Fuzzy Strong Nash Equilibria in Generalized Fuzzy Games with Application in Urban Public-Sports Services

Abstract

In this paper, we apply the existence of solutions of the Ky Fan minimax inequality to establish the existence of fuzzy strong Nash equilibria in generalized fuzzy games and strong Nash equilibria in fuzzy coalition generalized games. We then show some applications of our existence results about the strong Nash equilibrium in urban public-sports services.

1. Introduction

The Ky Fan minimax inequality [1] is one of the most important results in mathematical sciences that is equivalent to many important mathematical theorems, such as the classical Brouwer’s fixed point theorem and Kakutani fixed point theorem. It is a useful tool in solving existence problems in many fields such as game theory, mathematical economy, and optimizations. So far, there have been many known variations and generalizations of Ky Fan’s original minimax inequality in [1]. For example, Briec and Yesilce [2] studied Ky Fan inequality in some idempotent and harmonic convex structure, Lin and Tian [3] provided generalizations of the Ky Fan minimax inequality by relaxing the compactness and convexity of sets and the quasi-concavity of the functional, and Qiu [4] proved the existence of pseu-Ky Fan’s points for vector Ky Fan inequalities using the FKKM principle.

The Nash equilibrium is one of the most important concepts in game theory that has been studied extensively in the literature. In 1954, Nash [5] established the famous existence theorem for equilibrium of an n-person non-cooperative game. Since then, there have been many variations and generalizations of Nash’s theorem that have appeared in the literature, including those for discontinuous games by Dasgupta and Maskin [6] and Reny [7], and those for generalized games by Debreu [8], Ding and Yuan [9], Harker [10], Scalzo [11], Wu [12], and Wu and Shen [13]. Baye et al. [14] and Hou [15] provided characterizations for the existence of Nash equilibria. For fuzzy games, Dutta and Gupta [16] investigated the Nash equilibrium strategy of two-person zero-sum games with fuzzy payoffs and established a relation between the Pareto Nash equilibrium strategy and the parametric bi-matrix game, Liu and Yu [17] derived an existence theorem for Nash equilibria in generalized fuzzy games with locally convex Hausdorff topological vector spaces for strategy spaces and/or discontinuous payoff functions, Maeda [18] provided a characterization of equilibrium strategy of two-person zero-sum games with fuzzy payoffs through Nash equilibrium strategies of a family of parametric bi-matrix games with crisp payoffs, Wang and Teo [19] developed an algorithm to solve the fuzzy Nash equilibrium problem, and Praveena and Devi [20] provided a survey on fuzzy-based game-theory approaches for supply-chain uncertainties in E-Commerce applications. The existence of Strong Nash equilibria has also been studied, see Holman and Law-Yone [21], Konishi et al. [22], and Nessah and Tian [23]. However, unlike standard Nash equilibria, strong Nash equilibria rarely exist, even if they are highly relevant for certain decision-making problems. Thus, providing the sufficient conditions for their existence is desirable in both classical and fuzzy game theory. It is our main purpose here to study the existence of strong Nash equilibria, in particular in the fuzzy setting, through the Ky Fan minimax inequality.

In this paper, we apply the existence of solutions of the Ky Fan minimax inequality to establish the existence of fuzzy strong Nash equilibria in generalized fuzzy games and strong Nash equilibria in fuzzy coalition generalized games in Section 3 and Section 4. We then show some applications of our existence results about strong Nash equilibrium in urban public-sports services in Section 5. The main contribution and novelty here is the connection between the solutions of the appropriate Ky Fan minimax inequalities and the fuzzy strong Nash equilibria we provided, which leads to sufficient conditions for the existence of strong Nash equilibria that generalize the existence theorem by Nessah and Tian [23].

2. Necessary and Sufficient Conditions for Solutions of the Ky Fan Minimax Inequality

Let X and C be topological spaces and $\phi (x,z):X\times C⟶R$ be a function. The Ky Fan minimax inequality is the following problem:

$$\left\{\begin{array}{cc}\mathrm{Find} x^{*}\in C\hfill & \mathrm{such} \mathrm{that}\hfill \\ \phi (x,x^{*})\le 0\hfill & \forall x\in X.\hfill \end{array}\right.$$

(1)

The following concepts are given in [14], with $\phi (x,z)=U(x,z)-U(z,z)$.

Definition 1.

Let X be a non-empty subset of a topological vector space. The function $\phi (x,z):X\times X⟶R$ is diagonally transfer-continuous in z if $\phi (x,z)>0$ implies that there is an open neighborhood $O_z$ of z and $x^{\prime }\in X$ such that $\phi (x^{\prime },z^{\prime })>0$ for all $z^{\prime }\in O_z$.

Definition 2.

Let X be a non-empty convex subset of a topological vector space and C be a non-empty convex subset of X. A function $\phi (x,z):X\times C⟶R$ is diagonally transfer quasi-concave in x if, for any $\{x_1,\dots ,x_k\}\subseteq X$, there is $\{z_1,\dots ,z_k\}\subseteq C$, where $x_h\mapsto z_h$, such that, for each $z\in co\{z_1,\dots ,z_k\}$ and $z={\sum }_{j=1}^l{\lambda }_j{z}_{h_j}$ with each ${\lambda }_j>0$ and ${\sum }_{j=1}^l{\lambda }_j=1$, one has $\phi (x,z)\le 0$ for some $x\in \{x_{h_1},\dots ,x_{h_l}\}$.

