On Nash Equilibria in a Finite Game for Fuzzy Sets of Strategies

Abstract

The present paper investigates a finite game with fuzzy sets of player strategies. It is proven that Nash equilibria constitute a type-2 fuzzy set defined on the universal set of strategy profiles. Furthermore, the corresponding type-2 membership function is provided. This paper demonstrates that the Nash equilibria type-2 fuzzy set of the game can be decomposed based on the secondary membership grades into a finite collection of crisp sets. Each of these crisp sets represents the Nash equilibria set of the corresponding game with crisp sets of player strategies. A characteristic feature of the proposed decomposition approach is its independence from the chosen method for calculating the Nash equilibria in crisp subgames. Some properties of game equilibria T2FSs are studied. These sets correspond to specific partitions or cuts of the original fuzzy sets of player strategies. An illustrative example is also included for clarity.

1. Introduction

In classical game theory, it is traditionally assumed that all the data of a game are precisely known. However, in real-world scenarios, a notable feature of games is the inherent uncertainty and inaccuracy of available information. To address this issue, one powerful tool for modeling uncertainty is the theory of fuzzy sets (FSs).

Fuzzy sets allow for the representation of imprecise or vague data within various components of a game model. These components include the sets of players, sets of strategies, payoffs of players, and more. Early pioneers such as Orlovskii [1], Butnariu [2,3,4], and Billot [5] were among the first to introduce fuzzy sets into the realm of non-cooperative games. Orlovskii, for instance, leveraged the principle of decision-making in a fuzzy environment, as outlined by Zadeh and Bellman [6], to defuzzify game-related concepts. Building on the work of Butnariu [2,3,4] and Billot [5], other researchers have explored the modeling of each player’s beliefs about the actions of other players in a fuzzy set form.

Additionally, Campos [7] made significant contributions by delving into non-cooperative games with fuzzy payoffs. Campos’ approach is founded on a ranking method of fuzzy numbers to defuzzify the game. By utilizing Yager’s ordering method [8] for fuzzy numbers, Campos transformed the challenge of finding a solution for a fuzzy matrix game into a linear programming problem.

In the context of fuzzy non-cooperative games, the Nash equilibrium, much like in scenarios with crisp information, remains a fundamental principle of optimality. Notably, this article focuses on aspects other than fuzzy matrix games; for those interested in exploring this subject further, a detailed review is available in [9]. Generalizations of the Nash equilibrium concept have primarily been cultivated within the domain of bimatrix games featuring fuzzy payoffs. This inclination can be attributed to the wealth of well-established methods for constructing Nash equilibria in the realm of crisp bimatrix games.

One notable approach, advanced by Vijay et al. [10], leverages the theory of fuzzy duality and employs a ranking function for fuzzy numbers to calculate equilibria in bimatrix games with fuzzy payoffs. This innovative methodology ultimately leads to the formulation of a fuzzy nonlinear programming problem, which is subsequently defuzzified.

When player payoffs are expressed as triangular fuzzy numbers, Maeda [11] introduces a method employing fuzzy number ranking to convert the computation of Nash equilibria into a crisp optimization problem whenever feasible. In the context of games involving n-players, there has been a development that delves into non-cooperative games featuring fuzzy parameters. Notably, in [12], a game is examined wherein payoffs depend on certain parameters expressed as fuzzy numbers. These parameters reflect the influence of nature, and the players possess full information, including knowledge of the membership functions of the fuzzy parameters. The approach to solving such a game draws upon methods designed for addressing multicriteria problems with fuzzy parameters, as originally proposed in [13].

It is essential to acknowledge that the body of research in the domain of games with fuzzy payoffs significantly outweighs the research conducted in games featuring fuzzy sets of strategies. This disparity, in our view, can be attributed to the advancements in fuzzy arithmetic, which have made it relatively straightforward to introduce fuzzification in games with fuzzy payoffs. However, it is important to note that, in both fuzzy optimization problems and games, models with fuzzy parameters cannot entirely supplant models utilizing fuzzy sets of strategies. The exploration of game models involving fuzzy sets of strategies is warranted when such sets more naturally and effectively formalize the strategic choices made by players.

Additionally, there are instances where players are unable to precisely formulate their sets of strategies. The foundations of the approach to studying games with fuzzy strategies and fuzzy sets of strategies were established by Orlovskii [1], Butnariu [2,3,4], and Billot [5]. In [1], a two-person game with FSs of strategies and players’ goals is examined. For each player, crisp numerical assessments of game strategic profiles are provided. The players’ goals are expressed in the form of FSs over the set of assessments. The game model is grounded in Bellman and Zade’s [6] principle of decision-making in a fuzzy environment. For each player, a decision FS is defined as the intersection of the FSs representing the goal and strategies. In a defuzzified game, the membership functions (MFs) of decision FSs serve as the players’ payoff functions. A similar fuzzy game model is also explored in Aristidou and Sarangi [14]. The concept of equilibrium in this context aligns with the Nash equilibrium, with the sole distinction that it is defined within a fuzzy extension of the game. The existence of an equilibrium in a fuzzy game is demonstrated. Garazic and Cruz [15] propose an approach grounded in the development of fuzzy controllers. According to Arfi [16], a linguistic fuzzy game is defined utilizing linguistic fuzzy strategies, linguistic fuzzy preferences, and various forms of reasoning and inference. While it is important to acknowledge that this review is not exhaustive in its coverage of the existing literature, it is evident that models involving FSs of strategies and/or fuzzy strategies incorporate them indirectly by leveraging the Bellman and Zadeh approach or various types of preference relations in tandem with fuzzy goals.

We believe that the adoption of these alternative approaches arises from a lack of mathematical methods that directly facilitate the study of the impact of fuzzy sets of players’ strategies on a set of Nash equilibria. This research is motivated by the aspiration to develop the requisite methodology and derive the corresponding outcomes. It is important to note that the research conducted is fundamentally theoretical in nature. Practical applications of these findings warrant a separate investigation and fall beyond the scope of this article. The primary objectives of this article can be summarized as follows:

• To establish a rationale for the assertion that fuzzy sets (FSs) of players’ strategies in a finite game give rise to a type-2 fuzzy set (T2FS) of Nash equilibria, characterized by a specific, simplified form that is practical for real-world applications, as opposed to the general form T2FS.
• To conduct an in-depth examination of the properties of this T2FS.
• To develop a decomposition method for the construction of Nash equilibria T2FS.
• The practical significance of this research lies in its capacity to:
• Explore game scenarios in which the adoption of crisp strategies constrains players’ abilities to effectively resolve conflicts. Example 1 in Section 5.4 provides insight into such a scenario.
• Enable the modeling of uncertainty inherent in human judgments and the uncertainty associated with determining acceptable strategies. Such FSs of strategies might encompass categories like ‘Proven strategies’, ‘Robust strategies’, ‘Acceptable strategies’, and similar distinctions.

