Birough programming approach for solving bi-matrix games with birough payoff elements

Abstract

This paper analyzes the bi-matrix games under the light of birough programming. The combination of bi-matrix games and birough programming produces a new form which is defined here as birough bi-matrix games. In birough bi-matrix games, the payoff elements are characterized by birough variables and uncertainty of the birough variable is measured by birough measure which known as Chance (Ch). Utilizing the Chance measure, we have defined the expected birough Nash equilibrium strategy for the players which depends upon the confidence level and outcome of the birough bi-matrix game. In order to show the applicability and feasibility of our proposed method, a real-life example on birough bi-matrix game is presented and then solved.

Keywords

Bi-matrix game birough variable birough constraint expected birough nash equilibrium strategy genetic algorithm

1 Introduction

Game theory [10] is the theory of independent and interdependent model of decision making. It is concerned with decision making in organizations where the outcome depends on the decisions of two players, one of which may be nature itself and where no single decision maker has full control over the outcomes. A game model is constructed around the strategic choices available to players where the preferred outcomes (known as payoffs) are clearly defined and known. Game theory aims to find optimal solutions to situations of conflict and cooperation such as those outlined above, under the assumption that players are instrumentally rational and act in their own best interests. In traditional game theory, the payoff elements are real numbers and it has well proven solution concept. In real scenario, some times the payoff elements cannot be defined as crisp numbers rather have boundary values and these can be represented as birough variables. In this paper, the elements of the payoff matrix are treated as birough variables and the game defined as a birough bi-matrix game.

The rough set concept based on knowledge was proposed by Palwak [11] and its uncertainty measured by the rough measure named as trust which was defined by Liu in 2004. Roughly speaking, a birough variable is a rough variable defined on the universal set of rough variables or a rough variable taking rough variable values. The birough variable and its uncertainty measured by the uncertain measurable function known as Chance measure which is defined by Liu [1]. The birough variable is another type of rough variables whose boundary values are rough variables but in rough variable, boundary is real number. It has been applied to decision analysis, data analysis, electric power systems, etc. It is also an efficient mathematical tool to deal with imprecise, inconsistent and incomplete data. In birough-set theory, a fundamental assumption is that any object from universe is perceived through available information and such information may not be sufficient to characterize the object exactly. Due to the complexity of the real world, many researchers have considered some generalizations of basic rough set so that rough set theory has more applications. These extensions include variable precision birough set which extended the basic rough set to incorporate probabilisticinformation.

In the recent years, there have been attempted to extend the conversion technique of various problems on crisp game theory, which are credibilistic matrix games [9] and bi-matrix games [3], bifuzzy matrix games [4] and bi-matrix games [8], rough bi-matrix games [7] and so on. But the solution concepts are mainly considered as crisp scenario. In all the uncertain types of games firstly, converting into crisp scenario, which depends upon the confidence level and solving it with the help of traditional method. There have been various solutions which depend upon the confidence level and decision maker can find the optimal solution for appropriate confidence level. Firstly, describing the properties of rough set theory and its extension in birough set theory. Then considering the game whose payoff elements are characterized as birough variables and its properties like expected birough Nash equilibrium strategy, birough expected value operator and birough constraints. Finally, the birough bi-matrix game converts into crisp model depending upon the confidence levels and a numerical example has been demonstrated to illustrate the methodology.

The remainder of the paper is organized as follows: the next section described the preliminaries on the definitions of rough and birough theory and its properties. Section 2 discusses the properties of bi-matrix game theory. Section 3 defines the birough bi-matrix game and its properties with existence of strategies for the players and describing the conversion procedure from birough programming problem to the crisp programming problem. In Section 4, a brief description on genetic algorithm is presented. In Section 5, a real-life example has been illustrated to validate our proposed methodology and finally, conclusion of our paper is presented in Section 6.

2 Preliminaries

Definition 2.1. ([11]) Let Θ be the universal set, $R$ be an equivalence relation on Θ, $[θ]_{R}$ be the set of equivalence class of $R$ and Ω be a nonempty subset of Θ. The upper and lower approximations of the set Ω are defined as follows: $\begin{matrix} \bar{R Ω} & = & {θ \in Θ | [θ]_{R} \cap Ω \neq \emptyset} \\ \underline{R Ω} & = & {θ \in Θ | [θ]_{R} \subseteq Ω} \\ ℵ Ω & = & \bar{R Ω} - \underline{R Ω} \end{matrix}$

If ℵΩ≠ ∅, the set Ω is called the rough set.

Definition 2.2. Let Ω be a nonempty subset of Θ, Λ is a σ-algebra of Ω, θ is an element in Λ, and π is a nonnegative, real-valued set function. Then (Ω, θ, Λ, π) is called a rough space.

Definition 2.3. The rough variable ρ is a function from rough space (Ω, θ, Λ, π) to the set of real numbers. Let a, b, c and d are any real numbers such that c ≤ a < b ≤ d and ρ = ([a, b] , [c, d]) is called a rough variable if, $\begin{matrix} {\begin{matrix} a \leq ρ (θ) \leq b, iff θ \in \underline{R Ω}, for exact case . \\ c \leq ρ (θ) < a or b < ρ (θ) \leq d, iff θ \in ℵ Ω, \\ for uncertainty . \end{matrix} \end{matrix}$

Let Tr be the trust measure which deals with rough variables. It is a measure defined on rough space just as probability of random variable in the probability space.

Definition 2.4. Let Tr be the trust which is a measure on rough space (i.e, Tr is a mapping from rough space to [0, 1]) . The measure Tr is defined on set A ∈ Λ as follows: $\begin{matrix} Tr {A} = \frac{1}{2} [\bar{Tr} {A} + \underline{Tr} {A}] \\ where \bar{Tr} {A} = \frac{π {A}}{π {Ω}} and \underline{Tr} {A} = \frac{π {A \cap θ}}{π {θ}} \end{matrix}$

Definition 2.5. ([1]) Let ρ be a rough variable, then the expected value of ρ is defined as follows: $\begin{matrix} E [ρ] = \int_{0}^{\infty} Tr {ρ \geq r} dr - \int_{- \infty}^{0} Tr {ρ \leq r} dr \end{matrix}$ provided that at least one of the two integrals is finite.

Example 2.1. Let ρ = ([a, b] , [c, d]) be a rough variable with c ≤ a < b ≤ d, Then we have, $\begin{matrix} E [ρ] = \frac{1}{4} (a + b + c + d) \\ E [x ρ + y ϱ] = xE [ρ] + yE [ϱ] \end{matrix}$ where x, y are real numbers and ρ, ϱ are rough variables on the same rough space (Ω, θ, Λ, π).

Definition 2.6. ([1]) Let ρ be a rough variable and r ∈ (0, 1). Then $ρ_{sup} (r) = sup {w | Tr {ρ \geq w} \geq r}$ is called the r-optimistic value of ρ.

Example 2.2. Let ρ = ([a, b] , [c, d]) be a rough variable such that c ≤ a < b ≤ d. Then r-optimistic value of ρ is $\begin{matrix} ρ_{sup} (r) = {\begin{matrix} (1 - 2 r) d + 2 rc & if r \leq \frac{d - b}{2 (d - c)} \\ 2 (1 - r) d + (2 r - 1) c & if r \geq \frac{2 d - a - c}{2 (d - c)} \\ \frac{d (b - a) + b (d - c) - 2 r (b - a) (d - c)}{(b - a) + (d - c)} & if \frac{d - b}{2 (d - c)} < r \\ < \frac{2 d - a - c}{2 (d - c)} \end{matrix} \end{matrix}$

Definition 2.7. ([1]) Let ρ be a rough variable andr ∈ (0, 1). Then $ρ_{inf} (r) = inf {w | Tr {ρ \leq w} \geq r}$ is called the r-pessimistic value of ρ.

Example 2.3. Let ρ = ([a, b] , [c, d]) be a rough variable such that c ≤ a < b ≤ d, then r-pessimistic value of ρ is $\begin{matrix} ρ_{inf} (r) = {\begin{matrix} (1 - 2 r) c + 2 rd & if r \leq \frac{a - c}{2 (d - c)} \\ 2 (1 - r) c + (2 r - 1) d & if r \geq \frac{b + d - 2 c}{2 (d - c)} \\ \frac{c (b - a) + a (d - c) + 2 r (b - a) (d - c)}{(b - a) + (d - c)} & if \frac{a - c}{2 (d - c)} < r \\ < \frac{b + d - 2 c}{2 (d - c)} \end{matrix} \end{matrix}$

Definition 2.8. ([1]) A birough variable is a function ϱ from a rough space (Ω, θ, Λ, π) to the set of rough variables such that Tr {ϱ (θ) ∈ B} is a measurable function of θ for any Borel set B of $R$ .

