose that 𝑎𝐴=¹¹𝑎₁₂⋯𝑎_1𝑛𝑎₂₁𝑎₂₂⋯𝑎_2𝑛𝑎⋮⋮⋱⋮_𝑚1𝑎_𝑚2⋯𝑎_𝑚𝑛𝑏,𝐵=¹¹𝑏₁₂⋯𝑏_1𝑛𝑏₂₁𝑏₂₂⋯𝑏_2𝑛𝑏⋮⋮⋱⋮_𝑚1𝑏_𝑚2⋯𝑏_𝑚𝑛,𝑐𝐶=¹¹𝑐₁₂⋯𝑐_1𝑛𝑐₂₁𝑐₂₂⋯𝑐_2𝑛𝑐⋮⋮⋱⋮_𝑚1𝑐_𝑚2⋯𝑐_𝑚𝑛𝑑,𝐷=¹¹𝑑₁₂⋯𝑑_1𝑛𝑑₂₁𝑑₂₂⋯𝑑_2𝑛𝑑⋮⋮⋱⋮_𝑚1𝑑_𝑚2⋯𝑑_𝑚𝑛, (3.19) then 𝑥^𝑇𝑃𝑦=([𝑥^𝑇𝐴𝑦,𝑥^𝑇𝐵𝑦],[𝑥^𝑇𝐶𝑦,𝑥^𝑇𝐷𝑦]). (3.20) thus we have Tr{𝑥^𝑇𝑃𝑦≥𝑟}=0if𝑥^𝑇𝑥𝐷𝑦≤𝑟,_𝑇𝐷𝑦−𝑟2(𝑥^𝑇𝐷𝑦−𝑥^𝑇𝐶𝑦)if𝑥^𝑇𝐵𝑦≤𝑟≤𝑥^𝑇1𝐷𝑦,2(𝑥^𝑇𝐷𝑦−𝑟𝑥^𝑇𝐷𝑦−𝑥^𝑇+𝑥𝐶𝑦^𝑇𝐵𝑦−𝑟𝑥^𝑇𝐵𝑦−𝑥^𝑇𝐴𝑦)if𝑥^𝑇𝐴𝑦≤𝑟≤𝑥^𝑇1𝐵𝑦,2(𝑥^𝑇𝐷𝑦−𝑟𝑥^𝑇𝐷𝑦−𝑥^𝑇𝐶𝑦+1)if𝑥^𝑇𝐶𝑦≤𝑟≤𝑥^𝑇𝐴𝑦,1if𝑟≤𝑥^𝑇𝐶𝑦. (3.21) This completes the proof.

Because of 𝑐_𝑖𝑗≤𝑏_𝑖𝑗≤𝑎_𝑖𝑗≤𝑑_𝑖𝑗, we can easily have 𝑥^𝑇𝐶𝑦≤𝑥^𝑇𝐴𝑦≤𝑥^𝑇𝐵𝑦≤𝑥^𝑇𝐷𝑦. Then let us discuss the existence of 𝑟-trust maximin equilibrium strategy and consider two simple cases firstly.

Theorem 3.7. If 𝑟<min{𝑐_𝑖𝑗} for all 𝑖=1,2…,𝑚,𝑗=1,2,…,𝑛, then all strategies (𝑥,𝑦) are 𝑟-trust maximin equilibrium strategies.

Proof. Suppose 𝑙=min{𝑐_𝑖𝑗}, then 𝑥^𝑇𝐶𝑦=^𝑚𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝑥_𝑖𝑦_𝑗≥^𝑚𝑛_𝑖=1𝑗=1𝑙𝑥_𝑖𝑦_𝑗=𝑙. (3.22)
Because 𝑟<min{𝑐_𝑖𝑗}=𝑙, then all strategies (𝑥,𝑦) satisfy 𝑥^𝑇𝐶𝑦≥𝑟, according to Lemma 3.6, we have Tr{𝑥^𝑇𝑃𝑦≥𝑟}=1∀(𝑥,𝑦). (3.23) We choose any strategy (𝑥^∗,𝑦^∗), for all strategy (𝑥,𝑦), Tr{𝑥^𝑇𝑃𝑦^∗≥𝑟}=Tr{𝑥^∗𝑇𝑃𝑦^∗≥𝑟}=Tr{𝑥^∗𝑇𝑃𝑦≥𝑟}=1. (3.24) Thus, all strategies (𝑥,𝑦) are 𝑟-trust maximin equilibrium strategies. This completes the proof.

