Bertrand game under a fuzzy environment

Abstract

Players are often assumed to be perfect rational in Bertrand game. However, the decision-making of players is influenced by their behavioral characteristic, psychology preferences and other uncertain factors. Therefore, this paper focuses on investigating the price strategies of players with risk aversion in a Bertrand game under a fuzzy environment. Based on the credibility theory, a fuzzy Bertrand game model with risk aversion is constructed, where optimistic value criterion is applied to model players’ risk aversion. The market demand for each player is assumed as a fuzzy variable. Then, a solution concept of the (α₁, α₂)-optimistic equilibrium price is defined and investigated. Finally, some numerical studies are applied to analyze how the risk aversion behavior of players (the confidence levels of players) affects the (α₁, α₂)-optimistic equilibrium prices and the profits of the fuzzy Bertrand game.

Keywords

Bertrand game risk aversion credibility theory optimistic value criterion

1 Introduction

Bertrand game is a model about price competition between players, which presents the interactive behaviors among players that make price decisions and their consumers that choose quantities at a set price [2]. There are a lot of researches about the Bertrand game, including equilibrium of game models [16 , 29], strategic choice of timing [1 , 45], asymmetric costs [3, 36], and some applications to the economic problems [8 , 41]. These above studies are assumed to be in a certain environment. However, many uncertain factors, such as consumer demand and fluctuation in production, may limit the application in the practice, which can’t be ignored.

Fuzzy set theory proposed by Zadeh [50] offers a proper approach to handle these uncertain problems, which has already been adopted to describe the uncertainty in games. Maeda [26] explored the bimatrix games with fuzzy payoffs. Nishizaki and Sakawa [33] proved the equilibrium of multi-objective bimatrix games with fuzzy payoffs and goals. Campos [4] constructed a Fuzzy Linear Programming to obtain the optimal solutions of a two-person zero-sum game with uncertain payoff matrix. Sakawa and Nishizaki [37 –39] explored the Max-min solutions and Pareto-equilibrium solutions of two-person zero-sum games with fuzzy multiple payoff matrices. Wu and Soo [47] defined a fuzzy Nash equilibrium and explored an n-person matrix game with fuzzy strategies. Garagic and Cruz [15] analyzed an n-person non-cooperative fuzzy game. Fuzzy set theory is very popular in fuzzy games. Based on fuzzy set theory, the existence of fuzzy Nash equilibrium for fuzzy games can be proved.

Recently, abundant researches have investigated the oligopolistic competition game under a fuzzy environment, such as Cournot game and Bertrand game. Dang and Hong [6] studied duopoly competition with fuzzy demand and costs. Based on Dang and Hong [6], Dang et al. [7] further extended duopoly competition to multioligopoly competition. Tan et al. [43] employed the rank-dependent utility theory to analyze Cournot competition with fuzzy market demand. The significance of fuzzy profit function was emphasized strongly in the above papers. Meanwhile, it was also introduced into a fuzzy supply chain with Bertrand competition. Zhao et al. [52] explored the pricing problem of two substitutable products in a fuzzy supply chain with fuzzy market demand and costs, where a retailer purchased the products from two manufacturers competing in a Bertrand game. In a fuzzy environment where market demand and costs are fuzzy, the optimal pricing strategies of two substitutable and complementary products in Bertrand price competition were analyzed respectively [46, 53]. Based on the expected value criterion that denotes the player’s desire to maximize his/her expected payoff, all these studies examined the effect of the fuzzy parameters on the optimal pricing decision. The expected value criterion is a credibilistic method that credibility theory offers. Despite much attention on price competition between two players in a fuzzy environment, research on the Bertrand game composed of two competitive players under a fuzzy environment is inadequate. Thus, according to these achievements, we incorporate credibility theory into this paper to depict the uncertainty in Bertrand game.

Additionally, many works on game under a fuzzy environment are based on the assumption that players are full rational and desire to maximize their own expected profits. However, people in reality are often tend to show the behavioral characteristics of bounded rationality with risk aversion and do not seek to maximize their profits as they are in the face of gain. For example, a risk averse stakeholder would take a conservative watershed development plan to increase chances of system feasibility, which might reduce the agricultural profits [38, 44]. Under the fuzzy environment, optimistic value criterion, another credibilistic method that credibility theory offers, is suitable for its application to describe the situation where players desire to maximize the α-optimistic value of their profits at a predetermined confidence level α [12]. Each confidence level corresponds to a level of players’ risk aversion. Nowadays, optimistic value criterion has been incorporated into fuzzy games to model players’ risk aversion behavior, such as bimatrix game [9, 12], coalitional game [14, 40], extensive game [13], two-person nonzero-sum game [11] and n-person non-cooperative game [42]. However, little attention has been paid to analyze the risk aversion behavior in a fuzzy Bertrand game through optimistic value criterion. We thus adopt optimistic value criterion to model risk aversion of players in fuzzy Bertrand game and investigate the effect of risk aversion on the optimal price strategies.

The purpose of this paper is to explore the effect of risk aversion in a Bertrand game under a fuzzy environment. The market demand for each product are posited as a fuzzy variable. Based on credibility theory, a fuzzy Bertrand game model with risk aversion is established. To model risk aversion of players, optimistic value criterion is applied. By game-theoretical approach, the (α₁, α₂)-optimistic equilibrium price is obtained. Some numerical studies are conducted to analyze how the (α₁, α₂)-optimistic equilibrium prices and the profits of players change when both two players are risk averse. There are some interesting findings. For example, under a fuzzy environment, a higher risk aversion leads to a lower price and a lower profit. Moreover, the price and profit of a player are more greatly affected by his own risk aversion than the other player’s risk aversion. Our study extends the classical Bertrand game to a fuzzy Bertrand game with risk aversion and provides a multi-perspective analysis of the impact of risk aversion on the prices and profits to make results more reasonable and closer to reality.

The rest of this paper is organized as follows. In Section 2, the basic concepts of fuzzy variable, possibility theory, and credibility theory are discussed. In Section 3, a fuzzy Bertrand game model with risk aversion is established, and a solution concept of (α₁, α₂)-optimistic equilibrium price is defined, and four cases of (α₁, α₂)-optimistic equilibrium price is solved. In Section 4, some numerical studies are used to analyze the influence of risk aversion on the (α₁, α₂)-optimistic equilibrium prices and the profits in the fuzzy Bertrand game. In Section 5, some conclusions are summarized.

2 Preliminaries

In this section, some basic concepts of fuzzy variable, possibility theory, and credibility theory are presented, respectively. The classical Bertrand game under crisp environment is also introduced.

2.1 Fuzzy environment

Fuzzy set theory put forward by Zadeh [50] offers an alternative approach to solve uncertain problems about internal ambiguity and imprecision of human cognitive processes. Fuzzy variable is one of the most important concepts in fuzzy set theory [18 , 51]. As mathematical tools in fuzzy set theory, possibility theory [34, 51] and credibility theory [22, 23] are applied to explain and handle fuzzy phenomena, and have been widely used in many fields, such as option pricing [10] and transportation planning [19 , 49]. Hence, some basic concepts of fuzzy variable, possibility theory and credibility theory are introduced in this subsection.

Definition 1. [22] Let Θ be a nonempty set, and let P(Θ) be the power set of Θ. Then Pos: P(Θ) ⟶ [0, 1] is called a possibility measure if it satisfies the following three axioms:

Axiom 1. Pos {Θ} = 1.

Axiom 2. Pos {A} ≥0 for any event A.

Axiom 3. For every countable sequence of mutually disjoint events {A_i} of P(Θ), $Pos {\cup_{i = 1}^{\infty} A_{i}} = \sum_{i = 1}^{\infty} Pos {A_{i}}$ .

Definition 2. [22] Let Θ be a nonempty set, P(Θ) the power set of Θ, and Pos a possibility measure. Then, the triplet (Θ, P(Θ), Pos) is called a possibility space.

Definition 3. [22] A fuzzy variable is defined as a function from the possibility space (Θ, P(Θ), Pos) to the set of real numbers.

Definition 4. [22] Let $\tilde{ξ}$ be a fuzzy variable, and its membership function on the possibility space (Θ, P(Θ), Pos) is defined as $μ (x) = Pos (θ \in Θ | \tilde{ξ} (θ) = x), \forall x \in R .$

A fuzzy variable $\tilde{ξ}$ is nonnegative (or positive) if $Pos ({\tilde{ξ} < 0}) = 0$ (or $Pos ({\tilde{ξ} \leq 0}) = 0$ ).

Definition 5. [22] Let (Θ, P(Θ), Pos) be a possibility space, and A be a set in P(Θ). Then, the credibility measure of A can be defined as Cr { A } = (Pos { A } + 1 - Pos { A^c })/2, where A^c denotes the complement of A and Pos { A } ∨ Pos { A^c } = 1.

Then, the triplet (Θ, P(Θ), Cr) is called as a credibility space, and the function Cr is called as a credibility measure.