Let ${\mathsf{\Delta }}_n=\left\{({\lambda }_0,{\lambda }_1,\dots ,{\lambda }_n)\right|{\lambda }_i\ge 0 \mathrm{for} 0\le i\le n \mathrm{and} {\sum }_{i=0}^n{\lambda }_i=1\}$. The next concept from [15] (with $\phi (x,z)=f(x,z)-f(z,z)$) is weaker than diagonal transfer quasi-concavity.

Definition 3.

Let X be a topological space, and $A,Y\subseteq X$. A function $\phi :X\times Y⟶R$ is called $\mathcal{C}$-quasi-concave on A if, for any finite subset $\{x^0,x^1,\dots ,x^n\}\subseteq A$, there is a continuous mapping ${\varphi }_n:{\mathsf{\Delta }}_n⟶Y$ such that

$$min\left\{\phi (x^i,{\varphi }_n({\lambda }_0,{\lambda }_1,\dots ,{\lambda }_n))\right|i\in J\}\le 0$$

for all $({\lambda }_0,{\lambda }_1,\dots ,{\lambda }_n)\in {\mathsf{\Delta }}_n$, where $J=\{j\in \{0,1,\dots ,n\}|{\lambda }_j\ne 0\}$.

Note that function $\phi (x,z)$ is diagonally transfer quasi-concave on X implies that $\phi$ is $\mathcal{C}$-quasi-concave on X: For any $\{x^0,x^1,\dots ,x^n\}\subseteq X$, there is $\{z^0,z^1,\dots ,z^n\}\subseteq C$ such that for each $z\in co\{z^0,z^1,\dots ,z^k\}$, $z={\sum }_{j=0}^n{\lambda }_j{z}^j$, there is $x^i\in \{x^0,\dots ,x^n\}$ satisfying $\phi (x^i,z)\le 0$. Then, one can see that $\phi (x,z)$ is $\mathcal{C}$-quasi-concave by taking ${\varphi }_n({\lambda }_0,{\lambda }_1,\dots ,{\lambda }_n)={\sum }_{j=0}^n{\lambda }_j{z}^j$.

The following characterization for solutions of the Ky Fan minimax inequality follows easily from Lemma 1 in [24].

Theorem 1.

Assume that X is a subset of a Hausdorff topological vector space and C is a non-empty convex subset of X. Then, the Ky Fan minimax inequality (1) has a solution if and only if there is a non-empty convex compact subset D of C such that the restricted mapping ${\phi |}_{X\times D}:X\times D⟶\mathbb{R}$ is diagonally transfer-continuous on D and diagonally transfer quasi-concave on X.

In the following characterization, a finite-dimensional subset means a subset of a finite-dimensional subspace. The proof adopts a similar approach for the proof of Theorem 1 in [15], and we omit the detailed proof.

Theorem 2.

Let X be a Hausdorff topological vector space and C be a non-empty convex subset of X. Then, the Ky Fan minimax inequality (1) has a solution if and only if there is a non-empty convex compact finite-dimensional subset D of C such that the restricted mapping ${\phi |}_{X\times D}:X\times D⟶\mathbb{R}$ is diagonally transfer-continuous on D and $\mathcal{C}$-quasi-concave on X.

3. Necessary and Sufficient Conditions for the Existence of Fuzzy Strong Nash Equilibria in Generalized Fuzzy Games

Throughout this section, we let $N=\{1,2,\dots ,n\}$ be the set of n players and denote by $\mathcal{N}$ the set of all coalitions (subsets of N). For each $i\in N$, $X_i$ is the strategy space for player i and let $X={\prod }_{i\in N}{X}_i$ be the set of strategy profiles. A game G on player set N is ${(X_i,u_i)}_{i\in N}$ with the strategy space $X_i$ and the payoff function $u_i:X⟶\mathbb{R}$ for each $i\in N$.

A generalized game $\mathsf{\Gamma }={(X_i,F_i,u_i)}_{i\in N}$ is a game with player set N such that each player i has strategy space $X_i$, each player i has a payoff function $u_i:X⟶\mathbb{R}$ that depends on his or her own variable $x_i$ as well as on the variables $x_{-i}$ of all other players, and each player i’s strategy must belong to a set $F_i\left(x\right)=F_i\left(x_{-i}\right)\subseteq X_i$ that depends on the rival players’ strategies $x_{-i}$. Clearly, a game is a special generalized game with $F_i\left(x\right)=X_i$ for all $x\in X$ and each $i\in N$. A Nash equilibrium of a generalized game is defined as follows (note that the following concept is stronger than the one given in [10,25] where $x_i\in F_i\left(x^{*}\right)$ there is replaced by $x_i\in F_i\left(X\right)$ below).

Definition 4.