Theorem 2.1. ([1]) Assume that ϱ is a birough variable and B is a Borel set of $R$ . Then the trust Tr {ϱ (θ) ∈ B} is a rough variable.

Theorem 2.2. ([1]) Let ϱ be a birough variable. If the expected value of E [ϱ (θ)] is finite for each θ, then E [ϱ (θ)] is a rough variable. $\begin{matrix} E [ϱ (θ)] = \int_{0}^{\infty} Tr {ϱ (θ) \geq r} dr - \int_{- \infty}^{0} Tr {ϱ (θ) \leq r} dr \end{matrix}$ and $\begin{matrix} E [ϱ] = \int_{0}^{\infty} Tr {θ \in Θ ∣ E [ϱ (θ)] \geq r} dr \\ - \int_{- \infty}^{0} Tr {θ \in Θ ∣ E [ϱ (θ)] \leq r} dr \end{matrix}$ provided that at least one of the two integrals is finite.

Theorem 2.3. ([1]) Assume that ϱ and σ are birough variables with finite expected values. Then for any real numbers a and b, we have, $\begin{matrix} E [a ϱ + b σ] = aE [ϱ] + bE [σ] \end{matrix}$

Example 2.4. Let ϱ = ([ρ ₁, ρ ₂] , [ρ ₃, ρ ₄]) be a birough variable with independent rough variable ρ _i (i = 1, 2, 3, 4) Then we have, $\begin{matrix} E [ϱ] & = & \frac{1}{4} E [ρ_{1} + ρ_{2} + ρ_{3} + ρ_{4}] \\ = & \frac{1}{4} [E [ρ_{1}] + E [ρ_{2}] + E [ρ_{3}] + E [ρ_{4}]] \\ = & \frac{1}{4} [\frac{1}{4} (a_{1} + b_{1} + c_{1} + d_{1}) + \frac{1}{4} (a_{2} + b_{2} + c_{2} \\ + d_{2}) + \frac{1}{4} (a_{3} + b_{3} + c_{3} + d_{3}) \\ + \frac{1}{4} (a_{4} + b_{4} + c_{4} + d_{4})] \\ = & \frac{1}{16} (a_{1} + b_{1} + c_{1} + d_{1} + a_{2} + b_{2} + c_{2} + d_{2} \\ + a_{3} + b_{3} + c_{3} + d_{3} + a_{4} + b_{4} + c_{4} + d_{4}) \end{matrix}$ where the rough variable ρ _i = ([a _i, b _i] , [c _i, d _i]) such that c _i ≤ a _i < b _i ≤ d _i (i = 1, 2, 3, 4).

Definition 2.9. ([1]) Let ϱ be a birough variable and B be a Borel set of $R$ . Then the chance of birough event ϱ ∈ B is a function from (0, 1] to [0, 1] defined as, $\begin{matrix} Ch {ϱ \in B} (γ) = sup_{Tr {A} \geq γ} inf_{θ \in A} Tr {ϱ (θ) \in B} \end{matrix}$

Theorem 2.4. ([1]) Let ϱ be a birough variable and B be a Borel set of $R$ . Consider δ ^* = Ch {ϱ ∈ B} (γ ^*). Then we have, $\begin{matrix} Tr {θ \in Θ | Tr {ϱ (θ) \in B} \geq δ^{*}} \geq γ^{*} \end{matrix}$

Definition 2.10. ([1]) Let ϱ be a birough variable and γ, δ ∈ (0, 1]. Then we call $\begin{matrix} ϱ_{sup} (γ, δ) = sup {r | Ch {ϱ \geq r} (γ) \geq δ} \end{matrix}$ the (γ, δ)-optimistic value to ϱ, and $\begin{matrix} ϱ_{inf} (γ, δ) = inf {r | Ch {ϱ \leq r} (γ) \geq δ} \end{matrix}$ the (γ, δ)-pessimistic value to ϱ

Theorem 2.5. ([1]) Let ϱ be a birough variable and γ, δ ∈ (0, 1]. Assume that ϱ _sup (γ, δ) is the (γ, δ)- optimistic value and ϱ _inf (γ, δ) is the (γ, δ)- pessimistic value to ϱ . Then we have, $\begin{matrix} Ch {ϱ \leq ϱ_{inf} (γ, δ)} (γ) \geq δ}, \\ Ch {ϱ \geq ϱ_{sup} (γ, δ)} (γ) \geq δ} \end{matrix}$

Theorem 2.6. ([1]) Let ϱ _sup (γ, δ) and ϱ _inf (γ, δ) be the (γ, δ)-optimistic and (γ, δ)-pessimistic values of a birough variable ϱ, respectively. If γ ≤ 0.5, then we have, $\begin{matrix} ϱ_{inf} (γ, δ) \leq ϱ_{sup} (γ, δ) + δ_{1} \end{matrix}$ if γ ≥ 0.5, then we have, $\begin{matrix} ϱ_{inf} (γ, δ) + δ_{2} \geq ϱ_{sup} (γ, δ) \end{matrix}$ where δ ₁ and δ ₂ are defined by $\begin{matrix} δ_{1} & = & sup_{θ \in Θ} {ϱ (θ)_{sup} (1 - δ) - ϱ (θ)_{inf} (1 - δ)} \\ δ_{2} & = & sup_{θ \in Θ} {ϱ (θ)_{sup} (δ) - ϱ (θ)_{inf} (δ)} \end{matrix}$ and ϱ (θ) _sup (δ) and ϱ (θ) _inf (δ) are δ-optimistic and δ-pessimistic values of rough variable ϱ (θ) for each θ, respectively.

Theorem 2.7. Let ρ = ([a, b] , [c, d]) be a rough variable such that c ≤ a < b ≤ d and Tr be the trust which is measure on the rough space. For any given confidence level r ∈ (0, 1),

i) when $r \leq \frac{a - c}{2 (d - c)}, Tr {ρ \leq 0} \geq r$ if and only if $\begin{matrix} (1 - 2 r) c + 2 rd \leq 0 \end{matrix}$ ii) when $r \geq \frac{b + d - 2 c}{2 (d - c)}, Tr {ρ \leq 0} \geq r$ if and only if $\begin{matrix} 2 (1 - r) c + (2 r - 1) d \leq 0 \end{matrix}$ iii) when $\frac{a - c}{2 (d - c)} < r < \frac{b + d - 2 c}{2 (d - c)}, Tr {ρ \leq 0} \geq r$ if and only if $\begin{matrix} \frac{2 rbd + (1 - 2 r) ad + (1 - 2 r) bc + 2 (r - 1) ac}{(b - a) + (d - c)} \leq 0 \end{matrix}$

Proof: Since ρ = ([a, b] , [c, d]) be a rough variable such that c ≤ a < b ≤ d . Then the trust Tr is defined as follows, $\begin{matrix} Tr {ρ \leq x} = {\begin{matrix} 1 & if d \leq x \\ \frac{d - 2 c + x}{2 (d - c)} & if b \leq x \leq d \\ \frac{2 ac - ad - bc + x (b + d - c - a)}{2 (b - a) (d - c)} & if a \leq x \leq b \\ \frac{c - x}{2 (c - d)} & if c \leq x \leq a \\ 0 & if x \leq c \end{matrix} \end{matrix}$

Since the partition points are at x = c and x = d with boundary points c (i.e., initial point) and d (i.e., end point). In this theorem, x is considered as 0,

i) when $r \leq \frac{a - c}{2 (d - c)}, Tr {ρ \leq 0} \geq r,$ either c ≥ 0 or, $\frac{c}{2 (c - d)} \leq r so that (1 - 2 r) c + 2 rd \leq 0 .$

ii) when $r \geq \frac{b + d - 2 c}{2 (d - c)}, Tr {ρ \leq 0} \geq r,$ either d ≤ 0 or, $\frac{d - 2 c}{2 (d - c)} \geq r so that, 2 (1 - r) c + (2 r - 1) d \leq 0 .$

iii) when $\frac{a - c}{2 (d - c)} < r < \frac{b + d - 2 c}{2 (d - c)}, Tr {ρ \leq 0} \geq r,$ then, $\frac{2 ac - ad - bc}{2 (b - a) (d - c)} \geq r$

sothat, 2rbd + (1 - 2r) ad + (1 - 2r) bc + 2 (r - 1) ac ≤ 0 .

