Optimal Bayesian equilibrium for n -person credibilistic non-cooperative game with risk aversion

Abstract

In this paper, a n-person credibilistic non-cooperative game with risk aversion is investigated in the fuzzy environment. Firstly, optimistic value criterion is applied to modeling players’ risk aversion. We define a α-optimistic Nash equilibrium for this games. The existence of α-optimistic Nash equilibrium strategies is proposed. Then a Bayesian α-optimistic Nash equilibrium is given by assuming α to be not common knowledge and its existence theorem is also proved. Moreover, we present a sufficient and necessary condition to find the Bayesian α-optimistic Nash equilibrium. Finally, an example is given to illustrate the usefulness of the proposed game.

Keywords

risk aversion fuzzy payoff Bayesian α-optimistic Nash equilibrium credibility measure

1 Introduction

As a collection of mathematical model, game theory is often applied to analyzing interactive decision problem among rational players. The early research on game theory can go back to Borel [5, 6] and von Neumann [43]. Since established by von Neumann and Morgenstern [44], game theory has been received more and more attention [1 , 41]. And game theory has been widely applied in many areas, such as auctions, artificial intelligence, research and development races, e-commerce, biology and so on.

It is worthwhile noting that classical non- cooperative game theory is established by assuming players to be full rationality. However, for real game problems, player’s decision-making behavior is always subjective to some degree. That is, a player often shows bounded rational behavior characteristics in the actual game process. For example, in many situations, a player would rather seek sure gains than seek maximum of gains, that is, the player is risk averse in strategies involving sure gains. Risk aversion has been paid little attention in strategic (or normal form) games. Sabater-Grande and Georgantzis [40] studied repeated prisoners’ dilemma game with risk aversion by using experimental methods. Goeree et al. [19] considered 2×2 games with risk averse behavior by using experimental methods. Berden and Peters [2] investigated the effect of risk aversion in bimatrix game. Engelmann and Steiner [15] presented the sufficient condition that the expected payoffs of players increase or decrease with the degree of risk aversion in 2×2 games.

However, uncertainty plays a central role in many game situations. As pointed out by Harsanyi [21], each player often lacks of the information about her/his opponents’ or even her/his own payoffs. Generally speaking, this uncertainty of payoffs includes random payoffs and fuzzy payoffs [22, 45]. The Bayesian game model proposed by Harsanyi [21] is often used to deal with the games with random payoffs. Blau [4], Cassidy et al. [8] and Charnes et al. [11] investigated two-person zero-sum games with random payoffs, and Berg [3], Ein-Dor and Kanter [13] and Roberts [39] considered bimatrix games with random payoffs. Campos [7], Maeda [33] studied two-person zero-sum matrix games with fuzzy payoffs, respectively. Maeda [32] discussed bimatrix games with fuzzy payoffs. Multi-objective matrix games with fuzzy payoffs are studied by Nishizaki and Sakawa [38]. Vijay et al. [42] and Cevikel and Ahlatçioğlu [9] studied two-person zero-sum matrix games based on fuzzy goals and fuzzy payoffs, respectively. Li [25, 26] provided a solution to solve matrix games with payoffs of interval values and a solution to solve matrix games with payoffs of triangular fuzzy number, respectively. Nan et al. [34] provided a solution to solve matrix games with payoffs of triangular intuitionistic fuzzy numbers, and Chandra and Aggarwal [10] provided a solution to solve matrix games with payoffs of triangular fuzzy numbers. The existing literatures on games with fuzzy payoffs ignore bounded rational behavior characteristics of players. As one of behavior characteristics of bounded rationality, risk aversion has received relatively little attention in non-cooperative games with fuzzy payoffs, with the exception of credibilistic games proposed by Gao [16] and developed by Gao et al. [17] and Gao and Yang [18].

In this paper, we investigate n-person credibilistic non-cooperative game with risk aversion. Specifically, the optimistic value criterion is applied to dealing with the situation where players are risk averse and player i wants to optimize the α_i-optimistic values of her/his expected payoff, where α_i is a confidence level of player i. A solution concept of Bayesian α_i-optimistic Nash equilibrium is presented by assuming α_i to be not common knowledge and its existence theorem is proved. Finally, we give a sufficient and necessary condition for Bayesian α_i-optimistic Nash equilibrium.

This paper is organized as follows. In Section 2, we briefly review some concepts of n-person non-cooperative game and credibility theory. In Section 3, we give the n-person credibilistic non-cooperative game with α_i-optimistic Nash equilibrium. In Section 4, we present the concept of Bayesian α_i-optimistic Nash equilibrium when confidence levels are not common knowledge. Both its existence theorem and a sufficient and necessary condition are given. In Section 5, an example is given to illustrate the usefulness of the theory presented in this paper.

2 Preliminaries

2.1 Classical n-person non-cooperative games

Consider a game G = 〈N, (S_i) _i∈N, (v_i) _i∈N〉 with a finite player set N = {1, 2, …, n} and, player i ∈ N, with a finite pure strategy set S_i including m_i pure strategies and a payoff function v_i (s_i, s_-i) depending on the pure strategy combination (s_i, s_-i) played, where s_-i : = (s₁, …, s_i-1, s_i+1, …, s_n) indicates a pure strategy combination for all players except player i. Let S : = S₁ × S₂ × … × S_n be the set of pure strategy combinations for all players, and S_-i = S₁ × … × S_i-1 × S_i+1 × … × S_n be the set of pure strategy combinations which players other than i could choose. The set of probability measures over S_i is denoted by Λ_i. An element σ_i ∈ Λ_i is defined as a mixed strategy for player i ∈ N, where σ_i is a function σ_i : S_i → [0, 1]. Therefore, if player i ∈ N chooses σ_i, then she/he chooses pure strategy s_i with the probability σ_i (s_i). A mixed strategy combination is denoted by (σ_i, σ_-i), where σ_-i indicates a mixed strategy combination for all players except player i. Let Λ : = Λ₁ × Λ₂ × … × Λ_n be the set of mixed strategy combinations for all players and Λ_-i = Λ₁ × ⋯ × Λ_i-1 × Λ_i+1 × ⋯ × Λ_n be the set of mixed strategy combinations which players other than i could choose. A mixed strategy game is denoted by G₁ =〈 N, (Λ_i) _i∈N, (v_i) _i∈N 〉. The expected payoff u_i that player i obtains for a mixed strategy combination (σ_i, σ_-i) is given by

$\begin{matrix} u_{i} (σ_{i}, σ_{- i}) \\ = \sum_{s_{i} \in S_{i}, s_{- i} \in S_{- i}} σ_{1} (s_{1}) σ_{2} (s_{2}) \dots σ_{n} (s_{n}) v_{i} (s_{i}, s_{- i}) \end{matrix}$ (1)

Definition 2.1. A mixed strategy combination $σ^{*} = (σ_{i}^{*}, σ_{- i}^{*})$ is called a Nash equilibrium in game G₁, if it satisfies $u_{i} (σ^{*}) \geq u_{i} (σ_{i}, σ_{- i}^{*}), \forall i \in N, σ_{i} \in Λ_{i},$ (2) where $σ_{- i}^{*} = (σ_{1}^{*}, \dots, σ_{i - 1}^{*}, σ_{i + 1}^{*}, \dots, σ_{n}^{*}) \in Λ_{- i}$ .