Definition 6. [22] Let $\tilde{ξ}$ be a fuzzy variable with membership function u on the credibility space (Θ, P(Θ), Cr). Then for any real numbers, we have $Cr {\tilde{ξ} \in R} = \frac{1}{2} \underset{x \in R}{(sup} μ (x) + 1 - sup_{x \notin R} μ (x)) .$ (1)

As a special fuzzy variable, a triangular fuzzy variable [22] is denoted as $\tilde{ξ} = (ξ_{l}, ξ_{m}, ξ_{r})$ , where ξ_m is the mean of $\tilde{ξ}$ , ξ_l and ξ_r are the lower and upper limits of $\tilde{ξ}$ , respectively. Its membership function can be given as follows:

$\begin{matrix} μ_{\tilde{ξ}} (x) & = & Cr {θ \in Θ | \tilde{ξ} \leq x} \\ = & \frac{sup_{y \leq x} μ (y) - sup_{y > x} μ (y) + 1}{2} \\ = & {\begin{matrix} \begin{matrix} (x - ξ_{l}) / (ξ_{m} - ξ_{l}) \\ 1 \\ (ξ_{r} - x) / (ξ_{r} - ξ_{m}) \\ 0 \end{matrix} & \begin{matrix} ξ_{l} \leq x < ξ_{m} \\ x = ξ_{m} \\ ξ_{m} < x \leq ξ_{r} \\ otherwise . \end{matrix} \end{matrix} \end{matrix}$ (2)

Definition 7. [22] For any sets B₁, B₂, …, B_n of real numbers, fuzzy variables ${\tilde{ξ}}_{1}, {\tilde{ξ}}_{2}, \dots, {\tilde{ξ}}_{n}$ are independent if and only if $Cr {\tilde{ξ} \in R} = \frac{1}{2} \underset{x \in R}{(sup} μ (x) + 1 - sup_{x \notin R} μ (x)) .$ (3)

Definition 8. A triangular fuzzy variable $\tilde{ξ} = (ξ_{l}, ξ_{m}, ξ_{r})$ is called as non-negative triangular fuzzy variable if ξ_l ≥ 0 and ξ_r > 0.

For two triangular fuzzy variables $\tilde{ξ} = (ξ_{l}, ξ_{m}, ξ_{r})$ and $\tilde{η} = (η_{l}, η_{m}, η_{r})$ , according to the extension principle proposed by Negoita [32], there are some primary fuzzy operations as follows: $\tilde{ξ} + \tilde{η} = (ξ_{l} + η_{l}, ξ_{m} + η_{m}, ξ_{r} + η_{r}),$ (4) $λ \tilde{ξ} = {\begin{matrix} \begin{matrix} (λ ξ_{l}, λ ξ_{m}, λ ξ_{r}) \\ (λ ξ_{r}, λ ξ_{m}, λ ξ_{l}) \end{matrix} & \begin{matrix} if λ \geq 0 \\ if λ < 0, \end{matrix} \end{matrix}$ (5) $\tilde{ξ} - \tilde{η} = (ξ_{l} - η_{r}, ξ_{m} - η_{m}, ξ_{r} - η_{l}) .$ (6)

Definition 9. [22] For a fuzzy variable $\tilde{ξ}$ on the credibility space (Θ, P(Θ), Cr) and a confidence level $α \in (0, 1], {\tilde{ξ}}_{sup} (α) = sup {r | Cr {\tilde{ξ} \geq r} \geq α}$ and ${\tilde{ξ}}_{inf} (α) = inf {r | Cr {\tilde{ξ} \leq r} \geq α}$ are called the α-optimistic value and α-pessimistic value to $\tilde{ξ}$ , respectively.

It means that the fuzzy variable $\tilde{ξ}$ will reach upwards of the α-optimistic value ${\tilde{ξ}}_{sup} (α)$ with credibility α, and will be below the α-pessimistic value ${\tilde{ξ}}_{inf} (α)$ with credibility α. In other words, the α-optimistic value ${\tilde{ξ}}_{sup} (α)$ is the supremum value that $\tilde{ξ}$ achieves with credibility α, the α-pessimistic value ${\tilde{ξ}}_{inf} (α)$ is the infimum value that $\tilde{ξ}$ achieves with credibility α.

Then, the α-optimistic value and α-pessimistic value of a triangular fuzzy variable $\tilde{ξ} = (ξ_{l}, ξ_{m}, ξ_{r})$ on the credibility space (Θ, P(Θ), Cr) are given as follows, respectively,

$\begin{matrix} {\tilde{ξ}}_{sup} (α) \\ = {\begin{matrix} \begin{matrix} 2 α ξ_{m} + (1 - 2 α) ξ_{r} \\ (2 α - 1) ξ_{l} + 2 (1 - α) ξ_{m} \end{matrix} & \begin{matrix} α \leq 0.5 \\ α > 0.5, \end{matrix} \end{matrix} \\ {\tilde{ξ}}_{inf} (α) \\ = {\begin{matrix} \begin{matrix} (1 - 2 α) ξ_{l} + 2 α ξ_{m} \\ 2 (1 - α) ξ_{m} + (2 α - 1) ξ_{r} \end{matrix} & \begin{matrix} α \leq 0.5 \\ α > 0.5 . \end{matrix} \end{matrix} \end{matrix}$ (7)

Lemma 1. [25] Let $\tilde{ξ}$ and $\tilde{η}$ be independent fuzzy variables, and α ∈ (0, 1] be the confidence level. Then for any real numbers m and n, we have ${(m \tilde{ξ} + n \tilde{η})}_{sup} (α) = m {\tilde{ξ}}_{sup} (α) + n {\tilde{η}}_{sup} (α) .$ (8)

Lemma 2. [22] Let ${\tilde{ξ}}_{1}, {\tilde{ξ}}_{2}, \dots, {\tilde{ξ}}_{m}$ be independent fuzzy variables, and f₁, f₂, … , f_m be real-values functions, then $f_{1} ({\tilde{ξ}}_{1}), f_{2} ({\tilde{ξ}}_{2}), \dots, f_{m} ({\tilde{ξ}}_{m})$ are independent fuzzy variables.

Definition 10. [21] For any two independent fuzzy variables $\tilde{ξ}$ and $\tilde{η}$ , and some predetermined confidence level α ∈ (0, 1], the optimistic value criterion is denoted by $\tilde{ξ} > \tilde{η}$ if and only if ${\tilde{ξ}}_{sup} (α) > {\tilde{η}}_{sup} (α)$ , where ${\tilde{ξ}}_{sup} (α)$ and ${\tilde{η}}_{sup} (α)$ are the α-optimistic value of $\tilde{ξ}$ and $\tilde{η}$ , respectively, and the confidence level α reflects the level of risk aversion of players.

2.2 The classical Bertrand game

In a classical Bertrand game, players make their own price decisions to maximize their profits respectively. There are two players with two substitutable products in the market. More specifically, product 1 is produced by player 1 and product 2 is produced by player 2. Both two players decide their price strategies independently and simultaneously. They also have a commitment to supply whatever the market demand is forthcoming at the price it sets.

For player i, let Q_i (i = 1, 2) be the market demand, p_i (i = 1, 2) be the market price, and c_i (i = 1, 2) be the marginal production cost, and p_i > c_i. Then, the market demand function of player i is Q_i = a_i − b_ip_i +θp_j, i, j = 1, 2, where a_i represents the market potential, b_i denotes the price flexibility coefficient, and θ is the cross-price flexibility coefficients, and b_i > θ. To ensure that the market demand conform to reality, we assume that a_i > b_ip_i, namely p_i < a_i /b_i.

According to the above description, the profit of player i can be written as

$π_{i} = (p_{i} - c_{i}) Q_{i}, i = 1, 2 .$ (9)

The unique Nash equilibrium of Bertrand game is $p_{i}^{0} = \frac{2 b_{j} (a_{i} + b_{i} c_{i}) + θ (a_{j} + b_{j} c_{j})}{4 b_{i} b_{j} - θ^{2}}, i, j = 1, 2, i \neq j .$ And, the profit of player i is $π_{i}^{0} = b_{i} \times {(\frac{2 b_{j} (a_{i} - b_{i} c_{i}) + θ (a_{j} + b_{j} c_{j} + θ c_{i})}{4 b_{i} b_{j} - θ^{2}})}^{2}, i, j = 1, 2, i \neq j .$

3 The equilibrium analysis of the fuzzy Bertrand game

In this section, the market demand of Bertrand game is posited to be fuzzy. Then, based on credibility theory, a fuzzy Bertrand game model is established to analyze the optimal price strategies of two players. Under a fuzzy environment, both two players desire to pursuit their profits maximum. First, notations and assumptions are used to formulate the model. Second, a fuzzy Bertrand game model is constructed and the equilibrium analysis of this game is conducted.