Also, c ≤ a < b ≤ d, so that, (b - a) + (d - c) >0 . Hence the result follows.

Theorem 2.8. Let ρ _k = ([a _k, b _k] , [c _k, d _k]) are rough variables (k = 1, 2, ⋯ , n) and a function Γ (x, ρ) can be written as, $\begin{matrix} Γ (x, ρ) & = & h_{1} (x) ρ_{1} + h_{2} (x) ρ_{2} + \dots + h_{n} (x) ρ_{n} + h_{0} (x) \end{matrix}$

If h _k (x) be a nonnegative real valued function, then there exists a confidence level r ∈ (0, 1) , such that,

i) when $r \leq \frac{a_{k} - c_{k}}{2 (d_{k} - c_{k})},$ define $\begin{matrix} p_{k} = (1 - 2 r) c_{k} + 2 {rd}_{k} \end{matrix}$ ii) when $r \geq \frac{b_{k} + d_{k} - 2 c_{k}}{2 (d_{k} - c_{k})},$ define $\begin{matrix} p_{k} = 2 (1 - r) c_{k} + (2 r - 1) d_{k} \end{matrix}$ iii) when $\frac{a_{k} - c_{k}}{2 (d_{k} - c_{k})} < r < \frac{b_{k} + d_{k} - 2 c_{k}}{2 (d_{k} - c_{k})},$ define $\begin{matrix} p_{k} & = & \frac{2 {rb}_{k} d_{k} + (1 - 2 r) a_{k} d_{k} + (1 - 2 r) b_{k} c_{k}}{(b_{k} - a_{k}) + (d_{k} - c_{k})} \\ + \frac{2 (r - 1) a_{k} c_{k}}{(b_{k} - a_{k}) + (d_{k} - c_{k})} . \end{matrix}$ Then for a given confidence level r ∈ (0, 1) , when the rough measure Tr {Γ (x, ρ) ≤0} ≥ r if and only if, $\begin{matrix} \sum_{k = 1}^{n} p_{k} h_{k} (x) + h_{0} (x) \leq 0 \end{matrix}$

Proof: The function Γ (x, ρ) is written into the following form,

$Γ (x, ρ) = \sum_{k = 1}^{n} h_{k} (x) ρ_{k} + h_{0} (x)$ Using the Theorem 2.7, it is clear that when rough measurable Tr {Γ (x, ρ) ≤0} ≥ r then, p _k ≤ 0, also h _k (x) be the nonnegative real valued function. So the result holds.

Theorem 2.9. Let ϱ _k = ([ρ _1k, ρ _2k] , [ρ _3k, ρ _4k]) be birough variables (k = 1, 2, ⋯ , n) with rough variable ρ _mk = ([a _mk, b _mk] , [c _mk, d _mk]) (m = 1, 2, 3, 4) and a function g (x, ϱ) can be written as, $\begin{matrix} g (x, ϱ) & = & h_{1} (x) ϱ_{1} + h_{2} (x) ϱ_{2} + \dots + h_{n} (x) ϱ_{n} + h_{0} (x) \end{matrix}$ If two functions defined as $h_{k}^{+} (x) = h_{k} (x) \lor 0$ and $h_{k}^{-} (x) = - h_{k} (x) \land 0$ (k = 1, 2, ⋯ , n). Then there exists a confidence level (γ, δ) ∈ (0, 1] , the chance measure Ch {g (x, ϱ) ≤ 0} (γ) ≥ δ if and only if $\begin{matrix} \sum_{k = 1}^{n} Q_{k} h_{k} (x) + h_{0} (x) \leq 0 \end{matrix}$ where Q _k defined as,

i) when $δ \leq \frac{P_{1 k} - P_{3 k}}{2 (P_{4 k} - P_{3 k})},$ define $\begin{matrix} Q_{k} = (1 - 2 δ) P_{3 k} + 2 δ P_{4 k} \end{matrix}$

ii) when $δ \geq \frac{P_{2 k} + P_{4 k} - 2 P_{3 k}}{2 (P_{4 k} - P_{3 k})},$ define $\begin{matrix} Q_{k} = 2 (1 - δ) P_{3 k} + (2 δ - 1) P_{4 k} \end{matrix}$

iii) when $\frac{P_{1 k} - P_{3 k}}{2 (P_{4 k} - P_{3 k})} < δ < \frac{P_{2 k} + P_{4 k} - 2 P_{3 k}}{2 (P_{4 k} - P_{3 k})},$ define $\begin{matrix} Q_{k} & = & \frac{2 δ P_{2 k} P_{4 k} + (1 - 2 δ) P_{1 k} P_{4 k}}{(P_{2 k} - P_{1 k}) + (P_{4 k} - P_{3 k})} \\ + \frac{(1 - 2 δ) P_{2 k} P_{3 k} + 2 (δ - 1) P_{1 k} P_{3 k}}{(P_{2 k} - P_{1 k}) + (P_{4 k} - P_{3 k})} \end{matrix}$ where P _mk defined as,

i) when $γ \leq \frac{a_{mk} - c_{mk}}{2 (d_{mk} - c_{mk})},$ define $\begin{matrix} P_{mk} = (1 - 2 γ) c_{mk} + 2 γ d_{mk} \end{matrix}$ ii) when $γ \geq \frac{b_{mk} + d_{mk} - 2 c_{mk}}{2 (d_{mk} - c_{mk})},$ define $\begin{matrix} P_{mk} = 2 (1 - γ) c_{mk} + (2 γ - 1) d_{mk} \end{matrix}$ iii) when $\frac{a_{mk} - c_{mk}}{2 (d_{mk} - c_{mk})} < γ < \frac{b_{mk} + d_{mk} - 2 c_{mk}}{2 (d_{mk} - c_{mk})},$ define $\begin{matrix} P_{mk} & = & \frac{2 γ b_{mk} d_{mk} + (1 - 2 γ) a_{mk} d_{mk}}{(b_{mk} - a_{mk}) + (d_{mk} - c_{mk})} \\ + \frac{(1 - 2 γ) b_{mk} c_{mk} + 2 (γ - 1) a_{mk} c_{mk}}{(b_{mk} - a_{mk}) + (d_{mk} - c_{mk})} \end{matrix}$ where m = 1, 2, 3, 4

Proof: The function g (x, ϱ) is written into the following form $Γ (x, ϱ) = \sum_{k = 1}^{n} h_{k} (x) ϱ_{k} + h_{0} (x)$ using Theorem 2.7, it is clear that when birough measurable Ch {g (x, ϱ) ≤0} ≥ δ, then p _k ≤ 0, also h _k (x) be the nonnegative real valued function. So the results holds.

2.1 Bi-Matrix game

In this subsection, let us consider the bi-matrix game whose pay-off elements are characterized by real numbers. Let X ≡ {1, 2, ⋯ , m} be a set of strategies for the player I and Y ≡ {1, 2, ⋯ , n} be a set of strategies for player II. Let R ⁿ be the n-dimensional Euclidean space and $R_{+}^{n}$ be its non-negative orthant. Here e ^T be a vector of elements ‘1’ whose dimension is specified as per specific context. Mixed strategies of players I and II are represented by $S^{X} = {x \in R_{+}^{m} | e^{T} x = 1}$ and $S^{Y} = {y \in R_{+}^{n} | e^{T} y = 1}$ respectively.