Lemma 2.1. [35] There exists at least one mixed strategy Nash equilibrium in every finite n-person non-cooperative game.

2.2 Credibility theory

As a branch of mathematics, credibility theory applies to dealing with the behavior of fuzzy phenomena. Credibility theory has received extensively in many fields, such as portfolio selection [12, 31] and transportation planning [27 , 46].

Definition 2.2. [29] Let Θ be a nonempty set and Ξ (Θ) be the power set of Θ, a set function Cr {·} is called a credibility measure if it satisfies the following four axioms.

Axiom 1 (Normality) CrΘ = 1.

Axiom 2 (Monotonicity) Cr {A} ≤ Cr {B}, where A ⊂ B.

Axiom 3 (Self-Duality) Cr {A} + Cr {A^c} =1 for any A ∈ Ξ (Θ).

Axiom 4 (Maximality) $Cr {\cup_{i} A_{i}} = sup_{i} Cr {A_{i}}$ for any {A_i} with $sup_{i} Cr {A_{i}} < 0.5$ .

Definition 2.3. [29] Let Θ be a nonempty set, Ξ (Θ) be the power set of Θ and Cr be a credibility measure, then the triplet (Θ, Ξ (Θ) , Cr) is called a credibility space.

Definition 2.4. [29] A fuzzy variable $\tilde{v}$ is a measurable function from a credibility space (Θ, Ξ (Θ) , Cr) to the set of real numbers.

Lemma 2.2. [29] Let $\tilde{v}$ be a fuzzy variable with membership function μ, then for any set B of real numbers, we have $Cr {\tilde{v} \in B} = \frac{1}{2} (sup_{x \in B} μ (x) + 1 - sup_{x \notin B^{c}} μ (x))$ (3)

As a special fuzzy variable, a triangular fuzzy variable is denoted by $\tilde{v} = (v^{L}, v^{M}, v^{U})$ [14, 29], where v^M is the mean of $\tilde{v}$ , v^L and v^U are the low bound and the bound of $\tilde{v}$ , respectively. Its membership function is given by $μ (x) = {\begin{matrix} (x - v^{L}) / (v^{M} - v^{L}) & if v^{L} \leq x < v^{M} \\ 1 & if x = v^{M} \\ (v^{U} - x) / (v^{U} - v^{M}) & if v^{M} < x \leq v^{U} \\ 0 & else \end{matrix}$ (4)

For the triangular fuzzy number $\tilde{v} = (v^{L}, v^{M}, v^{U})$ , if v^L ≥ 0 and v^U ≥ 0, we call $\tilde{v}$ non-negative triangular fuzzy number. Let $\tilde{v} = (v^{L}, v^{M}, v^{U})$ and $\tilde{η} = (η^{L}, η^{M}, η^{U})$ be two triangular fuzzy numbers, then, according to extension principle proposed by Zadeh [47] and Negoita et al. [37], we have the following operation rules: $\tilde{v} + \tilde{η} = (v^{L} + η^{L}, v^{M} + η^{M}, v^{U} + η^{U})$ and $λ \tilde{v} = {\begin{matrix} (λ v^{L}, λ v^{M}, λ v^{U}) & if λ \geq 0 \\ (λ v^{U}, λ v^{M}, λ v^{L}) & if λ < 0 \end{matrix} .$

Definition 2.5. [29] The fuzzy variables ${\tilde{v}}_{1}, \dots, {\tilde{v}}_{m}$ are said to be independent if $Cr {{\cap}_{i = 1}^{m} ({\tilde{v}}_{i} \in B_{i})} = min_{1 \leq i \leq m} Cr {{\tilde{v}}_{i} \in B_{i}}$ (5) for any sets B₁, B₂, …, B_m of R.

Definition 2.6. [30] Let $\tilde{v}$ be a fuzzy variable and α ∈ (0, 1] be a confidence level, then for a real number r, ${\tilde{v}}_{sup} (α) = sup {r | Cr {\tilde{v} \geq r} \geq α}$ (6) is called the α-optimistic value to $\tilde{v}$ .

This means that the fuzzy variable $\tilde{v}$ will reach upwards of the α-optimistic value ${\tilde{v}}_{sup} (α)$ with credibility α. In other words, the α-optimistic value ${\tilde{v}}_{sup} (α)$ is the supremum value that $\tilde{v}$ achieves with credibility α.

Optimistic value criterion. [28] Let $\tilde{v}$ and $\tilde{η}$ be independent fuzzy variables and α ∈ (0, 1] be a confidence level, we say $\tilde{v} < \tilde{η}$ if and only if ${\tilde{v}}_{sup} (α) < {\tilde{η}}_{sup} (α)$ .

Under fuzzy decision-making environment, optimistic value criterion is applied to dealing with the situation where a player is risk averse and want to maximize the α-optimistic value of her/his fuzzy payoffs. Each confidence level corresponds one behavior type of players that reflects the risk aversion level the players can endure [18].

Lemma 2.3. [30] Let $\tilde{v}$ and $\tilde{η}$ be independent fuzzy variables and α ∈ (0, 1] be a confidence level, then for any nonnegative numbers a and b, we have $(a \tilde{v} + b \tilde{η})_{sup} (α) = a {\tilde{v}}_{sup} (α) + b {\tilde{η}}_{sup} (α)$ (7)

3 α-optimistic Nash equilibrium for n-person credibilistic game

Since decision environment is often characterized by a great many possible strategies, intricate relations between strategic choices and their influences to players’ payoffs, it is impossible that a player makes accurate or probabilistic estimation of her/his own payoffs. In such situations, we consider a n-person game with fuzzy payoffs. Specifically, for each player i ∈ N, the payoff ${\tilde{v}}_{i} (s_{i}, s_{- i})$ is modeled by a fuzzy variable for all (s_i, s_-i) ∈ S, then for any mixed strategy combination (σ_i, σ_-i), expected payoff of player i ∈ N is also fuzzy variable and given by $\begin{matrix} {\tilde{u}}_{i} (σ_{i}, σ_{- i}) \\ = \sum_{s_{i} \in S_{i}, s_{- i} \in S_{- i}} σ_{1} (s_{1}) σ_{2} (s_{2}) \dots σ_{n} (s_{n}) {\tilde{v}}_{i} (s_{i}, s_{- i}) . \end{matrix}$