Theorem 3.8. If 𝑟>max{𝑑_𝑖𝑗} for all 𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛, then all strategies (𝑥,𝑦) are 𝑟-trust maximin equilibrium strategies.

Proof. The proof is similar with that of Theorem 3.7.

After discussing two particular cases, let us consider the usual case if there exists 𝑟-trust maximin equilibrium strategy (𝑥,𝑦).

Theorem 3.9. In a two-person zero-sum game, rough variables 𝜉_𝑖𝑗 (𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛) represent the payoffs player I receives or player II loses, and the payoff matrix 𝑃 is defined by (3.1). For a predetermined number 𝑟, if for all (𝑥,𝑦), they cannot satisfy anyone of the following conditions: (1) 𝑥^𝑇𝐷𝑦≤𝑟, (2) 𝑥^𝑇𝐵𝑦≤𝑟≤𝑥^𝑇𝐷𝑦, (3) 𝑥^𝑇𝐴𝑦≤𝑟≤𝑥^𝑇𝐵𝑦, (4) 𝑥^𝑇𝐶𝑦≤𝑟≤𝑥^𝑇𝐴𝑦, (5) 𝑟≤𝑥^𝑇𝐶𝑦, then there does not exist one 𝑟-trust maximin equilibrium strategy.

Proof. Let us only discuss one of five cases, the others are considered similarly. Suppose 𝑆={(𝑥,𝑦)∣𝑥^𝑇𝐵𝑦≥𝑟≥𝑥^𝑇𝐴𝑦,∑^𝑚_𝑖=1𝑥_𝑖=1,∑^𝑛_𝑗=1𝑦_𝑗=1,0≤𝑥_𝑖,𝑦_𝑗≤1} and 𝑄={(𝑥,𝑦)∣𝑥^𝑇𝐴𝑦≥𝑟≥𝑥^𝑇𝐶𝑦,∑^𝑚_𝑖=1𝑥_𝑖=1,∑^𝑛_𝑗=1𝑦_𝑗=1,0≤𝑥_𝑖,𝑦_𝑗≤1}. If not all (𝑥,𝑦)∈𝑆, then without loss of generality we can suppose other (𝑥,𝑦)∈𝑄. If there exists a 𝑟-trust maximin equilibrium strategy (𝑥^∗,𝑦^∗) in 𝑆, according to Lemma 3.6, we have Tr{𝑥^∗𝑇𝑃𝑦^∗1≥𝑟}=2(𝑥^∗𝑇𝐷𝑦^∗−𝑟𝑥^∗𝑇𝐷𝑦^∗−𝑥^∗𝑇𝐶𝑦^∗+𝑥^∗𝑇𝐵𝑦^∗−𝑟𝑥^∗𝑇𝐵𝑦^∗−𝑥^∗𝑇𝐴𝑦^∗). (3.25)
Since 𝑄≠Φ, then for the strategy 𝑦^∗, there exists strategy (𝑥,𝑦^∗)∈𝑄 such that 𝑥^𝑇𝐶𝑦^∗<𝑟<𝑥^𝑇𝐴𝑦^∗, then according to Lemma 3.6, we have Tr{𝑥^𝑇𝑃𝑦^∗1≥𝑟}=2(𝑥^𝑇𝐷𝑦^∗−𝑟𝑥^𝑇𝐷𝑦^∗−𝑥^𝑇𝐶𝑦^∗+1). (3.26)
It is apparent that Tr{𝑥^∗𝑇𝑃𝑦^∗≥𝑟}≤Tr{𝑥^𝑇𝐶𝑦^∗>𝑟}. Namely, Tr{𝑥^∗𝑇𝑃𝑦^∗≥𝑟}≯Tr{𝑥^𝑇𝐶𝑦^∗>𝑟}. This is in conflict with the definition of 𝑟-trust maximin equilibrium strategy.
Similarly, if (𝑥^∗,𝑦^∗)∈𝑄, according to Lemma 3.6, we have Tr{𝑥^∗𝑇𝑃𝑦^∗1≥𝑟}=2(𝑥^∗𝑇𝐷𝑦^∗−𝑟𝑥^∗𝑇𝐷𝑦^∗−𝑥^∗𝑇𝐶𝑦^∗+1). (3.27)
Since 𝑆≠Φ, then for the strategy 𝑥^∗, there exists strategy (𝑥^∗,𝑦) such that 𝑥^∗𝑇𝐵𝑦≥𝑟≥𝑥^∗𝑇𝐴𝑦, then Tr{𝑥^∗𝑇1𝑃𝑦≥𝑟}=2(𝑥^∗𝑇𝐷𝑦−𝑟𝑥^∗𝑇𝐷𝑦−𝑥^∗𝑇+𝑥𝐶𝑦^∗𝑇𝐵𝑦−𝑟𝑥^∗𝑇𝐵𝑦−𝑥^∗𝑇𝐴𝑦). (3.28)
It is apparent that Tr{𝑥^∗𝑇𝑃𝑦^∗≥𝑟}≥Tr{𝑥^𝑇𝐶𝑦>𝑟}. Namely, Tr{𝑥^∗𝑇𝑃𝑦^∗≥𝑟}0𝑥0008𝑒≮Tr{𝑥^∗𝑇𝐶𝑦>𝑟}. This is in conflict with the definition of 𝑟-trust maximin equilibrium strategy too. Then there does not exist a 𝑟-trust maximin equilibrium strategy in this case.
The other cases can be proved in the same way. This completes the proof.