By considering real numbers a _ij and b _ij as the expected reward for the players I and II with their proposed strategies i and j respectively. Then the bi-matrix game can be defined as follows: $\begin{matrix} (A, B) = (\begin{matrix} (a_{11}, b_{11}) & (a_{12}, b_{12}) & . & . & . & (a_{1 n}, b_{1 n}) \\ (a_{21}, b_{21}) & (a_{22}, b_{22}) & . & . & . & (a_{2 n}, b_{2 n}) \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ (a_{m 1}, b_{m 1}) & (a_{m 2}, b_{m 2}) & . & . & . & (a_{mn}, b_{mn}) \end{matrix}) \end{matrix}$

Definition 2.11. A pair (x ^*, y ^*) ∈ S ^X × S ^Y is said to be Nash equilibrium strategy of the bi-matrix game BG = (S ^X, S ^Y, A, B) if, $\begin{matrix} x^{T} A y^{*} \leq {x^{*}}^{T} A y^{*}, x \in S^{X} \\ and {x^{*}}^{T} By \leq {x^{*}}^{T} B y^{*}, y \in S^{Y} \end{matrix}$

Theorem 2.10. Let BG = (S ^X, S ^Y, A, B) be the given bi-matrix game. A necessary and sufficient condition that (x ^*, y ^*) be an equilibrium strategy of BG is that it is a solution of the following quadratic programming problem, $\begin{matrix} max {x^{T} (A + B) y - v - w} \\ s . t . {\begin{matrix} Ay - ve & \leq & 0 \\ B^{T} x - we & \leq & 0 \\ e^{T} x - 1 & = & 0 \\ e^{T} y - 1 & = & 0 \\ v, w & \in & R \\ x, y & \geq & 0 \end{matrix} \end{matrix}$

Lemma 2.1. If v ^* be the value of the game for player I where v ^* is given by $v^{*} = {x^{*}}^{T} {Ay}^{*} = max_{x} {x^{T} {Ay}^{*} | e^{T} x = 1, x \geq 0}$ and w ^* is the value of the game for player II where w ^* is given by $w^{*} = {x^{*}}^{T} {By}^{*} = max_{y} {{x^{*}}^{T} By | e^{T} y = 1, y \geq 0}$

And if we assume that $x_{i}^{'} = \frac{x_{i}}{w}$ and $y_{j}^{'} = \frac{y_{j}}{v}$ (i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n). Then the aboveTheorem 2.10 reduces into the following quadratic programming problem, $\begin{array}{l} \max {x^{T} (A + B) y - v - w} \\ s.t. {A y - v e \leq 0 \\ B^{T} x - w e \leq 0 \\ e^{T} x - 1 = 0 \\ e^{T} y - 1 = 0 \\ v, w \in ℜ \\ x, y \geq 0 \end{array}$ where (x ^*, y ^*) and (v ^*, w ^*) are the Nash equilibrium strategy and outcome of the bi-matrix game BG.

Proof: (x ^*, y ^*) ∈ S ^X × S ^Y is the Nash equilibrium strategy of the bi-matrix game BG if and only if x ^* and y ^* are simultaneously solutions of the following two problems,

$\begin{matrix} max {x^{T} {Ay}^{*}} \\ s . t . {\begin{matrix} e^{T} x = 1 \\ x \geq 0 . \end{matrix} \end{matrix}$ (2.1) and

$\begin{matrix} max {{x^{*}}^{T} By} \\ s . t . {\begin{matrix} e^{T} y = 1 \\ y \geq 0 . \end{matrix} \end{matrix}$ (2.2)

Here (x ^*, y ^*) ∈ S ^X × S ^Y is an optimal strategy of BG that satisfies the conditions of (2.3) and (2.4) . Also and that is, and So the constraints of Theorem 2.10 is written into the following form:

$\begin{matrix} {Ay}^{*} & \leq & v^{*} e because \\ x^{T} {Ay}^{*} \leq v^{*} x^{T} e = v^{*} \\ Hence Ay & \leq & ve or {Ay}^{'} \leq e \\ i . e, \sum_{j = 1}^{n} a_{ij} y_{j}^{'} & \leq & 1 (i = 1, 2, \dots, m) \end{matrix}$ (2.3) and

$\begin{matrix} {x^{*}}^{T} B & \leq & w^{*} e because \\ {x^{*}}^{T} By \leq w^{*} y^{T} e = w^{*} \\ Hence B^{T} x & \leq & we or B^{T} x^{'} \leq e \\ i . e, \sum_{i = 1}^{m} b_{ij} x_{i}^{'} & \leq & 1 (j = 1, 2, \dots, n) \end{matrix}$ (2.4)

Therefore, the objective function of the Theorem 2.10 can be modified into the following form,

$\begin{matrix} max {x^{T} (A + B) y - v - w} \\ or max {{(\frac{x}{w})}^{T} (A + B) (\frac{y}{v}) + \frac{1}{w} + \frac{1}{v}} \\ Hence, max {x^{' T} (A + B) y^{'} + e^{T} x^{'} \\ + e^{T} y^{'}} \end{matrix}$ (2.5)

Now recall equations (2.3) , (2.4) and (2.5) with , So the theorem is obvious.

3 Birough bi-matrix game

In the crisp scenario, there exists a beautiful relationship on bi-matrix games. It is therefore natural to ask if something similar holds in birough scenario as well. In many applied situation, the elements of the bi-matrix game are not fixed. So the elements are imprecise. Therefore to introduce the bi-matrix game, we considered the elements as birough variables and then it is measured by Chance. Let the birough variable ϱ _ij denote the pay-off element that player I gains or the birough variable σ _ij be the pay-off element that player II gains when the players I and II play the pure strategies i and j respectively. Then the birough bi-matrix game can be represented as follows: $\begin{matrix} (ϱ, σ) = (\begin{matrix} (ϱ_{11}, σ_{11}) & (ϱ_{12}, σ_{12}) & . & . & . & (ϱ_{1 n}, σ_{1 n}) \\ (ϱ_{21}, σ_{21}) & (ϱ_{22}, σ_{22}) & . & . & . & (ϱ_{2 n}, σ_{2 n}) \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ (ϱ_{m 1}, σ_{m 1}) & (ϱ_{m 2}, σ_{m 2}) & . & . & . & (ϱ_{mn}, σ_{mn}) \end{matrix}) \end{matrix}$

Definition 3.1. Let ϱ _ij and σ _ij (i = 1, 2, . . . , m ; j = 1, 2, ⋯ , n) be independent birough variables. Then (x ^*, y ^*) is called an expected birough Nash equilibrium strategy to the birough bi-matrix game G ^ϱ,σ = {S ^X, S ^Y, ϱ, σ} if $\begin{matrix} v^{*} = E [{x^{*}}^{T} ϱ y^{*}] \geq E [x^{T} ϱ y^{*}], x \in S^{X} \\ w^{*} = E [{x^{*}}^{T} σ y^{*}] \geq E [{x^{*}}^{T} σ y], y \in S^{Y} \end{matrix}$

The pair (v ^*, w ^*) is called an optimum value of the birough bi-matrix game.

Definition 3.2. Let ϱ _ij and σ _ij (i = 1, 2, . . . , m ; j = 1, 2, ⋯ , n) are different independent birough variables, (γ, δ) ∈ (0, 1] and $w, v \in R$ be predetermined level of the birough pay-offs. Then (x ^*, y ^*) is called a (γ, δ)-birough equilibrium strategy to birough bi-matrix game G ^ϱ,σ = {S ^X, S ^Y, ϱ, σ} if $\begin{matrix} max {v | Ch {x^{T} ϱ y^{*} \geq v} (γ) \geq δ} \\ \leq max {v | Ch {x^{* T} ϱ y^{*} \geq v} (γ) \geq δ} \end{matrix}$ $\begin{matrix} max {w | Ch {{x^{*}}^{T} σ y \geq w} (γ) \geq δ} \\ \leq max {w | Ch {x^{* T} σ y^{*} \geq w} (γ) \geq δ} \end{matrix}$ Lemma 3.1. Let birough bi-matrix game G ^ϱ,σ = {S ^X, S ^Y, ϱ, σ} and the value of the game v ^* for player I is given by $\begin{matrix} v^{*} = E [{x^{*}}^{T} ϱ y^{*}] = max_{x} {x^{T} ϱ y^{*} | e^{T} x = 1, x \geq 0} \end{matrix}$ and w ^* is the value of the game for player II is given by $\begin{matrix} w^{*} = E [{x^{*}}^{T} σ y^{*}] = max_{y} {{x^{*}}^{T} σ y | e^{T} y = 1, y \geq 0} \end{matrix}$ where the max operator is defined as $\begin{matrix} max_{y \in S^{Y}} {x^{T} ϱ y} = x^{T} ϱ y_{max} \\ such that E [x^{T} ϱ y] \leq E [x^{T} ϱ y_{max}] \end{matrix}$

If we assume that $x_{i}^{'} = \frac{x_{i}}{w}$ and $y_{i}^{'} = \frac{y_{i}}{v} (i = 1, 2, \dots,$ m ; j = 1, 2, ⋯ , n) and the expected value operator (E) defined on the objective function and the uncertainty of the birough constraints are measured with birough measurable function Chance with confidence level (γ, δ) ∈ (0, 1]. Then the Lemma 2.1 reduces into the following quadratic programming problem (QPP), $\begin{array}{l} \max {e^{T} x^{'} + e^{T} y^{'} + E [x^{' T} (ϱ + σ) y^{'}]} \\ s .t . {{Ch {\sum_{j = 1}^{n} ϱ_{i j} y_{j}^{'} \leq 1} (γ) \geq δ (i = 1, 2, \dots, m) \\ Ch {\sum_{i = 1}^{m} σ_{i j} x_{i}^{'} \leq 1} (γ) \geq δ (j = 1, 2, \dots, n) \\ y^{'}, x^{'} \geq 0 \end{array}$ where (x ^*, y ^*) is the expected birough Nash equilibrium strategy and (v ^*, w ^*) is the equilibrium outcome of the birough bi-matrix game.