3.1 Notations and assumptions

Notations for the fuzzy Bertrand game model

p_i: The player i’s unit sale price, which is the decision variable, p_i > 0, i = 1, 2;

${\tilde{a}}_{i}$ : The potential market demand of player i, which is a triangular fuzzy variable, i = 1, 2;

${\tilde{b}}_{i}$ : The coefficient of price elasticity of player i, which is a triangular fuzzy variable, i = 1, 2;

$\tilde{θ}$ : The cross-price sensitivity coefficient of players, which is a triangular fuzzy variable;

c_i: The unit cost incurred by player i, c_i > 0, i = 1, 2;

${\tilde{Q}}_{i}$ : The market demand of player i, i = 1, 2;

${\tilde{π}}_{i}$ : Player i’s profit, i = 1, 2;

$\tilde{π}$ : The overall profit.

Based on the above descriptions, we have the following assumptions.

Assumptions

i) According to the demand function of the classical Bertrand game model, we have the fuzzy market demand function ${\tilde{Q}}_{i} (p_{i}, p_{j})$ of player i: ${\tilde{Q}}_{i} (p_{i}, p_{j}) = {\tilde{a}}_{i} - {\tilde{b}}_{i} p_{i} + {\tilde{θ}}_{i} p_{j}, i, j = 1, 2, i \neq j .$ (10)

The fuzzy market demand ${\tilde{Q}}_{i} (p_{i}, p_{j})$ must be nonnegative, thus $Pos ({{\tilde{a}}_{i} - {\tilde{b}}_{i} p_{i} < 0}) = 0$ .

The parameters ${\tilde{a}}_{i} = (a_{il}, a_{im}, a_{ir}), {\tilde{b}}_{i} = (b_{il}, b_{im}, b_{ir})$ and $\tilde{θ} = (θ_{l}, θ_{m}, θ_{r}), i = 1, 2$ , are independent and non-negative triangular fuzzy variables. Then, the fuzzy market demand function ${\tilde{Q}}_{i} (p_{i}, p_{j})$ of player i also can be written as: $\begin{matrix} {\tilde{Q}}_{i} (p_{i}, p_{j}) \\ = (a_{il} - b_{ir} p_{i} + θ_{l} p_{j}, a_{im} - b_{im} p_{i} + θ_{m} p_{j}, \\ a_{ir} - b_{il} p_{i} + θ_{r} p_{j}), i, j = 1, 2 . \end{matrix}$

ii) Since the decision variable of the fuzzy Bertrand game is price, the marginal cost c_i (i = 1, 2) is assumed to be a constant.

Therefore, the profit function ${\tilde{π}}_{i} (p_{i}, p_{j})$ of player i can be given as follow: ${\tilde{π}}_{i} (p_{i}, p_{j}) = (p_{i} - c_{i}) {\tilde{Q}}_{i} (p_{i}, p_{j}), i = 1, 2 .$ (11)

3.2 The fuzzy Bertrand game model analysis

Given that people always tend to have the behavioral characteristic of risk aversion in the face of gain, in this subsection, we consider a fuzzy Bertrand game where two players are risk aversion. Under a fuzzy environment, α-optimistic value is always used to depict the behavior that the decision-makers are risk aversion, where α denotes the confidence level of players. At a certain confidence level α, two players maximize their α-optimistic value of their profits.

Thus, two players are posited to obey optimistic value criterion. If player 2 choose a price strategy $p_{2}^{*}$ , then the best price strategy selection of player 1 is the optimal solution of the following fuzzy dependent-chance programming model

$\begin{matrix} max_{p_{1}} max_{π_{1}} Cr {{\tilde{π}}_{1} (p_{1}, p_{2}^{*}) \geq π_{1}} \geq α_{1} \\ s . t . {\begin{matrix} Pos ({{\tilde{a}}_{1} - {\tilde{b}}_{1} p_{1} < 0}) = 0 \\ Pos ({{\tilde{a}}_{2} - {\tilde{b}}_{2} p_{2} < 0}) = 0 \end{matrix} \end{matrix}$ (12) where α₁ denotes the confidence level of player 1 and π₁ is the real number, and $Pos ({{\tilde{a}}_{1} - {\tilde{b}}_{1} p_{1} < 0}) = 0, Pos ({{\tilde{a}}_{2} - {\tilde{b}}_{2} p_{2} < 0}) = 0$ guarantee that the fuzzy market demand conforms to reality.

Similarly, if player 1 choose a price strategy $p_{1}^{*}$ , then the best price strategy selection of player 2 is the optimal solution of the following fuzzy dependent-chance programming model

$\begin{matrix} max_{q_{2}} max_{π_{2}} Cr {{\tilde{π}}_{2} (p_{1}^{*}, p_{2}) \geq π_{2}} \geq α_{2} \\ s . t . {\begin{matrix} Pos ({{\tilde{a}}_{1} - {\tilde{b}}_{1} p_{1} < 0}) = 0 \\ Pos ({{\tilde{a}}_{2} - {\tilde{b}}_{2} p_{2} < 0}) = 0 \end{matrix} \end{matrix}$ (13) where α₂ denotes the confidence level of player 2, π₂ is the real number, and $Pos ({{\tilde{a}}_{1} - {\tilde{b}}_{1} p_{1} < 0}) = 0$ and $Pos ({{\tilde{a}}_{2} - {\tilde{b}}_{2} p_{2} < 0}) = 0$ guarantee that the fuzzy market demand conforms to reality.

Then, we define a (α₁, α₂)-optimistic equilibrium price of the fuzzy Bertrand game.

Definition 11. A strategy profile $(p_{1}^{*}, p_{2}^{*})$ is called as (α₁, α₂)-optimistic equilibrium price of the fuzzy Bertrand game if it satisfies $\begin{matrix} π_{1} (p_{1}^{*}, p_{2}^{*}) & = & max Cr {π_{1} | {\tilde{π}}_{1} (p_{1}^{*}, p_{2}^{*}) \geq π_{1}} \\ \geq & max Cr {π_{1} | {\tilde{π}}_{1} (p_{1}, p_{2}^{*}) \geq π_{1}}, \\ \forall p_{1} > 0, \\ π_{2} (p_{1}^{*}, p_{2}^{*}) & = & max Cr {π_{2} | {\tilde{π}}_{2} (p_{1}^{*}, p_{2}^{*}) \geq π_{2}} \\ \geq & max Cr {π_{2} | {\tilde{π}}_{2} (p_{1}^{*}, p_{2}) \geq π_{2}}, \\ \forall p_{2} > 0 . \end{matrix}$

When player 1 is risk aversion, according to optimistic value criterion and Equations (7) and (12), the profit of the player 1 is

$\begin{matrix} {\tilde{π}}_{1} (p_{1}, p_{2})_{sup} (α_{1}) \\ = ((p_{1} - c_{1}) (a_{1 l} - b_{1 r} p_{1} + θ_{l} p_{2}), \\ (p_{1} - c_{1}) (a_{1 m} - b_{1 m} p_{1} + θ_{m} p_{2}), \\ (p_{1} - c_{1}) (a_{1 r} - b_{1 l} p_{1} + θ_{r} p_{2}))_{sup} (α_{1}) . \end{matrix}$ (14)

Then,

$\begin{matrix} {\tilde{π}}_{1} (p_{1}, p_{2})_{sup} (α_{1}) \\ = {\begin{matrix} (p_{1} - c_{1}) & if α_{1} \leq 0.5 \\ [\begin{matrix} 2 α_{1} (a_{1 m} - b_{1 m} p_{1} + θ_{m} p_{2}) + \\ (1 - 2 α_{1}) (a_{1 r} - b_{1 l} p_{1} + θ_{r} p_{2}) \end{matrix}] \\ (p_{1} - c_{1}) & if α_{1} > 0.5 \\ [\begin{matrix} (2 α_{1} - 1) (a_{1 l} - b_{1 r} p_{1} + θ_{l} p_{2}) + \\ 2 (1 - α_{1}) (a_{1 m} - b_{1 m} p_{1} + θ_{m} p_{2}) \end{matrix}] \end{matrix} \end{matrix}$ (15)

So, we can have that

$\begin{matrix} {\tilde{π}}_{1} (p_{1}, p_{2})_{sup} (α_{1}) \\ = (p_{1} - c_{1}) ({\tilde{a}}_{1 sup} (α_{1}) - {\tilde{b}}_{1 inf} (α_{1}) p_{1} \\ + {\tilde{θ}}_{sup} (α_{1}) p_{2}) . \end{matrix}$ (16)

Similarly, when player 2 is risk aversion, according to optimistic value criterion and Equations (7) and (12), the profit of player 2 is

$\begin{matrix} {\tilde{π}}_{2} (p_{1}, p_{2})_{sup} (α_{2}) \\ = (p_{2} - c_{2}) ({\tilde{a}}_{2 sup} (α_{2}) - {\tilde{b}}_{2 inf} (α_{2}) p_{2} \\ + {\tilde{θ}}_{sup} (α_{2}) p_{1}) . \end{matrix}$ (17)