Finally, this article presents a compelling argument for the incorporation of FSs of player strategies in a game, highlighting that they give rise to Nash equilibria in the form of a T2FS defined over the set of strategy profiles. This approach not only enriches the modeling of strategic interactions but also provides a more nuanced perspective on decision-making in situations characterized by imprecision and ambiguity. Through this exploration, we contribute to the growing body of research at the intersection of game theory and fuzzy logic, further enhancing our understanding of strategic behavior in complex, uncertain environments.

2. Materials and Methods

2.1. A Classic Finite Game

A finite non-cooperative game can be formally represented in the normal form $\langle X_i,u_i:i\in N\rangle$, where $N=\{1,\dots ,n\}$ is the finite set of players; $n=\left|N\right|\ge 2$ is the cardinality of this set; $X_i$ is the finite set of strategies $x_i$ of the player $i\in N$; $u_i(x)$ is the payoff function of the player $i\in N$, which is defined on the set $X={\displaystyle \prod _{i\in N}{X}_i}$ of the strategies profiles $x=(x_1,\dots ,x_n)={(x_i)}_{i\in N}$.

A Nash equilibrium of the game $\langle X_i,u_i:i\in N\rangle$ is the strategies profile $\widehat{x}=({\widehat{x}}_1,\dots ,{\widehat{x}}_n)={({\widehat{x}}_i)}_{i\in N}$ for which the inequalities

$$u_i(\widehat{x})\ge u_i(x_i,{\widehat{x}}_{N\backslash i}) \mathrm{for} \mathrm{all} x_i\in X_i \mathrm{and} i\in N$$

(1)

hold, where ${\widehat{x}}_{N\backslash i}=({\widehat{x}}_1,\dots ,{\widehat{x}}_{i-1},{\widehat{x}}_{i+1}\dots ,{\widehat{x}}_n)={({\widehat{x}}_j)}_{j\in N\backslash \left\{i\right\}}$ is the collection of strategies of the players $j\in N\backslash \left\{i\right\}$. The choice of the strategy ${\widehat{x}}_i$ by each player $i\in N$ seems reasonable. Indeed, it is not profitable to deviate from these strategies for each of them individually. We denote by $NE(X)$ the set of Nash equilibria of the game $\langle X_i,u_i:i\in N\rangle$.

2.2. Type-2 Fuzzy Sets

The T2FS concept was proposed by Zadeh in [17] as an extension of the type-1 fuzzy sets (T1FSs). According to Mizumoto and Tanaka [18], a T2FS, denoted by $\tilde{C}$, on a crisp set $X$ is characterized by the fuzzy membership function $M_{\tilde{C}}:X\to {[0,1]}^{[0,1]}$. For fixed $x\prime \in X$, the value of $M_{\tilde{C}}(x\prime )$ is the T1FS $M_{\tilde{C}}(x\prime )=\{(u,{\mu }_{{\tilde{M}}_{\tilde{C}}(x\prime )}(u)):u\in U_{x\prime }\}$ on the set $U_{x\prime }\subseteq [0,1]$ of primary membership degrees $u$ of $x\prime$ to the T2FS $\tilde{C}$ with corresponding membership function ${\mu }_{{\tilde{M}}_{\tilde{C}}(x\prime )}(u)$, $u\in U_{x\prime }$. In [19], the representation of the T2FS $\tilde{C}$ in the form $\tilde{C}=\{(x,{\tilde{M}}_{\tilde{C}}(x\prime )):x\in X\}=\{(x,\{(u,{\mu }_{{\tilde{M}}_{\tilde{C}}(x)}(u)):u\in U_x\}):x\in X\}$ is called the vertical-slice manner.

Another definition, based on the ideas of Karnik and Mendel [20], was given by Mendel and John [21]. A T2FS $\tilde{C}$ on a crisp set $\tilde{X}$ is characterized by the type-2 membership function (T2MF) ${\eta }_{\tilde{C}}(x,u)$, that is $\tilde{C}=\{((x,u),{\eta }_{\tilde{C}}(x,u)): x\in X, u\in [0,1]\}$, where ${\eta }_{\tilde{C}}(x,u)={\mu }_{{\tilde{M}}_{\tilde{C}}(x)}(u)$ for all $u\in U_x$, and ${\eta }_{\tilde{C}}(x,u)=0$ for all $u\notin U_x$. The value ${\eta }_{\tilde{C}}(x,u)$ is a crisp number from the interval [0, 1], known as a secondary grade of pair $(x,u)$ to $\tilde{C}$.

Remark 1. 

The primary membership degree u is usually understood as the degree of manifestation of some property (that defines the given fuzzy set) for x ∈ X. The secondary grade is usually [19] associated with the degree of truth of the corresponding primary degree u of this property for x.

Following [21], we define embedded T2FSs and T1FSs for a T2FS $\tilde{C}=\{((x,u),{\eta }_{\tilde{C}}(x,u)):x\in X,u\in [0,1]\}$. Assume that $u_x={\mu }_{C^{e1}}(x)\in [0,1]$ is a unique primary degree of membership for each $x\in X$, where ${\mu }_{C^{e1}}(x)$, $x\in X$ is the MF of the T1FS $C^{e1}=\{(x,{\mu }_{C^{e1}}(x)):x\in X\}$. This T1FS is called embedded in the T2FS $\tilde{C}$. We define the embedded T2FS ${\tilde{C}}^{e2}$ in $\tilde{C}$ in the form ${\tilde{C}}^{e2}=\{((x,u_x),{\eta }_{{\tilde{C}}^{e2}}(x,u_x)):x\in X\}$ with ${\eta }_{{\tilde{C}}^{e2}}(x,u_x)={\eta }_{\tilde{C}}(x,{\mu }_{C^{e1}}(x))$ for all $x\in X$.

Remark 2. 

Each element of the type-2 fuzzy collection $\tilde{C}=\{((x,u),{\eta }_{\tilde{C}}(x,u)):x\in X,u\in [0,1]\}$ is interpreted as a subset. Thus, the collection is represented as the classical union of its elements in the sense of T1FSs.