Proof: (x ^*, y ^*) ∈ S ^X × S ^Y is an expected birough Nash equilibrium strategy of the birough bi-matrix game if and only if x ^* and y ^* are simultaneously solutions of the following two problems, $\begin{matrix} max {x^{T} ϱ y^{*}} \\ s . t . {\begin{matrix} e^{T} x = 1 \\ x \geq 0 . \end{matrix} \end{matrix}$ (3.6) $\begin{matrix} and max {{x^{*}}^{T} σ y} \\ s . t . {\begin{matrix} e^{T} y = 1 \\ y \geq 0 . \end{matrix} \end{matrix}$ (3.7)

Here (x ^*, y ^*) ∈ S ^X × S ^Y be an arbitrary strategy of the birough bi-matrix game that satisfy the conditions (3.6) and (3.7). Also $e^{T} x^{'} = \frac{1}{w}$ and $e^{T} y^{'} = \frac{1}{v},$ that is, and . So the constraints of Lemma 3.1 are the elements of birough variables for birough bi-matrix game. So we cannot directly use the inequality for birough pay-off. For the uncertainty of the pay-off elements, applying the birough measurable function Chance with confidence level (γ, δ) ∈ (0, 1] as follows: $\begin{matrix} Ch & {ϱ y^{*} \leq v^{*} e} (γ) \geq δ \\ because, Ch {x^{T} ϱ y^{*} \leq v^{*} x^{T} e = v^{*}} (γ) \geq δ \end{matrix}$ Hence, Ch {ϱ y ≤ ve} (γ) ≥ δ or, Ch {^ϱ y′ ≤ e} (γ) ≥ δ $\begin{matrix} Ch {\sum_{j = 1}^{n} ϱ_{ij} y_{j}^{'} \leq 1} (γ) \geq δ (i = 1, 2, \dots, m) ∥ \end{matrix}$ and $\begin{matrix} Ch & {{x^{*}}^{T} σ \leq w^{*} e} (γ) \geq δ \\ because, Ch {{x^{*}}^{T} σ y \leq w^{*} y^{T} e = w^{*}} (γ) \geq δ \end{matrix}$ Hence, Ch {σ ^T x ≤ we} (γ) ≥ δ or, Ch {σ ^T x′ ≤ e} (γ) ≥ δ $\begin{matrix} Ch & {\sum_{i = 1}^{m} σ_{ij} x_{i}^{'} \leq 1} (γ) \geq δ (j = 1, 2, \dots, n) \end{matrix}$

Theorem 3.1. In a bi-matrix game, the uncertain payoffs ϱ _ij and σ_ij (i = 1, 2, . . . , m ; j = 1, 2, ⋯ , n) are characterized as birough variables with finite expected values. Then there exists at least a birough expected Nash equilibrium strategy to the birough bi-matrix game G^ϱ,σ = {S^X, S^Y, ϱ, σ}

Proof: For any y ∈ S ^Y, let $Q (y) = {\bar{x} \in S^{X} | E [x^{T} ϱ y] \leq E [{\bar{x}}^{T} ϱ y], x \in S^{X}}$ then Q (y) ⊂ S ^X For any x ∈ S ^X, let $P (x) = {\bar{y} \in S^{Y} | E [x^{T} σ \bar{y}] \leq E [x^{T} σ y], y \in S^{Y}}$ then P (x) ⊂ S ^Y. We first prove that Q (y) and P (x) are both convex sets. For any x ₁, x ₂ ∈ Q (y) , it is clear that λx ₁ + (1 - λ) x ₂ ∈ S ^X with λ ∈ [0, 1] . Since the components of x ₁, x ₂ are all nonnegative real numbers, it follows that, $\begin{matrix} E [(λ x_{1} + (1 - λ) x_{2})^{T} ϱ y] & = & λ E [{x_{1}}^{T} ϱ y] \\ + (1 - λ) E [{x_{2}}^{T} ϱ y] \end{matrix}$

Moreover, it follows from above that E [x ₁ ^T ϱy] ≥ E [x ^T ϱy] and E [x ₂ ^T ϱy] ≥ E [x ^T ϱy] for any x ∈ S ^X . Thus for any λ ∈ [0, 1], λE [x ₁ ^T ϱy] ≥ λE [x ^T ϱy] and (1 - λ) E [x ₂ ^T ϱy] ≥ (1 - λ) E [x ^T ϱy] . Hence, we have, $E [(λ x_{1} + (1 - λ) x_{2})^{T} ϱ y] \geq E [x^{T} ϱ y], x \in S^{X}$

This implies that $λ x_{1} + (1 - λ) x_{2} \in Q (y)$

Hence Q (y) is a convex set. Similarly, we can prove that P (x) is a convex set. Let F : S ^X × S ^Y → P (S ^X × S ^Y) be a set-valued mapping defined by $F (z) = Q (y) \times P (x), z = (x^{T}, y^{T}) \in S^{X} \times S^{Y},$ where P (S ^X × S ^Y) is the power set of S ^X × S ^Y . Then we have, $\begin{matrix} (u, v) \in F (z) \Leftrightarrow u \in Q (y), v \in P (x) \end{matrix}$

Let z _n = (x _n ^T, y _n ^T) ^T and (u _n, v _n) ∈ F (z _n) , where u _n → u ₀, v _n → v ₀, x _n → x ₀ and y _n → y ₀ as n → ∞ . Since u _n ∈ Q (y _n) , v _n ∈ P (x _n) , for any x ∈ S ^X, y ∈ S ^Y we obtain $\begin{matrix} E [x^{T} ϱ y_{n}] & \leq & E [{u_{n}}^{T} ϱ y_{n}] \\ and E [{x_{n}}^{T} σ y] & \leq & E [{x_{n}}^{T} σ v_{n}] \\ Hence, lim_{n \to + \infty} E [x^{T} ϱ y_{n}] & \leq & lim_{n \to + \infty} E [{u_{n}}^{T} ϱ y_{n}] \\ lim_{n \to + \infty} E [{x_{n}}^{T} σ y] & \leq & lim_{n \to + \infty} E [{x_{n}}^{T} σ v_{n}] \\ It implies that, E [x^{T} ϱ y_{0}] & \leq & E [{u_{0}}^{T} ϱ y_{0}] \\ E [{x_{0}}^{T} σ y] & \leq & E [{x_{0}}^{T} σ v_{0}] \end{matrix}$

Thus u ₀ ∈ Q (y ₀) and v ₀ ∈ P (x ₀) . Hence Q (y) and P (x) are both convex closed sets and the graph of F is convex closed. It is clear that the set-valued maping F is upper semi-continuous with non-empty, convex closed values. It follows from Kakutani’s fixed-point theorem that there exists at least a point z ^* ∈ S ^X × S ^Y such that z ^* ∈ F (z ^*) , i.e, there exists at least a point (x ^*, y ^*) ∈ (S ^X, S ^Y) , such that $\begin{matrix} x^{*} \in Q (y^{*}) \Rightarrow E [x^{T} ϱ y^{*}] \leq E [{x^{*}}^{T} ϱ y^{*}], x \in S^{X} \\ y^{*} \in P (x^{*}) \Rightarrow E [{x^{*}}^{T} σ y^{*}] \leq E [{x^{*}}^{T} σ y], y \in S^{Y} \end{matrix}$ thus the theorem is proved.