Here, we consider the game with fuzzy payoffs where players are risk averse. Optimistic value criterion is used to model the situation where players are risk averse and player i ∈ N wants to optimize the optimistic value of her/his expected payoff ${\tilde{u}}_{i} (σ_{i}, σ_{- i})$ at given confidence level α_i. Then the best responses of player i ∈ N to her/his opponents’ strategy combination $σ_{- i}^{*} \in Λ_{- i}$ are the optimal solutions of the fuzzy chance-constrained programming model: $max_{σ_{i} \in Λ_{i}} \max_{{u^{‵}}_{i}} Cr {{\tilde{u}}_{i} (σ_{i}, σ_{- i}^{*}) \geq u_{i}^{'}} \geq α_{i}$ (8) where $u_{i}^{'}$ is a real number and $\begin{matrix} {\tilde{u}}_{i} (σ_{i}, σ_{- i}^{*}) \\ = \sum_{s_{i} \in S_{i}, s_{- i} \in S_{- i}} σ_{1}^{*} \dots σ_{i - 1}^{*} σ_{i} σ_{i + 1}^{*} \dots σ_{n}^{*} {\tilde{v}}_{i} (s_{i}, s_{- i}) . \end{matrix}$

Then a n-person credibilistic game is denoted by $G_{2} = 〈 N, (Λ_{i})_{i \in N}, ({\tilde{v}}_{i})_{i \in N}, (α_{i})_{i \in N} 〉$ , where α_i is a confidence level of player i.

Definition 3.1. A mixed strategy combination $σ^{*} = (σ_{i}^{*}, σ_{- i}^{*})$ is called a α_i-optimistic Nash equilibrium in game G₂, if for ∀i ∈ N, σ_i ∈ Λ_i, it satisfies $\begin{array}{l} u_{i}^{' *} = \max {u_{i}^{'} | C r {{\tilde{u}}_{i} (σ_{i}^{*}, σ_{- i}^{*}) \geq u_{i}^{'}} \geq α_{i}} \\ \geq \max {u_{i}^{'} | C r {{\tilde{u}}_{i} (σ_{i}, σ_{- i}^{*}) \geq u_{i}^{'}} \geq α_{i}}, \end{array}$ where $σ_{- i}^{*} = (σ_{1}^{*}, \dots, σ_{i - 1}^{*}, σ_{i + 1}^{*}, \dots, σ_{n}^{*}) \in Λ_{- i}$ and $u_{i}^{' *}$ is the optimistic value of player i’s expected payoff ${\tilde{u}}_{i} (σ_{i}^{*}, σ_{- i}^{*})$ at confidence level α_i ∈ (0, 1].

Theorem 3.1. Let payoffs ${\tilde{v}}_{1} (s_{1}, s_{- 1}), {\tilde{v}}_{2} (s_{2}, s_{- 2}), \dots, {\tilde{v}}_{n} (s_{n}, s_{- n})$ be independent fuzzy variables in game G₂, then there exists at least one α_i-optimistic Nash equilibrium.

Proof. For any mixed strategy combination (σ_i, σ_-i) ∈ Λ in game G₂, it follows from Definition 2.6 that

$\begin{matrix} max {u_{i}^{'} | Cr {{{\tilde{u}}_{i} (σ_{i}, σ_{-i}) \geq u_{i}^{'}} \geq α_{i}} \\ = ({\tilde{u}}_{i} (σ_{i}, σ_{- i}))_{sup} (α_{i}) \end{matrix}$ (9)

Since ${\tilde{u}}_{i} (σ_{i}, σ_{- i}) = \sum_{s_{i} \in S_{i}, s_{- i} \in S_{- i}} σ_{1} (s_{1}) σ_{2} (s_{2}) \dots σ_{n} (s_{n}) {\tilde{v}}_{i} (s_{i}, s_{- i})$ , we have that $\begin{matrix} ({\tilde{u}}_{i} (σ_{i}, σ_{- i}))_{sup} (α_{i}) \\ = {(\sum_{s_{i} \in S_{i}, s_{- i} \in S_{- i}} σ_{1} (s_{1}) σ_{2} (s_{2}) \dots σ_{n} (s_{n}) {\tilde{v}}_{i} (s_{i}, s_{- i}))}_{sup} (α_{i}) . \end{matrix}$

It follows from Lemma 2.3 that $\begin{matrix} ({\tilde{u}}_{i} (σ_{i}, σ_{- i}))_{sup} (α_{i}) \\ = {(\sum_{s_{i} \in S_{i}, s_{- i} \in S_{- i}} σ_{1} (s_{1}) σ_{2} (s_{2}) \dots σ_{n} (s_{n}) {\tilde{v}}_{i} (s_{i}, s_{- i}))}_{sup} (α_{i}) \\ = \sum_{s_{i} \in S_{i}, s_{- i} \in S_{- i}} σ_{1} (s_{1}) σ_{2} (s_{2}) \dots σ_{n} (s_{n}) ({\tilde{v}}_{i} (s_{i}, s_{- i}))_{sup} (α_{i}) \end{matrix}$ (10)

It follows from Equations (9) and (10) that $\begin{matrix} max {u_{i}^{'} | Cr {{{\tilde{u}}_{i} (σ_{i}, σ_{- i}) \geq u_{i}^{'}} \geq α_{i}} \\ = ({\tilde{u}}_{i} (σ_{i}, σ_{- i}))_{sup} (α_{i}) \\ = \sum_{s_{i} \in S_{i}, s_{- i} \in S_{- i}} σ_{1} (s_{1}) σ_{2} (s_{2}) \dots σ_{n} (s_{n}) ({\tilde{v}}_{i} (s_{i}, s_{- i}))_{sup} (α_{i}) \end{matrix}$ (11)

A game is denoted by $G_{2}^{'} = 〈 N, (Λ_{i})_{i \in N}, (({\tilde{v}}_{i} (s_{i}, s_{- i}))_{sup} (α_{i}))_{i \in N} 〉 .$

It is obvious that the $G_{2}^{'}$ is a classical n-person non-cooperative game. It follows from Definition 3.1 and Equation (11) that the existence of α_i-optimistic Nash equilibrium in game G₂ is equivalent to the existence of Nash equilibrium in game $G_{2}^{'}$ . It follows from Lemma 2.1 that there exists at least one mixed strategy Nash equilibrium in the game $G_{2}^{'}$ . Thus, there exists at least one α_i-optimistic Nash equilibrium in game G₂.

These complete the proof of Theorem 3.1. □

4 Bayesian α-optimistic Nash equilibrium for n-person credibilistic game

In a real game, the confidence level α_i for player i ∈ N is only known by herself/himself, that is, α_i is private information of player i. In such situations, the Bayesian game model is often used to model private information as a player’s type, i.e., each player is considered as Bayesian player and modeled by a random variable with a certain probability distribution on her/his behavior types. Then a n-person credibilistic game with private information is denoted by $G_{3} = 〈 N, (Λ_{i})_{i \in N}, ({\tilde{v}}_{i})_{i \in N}, (Φ_{i})_{i \in N} 〉$ , where Φ_i is a distribution function of the confidence level α_i over risk aversion of player i ∈ N.