According to Theorem 3.9, we know that this game exists 𝑟-trust maximin equilibrium strategy (𝑥^∗,𝑦^∗) only if all strategies (𝑥,𝑦) are in some section, for example, 𝑥^𝑇𝐵𝑦≤𝑟≤𝑥^𝑇𝐷𝑦. Next let us discuss the following case that all strategies (𝑥,𝑦) are in some section.

Theorem 3.10. For all strategies (𝑥,𝑦) satisfying 𝑥^𝑇𝐵𝑦≤𝑟≤𝑥^𝑇𝐷𝑦, the game exists a 𝑟-trust maximin equilibrium strategy if and only if linear programming problems (3.29) and (3.30) have optimal solutions, where problems (3.29) and (3.30) are separately characterized as follows: 1max2(𝑞^𝑇𝑞𝐷𝑦−𝑟𝑝)s.t.^𝑇𝑞𝐵𝑦−𝑟𝑝≤0,_𝑇𝑞𝐷𝑦−𝑟𝑝≥0,₁+𝑞₂+⋯+𝑞_𝑚𝑞=𝑝,_𝑖≥0,𝑖=1,2,…,𝑚, (3.29) where 𝑝=1/(𝑥^𝑇𝐷𝑦−𝑥^𝑇𝐶𝑦),𝑞^𝑇=𝑝𝑥^𝑇,𝑦 is any fixed vector, 1min2(𝑥^𝑇𝑥𝐷𝑡−𝑟𝑠)s.t.^𝑇𝑥𝐵𝑡−𝑟𝑠≤0,_𝑇𝑡𝐷𝑡−𝑟𝑠≥0,₁+𝑡₂+⋯+𝑡_𝑛𝑡=𝑠,_𝑖≥0,𝑖=1,2,…,𝑛, (3.30) where 𝑠=1/(𝑥^𝑇𝐷𝑦−𝑥^𝑇𝐶𝑦),𝑡=𝑝𝑦,𝑥 is any fixed vector.