In [21], Mendel and John stated that each T2FS can be represented as a collection of all possible embedded T2FSs. We shall need one special case of a T2FS to be defined according to [22,23,24,25]. Let $A=\{{\eta }_{\tilde{C}}(x,u):{\eta }_{\tilde{C}}(x,u)>0,x\in X,u\in [0,1]\}$ be the set of all possible positive values of secondary grades for the T2FS $\tilde{C}=\{((x,u),{\eta }_{\tilde{C}}(x,u)): x\in X, u\in [0,1]\}$. Assume that the set A is finite.

According to [22], an embedded T2FS ${\tilde{C}}^{e2}(\alpha )=\{((x,u_x),{\eta }_{{\tilde{C}}^{e2}(\alpha )}(x,u_x)):x\in X\}$ in the T2FS $\tilde{C}$ has a constant secondary grade $\alpha \in A$ if, for each $x\in X$, the unique primary degree $u_x={\mu }_{C^{e1}(\alpha )}(x)\in [0,1]$ exists for which ${\eta }_{{\tilde{C}}^{e2}(\alpha )}(x,u_x)\equiv \alpha$, i.e., ${\tilde{C}}^{e2}(\alpha )=\{((x,{\mu }_{C^{e1}(\alpha )}(x)),\alpha ): x\in X\}$. Here, ${\mu }_{C^{e1}(\alpha )}(x)$, $x\in X$ is the MF of the embedded T1FS $C^{e1}(\alpha )=\{(x,{\mu }_{C^{e1}(\alpha )}(x)):x\in X\}$ in the T2FS $\tilde{C}$.

Remark 3 

([25]). Obviously, for the T2FS $\tilde{C}$ and each $\alpha \in A$, there is the unique embedded T1FS $C^{e1}(\alpha )=\{(x,{\mu }_{C^{e1}(\alpha )}(x)): x\in X\}$, which is corresponding to the embedded T2FS ${\tilde{C}}^{e2}(\alpha )$ with the constant secondary grade $\alpha$. Hence, ${\tilde{C}}_{}^{e2}(\alpha )=\{(C_{}^{e1}(\alpha ),\alpha )\}=\{(\{(x,{\mu }_{C^{e1}(\alpha )}(x)): x\in X\},\alpha )\}=$$\{((x,{\mu }_{C^{e1}(\alpha )}(x)),\alpha ): x\in X\}$.

Further, we consider another special case of a T2FS.

Definition 1. 

We say that the T2FS $\tilde{C}$ is decomposable by secondary grades into a collection of embedded T2FSs with constant secondary grades if there are the T2FSs ${\tilde{C}}^{e2}(\alpha )=\{((x,{\mu }_{C^{e1}(\alpha )}(x)),\alpha ): x\in X\}=\{(C_{}^{e1}(\alpha ),\alpha )\}$ with constant secondary grades $\alpha \in A$, respectively, which are embedded in the T2FS $\tilde{C}$ satisfying $\tilde{C}=\{{\tilde{C}}_{}^{e2}(\alpha ): \alpha \in A\}$.

Remark 4. 

In view of Remark 3, if the T2FS $\tilde{C}$ is decomposable by secondary grades $\alpha \in A$ into the collection $\tilde{C}=\{C_{}^{e2}(\alpha ):\alpha \in A\}$ of embedded T2FSs with constant secondary grades, then the T2FS $\tilde{C}$ is represented as a collection $\tilde{C}=\{(C_{}^{e1}(\alpha ),\alpha ):\alpha \in A\}$ of embedded T1FSs $C_{}^{e1}(\alpha )$, $\alpha \in A$, each of which is assigned the constant secondary grade $\alpha \in A$, respectively.

3. Formulation of the Problem

Consider a finite non-cooperative game in the normal form $\langle X_i,u_i:i\in N\rangle$.

Assumption 1. 

Assume that the game $\langle X_i,u_i:i\in N\rangle$ has at least one Nash equilibrium in pure strategies, that is $NE(X)\ne \varnothing$.

Let ${\tilde{X}}_i=\{(x,{\mu }_{{\tilde{X}}_i}(x_i)):x_i\in X_i\}$, $i\in N$ be some FSs with the MFs ${\mu }_{{\tilde{X}}_i}(x_i)$, $x_i\in X_i$ $i\in N$ on the sets $X_i$ of pure strategies of players $i\in N$, respectively. We shall call ${\tilde{X}}_i$, $i\in N$ the FSs of strategies. We represent a game with FSs of strategies in the normal form $\langle {\tilde{X}}_i,u_i:i\in N\rangle$. A natural question is: ‘When is there a need for such game formulation?’ To answer this question, we consider the following examples. Suppose that some decision maker (DM) is trying to predict the outcome of a conflict between players, which can be formulated as some classical finite game with crisp sets of player strategies. The issue is that the DM only knows the degrees of membership of the player strategies to some FSs of their strategies. The following question arises: ‘What is the set of Nash equilibria in the case when the sets of players’ strategies are fuzzy?’

4. Main Idea

First, we generalize inequalities (1) for the case of arbitrary subsets $S_i\subseteq X_i$, $i\in N$ of strategies. They take the form

$$u_i(\widehat{x})\ge u_i(x_i,{\widehat{x}}_{N\backslash i}) \mathrm{for} \mathrm{all} x_i\in S_i \mathrm{and} i\in N.$$

(2)

We denote by $S={\displaystyle \prod _{i\in N}{S}_i}\subseteq X$ the set of the strategy profiles. The subsets $S_i\subseteq X_i$, $i\in N$ of strategies are parameters of inequalities (2) that the sets of constraints depend upon. In addition, for the game $G(S)=\langle S_i,u_i:i\in N\rangle$, we denote by

$$NE(S)=\{(x,{\mu }_{NE(S)}(x)):x\in X\}$$

(3)

the crisp set of Nash equilibria with the MF characteristic function

$${\mu }_{NE(S)}(x)=\{\begin{array}{cc}1,& u_i(x)\ge u_i(y_i,x_{N\backslash i}) \mathrm{for} \mathrm{all} y_i\in S_i \mathrm{and} i\in N;\\ 0,& \mathrm{otherwise};\end{array}$$

(4)

and with the support $\mathrm{supp}(NE(S))=\{x\in S:{\mu }_{NE(S)}(x)=1\}$ of the set $NE(S)$.

Remark 5. 

We use the MF (4) representation of a crisp set of Nash equilibria for the convenience of presenting the proposed method.