A vector $x^{*}={\left(x_i^{*}\right)}_{i\in N}\in X={\prod }_{i\in N}{X}_i$ is called a Nash equilibrium of a generalized game Γ if $x^{*}\in F\left(x^{*}\right)={\prod }_{i\in N}{F}_i\left(x^{*}\right)$ and

$$u_i(x_i^{*},x_{-i}^{*})\ge u_i(x_i,x_{-i}^{*})\mathrm{for}\mathrm{all}{x}_i\in F_i\left(X\right)\mathrm{and}\mathrm{for}\mathrm{every}i\in N.$$

(2)

The following concept of strong Nash equilibrium for generalized games extends the corresponding concept for games given in [23] (due to Aumann).

Definition 5.

A vector $x^{*}={\left(x_i^{*}\right)}_{i\in N}\in X={\prod }_{i\in N}{X}_i$ is called a strong Nash equilibrium of a generalized game $\mathsf{\Gamma }={(X_i,F_i,u_i)}_{i\in N}$ if $x^{*}\in F\left(x^{*}\right)={\prod }_{i\in N}{F}_i\left(x^{*}\right)$ and for all $S\in \mathcal{N}$ and all $x_S\in F_S\left(X\right)={\prod }_{i\in S}{F}_i\left(X\right)$,

$$u_i(x_S,x_{-S}^{*})\le u_i(x_S^{*},x_{-S}^{*})\mathrm{for}\mathrm{at}\mathrm{least}\mathrm{one}i\in S.$$

(3)

Clearly, a strong Nash equilibrium is a Nash equilibrium in a generalized game.

Let X be a topological space. A fuzzy set in X is a function with domain X and values in $[0,1]$. If A is a fuzzy set and $x\in X$, the function-value $A\left(x\right)$ (or ${\mu }_A\left(x\right)$) is called the grade of membership of x in A. The collection of all fuzzy sets in X is denoted by $\mathcal{F}\left(X\right)$. Let $A\in \mathcal{F}\left(X\right)$ and $\alpha \in [0,1]$. The $\alpha$-level set of A, denoted by $A_{\alpha }$, is defined by

$$A_{\alpha }=\{x\in X|A\left(x\right)\ge \alpha \} \mathrm{if} \alpha \in (0,1],$$

$$A_0=\{x\in X|A\left(x\right)>0\}.$$

A generalized fuzzy game $\mathsf{\Gamma }$ with player set N is a tuple $\mathsf{\Gamma }={(X_i,F_i,P_i,{\alpha }_i,{\beta }_i)}_{i\in N}$, where for each $i\in N$, $X_i$ is the nonempty strategy space of i, $F_i$ and $P_i$ are fuzzy mappings from $X={\prod }_{i\in N}{X}_i$ to the set $\mathcal{F}\left(X_i\right)$ of fuzzy sets in $X_i$. Any strategy profile $x\in X$ is denoted by $(x_i,x_{-i})$, where $x_i\in X_i$ and $x_{-i}\in X_{-i}={\prod }_{j\ne i}{X}_j$. For each $x\in X$, $F_i(·,x)$ is the fuzzy set of strategies feasible to player i and $P_i(·,x)$ is the fuzzy set of strategies in $X_i$ that are preferred to $x_i$ when the other players choose $x_{-i}$. The functions ${\alpha }_i,{\beta }_i:X⟶[0,1]$ provide the degree of feasibility ${\alpha }_i\left(x\right)$ on strategies and the level of satisfaction ${\beta }_i\left(x\right)$ on preferences for player i at the strategy profile $x\in X$, respectively.

Definition 6.

A fuzzy Nash equilibrium for a generalized fuzzy game ${(X_i,F_i,P_i,{\alpha }_i,{\beta }_i)}_{i\in N}$ is a profile $x^{*}={\left(x_i^{*}\right)}_{i\in N}\in X$ such that for every $i\in N$,

$$x_i^{*}\in F_i{(·,x^{*})}_{{\alpha }_i\left(x^{*}\right)}\mathrm{and}{F}_i{(·,x^{*})}_{{\alpha }_i\left(x^{*}\right)}\cap P_i{(·,x^{*})}_{{\beta }_i\left(x^{*}\right)}=\varnothing .$$

(4)

The following concept of fuzzy strong Nash equilibrium for a generalized fuzzy game extends the strong Nash equilibrium given in Definition 5. For each $S\in \mathcal{N}$, let $X={\prod }_{i\in N}{X}_i$ and $X_S={\prod }_{i\in S}{X}_i$ and denote

$${\left(F_S\left(X\right)\right)}_{\alpha \left(X\right)}=\prod _{i\in S}{\left(F_i\left(X\right)\right)}_{{\alpha }_i\left(X\right)}=\prod _{i\in S}\left[\bigcup _{x\in X}{\left(F_i\left(x\right)\right)}_{{\alpha }_i\left(x\right)}\right].$$

Definition 7.