Since $({\tilde{u}}_{i} (σ_{i}, σ_{- i}))_{sup} (α_{i})$ is optimistic value of player i’s expected payoff ${\tilde{u}}_{i} (σ_{i}, σ_{- i})$ at the confidence level α_i, where α_i is a random variable with a certain probability distribution. In game G₃, the best responses of player i ∈ N given other players’ strategies $σ_{- i}^{*}$ are the optimal solutions of the following fuzzy programming model $max_{σ_{i} \in Λ_{i}} \int_{Δ_{i}} {(({\tilde{u}}_{i} (σ_{i}, σ_{- i}^{*}))}_{sup} (t_{i})) d (Φ_{i} (t_{i}))$ (12) where Δ_i represents space of the confidence level α_i of player i.

Since ${\tilde{v}}_{1} (s_{1}, s_{- 1}), {\tilde{v}}_{2} (s_{2}, s_{- 2}), \dots, {\tilde{v}}_{n} (s_{n}, s_{- n})$ are in-dependent fuzzy variables, it follows from Lemma 2.3 that, for any player i,

$\begin{matrix} \int_{Δ_{i}} {(({\tilde{u}}_{i} (σ_{i}, σ_{- i}^{*}))}_{sup} (t_{i})) d (Φ_{i} (t_{i})) \\ = \int_{Δ_{i}} {(\sum_{s_{i} \in S_{i}, s_{- i} \in S_{- i}} σ_{1}^{*} \dots σ_{i - 1}^{*} σ_{i} σ_{i + 1}^{*} \dots σ_{n}^{*} {\tilde{v}}_{i} (s_{i}, s_{- i}))}_{sup} (t_{i})) d (Φ_{i} (t_{i}))) \\ = \sum_{s_{i} \in S_{i}, s_{- i} \in S_{- i}} σ_{1}^{*} \dots σ_{i - 1}^{*} σ_{i} σ_{i + 1}^{*} \dots σ_{n}^{*} (\int_{Δ_{i}} (({\tilde{v}}_{i} (s_{i}, s_{- i}))_{sup} (t_{i})) d (Φ_{i} (t_{i}))) . \end{matrix}$ (13)

For simplicity, let ${\tilde{v}}_{i}^{Δ_{i}} (s_{i}, s_{- i}) = \int_{Δ_{i}} ({({\tilde{v}}_{i} (s_{i}, s_{- i}))}_{sup} (t_{i})) d (Φ_{i} (t_{i})) .$ (14)

Definition 4.1. A mixed strategy combination $σ^{*} = (σ_{i}^{*}, σ_{- i}^{*})$ is called a Bayesian α_i-optimistic Nash equilibrium in game G₃, if for ∀i ∈ N, σ_i ∈ Λ_i, it satisfies

$\begin{array}{l} u_{i}^{'' *} = \sum_{s_{i} \in S_{i}, s_{- i} \in S_{- i}} σ_{1}^{*} σ_{2}^{*} \dots σ_{n}^{*} {\tilde{v}}_{i}^{Δ_{i}} (s_{i}, s_{- i}) \\ \geq \sum_{s \in S} σ_{1}^{*} \dots σ_{i - 1}^{*} σ_{i} σ_{i + 1}^{*} \dots σ_{n}^{*} {\tilde{v}}_{i}^{Δ_{i}} (s_{i}, s_{- i}), \end{array}$ (15) where $u_{i}^{'' *}$ μ^"* _i is the Bayesian α_i-optimistic value of player i ∈ N.

Theorem 4.1. Let payoff ${\tilde{v}}_{1} (s_{1}, s_{- 1}), {\tilde{v}}_{2} (s_{2}, s_{- 2}), \dots, {\tilde{v}}_{n} (s_{n}, s_{- n})$ be independent fuzzy variables in game G₃, then there exists at least one Bayesianα_i-optimistic Nash equilibrium.

Proof. For any player i and any mixed strategy combination (σ_i, σ_-i) ∈ Λ in the game G₃, according to Equations (13) and (14), we have

$\begin{matrix} \int_{Δ_{i}} {(({\tilde{u}}_{i} (σ_{i}, σ_{- i}))}_{sup} (t_{i})) d (Φ_{i} (t_{i})) \\ = \sum_{s_{i} \in S_{i}, s_{- i} \in S_{- i}} σ_{1} σ_{2} \dots σ_{n} {\tilde{v}}_{i}^{Δ_{i}} (s_{i}, s_{- i}) . \end{matrix}$ (16)

Assume a game is denoted by $G_{3}^{'} = 〈 N, (Λ_{i})_{i \in N}, ({\tilde{v}}_{i}^{Δ_{i}} (s_{i}, s_{- i}))_{i \in N} 〉 .$

It is obvious that the $G_{3}^{'}$ is a classical n-person non-cooperative game. It follows from Definition 4.1 and Equation (16) that the existence of a Bayesian

α_i-optimistic Nash equilibrium in the game G₃ is equivalent to the existence of a Nash equilibrium in the game $G_{3}^{'}$ . It follows from Lemma 2.1 that there exists at least one mixed strategy Nash equilibrium in game $G_{3}^{'}$ . Thus, there exists at least one Bayesian α_i-optimistic Nash equilibrium in game G₃.

These complete the proof of Theorem 4.1. □

If $(σ_{i}^{*}, σ_{- i}^{*}) \in Λ$ is a Bayesian α_i-optimistic Nash equilibrium in game G₃, then the Bayesianα_i-optimistic value of player i is.

Now, we present a sufficient and necessary condition of Bayesian α_i-optimistic Nash equilibrium in game G₃, which can be applied to finding Bayesian α_i-optimistic Nash equilibrium strategies.

Theorem 4.2. Let payoffs ${\tilde{v}}_{1} (s_{1}, s_{- 1}), {\tilde{v}}_{2} (s_{2}, s_{- 2}), \dots, {\tilde{v}}_{n} (s_{n}, s_{- n})$ be independent fuzzy variables, then a strategy combination $(σ_{i}^{*}, σ_{- i}^{*}) \in Λ$ is a Bayesian α_i-optimistic Nash equilibrium in game G₃ if and only if $(σ_{1}^{*}, σ_{2}^{*}, \dots, σ_{n}^{*}, u_{1}^{'' *}, u_{2}^{'' *}, \dots, u_{n}^{'' *})$ is an optimal solution to the following non-linear programming model