To obtain the (α₁, α₂)-optimistic equilibrium price, it is necessary to solve the following quadratic programming model. ${\begin{matrix} max_{p_{1}} {\tilde{π}}_{1} (p_{1}, p_{2})_{sup} (α_{1}) \\ max_{p_{2}} {\tilde{π}}_{2} (p_{1}, p_{2})_{sup} (α_{2}) \\ subject 1 pt 1 pt to \\ 1 pt 1 pt 1 pt 1 pt 1 pt 1 pt 1 pt Pos ({{\tilde{a}}_{1} - {\tilde{b}}_{1} p_{1} < 0}) = 0 \\ 1 pt 1 pt 1 pt 1 pt 1 pt 1 pt Pos ({{\tilde{a}}_{2} - {\tilde{b}}_{2} p_{2} < 0}) = 0 \end{matrix}$ (18)

Proposition 1. In the fuzzy Bertrand game, if $Pos ({{\tilde{a}}_{1} - {\tilde{b}}_{1} p_{1}^{*} < 0}) = 0$ and $Pos ({{\tilde{a}}_{2} - {\tilde{b}}_{2} p_{2}^{*} < 0}) = 0$ , the two players’ (α₁, α₂)-optimistic equilibrium price, denoted by $p_{1}^{*}$ and $p_{2}^{*}$ respectively, are given as follows $\begin{matrix} p_{1}^{*} & = & \frac{{\tilde{θ}}_{sup} (α_{1}) [{\tilde{a}}_{2 sup} (α_{2}) + {\tilde{b}}_{2 inf} (α_{2}) c_{2}]}{4 {\tilde{b}}_{1 inf} (α_{1}) {\tilde{b}}_{2 inf} (α_{2}) - {\tilde{θ}}_{sup} (α_{1}) {\tilde{θ}}_{sup} (α_{2})} \\ + \frac{2 {\tilde{b}}_{2 inf} (α_{2}) [{\tilde{a}}_{1 sup} (α_{1}) + {\tilde{b}}_{1 inf} (α_{1}) c_{1}]}{4 {\tilde{b}}_{1 inf} (α_{1}) {\tilde{b}}_{2 inf} (α_{2}) - {\tilde{θ}}_{sup} (α_{1}) {\tilde{θ}}_{sup} (α_{2})}, \end{matrix}$ (19) $\begin{matrix} p_{2}^{*} & = & \frac{2 {\tilde{b}}_{1 inf} (α_{1}) [{\tilde{a}}_{2 sup} (α_{2}) + {\tilde{b}}_{2 inf} (α_{2}) c_{2}]}{4 {\tilde{b}}_{1 inf} (α_{1}) {\tilde{b}}_{2 inf} (α_{2}) - {\tilde{θ}}_{sup} (α_{1}) {\tilde{θ}}_{sup} (α_{2})} \\ + \frac{{\tilde{θ}}_{sup} (α_{2}) [{\tilde{a}}_{1 sup} (α_{1}) + {\tilde{b}}_{1 inf} (α_{1}) c_{1}]}{4 {\tilde{b}}_{1 inf} (α_{1}) {\tilde{b}}_{2 inf} (α_{2}) - {\tilde{θ}}_{sup} (α_{1}) {\tilde{θ}}_{sup} (α_{2})} . \end{matrix}$ (20)

Proof. Firstly, the optimal price $p_{1}^{*}$ and $p_{2}^{*}$ have to meet the condition that $Pos ({{\tilde{a}}_{1} - {\tilde{b}}_{1} p_{1} < 0}) = 0$ and $Pos ({{\tilde{a}}_{1} - {\tilde{b}}_{2} p_{2} < 0}) = 0$ .

Secondly, taking the second-order derivatives of ${\tilde{π}}_{1} (p_{1}, p_{2})_{sup} (α 1)$ with respect to p₁, we have $\frac{\partial^{2} ({\tilde{π}}_{1} (p_{1}, p_{2})_{sup} (α_{1}))}{\partial p_{1}^{2}} = - 2 {\tilde{b}}_{1 inf} (α_{1}) < 0 .$

So ${\tilde{π}}_{1} (p_{1}, p_{2})_{sup} (α_{1})$ is jointly strictly concave in p₁, which means that an optimal solution for the quadratic programming model $max_{p_{1}} {\tilde{π}}_{1} (p_{1}, p_{2})_{sup} (α_{1})$ exists.

The first order derivatives of ${\tilde{π}}_{1} (p_{1}, p_{2})_{sup} (α_{1})$ with respect to p₁ is obtained as follow

$\begin{matrix} \frac{\partial ({\tilde{π}}_{1} (p_{1}, p_{2})_{sup} (α_{1}))}{\partial p_{1}} \\ = {\tilde{a}}_{1 sup} (α_{1}) - 2 {\tilde{b}}_{1 inf} (α_{1}) p_{1} \\ + {\tilde{θ}}_{sup} (α_{1}) p_{2} + {\tilde{b}}_{1 inf} (α_{1}) c_{1} . \end{matrix}$ (21)

Let Equation (21) equals to zero, we have

$\begin{matrix} {\tilde{a}}_{1 sup} (α_{1}) - 2 {\tilde{b}}_{1 inf} (α_{1}) p_{1} \\ + {\tilde{θ}}_{sup} (α_{1}) p_{2} + {\tilde{b}}_{1 inf} (α_{1}) c_{1} = 0 . \end{matrix}$ (22)

Similarly, taking the second-order derivatives of ${\tilde{π}}_{2} (p_{1}, p_{2})_{sup} (α^{2})$ with respect to p₂, we have $\frac{\partial^{2} ({\tilde{π}}_{2} (p_{1}, p_{2})_{sup} (α_{1}))}{\partial p_{2}^{2}} = - 2 {\tilde{b}}_{2 inf} (α_{2}) < 0 .$ (23)

So ${\tilde{π}}_{2} (p_{1}, p_{2})_{sup} (α_{2})$ is jointly strictly concave in p₂, which means that an optimal solution for the quadratic programming model $max_{p_{2}} {\tilde{π}}_{2} (p_{1}, p_{2})_{sup} (α_{2})$ exists.

The first order derivatives of ${\tilde{π}}_{2} (p_{1}, p_{2})_{sup} (α_{2})$ with respect to p₂ is obtained as follow

$\begin{matrix} \frac{\partial ({\tilde{π}}_{2} (p_{1}, p_{2})_{sup} (α_{2}))}{\partial p_{2}} \\ = {\tilde{a}}_{2 sup} (α_{2}) - 2 {\tilde{b}}_{2 inf} (α_{2}) p_{2} \\ + {\tilde{θ}}_{sup} (α_{2}) p_{1} + {\tilde{b}}_{1 inf} (α_{2}) c_{2} . \end{matrix}$ (24)

Let Equation (23) equals to zero, we have

$\begin{matrix} {\tilde{a}}_{2 sup} (α_{2}) - 2 {\tilde{b}}_{2 inf} (α_{2}) p_{2} + {\tilde{θ}}_{sup} (α_{2}) p_{1} \\ + {\tilde{b}}_{1 inf} (α_{2}) c_{2} = 0 . \end{matrix}$ (25)

Finally, according to Equation (22) and (24), the (α₁, α₂)-optimistic equilibrium prices are obtained as follows, $\begin{matrix} p_{1}^{*} & = & \frac{{\tilde{θ}}_{sup} (α_{1}) [{\tilde{a}}_{2 sup} (α_{2}) + {\tilde{b}}_{2 inf} (α_{2}) c_{2}]}{4 {\tilde{b}}_{1 inf} (α_{1}) {\tilde{b}}_{2 inf} (α_{2}) - {\tilde{θ}}_{sup} (α_{1}) {\tilde{θ}}_{sup} (α_{2})} \\ + \frac{2 {\tilde{b}}_{2 inf} (α_{2}) [{\tilde{a}}_{1 sup} (α_{1}) + {\tilde{b}}_{1 inf} (α_{1}) c_{1}]}{4 {\tilde{b}}_{1 inf} (α_{1}) {\tilde{b}}_{2 inf} (α_{2}) - {\tilde{θ}}_{sup} (α_{1}) {\tilde{θ}}_{sup} (α_{2})}, \end{matrix}$ and $\begin{matrix} p_{2}^{*} & = & \frac{2 {\tilde{b}}_{1 inf} (α_{1}) [{\tilde{a}}_{2 sup} (α_{2}) + {\tilde{b}}_{2 inf} (α_{2}) c_{2}]}{4 {\tilde{b}}_{1 inf} (α_{1}) {\tilde{b}}_{2 inf} (α_{2}) - {\tilde{θ}}_{sup} (α_{1}) {\tilde{θ}}_{sup} (α_{2})} \\ + \frac{{\tilde{θ}}_{sup} (α_{2}) [{\tilde{a}}_{1 sup} (α_{1}) + {\tilde{b}}_{1 inf} (α_{1}) c_{1}]}{4 {\tilde{b}}_{1 inf} (α_{1}) {\tilde{b}}_{2 inf} (α_{2}) - {\tilde{θ}}_{sup} (α_{1}) {\tilde{θ}}_{sup} (α_{2})} . \end{matrix}$

These complete the proof of Proposition 1. □

According to the different range values of the confidence levels α₁ and α₂, the (α₁, α₂)-optimistic equilibrium prices of the fuzzy Bertrand game are shown in Table 1.