Proof. For all strategies (𝑥,𝑦) satisfying 𝑥^𝑇𝐵𝑦≤𝑟≤𝑥^𝑇𝐷𝑦, the trust measure function of payoffs matrix 𝑃 is characterized by the following equation: Tr{𝑥^𝑇𝑥𝑃𝑦≥𝑟}=^𝑇𝐷𝑦−𝑟2(𝑥^𝑇𝐷𝑦−𝑥^𝑇.𝐶𝑦) (3.31) Suppose 𝑀={(𝑥,𝑦)∣𝑥^𝑇𝐵𝑦≤𝑟≤𝑥^𝑇𝐷𝑦,∑^𝑚_𝑖=1𝑥_𝑖=1,∑^𝑛_𝑗=1𝑦_𝑗=1,0≤𝑥_𝑖,𝑦_𝑗≤1}. According to the definition of 𝑟-trust maximin equilibrium strategy, whether the game equilibrium has an equilibrium strategy in 𝑀 is equal to the following two problems.
For any fixed 𝑦, 𝑥max^𝑇𝐷𝑦−𝑟2(𝑥^𝑇𝐷𝑦−𝑥^𝑇𝑥𝐶𝑦)s.t.^𝑇𝑥𝐵𝑦≤𝑟,_𝑇𝑥𝐷𝑦≥𝑟,₁+𝑥₂+⋯+𝑥_𝑚=1,0≤𝑥_𝑖≤1,𝑖=1,2,…,𝑚. (3.32)
For any fixed 𝑥, 𝑥min^𝑇𝐷𝑦−𝑟2(𝑥^𝑇𝐷𝑦−𝑥^𝑇𝑥𝐶𝑦)s.t.^𝑇𝑥𝐵𝑦≤𝑟,_𝑇𝑦𝐷𝑦≥𝑟,₁+𝑦₂+⋯+𝑦_𝑛=1,0≤𝑦_𝑖≤1,𝑖=1,2,…,𝑛. (3.33)
We know that only if the two problems have optimal solution, the game exists an equilibrium strategy. Because problems (3.32) and (3.33) are similar, then let us only discuss problem (3.32). For a fixed 𝑦, let 𝑝=1/2(𝑥^𝑇𝐷𝑦−𝑥^𝑇𝐶𝑦),𝑞^𝑇=𝑝𝑥^𝑇. Then problem (3.32) is converted into a linear programming problem 1max2(𝑞^𝑇𝑞𝐷𝑦−𝑟𝑝)s.t.^𝑇𝑞𝐵𝑦−𝑟𝑝≤0,_𝑇𝑞𝐷𝑦−𝑟𝑝≥0,₁+𝑞₂+⋯+𝑞_𝑚𝑞=𝑝,_𝑖≥0,𝑖=1,2,…,𝑚. (3.34) Similarly, problem (3.33) can be converted into the following problem, for any fixed 𝑥, 1min2(𝑥^𝑇𝑥𝐷𝑡−𝑟𝑠)s.t.^𝑇𝑥𝐵𝑡−𝑟𝑠≤0,_𝑇𝑡𝐷𝑡−𝑟𝑠≥0,₁+𝑡₂+⋯+𝑡_𝑛𝑡=𝑠,_𝑖≥0,𝑖=1,2,…,𝑛, (3.35) where 𝑠=1/(𝑥^𝑇𝐷𝑦−𝑥^𝑇𝐶𝑦),𝑡=𝑝𝑦.

For problems (3.29) and (3.30), we can make use of MATLAB to get optimal solution of the programming problem by turning them a bi-level programming problem. Here, we do not give the detail description. Then we go on to discuss another case that 𝑥^𝑇𝐶𝑦≤𝑟≤𝑥^𝑇𝐴𝑦. Similarly, we can get the following conclusion.

Theorem 3.11. For all strategies (𝑥,𝑦) satisfying 𝑥^𝑇𝐶𝑦≤𝑟≤𝑥^𝑇𝐴𝑦, the game exists a 𝑟-trust maximin equilibrium strategy if and only if linear programming problems (3.36) and (3.37) have optimal solutions, where problems (3.36) and (3.37) are characterized as follows: 1max2(𝑞^𝑇𝑞𝐷𝑦−𝑟𝑝+1)s.t.^𝑇𝑞𝐴𝑦−𝑟𝑝≥0,_𝑇𝑞𝐶𝑦−𝑟𝑝≤0,₁+𝑞₂+⋯+𝑞_𝑚𝑞=𝑝,_𝑖≥0,𝑖=1,2,…,𝑚, (3.36) where 𝑝=1/(𝑥^𝑇𝐷𝑦−𝑥^𝑇𝐶𝑦),𝑞^𝑇=𝑝𝑥^𝑇,𝑦 is any fixed vector, 1min2(𝑥^𝑇𝑥𝐷𝑡−𝑟𝑠+1)s.t.^𝑇𝑥𝐴𝑡−𝑟𝑠≥0,_𝑇𝑡𝐶𝑡−𝑟𝑠≤0,₁+𝑡₂+⋯+𝑡_𝑛𝑡=𝑠,_𝑖≥0,𝑖=1,2,…,𝑛, (3.37) where 𝑠=1/(𝑥^𝑇𝐷𝑦−𝑥^𝑇𝐶𝑦),𝑡=𝑠𝑦,𝑥 is any fixed vector.

Proof. It can be proved similarly as Theorem 3.10.