$\begin{array}{l} \max_{σ_{1}, \dots, σ_{n}, u_{1}^{''}, \dots, u_{n}^{''}} \sum_{i = 1}^{n} \\ (\sum_{s_{i} \in S_{i}, s_{- i} \in S_{- i}} σ_{1} σ_{2} \dots σ_{n} {\tilde{v}}_{i}^{Δ_{i}} (s_{i}, s_{- i})) - \sum_{i = 1}^{n} u_{i}^{''} \\ s . t . {\begin{cases} \sum_{s_{- 1} \in S_{- 1}} σ_{2} \dots σ_{n} {\tilde{v}}_{1}^{Δ_{1}} (s_{1}^{1}, s_{- 1}) \leq u_{1}^{''} \\ ⋮ \\ \sum_{s_{- 1} \in S_{- 1}} σ_{2} \dots σ_{n} {\tilde{v}}_{1}^{Δ_{1}} (s_{1}^{m_{1}}, s_{- 1}) \leq u_{1}^{''} \\ ⋮ \\ \sum_{s_{- i} \in S_{- i}} σ_{1} \dots σ_{i - 1} σ_{i + 1} \dots σ_{n} {\tilde{v}}_{i}^{Δ_{i}} (s_{i}^{1}, s_{- i}) \leq u_{i}^{''} \\ ⋮ \\ \sum_{s_{- i} \in S_{- i}} σ_{1} \dots σ_{i - 1} σ_{i + 1} \dots σ_{n} {\tilde{v}}_{i}^{Δ_{i}} (s_{i}^{m_{i}}, s_{- i}) \leq u_{i}^{''} \\ ⋮ \\ \sum_{s_{- n} \in S_{- n}} σ_{1} σ_{2} \dots σ_{n - 1} {\tilde{v}}_{n}^{Δ_{n}} (s_{n}^{1}, s_{- n}) \leq u_{n}^{''} \\ ⋮ \\ \sum_{s_{- n} \in S_{- n}} σ_{1} σ_{2} \dots σ_{n - 1} {\tilde{v}}_{n}^{Δ_{n}} (s_{n}^{m_{n}}, s_{- n}) \leq u_{n}^{''} \\ \forall i \in N, σ_{i} \in Λ_{i}, u_{i}^{''} \in R \end{cases} \end{array}$ (17) where $s_{i}^{j}$ is the pure strategy that player i ∈ N chooses jth j ∈ {1, 2, …, m_i} pure strategy from the pure strategy set S_i.

Proof. First, Assuming $(σ_{1}^{*}, σ_{2}^{*}, \dots, σ_{n}^{*}, u_{1}^{'' *}, u_{2}^{'' *}, \dots, u_{n}^{'' *})$ $(σ_{1}^{*}, σ_{2}^{*}, ..., σ_{n}^{*}, {μ^{″}}_{1}^{*}, {μ^{″}}_{2}^{*}, ..., {μ^{″}}_{n}^{*})$ be an optimal solution of the non-linear programming model (17), we prove that $(σ_{1}^{*}, σ_{2}^{*}, \dots, σ_{n}^{*})$ is a Bayesian α_i-optimistic Nash equilibrium in game G₃. Since the optimal solution of the non-linear programming model (17) is also a feasible solution of the model (17), we have that $\begin{array}{l} \sum_{s_{- 1} \in S_{- 1}} σ_{2}^{*} \dots σ_{n}^{*} {\tilde{v}}_{1}^{Δ_{1}} (s_{1}^{1}, s_{- 1}) \leq u_{1}^{'' *} \\ ⋮ \\ \sum_{s_{- 1} \in S_{- 1}} σ_{2}^{*} \dots σ_{n}^{*} {\tilde{v}}_{1}^{Δ_{1}} (s_{1}^{m_{1}}, s_{- 1}) \leq u_{1}^{'' *} \\ ⋮ \\ \sum_{s_{- i} \in S_{- i}} σ_{1}^{*} \dots σ_{i - 1}^{*} σ_{i + 1}^{*} \dots σ_{n}^{*} {\tilde{v}}_{i}^{Δ_{i}} (s_{i}^{1}, s_{- i}) \leq u_{i}^{'' *} \\ ⋮ \\ \sum_{s_{- i} \in S_{- i}} σ_{1}^{*} \dots σ_{i - 1}^{*} σ_{i + 1}^{*} \dots σ_{n}^{*} {\tilde{v}}_{i}^{Δ_{i}} (s_{i}^{m_{i}}, s_{- i}) \leq u_{i}^{'' *} \\ ⋮ \\ \sum_{s_{- n} \in S_{- n}} σ_{1}^{*} σ_{2}^{*} \dots σ_{n - 1}^{*} {\tilde{v}}_{n}^{Δ_{n}} (s_{n}^{1}, s_{- n}) \leq u_{n}^{'' *} \\ ⋮ \\ \sum_{s_{- n} \in S_{- n}} σ_{1}^{*} σ_{2}^{*} \dots σ_{n - 1}^{*} {\tilde{v}}_{n}^{Δ_{n}} (s_{n}^{m_{n}}, s_{- n}) \leq u_{n}^{'' *} \end{array}$

It follows from the definition of mixed strategy that $\begin{array}{l} u_{i}^{''} (σ_{i}, σ_{- i}^{*}) \\ = \sum_{s_{i} \in S_{i}, s_{- i} \in S_{- i}} σ_{1}^{*} \dots σ_{i - 1}^{*} σ_{i} σ_{i + 1}^{*} \dots \dots σ_{n}^{*} {\tilde{v}}_{i}^{Δ_{i}} (s_{i}, s_{- i}) \\ \leq \sum_{s_{i} \in S_{i}, s_{- i} \in S_{- i}} σ_{1}^{*} σ_{2}^{*} \dots σ_{n}^{*} {\tilde{v}}_{i}^{Δ_{i}} (s_{i}, s_{- i}) = u_{i}^{'' *}, \forall σ_{i} \in Λ_{i} \end{array}$

which means that the strategy combination $(σ_{i}^{*}, σ_{- i}^{*}) \in Λ, \forall i \in N$ is a Bayesian α_i-optimistic Nash equilibrium, and μ^ʺ* _i $u_{i}^{'' *}$ is the Bayesian α_i-optimistic value of player i ∈ N in the game G₃. The sufficient condition is proved.

Second, we assume that $(σ_{1}^{*}, σ_{2}^{*}, \dots, σ_{n}^{*})$ is a Bayesian α_i-optimistic Nash equilibrium in the game G₃. Here, we prove that $(σ_{1}^{*}, σ_{2}^{*}, ..., σ_{n}^{*}, {μ^{″}}_{1}^{*}, {μ^{″}}_{2}^{*}, ..., {μ^{″}}_{n}^{*})$ is an optimal solution of the non-linear programming model (17).

It follows from Definition 4.1 that $\begin{array}{l} u_{i}^{''} (σ_{i}, σ_{- i}^{*}) \\ = \sum_{s_{i} \in S_{i}, s_{- i} \in S_{- i}} σ_{1}^{*} \dots σ_{i - 1}^{*} σ_{i} σ_{i + 1}^{*} \dots σ_{n}^{*} {\tilde{v}}_{i}^{Δ_{i}} (s_{i}, s_{- i}) \\ \leq \sum_{s_{i} \in S_{i}, s_{- i} \in S_{- i}} σ_{1}^{*} σ_{2}^{*} \dots σ_{n}^{*} {\tilde{v}}_{i}^{Δ_{i}} (s_{i}, s_{- i}) = u_{i}^{'' *}, \forall σ_{i} \in Λ_{i} . \end{array}$