Table 1

(α₁, α₂)-optimistic equilibrium price of the fuzzy Bertrand game

Confidence level	(α₁, α₂)-optimistic equilibrium price
α₁ ≤ 0.5 and α₂ ≤ 0.5	$\begin{matrix} p_{1}^{*} = \frac{[2 α_{1} θ_{m} + (1 - 2 α_{1}) θ_{r}] {2 α_{2} a_{2 m} + (1 - 2 α_{2}) a_{2 r} + [(1 - 2 α_{2}) b_{2 l} + 2 α_{2} b_{2 m}] c_{2}}}{4 [(1 - 2 α_{1}) b_{1 l} + 2 α_{1} b_{1 m}] [(1 - 2 α_{2}) b_{2 l} + 2 α_{2} b_{2 m}] - [2 α_{1} θ_{m} + (1 - 2 α_{1}) θ_{r}] [2 α_{2} θ_{m} + (1 - 2 α_{2}) θ_{r}]}, \\ + \frac{2 [(1 - 2 α_{2}) b_{2 l} + 2 α_{2} b_{2 m}] {2 α_{1} a_{1 m} + (1 - 2 α_{1}) a_{1 r} + [(1 - 2 α_{1}) b_{1 l} + 2 α_{1} b_{1 m}] c_{1}}}{4 [(1 - 2 α_{1}) b_{1 l} + 2 α_{1} b_{1 m}] [(1 - 2 α_{2}) b_{2 l} + 2 α_{2} b_{2 m}] - [2 α_{1} θ_{m} + (1 - 2 α_{1}) θ_{r}] [2 α_{2} θ_{m} + (1 - 2 α_{2}) θ_{r}]}, \end{matrix}$
	$\begin{matrix} p_{2}^{*} = \frac{2 [(1 - 2 α_{2}) b_{1 l} + 2 α_{1} b_{1 m}] {2 α_{2} a_{2 m} + (1 - 2 α_{2}) a_{2 r} + [(1 - 2 α_{2}) b_{2 l} + 2 α_{2} b_{2 m}] c_{2}}}{4 [(1 - 2 α_{1}) b_{1 l} + 2 α_{1} b_{1 m}] [(1 - 2 α_{2}) b_{2 l} + 2 α_{2} b_{2 m}] - [2 α_{1} θ_{m} + (1 - 2 α_{1}) θ_{r}] [2 α_{2} θ_{m} + (1 - 2 α_{2}) θ_{r}]}, \\ + \frac{[2 α_{2} θ_{m} + (1 - 2 α_{2}) θ_{r}] {2 α_{1} a_{1 m} + (1 - 2 α_{1}) a_{1 r} + [(1 - 2 α_{1}) b_{1 l} + 2 α_{1} b_{1 m}] c_{1}}}{4 [(1 - 2 α_{1}) b_{1 l} + 2 α_{1} b_{1 m}] [(1 - 2 α_{2}) b_{2 l} + 2 α_{2} b_{2 m}] - [2 α_{1} θ_{m} + (1 - 2 α_{1}) θ_{r}] [2 α_{2} θ_{m} + (1 - 2 α_{2}) θ_{r}]} \end{matrix}$
α₁ ≤ 0.5 and α₂ > 0.5	$\begin{matrix} p_{1}^{*} = \frac{[2 α_{1} θ_{m} + (1 - 2 α_{1}) θ_{r}] {(2 α_{2} - 1) a_{2 l} + 2 (1 - α_{2}) a_{2 m} + [2 (1 - α_{2}) b_{2 m} + (2 α_{2} - 1) b_{2 r}] c_{2}}}{4 [(1 - 2 α_{1}) b_{1 l} + 2 α_{1} b_{1 m}] [2 (1 - α_{2}) b_{2 m} + (2 α_{2} - 1) b_{2 r}] - [2 α_{1} θ_{m} + (1 - 2 α_{1}) θ_{r}] [(2 α_{2} - 1) θ_{l} + 2 (1 - α_{2}) θ_{m}]} \\ + \frac{2 [2 (1 - 2 α_{1}) b_{2 m} + (2 α_{2} - 1) b_{2 r}] {2 α_{1} a_{1 m} + (1 - 2 α_{1}) a_{1 r} + [(1 - 2 α_{1}) b_{1 l} + 2 α_{1} b_{1 m}] c_{1}}}{4 [(1 - 2 α_{1}) b_{1 l} + 2 α_{1} b_{1 m}] [2 (1 - α_{2}) b_{2 m} + (2 α_{2} - 1) b_{2 r}] - [2 α_{1} θ_{m} + (1 - 2 α_{1}) θ_{r}] [(2 α_{2} - 1) θ_{l} + 2 (1 - α_{2}) θ_{m}]}, \end{matrix}$
	$\begin{matrix} p_{2}^{*} = \frac{2 [(1 - 2 α_{1}) b_{1 l} + 2 α_{1} b_{1 m}] {(2 α_{2} - 1) a_{2 l} + 2 (1 - α_{2}) a_{2 m} + [2 (1 - α_{2}) b_{2 m} + (2 α_{2} - 1) b_{2 r}] c_{2}}}{4 [(1 - 2 α_{1}) b_{1 l} + 2 α_{1} b_{1 m}] [2 (1 - α_{2}) b_{2 m} + (2 α_{2} - 1) b_{2 r}] - [2 α_{1} θ_{m} + (1 - 2 α_{1}) θ_{r}] [(2 α_{2} - 1) θ_{l} + 2 (1 - α_{2}) θ_{m}]} \\ + \frac{[(2 α_{2} - 1) θ_{l} + 2 (1 - α_{2}) θ_{m}] {2 α_{1} a_{1 m} + (1 - 2 α_{1}) a_{1 r} + [(1 - 2 α_{1}) b_{1 l} + 2 α_{1} b_{1 m}] c_{1}}}{4 [(1 - 2 α_{1}) b_{1 l} + 2 α_{1} b_{1 m}] [2 (1 - α_{2}) b_{2 m} + (2 α_{2} - 1) b_{2 r}] - [2 α_{1} θ_{m} + (1 - 2 α_{1}) θ_{r}] [(2 α_{2} - 1) θ_{l} + 2 (1 - α_{2}) θ_{m}]} \end{matrix}$
α₁ > 0.5 and α₂ ≤ 0.5	$\begin{matrix} p_{1}^{*} = \frac{[(2 α_{1} - 1) θ_{l} + 2 (1 - α_{1}) θ_{m}] {2 α_{2} a_{2 m} + (1 - 2 α_{2}) a_{2 r} + [(1 - 2 α_{2}) b_{2 l} + 2 α_{2} b_{2 m}] c_{2}}}{4 [2 (1 - α_{1}) b_{1 m} + (2 α_{1} - 1) b_{1 r}] [(1 - 2 α_{2}) b_{2 l} + 2 α_{2} b_{2 m}] - [(2 α_{1} - 1) θ_{l} + 2 (1 - α_{1}) θ_{m}] [2 α_{2} θ_{m} + (1 - 2 α_{2}) θ_{r}]} \\ + \frac{2 [(1 - 2 α_{2}) b_{2 l} + 2 α_{2} b_{2 m}] {(2 α_{1} - 1) a_{1 l} + 2 (1 - α_{1}) a_{1 m} + [2 (1 - α_{2}) b_{1 m} + (2 α_{1} - 1) b_{1 r}] c_{1}}}{4 [2 (1 - α_{1}) b_{1 m} + (2 α_{1} - 1) b_{1 r}] [(1 - 2 α_{2}) b_{2 l} + 2 α_{2} b_{2 m}] - [(2 α_{1} - 1) θ_{l} + 2 (1 - α_{1}) θ_{m}] [2 α_{2} θ_{m} + (1 - 2 α_{2}) θ_{r}]}, \end{matrix}$
	$\begin{matrix} p_{2}^{*} = \frac{2 [2 (1 - α_{1}) b_{1 m} + (2 α_{1} - 1) b_{1 r}] {2 α_{2} a_{2 m} + (1 - 2 α_{2}) a_{2 r} + [(1 - 2 α_{2}) b_{2 l} + 2 α_{2} b_{2 m}] c_{2}}}{4 [2 (1 - α_{1}) b_{1 m} + (2 α_{1} - 1) b_{1 r}] [(1 - 2 α_{2}) b_{2 l} + 2 α_{2} b_{2 m}] - [(2 α_{1} - 1) θ_{l} + 2 (1 - α_{1}) θ_{m}] [2 α_{2} θ_{m} + (1 - 2 α_{2}) θ_{r}]} \\ + \frac{[2 α_{2} θ_{m} + (1 - 2 α_{2}) θ_{r}] {(2 α_{1} - 1) a_{1 l} + 2 (1 - α_{1}) a_{1 m} + [2 (1 - α_{1}) b_{1 m} + (2 α_{1} - 1) b_{1 r}] c_{1}}}{4 [2 (1 - α_{1}) b_{1 m} + (2 α_{1} - 1) b_{1 r}] [(1 - 2 α_{2}) b_{2 l} + 2 α_{2} b_{2 m}] - [(2 α_{1} - 1) θ_{l} + 2 (1 - α_{1}) θ_{m}] [2 α_{2} θ_{m} + (1 - 2 α_{2}) θ_{r}]} \end{matrix}$
α₁ > 0.5 and α₂ > 0.5	$\begin{matrix} p_{1}^{*} = \frac{[(2 α_{1} - 1) θ_{l} + 2 (1 - α_{1}) θ_{m}] {(2 α_{2} - 1) a_{2 l} + 2 (1 - α_{2}) a_{2 m} + [2 (1 - α_{2}) b_{2 m} + (2 α_{2} - 1) b_{2 r}] c_{2}}}{4 [2 (1 - α_{1}) b_{1 m} + (2 α_{1} - 1) b_{1 r}] [2 (1 - α_{2}) b_{2 m} + (2 α_{2} - 1) b_{2 r}] - [(2 α_{1} - 1) θ_{l} + 2 (1 - α_{1}) θ_{m}] [(2 α_{2} - 1) θ_{l} + 2 (1 - α_{2}) θ_{m}]} \\ + \frac{2 [2 (1 - α_{2}) b_{2 m} + (2 α_{2} - 1) b_{2 r}] {(2 α_{1} - 1) a_{1 l} + 2 (1 - α_{1}) a_{1 m} + [2 (1 - α_{1}) b_{1 m} + (2 α_{2} - 1) b_{1 r}] c_{1}}}{4 [2 (1 - α_{1}) b_{1 m} + (2 α_{1} - 1) b_{1 r}] [2 (1 - α_{2}) b_{2 m} + (2 α_{2} - 1) b_{2 r}] - [(2 α_{1} - 1) θ_{l} + 2 (1 - α_{1}) θ_{m}] [(2 α_{2} - 1) θ_{l} + 2 (1 - α_{2}) θ_{m}]}, \end{matrix}$
	$\begin{matrix} p_{2}^{*} = \frac{2 [(1 - 2 α_{2}) b_{1 l} + 2 α_{1} b_{1 m}] {2 α_{2} a_{2 m} + (1 - 2 α_{2}) a_{2 r} + [(1 - 2 α_{2}) b_{2 l} + 2 α_{2} b_{2 m}] c_{2}}}{4 [(1 - 2 α_{1}) b_{1 l} + 2 α_{1} b_{1 m}] [(1 - 2 α_{2}) b_{2 l} + 2 α_{2} b_{2 m}] - [2 α_{1} θ_{m} + (1 - 2 α_{1}) θ_{r}] [2 α_{2} θ_{m} + (1 - 2 α_{2}) θ_{r}]}, \\ + \frac{[2 α_{2} θ_{m} + (1 - 2 α_{2}) θ_{r}] {2 α_{1} a_{1 m} + (1 - 2 α_{1}) a_{1 r} + [(1 - 2 α_{1}) b_{1 l} + 2 α_{1} b_{1 m}] c_{1}}}{4 [(1 - 2 α_{1}) b_{1 l} + 2 α_{1} b_{1 m}] [(1 - 2 α_{2}) b_{2 l} + 2 α_{2} b_{2 m}] - [2 α_{1} θ_{m} + (1 - 2 α_{1}) θ_{r}] [2 α_{2} θ_{m} + (1 - 2 α_{2}) θ_{r}]} . \end{matrix}$