For each fixed strategy profile $x\in X={\displaystyle \prod _{i\in N}{X}_i}$ of the initial game $\langle X_i,u_i:i\in N\rangle$, consider the mapping $V_{}^x:2^S\to [0,1]$ given by

$$V_{}^x(S)=\{\begin{array}{cc}1,& x\in \mathrm{supp}(NE(S));\\ 0,& \mathrm{otherwise};\end{array}$$

(5)

$S={\displaystyle \prod _{i\in N}{S}_i}, S_i\subseteq X_i, i\in N$. For any strategic profile $x\in X$, the mapping $V_{}^x$ associates each collection of the subsets $S_i\subseteq X_i, i\in N$ of strategies with the value of the MF

$${\mu }_{NE(S)}(x)=V_{}^x(S), x\in \mathrm{supp}(NE(S))=\{x\in X:V_{}^z(S)\ne 0\}$$

(6)

of the crisp set of Nash equilibria $NE(S)$. With Zadeh’s extension principle [26] at hand, we extend the domain of the mapping $V_{}^x$ to the collection of FSs ${\tilde{X}}_i=\{(x,{\mu }_{{\tilde{X}}_i}(x_i)):x_i\in X_i\}$, $i\in N$ of strategies that are defined on universal sets $X_i$, $i\in N$ of strategies, respectively, and generalize formulae (3) and (6) to this case. We denote by $\tilde{E}$ a set of Nash equilibria of the game $\langle {\tilde{X}}_i,u_i:i\in N\rangle$ for FSs of strategies, and we denote by $M_{\tilde{E}}(x)$, $x\in X$ the corresponding MF. In this case, for each fixed $x=x*$, the value of the MF $M_{\tilde{E}}(x)$ coincides with the image $V_{}^{x*}( \tilde{X})$ of the FS $\tilde{X}={\displaystyle \prod _{i\in N}{\tilde{X}}_i}$ of strategies profiles under the mapping $V_{}^{x*}$, that is,

$$M_{ \tilde{E}}(x*)=V_{}^{x*}( \tilde{X}).$$

(7)

According to Zadeh’s extension principle [26], the image of the FS $\tilde{X}$ of strategies profiles under the mapping $V_{}^{x*}:2^X\to [0,1]$ (see (5)) is the FS

$$V_{}^{x*}( \tilde{X})=\{(u,{\mu }_{V_{}^{x*}( \tilde{X})}(u)):u\in \{0,1\left\}\right\}$$

(8)

with the MF

$${\mu }_{V_{}^{x*}( \tilde{X})}(u)=\max \left\{\underset{i\in N}{\min }\right\{{\alpha }_i\}: {\alpha }_i\in [0,1]; u=V_{}^{x*}(X(\alpha ))\}$$

(9)

$u\in \mathrm{supp}(V_{}^{x*}(X(\alpha )))$, where

$$\mathrm{supp}(V_{}^{x*}(X(\alpha ))=\{u\in \{0,1\}: u=V_{}^{x*}(X(\alpha ));\alpha =({\alpha }_i), {\alpha }_i\in [0,1], i\in N\}$$

(10)

is the support of the FS $V_{}^{x*}(X(\alpha ))$;

$X(\alpha )={\displaystyle \prod _{i\in N}{X}_i}(\alpha )$ is the set of strategies profiles of the game $G(X(\alpha ))=\langle X_i(\alpha ),u_i:i\in N\rangle$;

$X_i({\alpha }_i)=\{x_i\in X_i:{\mu }_{{\tilde{X}}_i}(x_i)\ge {\alpha }_i\}$ is the ${\alpha }_i$-cut, ${\alpha }_i\in [0,1]$ of the FS ${\tilde{X}}_i=\{(x,{\mu }_{{\tilde{X}}_i}(x_i)):x_i\in X_i\}$ of strategies of the player $i\in N$;

$\alpha ={({\alpha }_i)}_{i\in N}$ is the vector of ${\alpha }_i$-cuts levels, ${\alpha }_i\in [0,1]$, $i\in N$ of strategies FSs;

$$V_{}^{x*}(X(\alpha ))={\mu }_{NE(X(\alpha ))}(x*)$$

(11)

is the image of the collection of cuts $X_i({\alpha }_i)=\{x_i\in X_i:{\mu }_{{\tilde{X}}_i}(x_i)\ge {\alpha }_i\}$, ${\alpha }_i\in [0,1]$, $i\in N$ of the FSs ${\tilde{X}}_i$, $i\in N$ of strategies under the mapping $V_{}^{x*}$ (see Equation (6)).

Remark 6. 

Let ${\mathsf{\Omega }}_i=\{{\mu }_{{\tilde{X}}_i}(x_i):x_i\in X_i\}$, $i\in N$ be the sets of membership degrees values ${\mu }_{{\tilde{X}}_i}(x_i), x_i\in X_i$, $i\in N$ of the FSs ${\tilde{X}}_i=\{(x,{\mu }_{{\tilde{X}}_i}(x_i)):x_i\in X_i\}$, $i\in N$ of strategies, respectively. Note that the cardinalities of the sets ${\mathsf{\Omega }}_i$, $i\in N$ are $\left|{\mathsf{\Omega }}_i\right|\le \left|X_i\right|$, $i\in N$, respectively. It is clear that when obtaining ${\alpha }_i$-cuts, ${\alpha }_i\in [0,1]$, $i\in N$ of the FS ${\tilde{X}}_i$, $i\in N$ we can assume that ${\alpha }_i\in {\mathsf{\Omega }}_i$, $i\in N$ rather than ${\alpha }_i\in {\mathsf{\Omega }}_i$, $i\in N$, respectively.

Thus, in view of (7)–(11) and Remark 6, for fixed $x=x*$, the values of the MF $M_{ \tilde{E}}(x*)$ form the FS $\{(u,{\mu }_{M_{ \tilde{E}}(x*)}(u)):u\in \{0,1\left\}\right\}$ on $\{0,1\}$ with the MF ${\mu }_{M_{ \tilde{E}}(x*)}(u)=\max \{\underset{i\in N}{\min }{\alpha }_i: {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N, u=V_{}^{x*}(X(\alpha ))\}$, $u\in \mathrm{supp}(M_{\tilde{E}}(x*))$, where $\mathrm{supp}(M_{\tilde{E}}(x*))=\{u\in \{0,1\}: u=V_{}^{x*}(X(\alpha )), {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N\}$ is the support of the FS $M_{\tilde{E}}(x*)$. Then, invoking (10) and (11) yields

$$\begin{array}{c}{\mu }_{M_{ \tilde{E}}(x*)}(u)=\max \{\underset{i\in N}{\min }{\alpha }_i: {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N, u={\mu }_{NE(X(\alpha ))}(x*)\},\\ u\in \mathrm{supp}(M_{\tilde{E}}(x*))=\{u\in \{0,1\}:u={\mu }_{NE(X(\alpha ))}(x*), {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N\}.\end{array}$$