Thanks to the arbitrariness of σ_i, we have that $\begin{matrix} \sum_{s_{- 1} \in S_{- 1}} σ_{2}^{*} \dots σ_{n}^{*} {\tilde{v}}_{1}^{Δ_{1}} (s_{1}^{1}, s_{- 1}) \leq u_{1}^{″ *} \\ ⋮ \\ \sum_{s_{- 1} \in S_{- 1}} σ_{2}^{*} \dots σ_{n}^{*} {\tilde{v}}_{1}^{Δ_{1}} (s_{1}^{m_{1}}, s_{- 1}) \leq u_{1}^{″ *} \\ ⋮ \\ \sum_{s_{- i} \in S_{- i}} σ_{1}^{*} \dots σ_{i - 1}^{*} σ_{i + 1}^{*} \dots σ_{n}^{*} {\tilde{v}}_{i}^{Δ_{i}} (s_{i}^{1}, s_{- i}) \leq u_{i}^{″ *} \\ ⋮ \\ \sum_{s_{- i} \in S_{- i}} σ_{1}^{*} \dots σ_{i - 1}^{*} σ_{i + 1}^{*} \dots σ_{n}^{*} {\tilde{v}}_{i}^{Δ_{i}} (s_{i}^{m_{i}}, s_{- i}) \leq u_{i}^{″ *} \\ ⋮ \\ \sum_{s_{- n} \in S_{- n}} σ_{1}^{*} σ_{2}^{*} \dots σ_{n - 1}^{*} {\tilde{v}}_{n}^{Δ_{n}} (s_{n}^{1}, s_{- n}) \leq u_{n}^{″ *} \\ ⋮ \\ \sum_{s_{- n} \in S_{- n}} σ_{1}^{*} σ_{2}^{*} \dots σ_{n - 1}^{*} {\tilde{v}}_{n}^{Δ_{n}} (s_{n}^{m_{n}}, s_{- n}) \leq u_{n}^{″ *}, \end{matrix}$ which means that $(σ_{1}^{*}, σ_{2}^{*}, ..., σ_{n}^{*}, {μ^{″}}_{1}^{*}, {μ^{″}}_{2}^{*}, ..., {μ^{″}}_{n}^{*})$ is a feasible solution of the non-linear programming model (17). According to the model (17), we have $\begin{matrix} \sum_{i = 1}^{n} (\sum_{s_{i} \in S_{i}, s_{- i} \in S_{- i}} σ_{1} σ_{2} \dots σ_{n} v_{i}^{Δ_{i}} (s_{i}, s_{- i})) \\ - \sum_{i = 1}^{n} u_{i}^{″} \leq 0 . \end{matrix}$

It follows from $\begin{matrix} \sum_{i = 1}^{n} (\sum_{s_{i} \in S_{i}, s_{- i} \in S_{- i}} σ_{1}^{*} σ_{2}^{*} \dots σ_{n}^{*} v_{i}^{Δ_{i}} (s_{i}, s_{- i})) \\ - \sum_{i = 1}^{n} u_{i}^{″ *} = 0 \end{matrix}$ that $(σ_{1}^{*}, σ_{2}^{*}, ..., σ_{n}^{*}, {μ^{″}}_{1}^{*}, {μ^{″}}_{2}^{*}, ..., {μ^{″}}_{n}^{*})$ is an optimal solution to the non-linear programming model (17). The necessary condition is proved.

These complete the proof of Theorem 4.2. □

5 Numerical example

With rapid economic development, during recent years, a great number of consumers have benefited from flat screens over conventional cathode ray tube (CRT) products, which was induced by aggressive marketing strategies and low-cost thin-film transistor liquid crystal display (TFT– LCD) production. The glass substrate is a major component of TFT– LCD, whose physical size for required for various TFT– LCD products plays an important role in the growing demand. There are three main global glass substrate suppliers for LCD: Corning Display Technologies (CDT), Asahi Glass (AG) and Nippon Electric Glass (NEG). we consider the situation where CDT, AG and NEG manufacture a new glass substrate, respectively. In order to sell as many glass substrates as possible, each glass substrate supplier has its own marketing strategy set including TV advertisements, cash rebates, free samples etc. Since different marketing strategies of the three glass substrate suppliers maybe lead to different total sales of the new glass substrate, the goal of each supplier is to sell as many glass substrates as possible by choosing one from its marketing strategy set given strategies of the other glass substrate suppliers.

Here, we denote CDT, AG and NEG by supplier 1, 2 and 3, respectively. The marketing strategy set of CDT is denoted by S₁, the marketing strategy set of AG is denoted by S₂, and the marketing strategy set of NEG is denoted by S₃, respectively. For simplicity, we assume that each glass substrate supplier has only two marketing strategies: TV advertisements and cash rebates. That is, the marketing strategy set of glass substrate supplier i is $S_{i} = {s_{i}^{1} : TV advertisements; s_{i}^{2} cash rebates}$ . Since the new glass substrate sold by the three glass substrate suppliers is a new product, there is no way to find past statistical historical records about sales of the new glass substrate. Thus, the three glass substrate suppliers have to depend on themselves experiences, subjective judgments and intuitions of experts to determine their own sales. We assume that the sales of the new glass substrate determined by different strategy combinations obey the triangular distributions under fuzzy information situation. Thus, the sales of the new glass substrate can be modeled as triangular fuzzy numbers. The fuzzy payoffs of the three suppliers are shown in the following game illustrated in Fig. 1, where CDT chooses rows, the AG chooses columns, while NEG chooses matrices and all payoffs are expressed in unit of thousand.

Fig.1

3-supplier game with triangular fuzzy payoffs.

Since the confidence level of each glass substrate supplier is its own private information. Here, as mentioned in Section 4, the three glass substrate suppliers are Bayesian suppliers, modeled by a random variable with a certain probability distribution on its risk aversion. For random variables with a certain probability distribution, there are two exhaustive cases: discrete random variables and random variables. Firstly, we assume that the confidence levels α₁, α₂ and α₃ are discrete random variables. Supplier i has three different risk aversion levels denoted by $α_{i}^{I}, α_{i}^{II}$ and $α_{i}^{III}$ , and chooses risk aversion level $α_{1}^{t}$ with probability p_it, where t ∈ {I, II, III}. Specifically, let $α_{1}^{I} = 0.7$ , $α_{1}^{II} = 0.8$ , $α_{1}^{III} = 0.9$ ; $α_{2}^{I} = 0.6$ , $α_{2}^{II} = 0.7$ , $α_{2}^{III} = 0.8$ and $α_{3}^{I} = 0.6$ , $α_{3}^{II} = 0.7$ , $α_{3}^{III} = 0.9$ . And let p_1I = 0.3, p_1II = 0.4, p_1III = 0.3; p_2I = 0.4, p_2II = 0.3, p_2III = 0.3; and p_3I = 0.2, p_3II = 0.4, p_3III = 0.4.