3.1 Quadratic programming problem

To derive the solution of the birough bi-matrix game, we have to solve the following birough quadratic programming problem, $\begin{array}{l} \max {e^{T} x^{'} + e^{T} y^{'} + x^{' T} (ϱ + σ) y^{'}} \\ s.t. {\sum_{j = 1}^{n} ϱ_{i j} y_{j}^{'} \leq 1 (i = 1, 2, \dots, m) \\ \sum_{i = 1}^{m} σ_{i j} x_{j}^{'} \leq 1 (j = 1, 2, \dots, n) \\ y^{'} \geq 0, x^{'} \geq 0 \end{array}$

Since the birough variables are presented in the above quadratic programming problem, so traditional method is not applicable. To find the solution of the above problem, we have to introduce the birough expected operator for the objective function and for the constraints, birough measurable function named as Chance with confidence level (γ, δ) ∈ (0, 1] . So the birough quadratic programming problem becomes, $\begin{array}{l} \max {E [e^{T} x^{'} + e^{T} y^{'} + x^{' T} (ϱ + σ) y^{'}]} \\ s.t. {Ch {\sum_{j = 1}^{n} ϱ_{i j} y_{j}^{'} \leq 1} (γ) \geq δ (i = 1, 2, \dots, m) \\ Ch {\sum_{i = 1}^{m} σ_{i j} x_{i}^{'} \leq 1} (γ) \geq δ (j = 1, 2, \dots, n) \\ y^{'} \geq 0, x^{'} \geq 0 \end{array}$

With the help of Theorem 2.9, we can easily find that h ^+(x)=x′, h ^+(y)=y′ and h ^- (x) =0, h ^- (y) =0 because , and using birough set theory, the above problem can be written into crisp quadratic programming problem which depends upon the confidence level (γ, δ) ∈ (0, 1] , $\begin{matrix} max [\frac{1}{16} \sum_{i = 1}^{m} \sum_{j = 1}^{n} {a_{ij 11} + a_{ij 12} + a_{ij 21} + a_{ij 22} + b_{ij 11} \\ + b_{ij 12} + b_{ij 21} + b_{ij 22} + c_{ij 11} + c_{ij 12} + c_{ij 21} + c_{ij 22} \\ + d_{ij 11} + d_{ij 12} + d_{ij 21} + d_{ij 22} + x_{ij 11} + x_{ij 12} + x_{ij 21} \\ + x_{ij 22} + y_{ij 11} + y_{ij 12} + y_{ij 21} + y_{ij 22} + z_{ij 11} + z_{ij 12} \\ + z_{ij 21} + z_{ij 22} + w_{ij 11} + w_{ij 12} + w_{ij 21} + w_{ij 22}} x_{i}^{'} y_{j}^{'} \\ + \sum_{i = 1}^{m^{x_{i}^{'}}} + \sum_{j = 1}^{n^{y_{j}^{'}}}] \\ s . t . {\begin{matrix} \sum_{j = 1}^{n} Q_{ij} y_{j}^{'} \leq 1 (i = 1, 2, \dots, m) \\ \sum_{i = 1}^{m} S_{ij} x_{i}^{'} \leq 1 (j = 1, 2, \dots, n) \\ y^{'}, x^{'} \geq 0 \end{matrix} \end{matrix}$ Where Q _ij, S _ij are derived from the following expressions:

If $δ \leq \frac{P_{ija} - P_{ijc}}{2 (P_{ijd} - P_{ijc})},$ $\begin{matrix} Q_{ij} = (1 - 2 δ) P_{ijc} + 2 δ P_{ijd} \end{matrix}$ If $δ \leq \frac{R_{ijx} - R_{ijz}}{2 (R_{ijw} - R_{ijz})},$ $\begin{matrix} S_{ij} = (1 - 2 δ) R_{ijz} + 2 δ R_{ijw} \end{matrix}$ If $δ \geq \frac{P_{ijb} + P_{ijd} - 2 P_{ijc}}{2 (P_{ijd} - P_{ijc})}$ $\begin{matrix} Q_{ij} = (2 - 2 δ) P_{ijc} + (2 δ - 1) P_{ijd} \end{matrix}$ If $δ \geq \frac{R_{ijy} + R_{ijw} - 2 R_{ijz}}{2 (R_{ijw} - R_{ijz})}$ $\begin{matrix} S_{ij} = (2 - 2 δ) R_{ijz} + (2 δ - 1) R_{ijw} \end{matrix}$ If $\frac{P_{ija} - P_{ijc}}{2 (P_{ijd} - P_{ijc})} < δ < \frac{P_{ijb} + P_{ijd} - 2 P_{ijc}}{2 (P_{ijd} - P_{ijc})}$ $\begin{matrix} Q_{ij} & = & \frac{2 δ P_{ijb} P_{ijd} + (1 - 2 δ) P_{ija} P_{ijd}}{(P_{ijb} - P_{ija}) + (P_{ijd} - P_{ijc})} \\ + \frac{(1 - 2 δ) P_{ijb} P_{ijc} + 2 (δ - 1) P_{ija} P_{ijc}}{(P_{ijb} - P_{ija}) + (P_{ijd} - P_{ijc})} \end{matrix}$ If $\frac{R_{ijx} - R_{ijz}}{2 (R_{ijw} - R_{ijz})} < δ < \frac{R_{ijy} + R_{ijw} - 2 R_{ijz}}{2 (R_{ijw} - R_{ijz})}$ $\begin{matrix} S_{ij} & = & \frac{2 δ R_{ijy} R_{ijw} + (1 - 2 δ) R_{ijx} R_{ijw}}{(R_{ijy} - R_{ijx}) + (R_{ijw} - R_{ijz})} \\ + \frac{(1 - 2 δ) R_{ijy} R_{ijz} + 2 (δ - 1) R_{ijx} R_{ijz}}{(R_{ijy} - R_{ijx}) + (R_{ijw} - R_{ijz})} \end{matrix}$ Where