(12)

Therefore, we conclude that the set of Nash equilibria $\tilde{E}$ is an FS on $X$ with the MF whose values form FSs on $\{0,1\}$ for each $x\in X$. Then, according to [17], $\tilde{E}$ is the T2FS on $X$. In the manner of vertical slices (see Section 2.2), the T2FS $\tilde{E}$ on $X$ has the form $\tilde{E}=\{(x,M_{\tilde{E}}(x)):x\in X\}=\{(x,\{(u,{\mu }_{M_{\tilde{E}}(x)}(u)): u\in U_x\}): x\in X\}$. In this formula, ${\mu }_{M_{\tilde{E}}(x)}(u)$, $u\in \{0,1\}$ is the MF of the FS $M_{\tilde{E}}(x)=\left\{\right\{(u,{\mu }_{M_{\tilde{E}}(x)}(u)): u\in \{0,1\}\}$ of values of fuzzy degree of membership of the strategies profile $x\in X$ to the T2FS $\tilde{E}$, and $U_x=\mathrm{supp}(M_{\tilde{E}}(x))$ is the set of primary membership degrees, where $\mathrm{supp}(M_{\tilde{E}}(z))$ is the support of the FS $M_{\tilde{E}}(x)$ for $x\in X$. According to Section 2.2, we can also characterize the T2FS $\tilde{E}$ by means of the T2MF ${\eta }_{\tilde{E}}(x,u)=0$ for $u\notin U_x$ and ${\eta }_{\tilde{E}}(x,u)=\max \{\underset{i\in N}{\min }{\alpha }_i: {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N, u={\mu }_{NE(X(\alpha ))}(x)\}$ for $u\in U_x$. This conclusion allows us to introduce the following notion.

Definition 2. 

By the set of Nash equilibria of the game $\langle {\tilde{X}}_i,u_i:i\in N\rangle$ for the FSs ${\tilde{X}}_i=\{(x,{\mu }_{{\tilde{X}}_i}(x_i)):x_i\in X_i\}$ of strategies is meant the T2FS

$$\tilde{E}=\{((x,u),{\eta }_{\tilde{E}}(x,u)): u\in \{0,1\}, x\in X\}$$

(13)

on $X$ with the T2MF

$${\eta }_{\tilde{E}}(x,u)=\{\begin{array}{c}\max \{\underset{i\in N}{\min }{\alpha }_i: {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N, u={\mu }_{NE(X(\alpha ))}(x*)\},\\ 0,\end{array}\begin{array}{c} u\in U_x;\\ u\notin U_x.\end{array}$$

(14)

In this definition,

$$U_x=\{u\in \{0,1\}: u={\mu }_{NE(X(\alpha ))}(x),{\alpha }_i\in {\mathsf{\Omega }}_i, i\in N\}$$

(15)

is the set of primary membership degrees $u\in \{0,1\}$ with strictly positive secondary grades ${\eta }_{\tilde{E}}(x,u)$, which coincides with the support $\mathrm{supp}(M_{\tilde{E}}(x))$ (see (12)) of the FS $M_{\tilde{E}}(x)$ of fuzzy membership degrees of the strategy profile $x\in X$;

$${\mu }_{NE(X(\alpha ))}(x)=\{\begin{array}{cc}1,& x\in NE(X(\alpha ));\\ 0,& \mathrm{otherwise}\end{array}$$

(16)

is the MF (characteristic function) of the crisp set

$$NE(X(\alpha ))=\{(x,{\mu }_{X(\alpha )}(x)):x\in X\}$$

(17)

of Nash equilibria of the game $G(X(\alpha ))=\langle X_i({\alpha }_i),u_i:i\in N\rangle$ for the sets $X_i({\alpha }_i)$, $i\in N$ of strategies (see (3),(4) with $S=X(\alpha )$ and $S_i=X_i({\alpha }_i)$, $i\in N$);

$$X_i({\alpha }_i)=\{x_i\in X_i:{\mu }_{{\tilde{X}}_i}(x_i)\ge {\alpha }_i\}, i\in N$$

(18)

is the ${\alpha }_i$-cuts, ${\alpha }_i\in {\mathsf{\Omega }}_i$ of the FS ${\tilde{X}}_i$ of strategies of the player $i\in N$;

$\alpha ={({\alpha }_i)}_{i\in N}$ is the vector of ${\alpha }_i$-cuts levels, ${\alpha }_i\in {\mathsf{\Omega }}_i$, $i\in N$ of the FSs ${\tilde{X}}_i$, $i\in N$ of strategies;

${\mathsf{\Omega }}_i=\{{\mu }_{{\tilde{X}}_i}(x_i):x_i\in X_i\}$, $i\in N$ are the sets of the membership degrees values ${\mu }_{{\tilde{X}}_i}(x_i), x_i\in X_i$, $i\in N$ of the FSs ${\tilde{X}}_i$, $i\in N$ of strategies (see Remark 6), respectively;

$X(\alpha )={\displaystyle \prod _{i\in N}{X}_i}({\alpha }_i)$ is the set of strategies profiles of the game $G(X(\alpha ))=\langle X_i({\alpha }_i),u_i:i\in N\rangle$, $\alpha ={({\alpha }_i)}_{i\in N}$, ${\alpha }_i\in {\mathsf{\Omega }}_i$, $i\in N$.

Remark 7. 

Since primary membership degrees $u\in \{0,1\}$ of the T2FS $\tilde{E}$ take only two values, 0 or 1, by Remark 2, this yields an interesting interpretation of the T2FS $\tilde{E}$. Similarly to a crisp set, there are only two options for each strategy profile $x\in X$: either $x$ completely belongs to the T2FS $\tilde{E}$ (the primary membership degree is $u=1$), or it does not belong completely ($u=0$). Unlike a crisp set, the degrees ${\eta }_{\tilde{E}}(x,0)$ and ${\eta }_{\tilde{E}}(x,1)$ of truth of the identification of these two facts can differ from 1 and take values in the closed interval $[0,1]$.

5. Nash Equilibria T2FS of a Game with Fuzzy Sets of Strategies

5.1. A Decomposition of Nash Equilibria T2FS

Proposition 1 justifies the decomposability (see Definition 1) of Nash equilibria T2FS of a game with FSs of strategies on a collection of embedded T2FSs with constant secondary grades.