If supplier i ∈ {1, 2, 3} chooses the strategy $s_{i}^{1}$ with the probability $σ_{i} (s_{i}^{1})$ , it chooses the strategy $s_{i}^{2}$ with the probability $1 - σ_{i} (s_{i}^{1})$ . For the fuzzy payoff of CDT ${\tilde{v}}_{1} (s_{1}^{1}, s_{2}^{1}, s_{3}^{1}) = (110, 120, 130)$ , according to Definition 2.7, we have $({\tilde{v}}_{1} (s_{1}^{1}, s_{2}^{1}, s_{3}^{1}))_{sup} (0.7) = 116$ for α₁ = 0.7. Similarly, we obtain $({\tilde{v}}_{1} (s_{1}^{1}, s_{2}^{1}, s_{3}^{1}))_{sup} (0.8) = 114$ and $({\tilde{v}}_{1} (s_{1}^{1}, s_{2}^{1}, s_{3}^{1}))_{sup} (0.9) = 112$ . It follows from Equation (14) that

$\begin{matrix} {\tilde{v}}_{1}^{Δ_{1}} (s_{1}^{1}, s_{2}^{1}, s_{3}^{1}) \\ = 0.3 \times 116 + 0.4 \times 114 + 0.3 \times 112 = 114 \end{matrix}$ for p₁ = (0.3, 0.4, 0.3). Similarly, we can obtain ${\tilde{v}}_{i}^{Δ_{i}} (s_{1}^{m_{1}}, s_{2}^{m_{2}}, s_{3}^{m_{3}})$ , where i ∈ {1, 2, 3} , m_i ∈ {1, 2}. Thus, we can obtain the following payoff matrices.

$\begin{matrix} \begin{matrix} s_{2}^{1} s_{2}^{2} \\ \begin{matrix} s_{1}^{1} \\ s_{1}^{2} \end{matrix} [\begin{matrix} 114, 118.6, 124.8 & 150, 154.8, 119.6 \\ 154, 141, 139.6 & 112, 106.2, 109.6 \end{matrix}] \\ s_{3}^{1} \\ s_{2}^{1} s_{2}^{2} \\ \begin{matrix} s_{1}^{1} \\ s_{1}^{2} \end{matrix} [\begin{matrix} 154, 132 . 4, 139 . 6 & 132, 156 . 2, 134 \\ 138, 144 . 8, 134 . 8 & 156, 134 . 8, 114 . 4 \end{matrix}] \\ s_{3}^{2} \end{matrix} \end{matrix}$ where CDT chooses rows, the AG chooses columns, while NEG chooses matrices.

It follows from Theorem 4.2 that $\begin{array}{l} \max_{σ_{1} (s_{1}^{1}), σ_{2} (s_{2}^{1}), σ_{3} (s_{3}^{1}), u_{1}^{''}, u_{2}^{''}, u_{3}^{''}} \\ (\begin{array}{l} - 165.2 σ_{1} (s_{1}^{1}) σ_{2} (s_{2}^{1}) σ_{3} (s_{3}^{1}) - 8.6 σ_{1} (s_{1}^{1}) σ_{2} (s_{2}^{1}) \\ + 79.6 σ_{1} (s_{1}^{1}) σ_{3} (s_{3}^{1}) + 94 .4 σ_{2} (s_{2}^{1}) σ_{3} (s_{3}^{1}) \\ + 17 σ_{1} (s_{1}^{1}) + 12.4 σ_{2} (s_{2}^{1}) - 77.4 σ_{3} (s_{3}^{1}) \\ + 405.2 - u_{1}^{''} - u_{2}^{''} - u_{3}^{''} \end{array}) \\ s t . {\begin{cases} - 58 σ_{2} (s_{2}^{1}) σ_{3} (s_{3}^{1}) + 22 σ_{2} (s_{2}^{1}) + 18 σ_{3} (s_{3}^{1}) + 132 \leq u_{1}^{"} \\ 60 σ_{2} (s_{2}^{1}) σ_{3} (s_{3}^{1}) - 18 σ_{2} (s_{2}^{1}) - 44 σ_{3} (s_{3}^{1}) + 156 \leq u_{1}^{"} \\ - 10 σ_{1} (s_{1}^{1}) σ_{3} (s_{3}^{1}) - 12.4 σ_{1} (s_{1}^{1}) - 3.8 σ_{3} (s_{3}^{1}) + 144.8 \leq u_{2}^{"} \\ 27.2 σ_{1} (s_{1}^{1}) σ_{3} (s_{3}^{1}) + 21.4 σ_{1} (s_{1}^{1}) - 28.6 σ_{3} (s_{3}^{1}) + 134.8 \leq u_{2}^{"} \\ - 24.8 σ_{1} (s_{1}^{1}) σ_{2} (s_{2}^{1}) + 10 σ_{1} (s_{1}^{1}) + 30 σ_{2} (s_{2}^{1}) + 109.6 \leq u_{3}^{"} \\ - 14.8 σ_{1} (s_{1}^{1}) σ_{2} (s_{2}^{1}) + 19.6 σ_{1} (s_{1}^{1}) + 20.4 σ_{2} (s_{2}^{1}) + 114.4 \leq u_{3}^{"} \\ 0 \leq σ_{1} (s_{1}^{1}) \leq 1 \\ 0 \leq σ_{2} (s_{2}^{1}) \leq 1 \\ 0 \leq σ_{3} (s_{3}^{1}) \leq 1 \\ 0 \leq u_{1}^{''} \\ 0 \leq u_{2}^{''} \\ 0 \leq u_{3}^{''} \end{cases} \end{array}$ (18)

The optimal solution of the programming model (18) is obtained as follows: $\begin{array}{l} σ_{1} (s_{1}^{1}) = 0.4134, σ_{2} (s_{2}^{1}) = 0.2197, σ_{3} (s_{3}^{1}) = 0.4217, \\ u_{1}^{''} = 139.0501, u_{2}^{''} = 136.3279, u_{3}^{''} = 122.4490. \end{array}$

Since $σ_{1} (s_{1}^{1}) = 0.4134$ , we have $1 - σ_{1} (s_{1}^{1}) = 0 . 5866$ . Similarly, we have $1 - σ_{2} (s_{2}^{1}) = 0.7803 and 1 - σ_{3} (s_{3}^{1}) = 0.5783 .$

Therefore, for i ∈ N, the Bayesian α_i-optimistic Nash equilibrium is (0.4134, 0.5866; 0.2197, 0.7803; 0.4217, 0.5783), which means that the optimal strategy for CDT is (0.4134, 0.5866) that yields a payoff 139.0501, the optimal strategy for AG is (0.2197, 0.7803) that yields a payoff 136.3279 and the optimal strategy for NEG is (0.4217, 0.5783) that yields a payoff 122.4990.