An element $x^{*}={\left(x_i^{*}\right)}_{i\in N}\in X$ is a fuzzy strong Nash equilibrium for a generalized fuzzy game ${(X_i,F_i,P_i,{\alpha }_i,{\beta }_i)}_{i\in N}$ if for every $i\in N$, $x_i^{*}\in {\left(F_i\left(x^{*}\right)\right)}_{{\alpha }_i\left(x^{*}\right)}$ and for all $S\in \mathcal{N}$, there is no $x_S\in {\left(F_S\left(X\right)\right)}_{\alpha \left(X\right)}={\prod }_{i\in S}{\left(F_i\left(X\right)\right)}_{{\alpha }_i\left(X\right)}$ satisfying

$$(x_S,x_{-S}^{*})\in P_i{\left(x^{*}\right)}_{{\beta }_i\left(x^{*}\right)}\mathrm{for}\mathrm{every}i\in S.$$

(5)

Note that a fuzzy game is a special generalized fuzzy game with $F_i\left(x\right)=X_i$ for all $x\in X$ and each $i\in N$.

For a generalized fuzzy game $\mathsf{\Gamma }={(X_i,F_i,u_i,{\alpha }_i,{\beta }_i)}_{i\in N}$ with payoff functions $u_i$ satisfying $|u_i\left(x\right)|\le M_i$ for every $i\in N$, we define the fuzzy preference mappings $P_i$ by

$$P_i\left(z\right)=\left\{\begin{array}{cc}\frac{u_i(x_i,z_{-i})-u_i(z_i,z_{-i})}{2M_i}\hfill & \mathrm{if} u_i(x_i,z_{-i})>u_i(z_i,z_{-i})\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(6)

Then an element $x^{*}={\left(x_i^{*}\right)}_{i\in N}\in X$ is a fuzzy strong Nash equilibrium for a generalized fuzzy game $\mathsf{\Gamma }={(X_i,F_i,u_i,{\alpha }_i,{\beta }_i)}_{i\in N}$ if for every $i\in N$, $x_i^{*}\in {\left(F_i\left(x^{*}\right)\right)}_{{\alpha }_i\left(x^{*}\right)}$ and for all $S\in \mathcal{N}$, there is no $x_S\in {\left(F_S\left(X\right)\right)}_{\alpha \left(X\right)}$ satisfying

$$\frac{u_i(x_S,x_{-S}^{*})-u_i(x_S^{*},x_{-S}^{*})}{2M_i}\ge {\beta }_i\left(x^{*}\right)\mathrm{for}\mathrm{every}i\in S.$$

(7)

Remark 1.

Denote by ${(X_i,F_i,P_i,{\alpha }_i)}_{i\in N}$ (or ${(X_i,F_i,u_i,{\alpha }_i)}_{i\in N}$) a generalized fuzzy game with ${\beta }_i=0$ for every $i\in N$. Then, it is clear from Definition 7 that if $x^{*}\in X$ is a fuzzy strong Nash equilibrium for a generalized fuzzy game ${(X_i,F_i,P_i,{\alpha }_i)}_{i\in N}$ (or ${(X_i,F_i,u_i,{\alpha }_i)}_{i\in N}$), then $x^{*}$ is a fuzzy strong Nash equilibrium for generalized fuzzy game ${(X_i,F_i,P_i,{\alpha }_i,{\beta }_i)}_{i\in N}$ (or ${(X_i,F_i,u_i,{\alpha }_i,{\beta }_i)}_{i\in N}$) for any ${\beta }_i:X⟶[0,1]$.

Let $\mathsf{\Gamma }={(X_i,F_i,P_i,{\alpha }_i,{\beta }_i)}_{i\in N}$ be a generalized fuzzy game. Denote $F={\prod }_{i\in N}{F}_i$ and let

$$C=\{x\in X|x\in {\left(F\left(x\right)\right)}_{\alpha \left(x\right)}\}.$$

Clearly, we have $C\subseteq X$.

Define the function $\mathsf{\Phi }:X\times C⟶\mathbb{R}$ as follows:

$$\mathsf{\Phi }(x,z)=\sum _{S\in \mathcal{N}}\underset{i\in S}{min}{\tilde{P}}_i((x_S,z_{-S}),z),$$

(8)

where ${\tilde{P}}_i((x_S,z_{-S}),z)=1$ if $x_S\in {\left(F_S\left(X\right)\right)}_{\alpha \left(X\right)}$ and $(x_S,z_{-S})\in {\left(P_i\left(z\right)\right)}_{{\beta }_i\left(z\right)}$, and ${\tilde{P}}_i((x_S,z_{-S}),z)=0$ otherwise.

Proposition 1.

An $x^{*}\in C$ is a fuzzy strong Nash equilibrium of the generalized fuzzy game $\mathsf{\Gamma }={(X_i,F_i,P_i,{\alpha }_i,{\beta }_i)}_{i\in N}$ if and only if $\mathsf{\Phi }(x,x^{*})\le 0$ for all $x\in X$.

Proof. 