If $γ \leq \frac{a_{ij 11} - c_{ij 11}}{2 (d_{ij 11} - c_{ij 11})}$ $\begin{matrix} P_{ija} = (1 - 2 γ) c_{ij 11} + 2 γ d_{ij 11} \end{matrix}$ If $γ \leq \frac{x_{ij 11} - z_{ij 11}}{2 (w_{ij 11} - z_{ij 11})}$ $\begin{matrix} R_{ijx} = (1 - 2 γ) z_{ij 11} + 2 γ z_{ij 11} \end{matrix}$ If $γ \geq \frac{b_{ij 11} + d_{ij 11} - 2 c_{ij 11}}{2 (d_{ij 11} - c_{ij 11})}$ $\begin{matrix} P_{ija} = (2 - 2 γ) c_{ij 11} + (2 γ - 1) d_{ij 11} \end{matrix}$ If $γ \geq \frac{y_{ij 11} + w_{ij 11} - 2 z_{ij 11}}{2 (w_{ij 11} - z_{ij 11})}$ $\begin{matrix} R_{ijx} = (2 - 2 γ) z_{ij 11} + (2 γ - 1) w_{ij 11} \end{matrix}$ If $\frac{a_{ij 11} - c_{ij 11}}{2 (d_{ij 11} - c_{ij 11})} < γ < \frac{b_{ij 11} + d_{ij 11} - 2 c_{ij 11}}{2 (d_{ij 11} - c_{ij 11})}$ $\begin{matrix} P_{ija} & = & \frac{2 γ b_{ij 11} d_{ij 11} + (1 - 2 γ) a_{ij 11} d_{ij 11}}{(b_{ij 11} - a_{ij 11}) + (d_{ij 11} - c_{ij 11})} \\ + \frac{(1 - 2 γ) b_{ij 11} c_{ij 11} + 2 (γ - 1) a_{ij 11} c_{ij 11}}{(b_{ij 11} - a_{ij 11}) + (d_{ij 11} - c_{ij 11})} \end{matrix}$ If $\frac{x_{ij 11} - z_{ij 11}}{2 (w_{ij 11} - z_{ij 11})} < γ < \frac{y_{ij 11} + w_{ij 11} - 2 z_{ij 11}}{2 (w_{ij 11} - z_{ij 11})}$ $\begin{matrix} R_{ijx} & = & \frac{2 γ y_{ij 11} w_{ij 11} + (1 - 2 γ) x_{ij 11} w_{ij 11}}{(y_{ij 11} - x_{ij 11}) + (w_{ij 11} - z_{ij 11})} \\ + \frac{(1 - 2 γ) y_{ij 11} z_{ij 11} + 2 (γ - 1) x_{ij 11} z_{ij 11}}{(y_{ij 11} - x_{ij 11}) + (w_{ij 11} - z_{ij 11})} \end{matrix}$ If $γ \leq \frac{a_{ij 12} - c_{ij 12}}{2 (d_{ij 12} - c_{ij 12})}$ $\begin{matrix} P_{ijb} = (1 - 2 γ) c_{ij 12} + 2 γ d_{ij 12} \end{matrix}$ If $γ \leq \frac{x_{ij 12} - z_{ij 12}}{2 (w_{ij 12} - z_{ij 12})}$ $\begin{matrix} R_{ijy} = (1 - 2 γ) z_{ij 12} + 2 γ w_{ij 12} \end{matrix}$ If $γ \geq \frac{b_{ij 12} + d_{ij 12} - 2 c_{ij 12}}{2 (d_{ij 12} - c_{ij 12})}$ $\begin{matrix} P_{ijb} = (2 - 2 γ) c_{ij 12} + (2 γ - 1) d_{ij 12} \end{matrix}$ If $γ \geq \frac{y_{ij 12} + w_{ij 12} - 2 z_{ij 12}}{2 (w_{ij 12} - z_{ij 12})}$ $\begin{matrix} R_{ijy} = (2 - 2 γ) z_{ij 12} + (2 γ - 1) w_{ij 12} \end{matrix}$ If $\frac{a_{ij 12} - c_{ij 12}}{2 (d_{ij 12} - c_{ij 12})} < γ < \frac{b_{ij 12} + d_{ij 12} - 2 c_{ij 12}}{2 (d_{ij 12} - c_{ij 12})}$ $\begin{matrix} P_{ijb} & = & \frac{2 γ b_{ij 12} d_{ij 12} + (1 - 2 γ) a_{ij 12} d_{ij 12}}{(b_{ij 12} - a_{ij 12}) + (d_{ij 12} - c_{ij 12})} \\ + \frac{(1 - 2 γ) b_{ij 12} c_{ij 12} + 2 (γ - 1) a_{ij 12} c_{ij 12}}{(b_{ij 12} - a_{ij 12}) + (d_{ij 12} - c_{ij 12})} \end{matrix}$ If $\frac{x_{ij 12} - z_{ij 12}}{2 (w_{ij 12} - z_{ij 12})} < γ < \frac{y_{ij 12} + w_{ij 12} - 2 z_{ij 12}}{2 (w_{ij 12} - z_{ij 12})}$ $\begin{matrix} R_{ijy} & = & \frac{2 γ y_{ij 12} w_{ij 12} + (1 - 2 γ) x_{ij 12} w_{ij 12}}{(y_{ij 12} - x_{ij 12}) + (w_{ij 12} - z_{ij 12})} \\ = & \frac{(1 - 2 γ) y_{ij 12} z_{ij 12} + 2 (γ - 1) x_{ij 12} z_{ij 12}}{(y_{ij 12} - x_{ij 12}) + (w_{ij 12} - z_{ij 12})} \end{matrix}$ If $γ \leq \frac{a_{ij 21} - c_{ij 21}}{2 (d_{ij 21} - c_{ij 21})}$ $\begin{matrix} P_{ijc} = (1 - 2 γ) c_{ij 21} + 2 γ d_{ij 21} \end{matrix}$ If $γ \leq \frac{x_{ij 21} - z_{ij 21}}{2 (w_{ij 21} - z_{ij 21})}$ $\begin{matrix} R_{ijz} = (1 - 2 γ) z_{ij 21} + 2 γ w_{ij 21} \end{matrix}$ If $γ \geq \frac{b_{ij 21} + d_{ij 21} - 2 c_{ij 21}}{2 (d_{ij 21} - c_{ij 21})}$ $\begin{matrix} P_{ijc} = (2 - 2 γ) c_{ij 21} + (2 γ - 1) d_{ij 21} \end{matrix}$ If $γ \geq \frac{y_{ij 21} + w_{ij 21} - 2 z_{ij 21}}{2 (w_{ij 21} - z_{ij 21})}$ $\begin{matrix} P_{ijz} = (2 - 2 γ) z_{ij 21} + (2 γ - 1) w_{ij 21} \end{matrix}$ If $\frac{a_{ij 21} - c_{ij 21}}{2 (d_{ij 21} - c_{ij 21})} < γ < \frac{b_{ij 21} + d_{ij 21} - 2 c_{ij 21}}{2 (d_{ij 21} - c_{ij 21})}$ $\begin{matrix} P_{ijc} & = & \frac{2 γ b_{ij 21} d_{ij 21} + (1 - 2 γ) a_{ij 21} d_{ij 21}}{(b_{ij 21} - a_{ij 21}) + (d_{ij 21} - c_{ij 21})} \\ + \frac{(1 - 2 γ) b_{ij 21} c_{ij 21} + 2 (γ - 1) a_{ij 21} c_{ij 21}}{(b_{ij 21} - a_{ij 21}) + (d_{ij 21} - c_{ij 21})} \end{matrix}$ If $\frac{x_{ij 21} - z_{ij 21}}{2 (w_{ij 21} - z_{ij 21})} < γ < \frac{y_{ij 21} + w_{ij 21} - 2 z_{ij 21}}{2 (w_{ij 21} - z_{ij 21})}$ $\begin{matrix} R_{ijz} & = & \frac{2 γ y_{ij 21} w_{ij 21} + (1 - 2 γ) x_{ij 21} w_{ij 21}}{(y_{ij 21} - x_{ij 21}) + (w_{ij 21} - z_{ij 21})} \\ + \frac{(1 - 2 γ) y_{ij 21} z_{ij 21} + 2 (γ - 1) x_{ij 21} z_{ij 21}}{(y_{ij 21} - x_{ij 21}) + (w_{ij 21} - z_{ij 21})} \end{matrix}$ If $γ \leq \frac{a_{ij 22} - c_{ij 22}}{2 (d_{ij 22} - c_{ij 22})}$ $\begin{matrix} P_{ijd} = (1 - 2 γ) c_{ij 22} + 2 γ d_{ij 22} \end{matrix}$ If $γ \leq \frac{x_{ij 22} - z_{ij 22}}{2 (w_{ij 22} - z_{ij 22})}$ $\begin{matrix} R_{ijw} = (1 - 2 γ) z_{ij 22} + 2 γ w_{ij 22} \end{matrix}$ If $γ \geq \frac{b_{ij 22} + d_{ij 22} - 2 c_{ij 22}}{2 (d_{ij 22} - c_{ij 22})}$ $\begin{matrix} P_{ijd} = (2 - 2 γ) c_{ij 22} + (2 γ - 1) d_{ij 22} \end{matrix}$ If $γ \geq \frac{y_{ij 22} + w_{ij 22} - 2 z_{ij 22}}{2 (w_{ij 22} - z_{ij 22})}$ $\begin{matrix} R_{ijw} = (2 - 2 γ) z_{ij 22} + (2 γ - 1) w_{ij 22} \end{matrix}$ If $\frac{a_{ij 22} - c_{ij 22}}{2 (d_{ij 22} - c_{ij 22})} < γ < \frac{b_{ij 22} + d_{ij 22} - 2 c_{ij 22}}{2 (d_{ij 22} - c_{ij 22})}$ $\begin{matrix} P_{ijd} & = & \frac{2 γ b_{ij 22} d_{ij 22} + (1 - 2 γ) a_{ij 22} d_{ij 22}}{(b_{ij 22} - a_{ij 22}) + (d_{ij 22} - c_{ij 22})} \\ + \frac{(1 - 2 γ) b_{ij 22} c_{ij 22} + 2 (γ - 1) a_{ij 22} c_{ij 22}}{(b_{ij 22} - a_{ij 22}) + (d_{ij 22} - c_{ij 22})} \end{matrix}$ If $\frac{x_{ij 22} - z_{ij 22}}{2 (w_{ij 22} - z_{ij 22})} < γ < \frac{y_{ij 22} + w_{ij 22} - 2 z_{ij 22}}{2 (w_{ij 22} - z_{ij 22})}$ $\begin{matrix} R_{ijw} & = & \frac{2 γ y_{ij 22} w_{ij 22} + (1 - 2 γ) x_{ij 22} w_{ij 22}}{(y_{ij 22} - x_{ij 22}) + (w_{ij 22} - z_{ij 22})} \\ + \frac{(1 - 2 γ) y_{ij 22} z_{ij 22} + 2 (γ - 1) x_{ij 22} z_{ij 22}}{(y_{ij 22} - x_{ij 22}) + (w_{ij 22} - z_{ij 22})} \end{matrix}$ where Q _ij and S _ij are real numbers derived from the birough variables, ϱ _ij = ([p _ija, p _ijb] , [p _ijc, p _ijd]) and σ _ij = ([r _ijx, r _ijy] , [r _ijz, r _ijw]) with particular confidence δ. The real numbers P _ija, P _ijb, P _ijc, P _ijd, R _ijx, R _ijy, R _ijz, R _ijware derived from the rough variables, p _ija = ([a _ij11, b _ij11] , [c _ij11, d _ij11]), p _ijb = ([a _ij12, b _ij12] , [c _ij12, d _ij12]), p _ijc = ([a _ij21, b _ij21] , [c _ij21, d _ij21]), p _ijd = ([a _ij22, b _ij22] , [c _ij22, d _ij22]) and r _ijx = ([x _ij11, y _ij11] , [z _ij11, w _ij11]), r _ijy = ([x _ij12, y _ij12] , [z _ij12, w _ij12]), r _ijz = ([x _ij21, y _ij21] , [z _ij21, w _ij21]), r _ijw = ([x _ij22, y _ij22] , [z _ij22, w _ij22]) with depends upon the confidence level γ.