Proposition 1. 

The Nash equilibria T2FS $\tilde{E}$ of the game $\langle {\tilde{X}}_i,u_i:i\in N\rangle$ for FSs of strategies is decomposable by secondary grades $\underset{i\in N}{\min }{\alpha }_i$ into the collection

$$\tilde{E}=\{{\tilde{E}}_{}^{e2}(X(\alpha )): \alpha =({\alpha }_i{)}_{i\in N}, {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N\}$$

(19)

of the embedded T2FSs

$${\tilde{E}}_{}^{e2}(X(\alpha ))=\{(NE(X(\alpha )),\underset{i\in N}{\min }{\alpha }_i)\}$$

(20)

where $NE(X(\alpha ))$ is the embedded T1FS. It is a crisp set (a crisp set is a special case of a T1FS), which is the set of Nash equilibria of the game $G(X(\alpha ))=\langle X_i({\alpha }_i),u_i:i\in N\rangle$ with the crisp sets $X_i({\alpha }_i)$, $i\in N$of strategies.

Proof of Proposition 1. 

By (13), the Nash equilibria T2FS is given by $\tilde{E}=\{((x,u),{\eta }_{\tilde{E}}(x,u)): u\in \{0,1\}, x\in X\}$. According to (14), $\tilde{E}=\{((x,u),0):u\notin U_x\}:x\in X\}\cup$ $\left\{\right\{((x,u),\max \{\underset{i\in N}{\min }{\alpha }_i: {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N, u={\mu }_{NE(X(\alpha ))}(x)\}):u\in U_x\}$. Since Remark 2 allows us to ignore the pairs $(x,u)$ that have secondary grades equal to 0, we conclude that $\tilde{E}=\left\{\right\{((x,u),\max \{\underset{i\in N}{\min }{\alpha }_i: {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N, u={\mu }_{NE(X(\alpha ))}(x)\}): u\in U_x, x\in X\}$, which is equivalent $\tilde{E}=\{(x,\{({\mu }_{NE(X(\alpha ))}(x),\underset{i\in N}{\min }{\alpha }_i): {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N\}), x\in X\}$ according to (14). Further, regrouping the elements leads to $\tilde{E}=\{(x,({\mu }_{NE(X(\alpha ))}(x),\underset{i\in N}{\min }{\alpha }_i)): {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N, x\in X\}=$ $\left\{\right\{((x,({\mu }_{NE(X(\alpha ))}(x)),\underset{i\in N}{\min }{\alpha }_i):x\in X\}: {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N\}$. Then, invoking (17) yields $\tilde{E}=\{(NE(X(\alpha )),\underset{i\in N}{\min }{\alpha }_i): {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N\}$ whence (19) comes by (20). □

A characteristic feature of the proposed decomposition approach is its independence from the chosen method for calculating the sets $NE(X(\alpha ))$ of Nash equilibria of games $G(X(\alpha ))=\langle X_i({\alpha }_i),u_i:i\in N\rangle$ for crisp sets $X_i({\alpha }_i)$, $i\in N$ of strategies. Denote by

$$\mathsf{\Omega }=\{\underset{i\in N}{\min }{\alpha }_i: {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N\}$$

(21)

the set of secondary grades of the Nash equilibria T2FS $\tilde{E}$ of the game $\langle {\tilde{X}}_i,u_i:i\in N\rangle$. Corollary 1 states that the T2FS $\tilde{E}$ can be directly decomposed by the set $\mathsf{\Omega }$ of secondary grades.

Corollary 1. 

The Nash equilibria T2FS $\tilde{E}$ of the game $\langle {\tilde{X}}_i,u_i:i\in N\rangle$ for FSs of strategies is decomposable by secondary grades $\gamma \in \mathsf{\Omega }$ into the collection

$$\tilde{E}=\{(E_{\gamma }^{e1},\gamma ): \gamma \in \mathsf{\Omega }\}$$

(22)

 of the embedded T2FSs ${\tilde{E}}_{\gamma }^{e2}=\{(E_{\gamma }^{e1},\gamma )\}$. For each $\gamma \in \mathsf{\Omega }$, the embedded T1FS

$$E_{\gamma }^{e1}=\underset{\alpha ={({\alpha }_i)}_{i\in N}, {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N: \underset{i\in N}{\min }{\alpha }_i=\gamma }{\cup }NE(X(\alpha )),\gamma \in \mathsf{\Omega }$$

(23)

is the union of the crisp sets $NE(X(\alpha ))$ (a crisp set is a special case of a T1FS). For each $\alpha ={({\alpha }_i)}_{i\in N}, {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N$ such that $\underset{i\in N}{\min }{\alpha }_i=\gamma$, the set $NE(X(\alpha ))$ is a Nash equilibria set of the game $G(X(\alpha ))=\langle X_i({\alpha }_i),u_i:i\in N\rangle$ for the sets $X_i({\alpha }_i)$, $i\in N$ of strategies.

Proof of Corollary 1. 

Formulae (19) and (20) imply that ${\tilde{E}}_{}^{e2}(X(\alpha ))=\{(NE(X(\alpha )),\underset{i\in N}{\min }{\alpha }_i): \alpha =({\alpha }_i{)}_{i\in N}, {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N\}$. According to Remark 3, each element of this collection can be interpreted as a subset. Thus, the collection is represented as the classical union of its elements in the sense of T1FSs. With this at hand, we conclude that by (21), associated to each $\gamma \in \mathsf{\Omega }$ is the subset $\underset{\alpha ={({\alpha }_i)}_{i\in N}, {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N: \underset{i\in N}{\min }{\alpha }_i=\gamma }{\cup }\{(NE(X(\alpha )),\gamma )\}$ in the collection $\tilde{E}$. Therefore, an appeal to (23) yields (22). □