Secondly, we assume that the confidence levels α₁, α₂ and α₃ are random variables, which are uniformly distributed on a certain interval. That is, AG and NEG believe that risk aversion of CDT modeled by α₁ is uniformly distributed on an interval [a₁₁, a₁₂]. Similarly, CDT and NEG believe that the risk aversion levels of AG modeled by the confidence level α₂ is uniformly distributed on an interval [a₂₁, a₂₂], and CDT and AG believe that the risk aversion levels of NEG modeled by the confidence level α₃ is uniformly distributed on an interval [a₃₁, a₃₂]. Let [a₁₁, a₁₂] = [0.6, 1] , [a₂₁, a₂₂] = [0.8, 1] and [a₃₁, a₃₂] = [0.7, 1]. For the fuzzy payoff of CDT ${\tilde{v}}_{1} (s_{1}^{1}, s_{2}^{1}, s_{3}^{1}) = (110, 120, 130)$ , It follows from Equation (14) and Definition 2.7 that $\begin{matrix} {\tilde{v}}_{1}^{Δ_{1}} (s_{1}^{1}, s_{2}^{1}, s_{3}^{1}) \\ = (\int_{0.6}^{1} (100 (2 t_{1} - 1) + 120 (2 - 2 t_{1})) d (t_{1})) / - \\ (1 - 0.6) = 114 \end{matrix}$ when α₁ is a random variable with uniform distribution [0.6, 1]. Similarly, we can obtain ${\tilde{v}}_{i}^{Δ_{i}} (s_{1}^{m_{1}}, s_{2}^{m_{2}}, s_{3}^{m_{3}})$ , where i ∈ {1, 2, 3}, m_i ∈ {1, 2}. Thus, we can obtain the following payoff matrices. $\begin{matrix} \begin{matrix} s_{2}^{1} s_{2}^{2} \\ \begin{matrix} s_{1}^{1} \\ s_{1}^{2} \end{matrix} [\begin{matrix} 114, 106, 123 & 150, 138, 116 \\ 154, 120, 136 & 112, 102, 106 \end{matrix}] \\ s_{3}^{1} \\ s_{2}^{1} s_{2}^{2} \\ \begin{matrix} s_{1}^{1} \\ s_{1}^{2} \end{matrix} [\begin{matrix} 154, 124, 136 & 132, 152, 125 \\ 138, 128, 133 & 156, 118, 109 \end{matrix}] \\ s_{3}^{2} \end{matrix} \end{matrix}$ where CDT chooses rows, the AG chooses columns, while NEG chooses matrices.

According to Theorem 4.2, we have $\begin{array}{l} \max_{σ_{1} (s_{1}^{1}), σ_{2} (s_{2}^{1}), σ_{3} (s_{3}^{1}), u_{1}^{''}, u_{2}^{''}, u_{3}^{''}} \\ (\begin{array}{l} - 140 σ_{1} (s_{1}^{1}) σ_{2} (s_{2}^{1}) σ_{3} (s_{3}^{1}) - 11 σ_{1} (s_{1}^{1}) σ_{2} (s_{2}^{1}) \\ + 58 σ_{1} (s_{1}^{1}) σ_{3} (s_{3}^{1}) + 74 σ_{2} (s_{2}^{1}) σ_{3} (s_{3}^{1}) \\ + 26 σ_{1} (s_{1}^{1}) + 16 σ_{2} (s_{2}^{1}) - 63 σ_{3} (s_{3}^{1}) \\ + 383 - u_{1}^{''} - u_{2}^{''} - u_{3}^{''} \end{array}) \\ s . t . {\begin{cases} - 58 σ_{2} (s_{2}^{1}) σ_{3} (s_{3}^{1}) + 22 σ_{2} (s_{2}^{1}) + 18 σ_{3} (s_{3}^{1}) + 132 \leq u_{1}^{''} \\ 60 σ_{2} (s_{2}^{1}) σ_{3} (s_{3}^{1}) - 18 σ_{2} (s_{2}^{1}) - 44 σ_{3} (s_{3}^{1}) + 156 \leq u_{1}^{''} \\ - 10 σ_{1} (s_{1}^{1}) σ_{3} (s_{3}^{1}) - 4 σ_{1} (s_{1}^{1}) - 8 σ_{3} (s_{3}^{1}) + 128 \leq u_{2}^{''} \\ 2 σ_{1} (s_{1}^{1}) σ_{3} (s_{3}^{1}) + 34 σ_{1} (s_{1}^{1}) - 16 σ_{3} (s_{3}^{1}) + 118 \leq u_{2}^{''} \\ - 23 σ_{1} (s_{1}^{1}) σ_{2} (s_{2}^{1}) + 10 σ_{1} (s_{1}^{1}) + 30 σ_{2} (s_{2}^{1}) + 106 \leq u_{3}^{''} \\ - 13 σ_{1} (s_{1}^{1}) σ_{2} (s_{2}^{1}) + 16 σ_{1} (s_{1}^{1}) + 24 σ_{2} (s_{2}^{1}) + 109 \leq u_{3}^{''} \\ 0 \leq σ_{1} (s_{1}^{1}) \leq 1 \\ 0 \leq σ_{2} (s_{2}^{1}) \leq 1 \\ 0 \leq σ_{3} (s_{3}^{1}) \leq 1 \\ 0 \leq u_{1}^{''} \\ 0 \leq u_{2}^{''} \\ 0 \leq u_{3}^{''} \end{cases} \end{array}$ (19)

The optimal solution of the programming model (19) is obtained as follows: $\begin{array}{l} σ_{1} (s_{1}^{1}) = 0.3106, σ_{2} (s_{2}^{1}) = 0.2174, σ_{3} (s_{3}^{1}) = 0.4210, \\ u_{1}^{''} = 139.0537, u_{2}^{''} = 122.0850, u_{3}^{''} = 116.5267 \end{array}$

Since $σ_{1} (s_{1}^{1}) = 0.3106$ , we have $1 - σ_{1} (s_{1}^{1}) = 0.6894$ . Similarly, we have $1 - σ_{2} (s_{2}^{1}) = 0.7826 and 1 - σ_{3} (s_{3}^{1}) = 0.5790 .$

Therefore, for i ∈ {1, 2, 3}, the Bayesian α_i-optimistic Nash equilibrium is (0.3106, 0.6894; 0.2174, 0.7826; 0.4210, 0.5790), which means that the optimal strategy for CDT is (0.3106, 0.6894) that yields a payoff 139.0537, the optimal strategy for AG is (0.2174, 0.7826) that yields a payoff 122.0850 and the optimal strategy for NEG is (0.4210, 0.5790) that yields a payoff 116.5267.

6 Conclusions

In many real games, the payoffs are often not given by crisp numbers but estimated by approximate values, some information are not common knowledge but private information. Players are often inclined to behavior character of risk aversion. In this paper, we have developed a new theoretical approach to n-person non-cooperative game with fuzzy payoffs. A n-person credibilistic non-cooperative game with risk aversion are investigated. Optimistic value criterion is used to model players’ risk aversion character. Based on the assumption that the confidence levels are players’ private information, we investigate n-person credibilistic non-cooperative games. A solution concept of Bayesian α_i-optimistic Nash equilibrium is proposed and its existence theorem is proved. Moreover, we present a sufficient and necessary condition that applies to finding the Bayesian optimistic Nash equilibrium.

Footnotes

Acknowledgments

This research is supported by National Natural Science Foundation of China (Nos. 71671188, 71271217, and 71571192), and Natural Science Foundation of Hunan (No. 2016JJ1024) and Priority Academic Program Development of Jiangsu Higher Education Institutions.

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