We have discussed many simple cases; there is still a more complicated case that 𝑥^𝑇𝐴𝑦≤𝑟≤𝑥^𝑇𝐵𝑦. For this case, according to Lemma 3.6, we have that Tr{𝑥^𝑇1𝑃𝑦≥𝑟}=2(𝑥^𝑇𝐷𝑦−𝑟𝑥^𝑇𝐷𝑦−𝑥^𝑇+𝑥𝐶𝑦^𝑇𝐵𝑦−𝑟𝑥^𝑇𝐵𝑦−𝑥^𝑇𝐴𝑦). (3.38) Right now, the problem to find if the game has the 𝑟-trust maximin equilibrium strategy is converted into the following two problems: 1max2(𝑥^𝑇𝐷𝑦−𝑟𝑥^𝑇𝐷𝑦−𝑥^𝑇+𝑥𝐶𝑦^𝑇𝐵𝑦−𝑟𝑥^𝑇𝐵𝑦−𝑥^𝑇)𝑥𝐴𝑦s.t.^𝑇𝑥𝐴𝑦≤𝑟,_𝑇𝑥𝐵𝑦≥𝑟,₁+𝑥₂+⋯+𝑥_𝑚𝑥=1,_𝑖≥0,𝑖=1,2,…,𝑚, (3.39) where 𝑦 is any fixed vector, and

1min2(𝑥^𝑇𝐷𝑦−𝑟𝑥^𝑇𝐷𝑦−𝑥^𝑇+𝑥𝐶𝑦^𝑇𝐵𝑦−𝑟𝑥^𝑇𝐵𝑦−𝑥^𝑇)𝑥𝐴𝑦s.t.^𝑇𝑥𝐴𝑦≤𝑟,_𝑇𝑦𝐵𝑦≥𝑟,₁+𝑦₂+⋯+𝑦_𝑛𝑦=1,_𝑖≥0,𝑖=1,2,…,𝑛, (3.40) where 𝑥 is any fixed vector. The two problems are nonlinear programming problems. We can make use of many traditional methods to solve them, for example, methods of feasible directions (see Polak [21]), Frank-Wolfe methods (see Meyer and Podrazik [22]). However, the solutions we got through these methods are usually local optimal solutions, not the global optimal solutions. In the following section, we will introduce an algorithm to solve the nonlinear programming such as problems (3.39) and (3.40). Then we can get the 𝑟-trust minmax equilibrium strategy of the game.

4. Genetic Algorithm

For many complex problems such as problem (3.39) and (3.40), it is difficult to obtain its optimal solution by the traditional technique. Therefore, GA is an efficient tool to obtain the efficient solution by its global searching ability. Take problem (3.39) as an example and we will list the detailed procedure to illustrate how the genetic algorithm introduced by Gen and Cheng [23] works. Let 𝐻(𝑥,𝑦)=(1/2)((𝑥^𝑇𝐷𝑦−𝑟)/(𝑥^𝑇𝐷𝑦−𝑥^𝑇𝐶𝑦)+(𝑥^𝑇𝐵𝑦−𝑟)/(𝑥^𝑇𝐵𝑦−𝑥^𝑇𝐴𝑦)) express the objective function and 𝑋={(𝑥,𝑦)∣𝑥^𝑇𝐴𝑦≤𝑟,𝑥^𝑇𝐵𝑦≥𝑟,∑^𝑚_𝑖=1𝑥_𝑖=1,∑^𝑚_𝑖=1𝑦_𝑖=1,𝑦_𝑖,𝑥_𝑖≥0,𝑖=1,2,…,𝑚} express the feasible set.

(1) Initializing process: the initial population is formed by 𝑁_pop-size chromosomes associated with basic feasible solutions of problem (3.39). Hence any general procedure to get them can be applied. Therefore, we can take the solution (𝑥,𝑦)=(𝑥₁,…,𝑥_𝑚;𝑦₁,…,𝑦_𝑛)^𝑇∈𝑋 as a chromosome. Randomly generate the feasible solution (𝑥,𝑦) in 𝑋. Repeat the above process 𝑁_pop-size times, then we have 𝑁 initial feasible chromosomes (𝑥¹,𝑦¹),(𝑥²,𝑦²),…,(𝑥^𝑁pop-size,𝑦^𝑁popsize).