From Table 1, if α₁ = α₂ = 0.5, then the (α₁, α₂)-optimistic equilibrium prices of two players are $\begin{matrix} p_{1}^{*} & = & \frac{θ_{m} (a_{2 m} + 2 b_{2 m} c_{2}) + 2 b_{2 m} (a_{1 m} + b_{1 m} c_{1})}{4 b_{1 m} b_{2 m} - θ_{m}^{2}}, \\ p_{2}^{*} & = & \frac{2 b_{1 m} (a_{2 m} + b_{2 m} c_{2}) + θ_{m} (a_{1 m} + b_{1 m} c_{1})}{4 b_{1 m} b_{2 m} - θ_{m}^{2}}, \end{matrix}$ which are the same as those of the classical Bertrand game in crisp environment; if α₁ ≠ 0.5 or α₂ ≠ 0.5, the (α₁, α₂)-optimistic equilibrium prices of two players are altered by different confidence levels. Then, risk aversion of two players influences the (α₁, α₂)-optimistic equilibrium prices. For the complication of the expressions of the (α₁, α₂)-optimistic equilibrium prices, some numerical studies are conducted to investigate how the (α₁, α₂)-optimistic equilibrium prices change as two players are risk averse.

4 Numerical studies

In China, fixed broadband is an important part of the Telecom industry as well as a representative oligopoly industry. There are two competitive companies in Chinese fixed broadband market, i.e., China Unicom and China Telecom. At present, two companies roughly enjoy 80 percent of the Chinese market for fixed broadband, and are famous for their favorable price and good quality [24]. So, China Unicom and China Telecom are engaged in the Bertrand competition game [5].

Recently, two companies are confronted with many different risks and a cloud of uncertainties. As facing these risks and uncertainties, most companies tend to show the characteristics behavior of risk aversion. To provide some management highlights for managers, the proposed fuzzy Bertrand game model is used to analyze the (α₁, α₂)-optimistic equilibrium prices of China Unicom and China Telecom.

For convenience, the fixed broadband products of China Telecom and China Unicom are called as product 1 and 2, respectively. The fixed broadband costs of China Telecom and China Unicom are $1 and $2 per unit, respectively. Under a fuzzy environment, the market demand is assumed as a triangular fuzzy variable. The confidence levels of China Telecom and China Unicom are denoted by α₁ and α₂ respectively. The corresponding fuzzy parameters are shown in Table 2.

Table 2
Values of triangular fuzzy variables

Triangular fuzzy variables Value

Potential market demand of product 1 a₁ (16, 20, 22)

Potential market demand of product 2 a₂ (14, 16, 20)

Price elasticity coefficient of product 1 b₁ (2, 3, 5)

Price elasticity coefficient of product 2 b₂ (0.6, 2, 2.8)

Cross-price elasticity coefficients θ (0.7, 1, 1.7)

Triangular fuzzy variables	Value
Potential market demand of product 1 a₁	(16, 20, 22)
Potential market demand of product 2 a₂	(14, 16, 20)
Price elasticity coefficient of product 1 b₁	(2, 3, 5)
Price elasticity coefficient of product 2 b₂	(0.6, 2, 2.8)
Cross-price elasticity coefficients θ	(0.7, 1, 1.7)

4.1 Results of numerical studies

According to the above analysis, there are four cases:

If α₁ ≤ 0.5 and α₂ ≤ 0.5, then we have the (α₁, α₂)-optimistic equilibrium prices are given by $\begin{matrix} p_{1}^{*} & = & \frac{2 (1.7 - 1.4 α_{1}) (10.6 - 1.2 α_{2}) + 4 (0.6 + 2.8 α_{2}) (12 - α_{1})}{8 (1 + α_{1}) (0.6 + 2.8 α_{2}) - (1.7 - 1.4 α_{1}) (1.7 - 1.4 α_{2})}, \\ p_{2}^{*} & = & \frac{8 (1 + α_{1}) (10.6 - 1.2 α_{2}) + 2 (1.7 - 1.4 α_{2}) (12 - α_{1})}{8 (1 + α_{1}) (0.6 + 2.8 α_{2}) - (1.7 - 1.4 α_{1}) (1.7 - 1.4 α_{2})} . \end{matrix}$

If α₁ ≤ 0.5 and α₂ > 0.5, then we have the (α₁, α₂)-optimistic equilibrium prices are shown as follow $\begin{matrix} p_{1}^{*} & = & \frac{2 (1.7 - 1.4 α_{1}) (10.2 - 0.4 α_{2}) + 4 (1.2 + 1.6 α_{2}) (12 - α_{1})}{8 (1 + α_{1}) (1.2 + 1.6 α_{2}) - (1.7 - 1.4 α_{1}) (1.3 - 0.6 α_{2})}, \\ p_{2}^{*} & = & \frac{8 (1 + α_{1}) (10.2 - 0.4 α_{2}) + 2 (1.3 - 0.6 α_{2}) (12 - α_{1})}{8 (1 + α_{1}) (1.2 + 1.6 α_{2}) - (1.7 - 1.4 α_{1}) (1.3 - 0.6 α_{2})} . \end{matrix}$