5.2. Calculation of a Nash equilibria T2FS

First, we construct the sets ${\mathsf{\Omega }}_i=\{{\mu }_{{\tilde{X}}_i}(x_i):x_i\in X_i\}$, $i\in N$ of membership degrees values of the FSs ${\tilde{X}}_i=\{(x,{\mu }_{{\tilde{X}}_i}(x_i)):x_i\in X_i\}$, $i\in N$ of strategies, respectively. For each ${\alpha }_i\in {\mathsf{\Omega }}_i, i\in N$, according to (18), we construct the ${\alpha }_i$-cuts $X_i({\alpha }_i)=\{x_i\in X_i:{\mu }_{{\tilde{X}}_i}(x_i)\ge {\alpha }_i\}$, $i\in N$ of the FS ${\tilde{X}}_i$, $i\in N$, respectively. Next, we construct the Nash equilibria sets $NE(X(\alpha ))$ of the games $G(X(\alpha ))=\langle X_i({\alpha }_i),u_i:i\in N\rangle$, $\alpha ={({\alpha }_i)}_{i\in N}$ for crisp sets $X_i({\alpha }_i)$, $i\in N$, ${\alpha }_i\in {\mathsf{\Omega }}_i, i\in N$. To construct the Nash equilibria sets NE(X(α)), one can use any known method. Further, we use the representation of the T2FS $\tilde{E}$ in the form of a collection of embedded T2FSs with constant secondary grades (see Corollary 1). To this end, we need to construct the embedded T1FS $E_{\gamma }^{e1}=\underset{\alpha ={({\alpha }_i)}_{i\in N}, {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N: \underset{i\in N}{\min }{\alpha }_i=\gamma }{\cup }NE(X(\alpha ))$ for each $\gamma \in \mathsf{\Omega }$ according to (23). Once all embedded T1FS $E_{\gamma }^{e1}$ with constant secondary grades $\gamma \in \mathsf{\Omega }$ have been obtained, the resulting Nash equilibria T2FS has the form $\tilde{E}=\{(E_{\gamma }^{e1},\gamma ): \gamma \in \mathsf{\Omega }\}$ according to (22) and (23).

By Remark 1, the T2FS $\tilde{E}$ can be interpreted as the collection of unions $E_{\gamma }^{e1}=\underset{\alpha ={({\alpha }_i)}_{i\in N}, {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N: \underset{i\in N}{\min }{\alpha }_i=\gamma }{\cup }NE(X(\alpha ))$, $\gamma \in \mathsf{\Omega }$ of Nash equilibria sets NE(X(α)) of the fuzzy games $G(X(\alpha ))=\langle X_i({\alpha }_i),u_i:i\in N\rangle$ for the corresponding crisp sets $X_i({\alpha }_i)$, $i\in N$ of strategies for $\alpha ={({\alpha }_i)}_{i\in N}, {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N$, such that $\underset{i\in N}{\min }{\alpha }_i=\gamma$ with the degree of truth of the set $E_{\gamma }^{e1}$ being equal to $\gamma$.

5.3. Properties of Constructing of the Nash equilibria T2FS

Propositions 2 and 3 point at some useful properties of the Nash equilibria T2FS of the game with FSs of strategies.

Proposition 2. 

If a strategy profile $x\in X$ is (is not) a Nash equilibrium of the game $G(X({\alpha }^{\ast }))=\langle X_i({\alpha }_i^{\ast }),u_i:i\in N\rangle$ for the collection of levels ${\alpha }_i^{\ast }\in {\mathsf{\Omega }}_i, i\in N$ of cuts $X_i({\alpha }_i^{\ast })$ of FSs ${\tilde{X}}_i$, $i\in N$ of strategies, then a primary degree of membership $u=1$ ($u=0$) of the strategy profile $x$ to the Nash equilibria T2FS has a secondary grade (degree of truth) not smaller than $\gamma *=\underset{i\in N}{\min }{\alpha }_i^{\ast }$, i.e., ${\eta }_{\tilde{E}}(x,1)\ge \gamma *$ (${\eta }_{\tilde{E}}(x,0)\ge \gamma *$).

Proof Proposition 2. 

Let ${\alpha }_i^{\ast }\in {\mathsf{\Omega }}_i, i\in N$ and $u={\mu }_{NE(X({\alpha }^{\ast }))}(x)=1$ ($u={\mu }_{NE(X({\alpha }^{\ast }))}(x)=0$). Then, by (15), $u\in U_x$. Therefore, in view of (14), ${\eta }_{\tilde{E}}(x,1)=\max \{\underset{i\in N}{\min }{\alpha }_i: {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N,$ ${\mu }_{NE(X({\alpha }^{\ast }))}(x)=1={\mu }_{NE(X(\alpha ))}(x)\}\ge \underset{i\in N}{\min }{\alpha }_i^{\ast }={\gamma }^{\ast }$ (${\eta }_{\tilde{E}}(x,0)=\max \{\underset{i\in N}{\min }{\alpha }_i: {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N$, (${\mu }_{NE(X({\alpha }^{\ast }))}(x)=0={\mu }_{NE(X(\alpha ))}(x)\}\ge \underset{i\in N}{\min }{\alpha }_i^{\ast }={\gamma }^{\ast })$. □

In other words, according to Proposition 2, the guaranteed value of the degree of truth $\gamma *\in \mathsf{\Omega }$ of the primary degree of membership $u=1$ ($u=0$) of the strategy profile $x\in X$ to the Nash equilibria T2FS is determined by the levels ${\alpha }_i^{\ast }\in {\mathsf{\Omega }}_i, i\in N$ of cuts of the FSs ${\tilde{X}}_i$, $i\in N$ of strategies, under which the strategy profile $x$ is (is not) a Nash equilibrium of the game $G(X({\alpha }^{\ast }))=\langle X_i({\alpha }_i^{\ast }),u_i:i\in N\rangle$.

Proposition 3. 

The Nash equilibria T2FS $\tilde{E}$ is not empty.

Proof Proposition 3. 

We denote by ${\alpha }^{\min }={({\alpha }_i^{\min })}_{i\in N}$ the vector of least levels ${\alpha }_i^{\min }=\underset{{\alpha }_i\in {\mathsf{\Omega }}_i}{\min }{\alpha }_i^{}$ of ${\alpha }_i^{}$-cuts of FSs ${\tilde{X}}_i$ of players strategies $i\in N$. Assume that $\tilde{E}=\varnothing$. Then, according to formulae (19) and (20), the equality $\tilde{E}=\{(NE(X(\alpha )),\underset{i\in N}{\min }{\alpha }_i): \alpha =({\alpha }_i{)}_{i\in N}, {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N\}=\varnothing$ is held. This entails $NE(X(\alpha ))=\varnothing$ for any $\alpha =({\alpha }_i{)}_{i\in N}, {\alpha }_i\in {\mathsf{\Omega }}_i, i\in N$ including for ${\alpha }^{\min }={({\alpha }_i^{\min })}_{i\in N}$. Therefore, $NE(X({\alpha }^{\min }))=\varnothing$. With the equalities $X_i({\alpha }_i^{\min })=X_i$, $i\in N$ at hand, we conclude that $NE(X)=\varnothing$ having utilized (18) and Remark 6, a contradiction to Assumption 1. □

5.4. Example of Constructing the Nash Equilibria T2FS

Assume that some DM analyzes a conflict of two players. The model of this conflict is a generalization of the well-known prisoner’s dilemma game to the case of fuzzy sets “Predictable Strategies” of players 1 and 2. The DM perceives sets $X_1=\{P_1,A_1,F_1\}$ and $X_2=\{P_2,A_2,F_2\}$ of players 1 and 2 strategies in the form of FSs ${\tilde{X}}_1=\{(P_1;1),(A_1;1),(F_1;0,7)\}$ and ${\tilde{X}}_2=\{(P_2;1),(A_2;1),(F_2;0,5)\}$, respectively. Here, the strategies $P_i,A_i,F_i$ have a sense of a peaceful, aggressive, and frenzied behavior, respectively, of the player $i=1,2$.