(2) Evaluation function: in this case, we only attempt to obtain the best solution, which is absolutely superior to all other alternatives by comparing the objective function. Then we can construct the evaluation function by the following procedure: (i) compute the objective value 𝐻(𝑥^𝑖,𝑦^𝑖), (ii) the evaluation function is constructed as follows: eval(𝑥^𝑖,𝑦^𝑖)=𝐻(𝑥^𝑖,𝑦^𝑖)∑^𝑁pop-size_𝑖=1𝐻(𝑥^𝑖,𝑦^𝑖) (4.1) which expresses the evaluation value of the 𝑖th chromosome in current generation.

(3) Selection process: The selection process is based on spinning the roulette wheel 𝑁_pop-size times. Each time a single chromosome for a new population is selected in the following way. Calculate the cumulative probability 𝑞_𝑖 for each chromosome (𝑥^𝑖,𝑦^𝑖): 𝑞₀=0,𝑞_𝑖=^𝑖_𝑗=1eval(𝑥^𝑗,𝑦^𝑗),𝑖=1,2,…,pop-size. (4.2) Generate a random number 𝑟 in [0,𝑞_pop-size] and select the chromosomes (𝑥^𝑖,𝑦^𝑖) such that 𝑞_𝑖−1<𝑟≤𝑞_𝑖 (1≤1≤𝑁_pop-size). Repeat the above process 𝑁_pop-size times and obtain 𝑁_pop-size copies of chromosomes.

(4) Crossover operation: the goal of crossover is to exchange information between two parent chromosomes in order to produce two new offspring for the next population. The uniform crossover of Genetic operator proposed by Li et al. [24] in this paper. The detail is as follows. Generate a random number 𝑐∈(0,1) and if 𝑐<𝑃_𝑐, then the chromosome (𝑥^𝑖,𝑦^𝑖) is selected as a parament, where the parameter 𝑃_𝑐 which is the probability of crossover operation. Repeat this process 𝑁_pop-size times and we get 𝑃_𝑐⋅𝑁_pop-size parent chromosomes to undergo the crossover operation. The crossover operator on (𝑥¹,𝑦¹) and (𝑥²,𝑦²) will produce two children as follows: 𝑋¹𝑌¹𝑥=𝑐¹𝑦¹𝑥+(1−𝑐)²𝑦²,𝑋²𝑌²𝑥=𝑐²𝑦²𝑥+(1−𝑐)¹𝑦¹. (4.3) The children of chromosomes (𝑋¹,𝑌¹) and (𝑋²,𝑌²) can be generated as above. They are feasible if they are both in 𝑋 and then we replace the parents with them. Or else we keep the feasible one if it exists. Redo the crossover operator until we obtaine two feasible children or a given number of cycles is finished.

(5) Mutation operation: a mutation operator is a random process where one genotype is replaced by another to generate a new chromosome. Each genotype has the probability of mutation, 𝑃_𝑚, to change from 0 to 1. Let (𝑥^𝑖,𝑦^𝑖) be selected as parent. Choose a mutation direction 𝐝∈𝐑^𝑚+𝑛 randomly. 𝑀 is an appropriate large positive number. We replace the parent (𝑥^𝑖,𝑦^𝑖) with the child 𝑋^𝑖𝑌^𝑖=𝑥^𝑖𝑦^𝑖+𝑀⋅𝐝. (4.4) If (𝑋^𝑖,𝑌^𝑖) are infeasible, we set 𝑀 as a random number between 0 and 𝑀 until it is feasible and then replace (𝑥^𝑖,𝑦^𝑖) with it.

Above all, it can be simply summarized in Procedure 1.

Input: GA parameters: 𝑁pop-size,𝑃𝑐,𝑃𝑚 and the cycle number Genmax Output: optimal sampling level, NPV begin  𝑔←0;  initialization( ) by checking the feasiblity;  evaluation(0);  while (𝑔≤Genmax) do    selection( );    crossover( );    mutation( );    evaluation(𝑔);    𝑔←𝑔+1;  end  Output: the optimal solution End