If α₁ > 0.5 and α₂ ≤ 0.5, then we have the (α₁, α₂)-optimistic equilibrium prices are given by $\begin{matrix} p_{1}^{*} & = & \frac{2 (1.3 - 0.6 α_{1}) (10.6 - 1.2 α_{2}) + 2 (0.6 + 2.8 α_{2}) (25 - 4 α_{1})}{4 (1 + 4 α_{1}) (0.6 + 2.8 α_{2}) - (1.3 - 0.6 α_{1}) (1.7 - 1.4 α_{2})}, \\ p_{2}^{*} & = & \frac{4 (1 + 4 α_{1}) (10.6 - 1.2 α_{2}) + (1.7 - 1.4 α_{2}) (25 - 4 α_{1})}{4 (1 + 4 α_{1}) (0.6 + 2.8 α_{2}) - (1.3 - 0.6 α_{1}) (1.7 - 1.4 α_{2})} . \end{matrix}$

If α₁ > 0.5 and α₂ > 0.5, then we have the (α₁, α₂)-optimistic equilibrium prices are shown as follow $\begin{matrix} p_{1}^{*} & = & \frac{2 (1.3 - 0.6 α_{1}) (10.2 - 0.4 α_{2}) + 2 (1.2 + 1.6 α_{2}) (25 - 4 α_{1})}{4 (1 + 4 α_{1}) (1.2 + 1.6 α_{2}) - (1.3 - 0.6 α_{1}) (1.3 - 0.6 α_{2})}, \\ p_{2}^{*} & = & \frac{4 (1 + 4 α_{1}) (10.2 - 0.4 α_{2}) + (1.3 - 0.6 α_{2}) (25 - 4 α_{1})}{4 (1 + 4 α_{1}) (1.2 + 1.6 α_{2}) - (1.3 - 0.6 α_{1}) (1.3 - 0.6 α_{2})} . \end{matrix}$

4.2 Sensitivity analysis of the confidence levels

In this section, we explore the effect of confidence levels on the (α₁, α₂)-optimistic equilibrium prices and maximal profits of China Telecom and China Unicom. First, suppose α₁ and α₂ are continuous variables with the range of 0 to 1, which reflects the degrees of two companies’ risk aversion. We define that the degree of a company’s risk aversion is low if the confidence level of a company is no more than 0.5 and vice versa. So, in this paper, four situations for sensitivity analysis of companies’ risk aversion degree on (α₁, α₂)-optimistic equilibrium price and profits are analyzed, including 1) both the risk aversion degree of two companies are low, i.e., α₁ ≤ 0.5 and α₂ ≤ 0.5; 2) the risk aversion degree α₁ of China Telecom is low and the risk aversion degree α₂ of China Unicom is high, i.e., α₁ ≤ 0.5 and α₂ > 0.5; 3) the risk aversion degree α₁ of China Telecom is high and the risk aversion degree α₂ of China Unicom is low, i.e., α₁ > 0.5 and α₂≤0.5; 4) both the risk aversion degrees of two companies are high, i.e., α₁ > 0.5 and α₂ > 0.5.

Low risk averse China Telecom and China Unicom (α₁≤0.5 and α₂≤0.5)

In this situation, the (α₁, α₂)-optimistic equilibrium prices and the profits of two companies are shown in Fig. 1. Since market demand must be positive, the confidence levels of two companies must be in the range of 0.2 to 0.5. Figure 1 shows that as the confidence levels of two companies increase, the prices and profits of two companies and the total profit decrease correspondingly. Moreover, the price and profit of one player decline faster than those of the other player as his own confidence level increases, which implies that player’s price and profit are greatly affected by his own risk aversion. China Unicom’s price is higher than China Telecom’s price.

Low risk averse China Telecom and high risk averse China Unicom (α₁≤0.5 and α₂ > 0.5)

In this situation, the results are shown in Fig. 2. Since market demand must be positive, the confidence level of China Telecom must be in the range of 0.2 to 0.5 and the confidence level of China Unicom must be in the range of 0.7 to 1. From Fig. 2, the change of the prices and profits of two companies with risk aversion in the second situation is similar to that in the first situation. However, the prices and profits of both two companies in the second situation are lower than those in the first situation. The profit of China Telecom is higher than that of China Unicom. China Telecom may benefit from his low risk aversion while China Unicom may be hurt by his high risk aversion.

High risk averse China Telecom is and Low risk averse China Unicom (α₁ > 0.5 and α₂≤0.5)

In this situation, the results are shown in Fig. 3. Since market demand must be positive, the confidence level of China Telecom must be in the range of 0.7 to 1 and the confidence level of China Unicom must be in the range of 0.2 to 0.5. Figure 3 shows that the changes in this situation are similar to those in the first situation. And, the price and profit of China Unicom are higher than those of China Telecom. China Unicom may benefit from his low risk aversion but China Telecom may be hurt by his high risk aversion.

High risk averse China Telecom and China Unicom (α₁ > 0.5, and α₂ > 0.5).

In this situation, the results are shown in Fig. 4. Since market demand must be positive, the confidence levels of two companies must be in the range of 0.7 to 1. Figure 4 shows that the changes are similar to those in the first situation. Moreover, the prices and profits of two companies in this situation are the lowest among the four situations, which means that a high risk aversion of player may hurt himself. The price of China Telecom is lower than that of China Unicom.

Fig.1

The impact of confidence level on the prices and profits of companies (α₁≤0.5, α₂≤0.5).

Fig.2

The impact of confidence level on the prices and profits of companies (α₁≤0.5, α₂ > 0.5).

Fig.3

The impact of confidence level on the prices and profits of companies (α₁ > 0.5, α₂≤0.5).

Fig.4

The impact of confidence level on the prices and profits of companies (α₁ > 0.5, α₂ > 0.5).

Comparing with the four situations, we can obtain the following conclusions. The prices and profits of companies in the first situation are the highest among the four situations, while the prices and profits of two companies in the last situation are the lowest. A higher risk aversion leads to a lower price and a lower profit. If two companies are lower risk averse, they may raise their prices to obtain more profits. Meanwhile, if two companies are higher risk averse, price competition between companies is more furious, and they may cut their price to attract more customers. Therefore, a risk averse player may benefit from his low risk aversion but be hurt by his high risk aversion. Moreover, the price of a player with lower risk aversion and higher cost is higher than that of a player with a higher risk aversion and a lower cost, while the price of a player with a higher risk aversion and a higher cost may be lower than that of a player with a lower risk aversion and a lower cost. The price and profit of a player are more greatly affected by his own risk aversion than the other player’s risk aversion.

4.3 Comparison analysis

In this section, we contrast the findings of our study with the results of Zhao et al. [50] and Wei and Zhao [44], who studied the pricing problem of substitutable products in a fuzzy supply chain with one retailer and two manufacturers. We use optimistic value criterion to model risk aversion of players and explore how the prices and profits change when players are risk averse in a fuzzy Bertrand game. We find that the increased risk aversion of players leads the prices and profits of the fuzzy Bertrand game to decline. However, Zhao et al. [50] and Wei and Zhao [44] adopted expected value criterion to analyze the effect of the fuzziness of market demand on the prices and expected profits, and found that the prices and expected profits of two manufactures are decreasing with the fuzziness of marked demand. There is no contradiction between our study and these two papers mentioned. In a fuzzy environment, the prices and profits of players may be affected by the fuzziness of market demand and risk aversion. Player with a lower risk aversion may prefer the upper limits of the fuzzy market demand and player with a higher risk aversion may prefer the lower limits of the fuzzy market demand. Therefore, combining our study and the works of Zhao et al. [50] and Wei and Zhao [44], both the fuzziness of market demand and the risk aversion of players may decline the prices and profits.

5 Conclusions

In this paper, we consider a Bertrand game between two players with two substitutable products under a fuzzy environment. The market demand for each product is posited as a triangular fuzzy variable. A fuzzy Bertrand game with risk aversion is investigated. Optimistic value criterion is applied to formulate risk aversion of players. The (α₁, α₂)-optimistic equilibrium prices of the fuzzy Bertrand game are analyzed. The results show that risk aversion of players has an important role in the price decisions of a fuzzy Bertrand game. First, a higher risk aversion leads to a lower price and profit. Therefore, a risk averse player may benefit from his low risk aversion but be hurt by his high risk aversion. Second, the price of a player with lower risk aversion and higher cost is higher than that of a player with a higher risk aversion and a lower cost, while the price of a player with a higher risk aversion and a higher cost may be lower than that of a player with a lower risk aversion and a lower cost. Finally, the price and profit of a player are more greatly affected by his own risk aversion than the other player’s risk aversion. This paper may offer a managerial insight into price competition for both theoretical researchers and practical practitioners.

For future research, an n-person fuzzy Bertrand game can be discussed; on the other hand, the degree of players’ risk aversion may be private information, so this game should be in a condition where player has no ideal of the other player’s risk aversion.

Footnotes

Acknowledgments

This research is supported by National Natural Science Foundation of China (Nos. 71671188, 71571192, and 71271217), the state key program of NSFC (No. 71431006), Natural Science Foundation of Hunan (No. 2016JJ1024) and the Postgraduate Innovation Project of Central South University (2017zzts048).