4 Genetic Algorithm(GA)

In this section, we introduce the genetic algorithm (GA) to find the optimal solution (strategies of the players and value of the birough bi-matrix game) of the birough bi-matrix game. The GA is a stochastic search method for optimization problems based on the mechanics of natural selection and natural genetics. One of the important concept in GA is chromosome which is a coding of a solution of an optimization problem, not necessarily the solution itself. In the birough bi-matrix game, strategies of the players consider as chromosomes (randomly selected value between 0 and 1 and satisfy the constraints of the game). GA starts with an initial set of random-generated chromosomes called a population. The number of individuals in the population is a predetermined integer which is called population size, considered here as 50. All the chromosomes are evaluated by so-called evaluation function, which satisfied the constraints of the birough bi-matrix game. A new population will be formulated by a selection process using sampling mechanism based on objective function of the birough bi-matrix game. The cycle from one population to the next one is called a generation. In each new generation, all chromosomes will be updated by crossover and mutation operations. The selection process selects chromosomes to form a new population and the genetic system enters a new generation. After performing the genetic system a given of cycles, we decode the best chromosome into a solution which is regarded as the optimal solution of the objective function.

Crossover: Generate the new chromosome using crossover of the pair of chromosomes and the new chromosome must satisfy the constraints of the birough bi-matrix game. We define a parameter P _c = 0.3 of a genetic system as the probability of the crossover. This probability gives the expected number P _c * population size (=50) of the chromosomes undergoing the crossover operation. In order to determine the parents from crossover operation, process repeatedly from i = 1 to 50 generating random number r from the interval [0, 1], the chromosome selects as a parent if r < P _c.

Mutation: Mutation is a genetic operator used to maintain genetic diversity from one generation of a population of genetic chromosomes to the next. Mutation alters one or more gene values in a chromosome from its initial state. In mutation, the solution may change entirely from the previous solution. Hence GA can come to better solution by using mutation. We have used the probability of the mutation is P _m = 0.2.

We now summarize the GA for applying to the proposed problem in our paper as follows:

Step 1: Initialize population size chromosomes at random and the constrains of the birough bi-matrix game must satisfy.

Step 2: Modify the chromosomes by crossover and mutation operations.

Step 3: Calculate the objective value for all the chromosomes.

Step 4: Compute the fitness of each chromosome via constrains of the birough bi-matrix game.

Step 5: Select the chromosomes which satisfy the constraints of the birough bi-matrix game.

Step 6: Repeat the second to fifth steps for an iteration of 500.

5 A Numerical example

In order to show the applicability of the proposed method, let us consider a real-life example on bi-matrix games. The two reputed insurance companies, both have four types of insurances such as life insurance, property insurance, vehicle insurance, medical insurance. The company will do their business through their agent. So the profit of the company depends upon their agent popularity, advertisement and social work in the insurance field. The profit of the agent is described by rough variables. The profit of the company depends upon the agent how much gains from the insurance can be defined as birough variables. Let us consider the first company’s profits through their agent is defined as on Table 1 .

Similarly, the second company’s profits through their agent is defined on Table 2 .

The company profit depends upon the agent profits with their commission which varies on company’s policy. For the first company, the agent’s profit is average -1 to 1 and the minimum profit is -2 and maximum profit is 2. Similarly, the second company, agent’s profit is -2 to 1 and the minimum profit is -3 and maximum profit is 2.

Hence, the profit of the companies through their agents and agents’ profit can be represented as birough bi-matrix game, where the payoff elements for the first company represents as ρ ₁₁ = ([180, 190] , [175, 195]), ρ ₁₂ = ([156, 158] , [150, 160]), ρ ₂₁ = ([90, 95] , [80, 100]) and ρ ₂₂ = ([120, 130] , [100, 140]). And that of the second company are ς ₁₁ = ([165, 170] , [160, 175]), ς ₁₂ = ([145, 148] , [140, 150]), ς ₂₁ = ([75, 78] , [70, 82]) and ς ₂₂ = ([100, 110] , [90, 130]). Finally, the first company payoff elements for the birough bimatrix game is defined as birough variables, ϱ _ij = ([ρ _ij - 1, ρ _ij + 1] , [ρ _ij - 2, ρ _ij + 2]) and that for the second company is σ _ij = ([ς _ij - 2, ς _ij + 1] , [ς _ij - 3, ς _ij + 2]).

Using the proposed method described earlier and choosing the particular confidence level γ = 0.75, the birough payoffs for the insurance companies can be converted into rough variables and the payoff matrix is defined in the following way.

Payoff matrices for the first company and that of the second company are defined respectively as follows: $\begin{matrix} P_{2 \times 2} = (\begin{matrix} ([187.333, 189.333], & [186.333, 190.333]) \\ ([156.500, 158.500], & [155.500, 159.500]) \\ ([093.000, 095.000], & [092.000, 096.000]) \\ ([127.000, 129.000], & [126.000, 130.000]) \end{matrix}) \end{matrix}$

$\begin{matrix} R_{2 \times 2} = (\begin{matrix} ([167.375, 170.375], & [166.375, 171.375]) \\ ([075.600, 078.600], & [074.600, 079.600]) \\ ([145.308, 148.308], & [144.308, 149.308]) \\ ([108.000, 111.000], & [107.000, 112.000]) \end{matrix}) \end{matrix}$ Choosing the confidence level δ = 0.25, the birough bi-matrix game is converted into crisp payoff matrix as follows: $\begin{matrix} Q = (\begin{matrix} 187.666667 & 156.833333 \\ 93.333333 & 127.333333 \end{matrix}) \end{matrix}$ $\begin{matrix} S = (\begin{matrix} 167.937500 & 76.162500 \\ 145.870192 & 108.562500 \end{matrix}) \end{matrix}$

The crisp quadratic programming for the particular confidence level (0.75, 0.25) is

$\begin{array}{l} max {352 x_{1}^{'} y_{1}^{'} + 167 x_{1}^{'} y_{2}^{'} + 323.25 x_{2}^{'} y_{1}^{'} \\ + 239.9375 x_{2}^{'} y_{2}^{'} + x_{1}^{'} + x_{2}^{'} + y_{1}^{'} + y_{2}^{'}} \\ s.t. {187.666667 y_{1}^{'} + 156.833333 y_{2}^{'} \leq 1 \\ 93.333333 y_{1}^{'} + 127.333333 y_{2}^{'} \leq 1 \\ 167.9375 x_{1}^{'} + 76.1625 x_{2}^{'} \leq 1 \\ 145.870192 x_{1}^{'} + 108.5625 x_{2}^{'} \leq 1 \\ y_{1}^{'}, y_{2}^{'}, x_{1}^{'} x_{2}^{'} \geq 0 \end{array}$ Here, the chromosomes are considered as for the player I and for player II is , randomly generated numbers between 0 and 1 which satisfy the constraints of the birough bi-matrix game. We have obtained the optimal value of the objective function and the values of and Hence the values of the game of the players are and with their strategies are and for the players I and II respectively.

The solution of the above problem with the help of Genetic Algorithm is and with values of the game are w ^* = 187.666660 and v ^* = 108.562498 with strategies (0, 1) and (1, 0) for the players I and II respectively.

Different values of confidence level, Table 3 is generated with the solution of the birough bi-matrix game. From the Table 3 , is shown that the profit (v ^*, w ^*) for the insurance companies achieved with strategy (x ^*, y ^*) for the different confidence levels. The maximum profit can be earned (form the tabulated values) by the insurance companies when the confidence level is (1, 1).

6 Conclusion

The birough bi-matrix game has wide applications in competitive systems, business planning and strategic management. In real world application, the elements of the payoff matrix in game problems may not be known precisely due to uncertain factors. If the payoff elements are considered as crisp values, then some helpful information may be lost. Since the payoff elements are expressed by birough variables rather than crisp values, so more information is provided to take right decision for the decision maker. Corresponding to different decision criteria, various uncertain equilibrium strategies are proposed to deal different situations with confidence level. Moreover, the existence theorem ensures the significance of the solution concepts, and the sufficient and necessary condition provided a way to find the uncertain equilibrium strategies. In addition, an application is given to illustrate the usefulness of the theory developed in this paper.

Studying more concepts of the presented method may be a paradigm for future research. Moreover, applying this new concept is a challenging task to deal with real world decision making problems for further research.

References

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