Table 1. Payoffs vectors of players.

Strategy	F2	A2	P2
F1	(0, 0)	(2, −1)	(4, −2)
A1	(−1, 2)	(1, 1)	(3, 0)
P1	(−2, 4)	(0, 3)	(2, 2)

contains the payoffs vectors $(u_1(x_1,x_2),u_2(x_1,x_2))$, $x_1\in \{P_1,A_1,F_1\}$, $x_2\in \{P_2,A_2,F_2\}$ of players. The DM intends to predict Nash equilibria.

According to Remark 6, the sets of membership degrees of FSs ${\tilde{X}}_1$ and ${\tilde{X}}_2$ of strategies are given by ${\mathsf{\Omega }}_1=\{0,7;1\}$ and ${\mathsf{\Omega }}_2=\{0,5;1\}$, respectively. For each pair ${\alpha }_1=\{0,7;1\}$ and ${\alpha }_2=\{0,5;1\}$, we use (18) to construct the corresponding cuts $X_i({\alpha }_i)=\{x_i\in X_i:{\mu }_{{\tilde{X}}_i}(x_i)\ge {\alpha }_i\}$, $i=1,2$ of FSs ${\tilde{X}}_i$, $i=1,2$, respectively. Then, we construct the sets $NE(X({\alpha }_1;{\alpha }_2))$ of Nash equilibria of the games $G(X({\alpha }_1;{\alpha }_2))=\langle X_1({\alpha }_1),u_1;X_2({\alpha }_2),u_2\rangle$, ${\alpha }_1=\{0,7;1\}$, ${\alpha }_2=\{0,5;1\}$ in the form (3)–(4). We get

$$NE(X(0,7;0,5))=\{((F_1,F_2);1)\}\cup \{((x_1,x_2);0):(x_1,x_2)\in \{P_1,A_1,F_1\}\times \{P_2,A_2,F_2\}\backslash \{(F_1,F_2)\left\}\right\},$$

(24)

$$NE(X(0,7;1))=\{((F_1,A_2);1)\}\cup \{((x_1,x_2);0):(x_1,x_2)\in \{P_1,A_1,F_1\}\times \{P_2,A_2,F_2\}\backslash \{(F_1,A_2)\left\}\right\},$$

(25)

$$NE(X(1;0,5))=\{((A_1,F_2);1)\}\cup \{((x_1,x_2);0):(x_1,x_2)\in \{P_1,A_1,F_1\}\times \{P_2,A_2,F_2\}\backslash \{(A_1,F_2)\left\}\right\},$$

(26)

$$NE(X(1;1))=\{((A_1,A_2);1)\}\cup \{((x_1,x_2);0):(x_1,x_2)\in \{P_1,A_1,F_1\}\times \{P_2,A_2,F_2\}\backslash \{(A_1,A_2)\left\}\right\}.$$

(27)

According to Remark 2, the Nash equilibria T2FS has the form

$$\begin{array}{l}\tilde{E}=\{(NE(X(0,7;0,5))\cup NE(X(1;0,5));0,5),(NE(X(0,7;1));0,7),(NE(X(1;1));1)\}=\\ \{(((F_1,F_2);1);0,5),(((A_1,F_2);1);0,5),(((F_1,A_2);1);0,7),(((A_1,A_2);1);1)\}\cup \\ \{(((x_1,x_2);0);1):(x_1,x_2)\in \{P_1,A_1,F_1\}\times \{P_2,A_2,F_2\}\backslash \{(A_1,A_2)\left\}\right\}.\end{array}$$

(28)

6. Discussion

In this section, we discuss the Nash equilibria T2FS obtained in Section 5.4 in comparison with solutions for crisp game settings. The DM who is analyzing a two-player conflict in the example might interpret the T2FS $\tilde{E}$ as follows:

• The strategic profile (A₁, A₂) is a Nash equilibrium with the degree of truth being equal to 1;
• The strategic profile (F₁, A₂) is a Nash equilibrium with the degree of truth being equal to 0,7;
• The strategic profiles (F₁, F₂) and (A₁, F₂) are Nash equilibria with degrees of truth being equal to 0,5;
• Any strategic profile other than the above is a Nash equilibrium with the degree of truth being equal to 0.

If this DM perceives sets of players’ strategies crisply, then the following Nash equilibria would exist for him:

• (A₁, A₂) in the case of a pessimistic assessment of possible strategies of players in the form of crisp sets {P₁, A₁} and {P₂, A₂} of player 1 and 2, respectively;
• (F₁, F₂) in the case of a pessimistic assessment of possible strategies of players in the form of crisp sets {P₁, A₁, F₁} and {P₂, A₂, F₂} of player 1 and 2, respectively.

At the same time, the DM does not know degrees of truth of all Nash equilibria and misses the equilibria (A₁, F₂) and (F₁, A₂). Thus, we conclude that game models with fuzzy sets of strategies are more informative.

7. Conclusions

According to the proposed approach, the set of Nash equilibria for the game with fuzzy sets of strategies can be decomposed into a collection of embedded T2FSs with constant secondary grades. These sets are relatively simple to use in practice, in contrast to general T2FSs. The results we have obtained allow us to break down the Nash equilibria T2FS based on secondary grades into finite collections of sets. These collections represent Nash equilibria sets for games corresponding to different cuts of FSs of players’ strategies. Thus, the proposed approach allows us to determine several solutions to the same game depending on the required degree of truth. The properties of the Nash equilibria T2FS are studied.

One possible avenue for future research could involve developing a similar approach for coalition games. Along with other studies of fuzzy games, the authors hope that the proposed approach certainly expands the scope of game theory in social sciences, artificial intelligence, and many other fields.