Procedure 1 

Procedure of genetic algorithm

5. Numerical Example

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 𝑖 and player II gives strategy 𝑗, player II will give some money to player I which is at least between 𝑐_𝑖𝑗 and 𝑎_𝑖𝑗, or at most between 𝑏_𝑖𝑗 and 𝑑_𝑖𝑗. The payoff matrix of player I is as follows 𝜉𝑃=¹¹𝜉₁₂𝜉₁₃𝜉₂₁𝜉₂₂𝜉₂₃𝜉₃₁𝜉₃₂𝜉₃₃, (5.1) where rough variables 𝜉_𝑖𝑗 (𝑖=1,2,3,𝑗=1,2,3) are characterized as 𝜉₁₁=([15,25],[10,28]),𝜉₁₂=([13.5,22],[8,25]),𝜉₁₃𝜉=([15,20],[11.2,21]),₂₁=([17,30],[9,35]),𝜉₂₂=([16.2,26],[12,28]),𝜉₂₃𝜉=([13,27],[10,30]),₃₁=([18,20],[11,24]),𝜉₃₂=([18,24],[12,29]),𝜉₃₃=([13,20],[12,25]). (5.2)

Firstly, let us consider the expected maximin equilibrium strategy of this game. According to Remarks 2.9 and 3.2, we have that 𝐸[𝑥^𝑇𝑃𝑦]=𝐸[^𝑛𝑚_𝑗=1𝑖=1𝜉_𝑖𝑗𝑥_𝑖𝑦_𝑗]=^𝑛𝑚_𝑗=1𝑖=1𝐸[𝜉_𝑖𝑗]𝑥_𝑖𝑦_𝑗=𝑥^𝑇𝑃^′𝑦, (5.3) where 𝑃^′=𝐸[𝜉₁₁]𝐸[𝜉₁₂]𝐸[𝜉₁₃]𝐸[𝜉₂₁]𝐸[𝜉₂₂]𝐸[𝜉₂₃]𝐸[𝜉₃₁]𝐸[𝜉₃₂]𝐸[𝜉₃₃]=19.517.12516.822.7520.552018.2520.7517.5. (5.4)

Then, we can get the equilibrium strategy that when player I gives the mixed strategy 𝑥=(0,0,1) and player II gives the mixed strategy 𝑦=(0,1,0), player I gets the most payoff 20 which is the least payoffs player II loses.

Next, let us consider if this game has the 𝑟-trust maximin equilibrium strategy. According to Lemma 3.6, we have ,.𝐴=1513.5151716.213181813,𝐵=252220302627202420𝐶=10811.291210111212,𝐷=282521352830242925 (5.5)

Then we give five predetermined numbers 𝑟 and discuss if the game exists a 𝑟-trust maximin equilibrium strategy.

Case 1 (𝑟=5). Apparently, min𝑐_𝑖𝑗=8 and 0≤𝑥,𝑦≤1. Thus, for all (𝑥,𝑦), they satisfy 𝑥^𝑇𝐶𝑦≥5. Based on Theorem 3.7, we know that all (𝑥,𝑦) are 5-trust maximin equilibrium strategy of this game.

Case 2 (𝑟=40). Similarly, max𝑑_𝑖𝑗=35 and 0≤𝑥,𝑦≤1. Thus, for all (𝑥,𝑦), they satisfy 𝑥^𝑇𝐶𝑦≤40. Based on Theorem 3.8, we know that all (𝑥,𝑦) are 40-trust maximin equilibrium strategy of this game.

Case 3 (𝑟=25). Because max𝑏_𝑖𝑗=30 and min𝑑_𝑖𝑗=21, for 0≤𝑥,𝑦≤1, not all (𝑥,𝑦) satisfy 𝑥^𝑇𝐵𝑦≤25≤𝑥^𝑇𝐷𝑦. According to Theorem 3.9, we know that this game does not exist a 25-trust maximin equilibrium strategy.