The necessity follows directly from the definition. For the sufficiency, let $x^{*}\in C$ be such that $\mathsf{\Phi }(x,x^{*})\le 0$ for all $x\in X$. Then, by (5) and (8), we have for all $S\in \mathcal{N}$ and all $x_S\in {\left(F_S\left(X\right)\right)}_{\alpha \left(X\right)}$,

$$(x_S,x_{-S}^{*})\notin {\left(P_i\left(x^{*}\right)\right)}_{{\beta }_i\left(x^{*}\right)} \mathrm{for} \mathrm{at} \mathrm{least} \mathrm{one} i\in S,$$

that is, $x^{*}$ is a fuzzy strong Nash equilibrium of $\mathsf{\Gamma }$. □

Clearly, Theorems 1 and 2, together with Proposition 1, imply the following two characterizations for the existence of fuzzy strong Nash equilibria in generalized fuzzy games.

Theorem 3.

Assume that $\mathsf{\Gamma }={(X_i,F_i,P_i,{\alpha }_i,{\beta }_i)}_{i\in N}$ is a generalized fuzzy game such that for each $i\in N$, $X_i$ is a non-empty convex subset of a Hausdorff topological vector space, and let C be non-empty convex. Then, Γ has a fuzzy strong Nash equilibrium if and only if there is a non-empty convex compact subset D of C such that the restricted mapping ${\mathsf{\Phi }|}_{X\times D}:X\times D⟶\mathbb{R}$ is diagonally transfer-continuous on D and diagonally transfer quasi-concave on X.

Theorem 4.

Assume that $\mathsf{\Gamma }={(X_i,F_i,P_i,{\alpha }_i,{\beta }_i)}_{i\in N}$ is a generalized fuzzy game such that for each $i\in N$, $X_i$ is a non-empty convex subset of a Hausdorff topological vector space, and let C be non-empty convex. Then, Γ has a fuzzy strong Nash equilibrium if and only if there is a non-empty convex compact finite-dimensional subset D of C such that the restricted mapping ${\mathsf{\Phi }|}_{X\times D}:X\times D⟶R$ is diagonally transfer-continuous on D and $\mathcal{C}$-quasi-concave on X.

Next, we establish the existence of fuzzy strong Nash equilibria in generalized games by applying Theorem 3.

Proposition 2.

Let $\mathsf{\Gamma }={(X_i,F_i,u_i,{\alpha }_i,{\beta }_i)}_{i\in N}$ be a generalized fuzzy game with ${\beta }_i=0$ for every $i\in N$. If $u_i$ is continuous for each $i\in N$, then Φ is diagonally transfer-continuous.

Proof. 

Let $\mathsf{\Phi }(x,z)>0$. Then, by the definition, there exist $S\in \mathcal{N}$ and $x_S\in {\left(F_S\left(X\right)\right)}_{\alpha \left(X\right)}$ such that $(x_S,z_{-S})\in {\left(P_i\left(z\right)\right)}_0$, i.e., $u_i(x_S,z_{-S})>u_i\left(z\right)$ or $u_i(x_S,z_{-S})-u_i\left(z\right)>0$ for all $i\in S$. For each $i\in S$, since $u_i$ is continuous, $u_i(x_S,z_{-S})-u_i\left(z\right)$ is continuous and so there is an open neighborhood $O_z^i$ of z such that $u_i(x_S,z_{-S}^{\prime })-u_i\left(z^{\prime }\right)>0$, i.e., $u_i(x_S,z_{-S}^{\prime })>u_i\left(z^{\prime }\right)$ for all $z^{\prime }\in O_z^i$. Take $O_z={\cap }_{i\in S}{O}_z^i$. Then, $O_z$ is an open neighborhood of z such that $u_i(x_S,z_{-S}^{\prime })>u_i\left(z^{\prime }\right)$, i.e., $(x_S,z_{-S}^{\prime })\in P_i\left(z^{\prime }\right)$, for all $z^{\prime }\in O_z$ and every $i\in S$. Since $x_S\in {\left(F_S\left(X\right)\right)}_{\alpha \left(X\right)}$ and $(x_S,z_{-S}^{\prime })\in P_i\left(z^{\prime }\right)$ for all $i\in S$, we have ${\tilde{P}}_S(x,z^{\prime })=1$ for all $z^{\prime }\in O_z$. It follows that $\mathsf{\Phi }(x,z^{\prime })>0$ for all $z^{\prime }\in O_z$. Thus, $\mathsf{\Phi }$ is diagonally transfer continuous. □

Proposition 3.

Let $\mathsf{\Gamma }={(X_i,F_i,u_i,{\alpha }_i,{\beta }_i)}_{i\in N}$ be a generalized fuzzy game with ${\beta }_i=0$ for every $i\in N$ and $C=X$. If $u_i$ is quasi-concave for each $i\in N$, then Φ is diagonally transfer quasi-concave on X.

Proof. 