References

Barcena-Ruiz

J.C.

and Garzon

M.B.

, Endogenous timing in a mixed duopoly with capacity choice, The Manchester School 78(2) (2010), 93–109.

Bertrand

, Review of Walras’s th&orie mathématique de la richesse so including ciale and Cournot’s recherches sur les principes mathematiques de la theorie des richesses in Cournot oligopoly: Characterization and applications. edited by Daughety

A.F.

, Cambridge University Press, 1883.

Blume

, Bertrand without fudge, Economics Letters 78(2) (2003), 167–168.

Campose

, Fuzzy linear programming models to solve fuzzy matrix games, Fuzzy Sets and Systems 32(3) (1989), 275–289.

Chen

, Ma

J.H.

and Chen

X.Q.

, The study of dynamic process of the triopoly games in Chinese 3G telecommunication market, Chaos, Solitons & Fractals 42(3) (2009), 1542–1551.

Dang

J.F.

and Hong

I.H.

, The Cournot game under a fuzzy decision environment, Computers & Mathematics with Applications 59(9) (2010), 3099–3109.

Dang

J.F.

, Hong

I.H.

and Lin

J.M.

, The Cournot production game with multiple firms under an ambiguous decision environment, Information Sciences 266 (2014), 186–198.

Fanti

, Endogenous timing under price competition and unions, International Journal of Economic Theory 12(4) (2016), 401–413.

Gao

, Credibilistic game with fuzzy information, Journal of Uncertain Systems 1(1) (2007), 74–80.

10.

Gao

and Gao

, A new stock model for credibilistic option pricing, Journal of Uncertain Systems 2(4) (2008), 243–247.

11.

Gao

, Liu

Z.Q.

and Liu

Y.K.

, Equilibrium strategies of two-player nonzero-sum games with fuzzy payoffs, In Proceedings of the IEEE international conference on Fuzzy systems, (July 23–26, London, 2007), pp. 883–887.

12.

Gao

and Yang

, Credibilistic bimatrix game with asymmetric information: Bayesian optimistic equilibrium strategy, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 21(supp01) (2013), 89–100.

13.

Gao

and Yu

, Credibilistic extensive game with fuzzy payoffs, Soft Computing 17(4) (2013), 557–567.

14.

Gao

, Zhang

and Shen

, Coalitional game with fuzzy payoffs and credibilistic Shapley value, Iranian Journal of Fuzzy Systems 8(4) (2011), 107–117.

15.

Garagic

and Cruz

J.B.

, An approach to fuzzy noncooperative Nash games, Journal of Optimization Theory and Applications 118(3) (2003), 475–491.

16.

Hoernig

S.H.

, Bertrand games and sharing rules, Economic theory 31(3) (2007), 573–585.

17.

Kaplan

T.R.

and Wettstein

, The possibility of mixed-strategy equilibria with constant-returns-to-scale technology under Bertrand competition, Spanish Economic Review 2(1) (2000), 65–71.

18.

Kaufmann

, Introduction to the theory of fuzzy subsets, New York: Academic press, 1975.

19.

, Chien

, Li

, Gao

Z.Y.

and Yang

, Energy-constraint operation strategy for high-speed railway, Int J Innov Comput Inf Control 8(10) (2012), 6569–6583.

20.

, Wang

, Li

and Gao

, A green train scheduling model and fuzzy multi-objective optimization algorithm, Applied Mathematical Modelling 37(4) (2013), 2063–2073.

21.

Liu

, Theory and practice of uncertain programming, Heidelberg: Physica-verlag, 2002.

22.

Liu

, Uncertainty Theory, 2nd edn. Springer-Verlag: Berlin, 2007.

23.

Liu

and Liu

Y.K.

, Expected value of fuzzy variable and fuzzy expected value models, IEEE transactions on Fuzzy Systems 10(4) (2002), 445–450.

24.

Liu

, Liu

, Li

and Xu

H.L.

, Dynamics analysis of game and chaotic control in the Chinese fixed broadband telecom market, Discrete Dynamics in Nature and Society 2014 (2014), 1–8.

25.

Liu

Y.K.

and Liu

, Expected value operator of random fuzzy variable and random fuzzy expected value models, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 11(02) (2003), 195–215.

26.

Maeda

, Characterization of equilibrium strategy of the bimatrix game with fuzzy Variables, Journal of Mathematical Analysis and Applications 251(2) (2000), 885–896.

27.

Modak

N.M.

, Panda

and Sana

S.S.

, Two-echelon supply chain coordination among manufacturer and duopolies retailers with recycling facility, The International Journal of Advanced Manufacturing Technology 87(5-8) (2016), 1531–1546.

28.

Modak

N.M.

, Panda

and Sana

S.S.

, Pricing policy and coordination for a two-layer supply chain of duopolistic retailers and socially responsible manufacturer, International Journal of Logistics Research and Applications 19(6) (2016), 487–508.

29.

Morrow

W.R.

, Finite purchasing power and computations of Bertrand-Nash equilibrium prices, Computational Optimization and Applications 62(2) (2015), 477–515.

30.

Nahmias

, Fuzzy variables, Fuzzy Sets and Systems 1(2) (1978), 97–110.

31.

Naya

J.M.

, Mixed oligopoly, endogenous timing and mergers, International Journal of Economic Theory 7(3) (2011), 283–291.

32.

Negoita

, Zadeh

and Zimmermann

, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1(3-28) (1978), 61–72.

33.

Nishizaki

and Sakawa

, Equilibrium solutions in multiobjective bimatrix games with fuzzy payoffs and fuzzy goals, Fuzzy Sets and Systems 111(1) (2000), 99–116.

34.

Prade

and Dubois

, Possibility theory: An approach to computerized processing of uncertainty, NY: Plenum, 1988.

35.

Panda

and Modak

N.M.

, Exploring the effects of social responsibility on coordination and profit division in a supply chain, Journal of Cleaner Production 139 (2016), 25–40.

36.

Routledge

R.R.

, Bertrand competition with cost uncertainty, Economics Letters 107(3) (2010), 356–359.

37.

Sakawa

and Nishizaki

, A solution concept based on fuzzy decision in n-person cooperative games, Cybernetics and Systems Research 92 (1992), 423–430.

38.

Sakawa

and Nishizaki

, Max-min solutions for fuzzy multiobjective matrix games, Fuzzy Sets and Systems 67(1) (1994), 53–69.

39.

Sakawa

and Nishizaki

, A solution concept in multiobjective matrix games with fuzzy payoffs and fuzzy goals, Fuzzy Logic and its Applications to Engineering, Information Sciences, and Intelligent Systems. Springer Netherlands (1995), 417–426.

40.

Shen

and Gao

, Coalitional game with fuzzy payoffs and credibilistic core, Soft Computing 15(4) (2010), 781–786.

41.

Szech

and Weinschenk

, Rebates in a Bertrand game, Journal of Mathematical Economics 49(2) (2013), 124–133.

42.

Tan

, Feng

, Li

and Yi

, Optimal bayesian equilibrium for n-person credibilistic non-cooperative game with risk aversion, Journal of Intelligent & Fuzzy Systems 33(1) (2017), 1–11.

43.

Tan

, Liu

, Wu

D.D.

and Chen

, Cournot game with incomplete information based on rank-dependent utility theory under a fuzzy environment, International Journal of Production Research (2016), 1–17.

44.

Tan

, Huang

, Cai

and Yang

, A non-probabilistic programming approach enabling risk-aversion analysis for supporting sustainable watershed development, Journal of Cleaner Production 112 (2016), 4771–4788.

45.

Tremblay

V.J.

, Tremblay

C.H.

and Isariyawongse

, Endogenous timing and strategic choice: The Cournot-Bertrand model, Bulletin of Economic Research 65(4) (2013), 332–342.

46.

Wei

and Zhao

, Pricing decisions for substitutable products with horizontal and vertical competition in fuzzy environments, Annals of Operations Research 242(2) (2016), 505–528.

47.

S.H.

and Soo

V.W.

, A fuzzy theoretic approach to multi-agent coordination, Multiagent Platform, New York: Springer Verlag (1998), 76–87.

48.

Yang

, Li

, Gao

and Li

, Optimizing trains movement on a railway network, Omega 40(5) (2012), 619–633.

49.

Yang

, Gao

and Li

, Railway freight transportation planning with mixed uncertainty of randomness and fuzziness, Applied Soft Computing 11(1) (2011), 778–792.

50.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

51.

Zadeh

L.A.

, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1(1) (1978), 3–28.

52.

Zhao

, Tang

, Zhao

and Wei

, Pricing decisions for substitutable products with a common retailer in fuzzy environments, European Journal of Operational Research 216(2) (2012), 409–419.

53.

Zhao

, Wei

and Li

, Pricing decisions of complementary products in a two-level fuzzy supply chain, International Journal of Production Research (2016), 1–22.