Case 4 (𝑟=12.5). Apparently, max𝑐_𝑖𝑗=12 and min𝑎_𝑖𝑗=13. Thus, for 0≤𝑥,𝑦≤1, all (𝑥,𝑦) satisfy 𝑥^𝑇𝐶𝑦≤12.5≤𝑥^𝑇𝐴𝑦. Based on Theorem 3.11, we can get the following two problems 1max2(³³_𝑖=1𝑗=1𝑞_𝑖𝑦_𝑗𝑑_𝑖𝑗−12.5𝑝+1)s.t.^{33𝑖=1𝑗=1}𝑞_𝑖𝑦_𝑗𝑎_𝑖𝑗−12.5𝑝≥0,_{33𝑖=1𝑗=1}𝑞_𝑖𝑦_𝑗𝑐_𝑖𝑗𝑞−12.5𝑝≤0,₁+𝑞₂+𝑞₃𝑞=𝑝,₁,𝑞₂,𝑞₃≥0, (5.6) where 𝑦 is any fixed strategy, 𝑝=1/(𝑥^𝑇𝐷𝑦−𝑥^𝑇𝐶𝑦),𝑞^𝑇=𝑝𝑥^𝑇,1min2(³³_𝑖=1𝑗=1𝑥_𝑖𝑡_𝑗𝑑_𝑖𝑗−12.5𝑠+1)s.t.^{33𝑖=1𝑗=1}𝑥_𝑖𝑡_𝑗𝑎_𝑖𝑗−12.5𝑠≥0,_{33𝑖=1𝑗=1}𝑥_𝑖𝑡_𝑗𝑐_𝑖𝑗𝑡−12.5𝑠≤0,₁+𝑡₂+𝑡₃𝑡=𝑠,₁,𝑡₂,𝑡₃≥0, (5.7) where 𝑥 is any fixed strategy, 𝑠=1/(𝑥^𝑇𝐷𝑦−𝑥^𝑇𝐶𝑦),𝑡=𝑠𝑦.
Then we can make use of simplex method to get the optimal solutions of problems (5.6) and (5.7). The optimal solutions is (𝑥,𝑦)^𝑇=(0.378,0.124,0.498,0.397,0.469,0.194)^𝑇.

Case 5 (𝑟=19). Apparently, max𝑎_𝑖𝑗=18 and min𝑏_𝑖𝑗=20. Thus, for 0≤𝑥,𝑦≤1, all (𝑥,𝑦) satisfy 𝑥^𝑇𝐴𝑦≤19≤𝑥^𝑇𝐵𝑦. Then we can get two nonlinear programming problems.
For any fixed 𝑦1max2(∑³_𝑖=1∑³_𝑗=1𝑥_𝑖𝑦_𝑗𝑑_𝑖𝑗−19∑³_𝑖=1∑³_𝑗=1𝑥_𝑖𝑦_𝑗(𝑑_𝑖𝑗−𝑐_𝑖𝑗)+∑³_𝑖=1∑³_𝑗=1𝑥_𝑖𝑦_𝑗𝑏_𝑖𝑗−19∑³_𝑖=1∑³_𝑗=1𝑥_𝑖𝑦_𝑗(𝑏_𝑖𝑗−𝑎_𝑖𝑗))s.t.^{33𝑖=1𝑗=1}𝑥_𝑖𝑦_𝑗𝑎_𝑖𝑗≤19,_{33𝑖=1𝑗=1}𝑥_𝑖𝑦_𝑗𝑏_𝑖𝑗𝑥≥19,₁+𝑥₂+𝑥₃𝑥=1,_𝑖,𝑦_𝑗≥0. (5.8)
For any fixed 𝑥, 1min2(∑³_𝑖=1∑³_𝑗=1𝑥_𝑖𝑦_𝑗𝑑_𝑖𝑗−19∑³_𝑖=1∑³_𝑗=1𝑥_𝑖𝑦_𝑗(𝑑_𝑖𝑗−𝑐_𝑖𝑗)+∑³_𝑖=1∑³_𝑗=1𝑥_𝑖𝑦_𝑗𝑏_𝑖𝑗−19∑³_𝑖=1∑³_𝑗=1𝑥_𝑖𝑦_𝑗(𝑏_𝑖𝑗−𝑎_𝑖𝑗))s.t.^{33𝑖=1𝑗=1}𝑥_𝑖𝑦_𝑗𝑎_𝑖𝑗≤19,_{33𝑖=1𝑗=1}𝑥_𝑖𝑦_𝑗𝑏_𝑖𝑗𝑦≥19,₁+𝑦₂+𝑦₃𝑥=1,_𝑖,𝑦_𝑗≥0. (5.9) For problems (5.8) and (5.9), we can solve them by using the rough simulation-based genetic algorithm. Let the number of chromosomes in every generation be 20, the probability of crossover 0.85 and of mutation 0.1. Then after a run of a genetic algorithm computer program (5000 generations), we get the optimal solution (𝑥,𝑦)^𝑇=(0.178,0.251,0.571,0.349,0.479,0.172)^𝑇 and the objective value as shown in Figure 1.

Figure 1 

The results by GA.

6. Conclusion

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.

Acknowledgments

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).