Assume that $\mathsf{\Gamma }={(X_i,F_i,u_i,{\alpha }_i,{\beta }_i)}_{i\in N}$ is a generalized fuzzy game with ${\beta }_i=0$ for every $i\in N$. Then, by (6), we have

$${\left(P_i\left(z\right)\right)}_0=\{x\in X|u_i(x_i,z_{-i})>u_i(z_i,z_{-i})\}.$$

Let ${\psi }_i^S(x,z)=u_i(x_S,z_{-S})$ for each $S\in \mathcal{N}$. For each $i\in N$ and every $S\in \mathcal{N}$, since $u^i$ is quasi-concave, ${\psi }_i^S(x,z)$ is quasi-concave in x. Let $\{x_1,\dots ,x_k\}$ be any subset of X, take $\{z_1,\dots ,z_k\}=\{x_1,\dots ,x_k\}\subseteq C=X$, and let $z\in co\{x_1,\dots ,x_k\}$. Then, $z={\sum }_{j=1}^l{\lambda }_j{x}_{h_j}$ with each $1\le h_j\le k$, ${\lambda }_j>0$ and ${\sum }_{j=1}^l{\lambda }_j=1$. It follows that for each $i\in N$ and every $S\in \mathcal{N}$,

$${\psi }_i^S(z,z)={\psi }_i^S\left(\sum _{j=1}^l{\lambda }_j{x}_{h_j},z\right)\ge \underset{1\le j\le l}{min}{\psi }_i^S(x_{h_j},z)={\psi }_i^S(x_i^S,z),$$

for some $x_i^S\in \{x_{h_{j_1}},\dots ,x_{h_{j_n}}\}$. Let $x\in \{x_{h_{j_1}},\dots ,x_{h_{j_n}}\}$ be such that ${\psi }_i^S(x,z)\le {\psi }_i^S(x_i^S,z)$ for all $i\in N$ and all $S\in \mathcal{N}$. Then, it follows that for all $S\in \mathcal{N}$ and every $i\in N$,

$${\psi }_i^S(x,z)\le {\psi }_i^S(z,z) \mathrm{or} u_i(x_S,z_{-S})\le u_i\left(z\right).$$

This implies that $\mathsf{\Phi }(x,z)\le 0$. □

Theorem 3 and Propositions 2 and 3 together give the following theorem for the existence of fuzzy strong Nash equilibria of generalized fuzzy games.

Theorem 5.

Assume that $\mathsf{\Gamma }={(X_i,F_i,u_i,{\alpha }_i,{\beta }_i)}_{i\in N}$ is a generalized fuzzy game such that for each $i\in N$, $X_i$ is a non-empty convex compact subset of a Hausdorff topological vector space, $u_i\left(x\right)$ is continuous in x and quasi-concave, and $C=X$. Then, there is a fuzzy strong Nash equilibrium for Γ.

Proof. 

Since $X_i$ is compact for each $i\in N$, it follows from the Tychonoff theorem that $X={\prod }_{i\in N}{X}_i$ is compact. By Remark 1, we only need to consider the case ${\beta }_i=0$ for every $i\in N$, i.e., the special generalized fuzzy game ${(X_i,F_i,u_i,{\alpha }_i)}_{i\in N}$. It follows from Theorem 3 with $C=D$ and Propositions 2 and 3 that there is a fuzzy strong Nash equilibrium for the special generalized fuzzy game ${(X_i,F_i,u_i,{\alpha }_i)}_{i\in N}$. Thus, $\mathsf{\Gamma }$ has a fuzzy strong Nash equilibrium by Remark 1. □

Recall that a game $\mathsf{\Gamma }={(X_i,u_i)}_{i\in N}$ is a special generalized game $\mathsf{\Gamma }={(X_i,F_i,u_i)}_{i\in N}$ with $F_i\left(x\right)=X_i$ for all $x\in X$ and each $i\in N$. By taking $F_i$ to be a crisp mapping (i.e., $F_i(·,x)$ takes values 0 or 1) with $F_i\left(x\right)=X_i$ for all $x\in X$ and ${\alpha }_i={\beta }_i=0$ for each $i\in N$, Theorem 5 provides the following existence theorem of strong Nash equilibrium, which generalizes Theorem 3.1 in [23].

Theorem 6.

Assume that $\mathsf{\Gamma }={(X_i,u_i)}_{i\in N}$ is a game such that for each $i\in N$, $X_i$ is a non-empty convex compact subset of a Hausdorff topological vector space and $u_i\left(x\right)$ is continuous in x and quasi-concave. Then, there is a strong Nash equilibrium for Γ.

4. Existence of Strong Nash Equilibria in Fuzzy Coalition Generalized Games

Recall from Aubin [26] that a $fuzzycoalition$ is a vector $s\in {[0,1]}^n$, namely, $s=(s_1,s_2,\dots ,s_n)$ with $0\le s_i\le 1$ for each $1\le i\le n$ (where $s_i$ is the participation level of agent i). A crisp coalition $S\subset N$ corresponds to a special fuzzy coalition $s=(s_1,s_2,\dots ,s_n)$ with $s_i=1$ if $i\in S$ and $s_i=0$ if $i\notin S$. Denote $e^N=(1,1,\dots ,1)$. We will use ${\mathcal{F}}^N$ for the set of all non-zero fuzzy coalitions on player set N, that is,

$${\mathcal{F}}^N=\{s=(s_1,s_2,\dots ,s_n)\ne 0|0\le s_i\le 1 \mathrm{for} \mathrm{each} i\le n\}.$$

The following concept of fuzzy coalition generalized games is a natural extension of the corresponding concept of generalized games given in [27], with payoff functions $u_i$ instead of preferences $P^i$.

A fuzzy coalition generalized game $\mathsf{\Gamma }=({(X_i,u_i)}_{i\in N},{\left(F^s\right)}_{s\in {\mathcal{F}}^N})$ is a game with a player set N such that each player i has strategy space $X_i$; each player i has a payoff function $u_i:X={\prod }_{i\in N}{X}_i⟶\mathbb{R}$ that depends on his or her own variable $x_i$ as well as on the variables $x_{-i}$ of all other players, and for each fuzzy coalition $s\in {\mathcal{F}}^N$, $F^s\left(X\right)\subseteq X_S={\prod }_{i\in S}{X}_i$ represents the set of feasible strategies corresponding to the fuzzy coalition s, where $S=car\left(s\right)$.

The following concept extends the strong Nash equilibrium given in Definition 5, where $F^i=F^s$ with $s_i=1$ and $s_j=0$ for $j\ne i$.

Definition 8.

A vector $x^{*}={\left(x_i^{*}\right)}_{i\in N}\in X={\prod }_{i\in N}{X}_i$ is called a strong Nash equilibrium of a fuzzy coalition generalized game $\mathsf{\Gamma }=({(X_i,u_i)}_{i\in N},{\left(F^s\right)}_{s\in {\mathcal{F}}^N})$ if $x^{*}\in F\left(x^{*}\right)={\prod }_{i\in N}{F}^i\left(x^{*}\right)$ and for all $s\in {\mathcal{F}}^N$ and all $x_S\in F^s\left(X\right)$ with $S=car\left(s\right)$,

$$u_i(x_S,x_{-S}^{*})\le u_i(x_S^{*},x_{-S}^{*})\mathrm{for}\mathrm{at}\mathrm{least}\mathrm{one}i\in S.$$

(9)

Clearly, any strong Nash equilibrium of a fuzzy coalition generalized game is a Nash equilibrium.

Let $\mathsf{\Gamma }=({(X_i,u_i)}_{i\in N},{\left(F^s\right)}_{s\in {\mathcal{F}}^N})$ be a fuzzy coalition generalized game. Denote

$$C=\{x\in X|x\in F(x)=\prod _{i\in N}{F}^i(x)\}.$$

For each $S\in \mathcal{N}$, denote

$$F_S^{*}={\cup }_{s\in {\mathcal{F}}^N:car\left(s\right)=S}{F}^s\left(X\right).$$

Clearly, $F_S^{*}\subseteq X_S={\prod }_{i\in S}{X}_i$.

Let

$$P_i\left(z\right)=\{x\in X|u_i\left(x\right)>u_i\left(z\right)\}.$$

Define the function $\mathsf{\Psi }:X\times C⟶\mathbb{R}$ as follows:

$$\mathsf{\Psi }(x,z)=\sum _{S\in \mathcal{N}}\underset{i\in S}{min}{\tilde{P}}_i((x_S,z_{-S}),z),$$

(10)

where ${\tilde{P}}_i((x_S,z_{-S}),z)=1$ if $x_S\in F_S^{*}$ and $(x_S,z_{-S})\in P_i\left(z\right)$, and ${\tilde{P}}_i((x_S,z_{-S}),z)=0$ otherwise.

Proposition 4.

An $x^{*}\in C$ is a strong Nash equilibrium of the fuzzy coalition generalized game $\mathsf{\Gamma }=({(X_i,u_i)}_{i\in N},{\left(F^s\right)}_{s\in {\mathcal{F}}^N})$ if and only if $\mathsf{\Psi }(x,x^{*})\le 0$ for all $x\in X$.

Proof. 

The necessity follows directly from the definition. For the sufficiency, let $x^{*}\in C$ be such that $\mathsf{\Psi }(x,x^{*})\le 0$ for all $x\in X$. By (10), we have for all $S\in \mathcal{N}$ and all $x_S\in F_S^{*}\subseteq X_S$,

$$(x_S,x_{-S}^{*})\notin P_i\left(x^{*}\right) \mathrm{for} \mathrm{at} \mathrm{least} \mathrm{one} i\in S,$$

that is, $x^{*}$ is a strong Nash equilibrium of $\mathsf{\Gamma }$ by (9). □

The following two characterizations follow from Theorems 1 and 2 and Proposition 4.

Theorem 7.

Assume that $\mathsf{\Gamma }=({(X_i,u_i)}_{i\in N},{\left(F^s\right)}_{s\in {\mathcal{F}}^N})$ is a fuzzy coalition generalized game such that for each $i\in N$, $X_i$ is a non-empty convex subset of a Hausdorff topological vector space, and let C be non-empty convex. Then, Γ has a strong Nash equilibrium if and only if there is a non-empty convex compact subset D of C such that the restricted mapping ${\mathsf{\Psi }|}_{X\times D}:X\times D⟶\mathbb{R}$ is diagonally transfer-continuous on D and diagonally transfer quasi-concave on X.

Theorem 8.

Assume that $\mathsf{\Gamma }=({(X_i,u_i)}_{i\in N},{\left(F^s\right)}_{s\in {\mathcal{F}}^N})$ is a fuzzy coalition generalized game such that for each $i\in I$, $X_i$ is a non-empty convex subset of a Hausdorff topological vector space, and let C be non-empty convex. Then, Γ has a strong Nash equilibrium if and only if there is a non-empty convex compact finite-dimensional subset D of C such that the restricted mapping ${\mathsf{\Psi }|}_{X\times D}:X\times D⟶R$ is diagonally transfer-continuous on D and $\mathcal{C}$-quasi-concave on X.

Similar to Propositions 2 and 3, we have the next two facts.

Proposition 5.

Let $\mathsf{\Gamma }=({(X_i,u_i)}_{i\in N},{\left(F^s\right)}_{s\in {\mathcal{F}}^N})$ be a fuzzy coalition generalized game. If $u_i$ is continuous for each $i\in N$, then Ψ is diagonally transfer-continuous.

Proposition 6.

Let $\mathsf{\Gamma }=({(X_i,u_i)}_{i\in N},{\left(F^s\right)}_{s\in {\mathcal{F}}^N})$ be a fuzzy coalition generalized game and $C=X$. If $u_i$ is quasi-concave for each $i\in N$, then Ψ is diagonally transfer quasi-concave on X.

By Theorem 7 and Propositions 5 and 6, one has the following existence theorem for the strong Nash equilibrium of a fuzzy coalition generalized game.

Theorem 9.

Assume that $\mathsf{\Gamma }=({(X_i,u_i)}_{i\in N},{\left(F^s\right)}_{s\in {\mathcal{F}}^N})$ is a fuzzy coalition generalized game such that for each $i\in N$, $X_i$ is a non-empty convex compact subset of a Hausdorff topological vector space, $u_i\left(x\right)$ is continuous in x and quasi-concave, and $C=X$. Then, Γ has a strong Nash equilibrium.

5. Applications

By viewing a sport public service or sport facility as a resource, we present the following example through congestion games to show applications in sport public services. Congestion games have been used to model situations such as rush-hour traffic and the demand for factors of production, see [21] for more on congestion games. Additionally, please see [28] for optimal resource allocations in cloud manufacturing platforms.

Example 1.

A congestion game is described as follows: $N=\{1,2,...,n\}$ is the set of players, and $M=\{1,2,...,m\}$ is the set of facilities. Each player j has a non-empty set of strategies $X_i$. Every strategy $A_i\subseteq X_i$ is a subset of M. With every facility a and every integer $1\le k\le n$, a real number $u_a\left(k\right)$ is associated, which can be interpreted as the utility to each user of a if the total number of users of a is k. Let $X={\prod }_{i=1}^n{X}_i$ and let $A=(A_1,\dots ,A_n)\in X$. The m-dimensional congestion vector corresponding to A is $\sigma \left(A\right)={\left({\sigma }_a\left(A\right)\right)}_{a\in M}$, where

$${\sigma }_a\left(A\right)=\left|\{i\in N|a\in A_i\}\right|.$$

The payoff function of player i is defined by

$${\pi }_i\left(A\right)=\sum _{a\in A_i}{u}_a\left({\sigma }_a\left(A\right)\right).$$

When each player tries to make strategy choices to maximize payoffs, a strong Nash equilibrium is a very good solution. One can replace $A_i$ by a mixed action $x_i=(x_{i,1},x_{i,2},\dots ,x_{i,m})$, with $0\le x_{i,j}\le 1$ being the participation level of player i in facility j for $1\le j\le m$. The set of all mixed actions is denoted by $\mathsf{\Delta }\left(X\right)$. Clearly, $\mathsf{\Delta }\left(X\right)$ is convex and compact. Define

$${\sigma }_j\left(x\right)=\sum _{i=1}^n{x}_{i,j} for j\in M.$$

For each $x\in \mathsf{\Delta }\left(X\right)$, define the payoff function of player i by

$${\pi }_i\left(x\right)=\sum _{j=1}^m{u}_j\left({\sigma }_j\left(x\right)\right).$$

By Theorem 6, when payoff functions ${\pi }_i\left(x\right)$ is continuous and quasi-concave, there is a strong Nash equilibrium for the game.

6. Concluding Remarks

In this paper, we apply the existence of solutions of the Ky Fan minimax inequality to establish the existence of fuzzy strong Nash equilibria in generalized fuzzy games and strong Nash equilibria in fuzzy coalition generalized games in Section 3 and Section 4, and we provide an application of our existence results about strong Nash equilibrium in urban public-sports services in Section 5.

As strong Nash equilibria rarely exist, providing sufficient conditions for their existence is a relevant contribution to game theory. As a consequence of our main result (Theorem 5), Theorem 6 improves the existence theorem by Nessah and Tian [23] through a different approach—the Ky Fan minimax inequality.

It would be desirable to obtain more sufficient conditions for the existence of strong Nash equilibria and fuzzy strong Nash equilibria and explore more applications involving strong Nash equilibria in future studies.