The Cournot game under a fuzzy decision environment

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Computers and Mathematics with Applications

journal homepage:www.elsevier.com/locate/camwa

The Cournot game under a fuzzy decision environment

Jr-Fong Dang

a

, I-Hsuan Hong

b,c,∗

a_{Department of Industrial Engineering & Management, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan} b_{Institute of Industrial Engineering, National Taiwan University, 1, Sec. 4, Roosevelt Road, Taipei 106, Taiwan}

c_{Department of Mechanical Engineering, National Taiwan University, Taiwan}

a r t i c l e i n f o Article history:

Received 19 February 2010 Accepted 19 February 2010 Keywords:

Fuzzy set theory Cournot game Equilibrium quantity

a b s t r a c t

The conventional precision-based decision analysis methodology is invalid for business decision analysis when precise assessment data seldom exist. This paper considers the Cournot game with fuzzy demand and fuzzy costs that are assumed to be triangular fuzzy numbers. Our model utilizes the weighted center of gravity (WCoG) method to defuzzify the fuzzy profit function into a crisp value. We derive the equilibrium Cournot quantity of each firm by simultaneously solving the first-order condition of each firm. Our model explicitly derives the necessary condition to avoid an unreasonable outcome of negative equilibrium quantities and lack of flexibility for modification of the ranking method. In addition, we examine the standard deviation of the fuzzy profit resulting from the fuzziness of each firm’s cost and market demand functions. We conduct sensitivity analysis to investigate the effect of parameter perturbations on firms’ outcomes. The results indicate that the center of parameter plays an important role in sensitivity analysis and dominates over variations in equilibrium quantity due to a perturbation of fuzzy parameters.

Game Theory models and analyzes situations in which multiple agents have their own profit functions and independently make their decisions. Often, the decisions of multiple agents affect the payoffs of the others. The classic Cournot model is one

of several models commonly applied to analyze such precise scenarios. Liang et al. [1] categorize them as precision-based

models since all data and required parameters are precise. However, in the real world, many uncertain factors that exist in the decision environment (e.g., customer demand, production fluctuation, etc.) tend to restrict the usage of the classic Cournot model to real problems. This motivates us to develop the Cournot model with fuzzy parameters instead of precise information, which is usually difficult to obtain, even unavailable, in reality.

The literature has focused on the randomness aspect of uncertainty and many stochastic models have been developed to

account for uncertainty in game-theoretical models [2,3] where uncertain parameters are typically modeled by probability

distributions. However, the probability distribution may not be available in practice or may be difficult to estimate from limited data points. For instance, it is difficult to provide exact estimates of a manufacturer’s variable cost because procurement costs may fluctuate. Under this scenario, the fuzzy set theory is an appropriate modeling tool when uncertain parameters cannot be described in distributions. In addition, the fuzzy set theory provides a mathematical approach to model the intrinsic vagueness and imprecision of human cognitive processes, e.g., the phrase ‘‘around x dollars’’ to describe

a cost that can be regarded as a fuzzy number

_e

x.

Since Zadeh [4] introduced the concept of fuzzy sets, applications to game theory have been proposed in the literature. In

general, there are two streams of fuzzy games: fuzzy matrix games and fuzzy non-cooperative games. Most of them consider ∗_{Corresponding author. Tel.: +886 2 3366 9507; fax: +886 2 2362 5856.}

E-mail address:ihong@ntu.edu.tw(I-H. Hong).

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Table 1

Relative papers of fuzzy games.

Fuzzy parameters Model type

Economic surplus Revenue Mutual price Non-cooperative game

Fuzzy demand Chang and Yao [11] Yao and Wu [12] Yao and Shih [14]

Yao and Chang [13] Xu and Zhai [15]

Fuzzy cost Liang et al. [1]

Wang et al. [16] Fuzzy demand and fuzzy cost Yao and Wu [8]

only at zero-sum games. Campos [5] proposes the fuzzy matrix game which is based on the establishment of a fuzzy linear

programming for each player, but does not define explicit concepts of the equilibrium strategy. For bi-matrix games with

fuzzy payoffs, Maeda [6] defines two types of concepts of equilibrium strategies and investigates their relations. Liu and

Kao [7] apply the extension principle and a pair of two-level mathematical programming to construct the membership

function of matrix games.

Yao and Wu [8] probably initiated the fuzzy non-cooperative games involving fuzzy data. They apply the ranking method

transforming fuzzy numbers into crisp numbers for comparison to defuzzify the demand and supply functions so that consumer surplus and producer surplus can be calculated in a conventional manner. Their concept is also utilized to

construct the inventory models by Ouyang and Yao [9] and Wu and Yao [10]. Chang and Yao [11] optimize the revenue

of monopoly when the parameters of the demand function are fuzzy numbers. Yao and Wu [12] discuss the best price of

two mutual complementary merchandises in fuzzy sense. Yao and Chang [13] obtain the optimal quantity for maximizing

the profit function whose parameters are fuzzy numbers. Yao and Shih [14] derive the membership function of the profit

function when the optimal quantity occurs. Xu and Zhai [15] develop an optimal technique for dealing with the fuzziness

aspect of demand uncertainties for supply chains. Liang et al. [1] propose their duopoly model involving fuzzy cost to obtain

the optimal quantity of each firm. Recently, Wang et al. [16] introduce their fuzzy Bayesian game where the decision rules

for players are based on the creditability theory.Table 1shows the classification of fuzzy parameters and model types where

fuzzy parameters include the demand, cost or both, and the model types contain economic surplus, revenue, mutual price and non-cooperative games. It should be noted that the approaches described in existing papers may lead to an unreasonable outcome where the model returns a negative optimal quantity in some circumstances due to fuzzy parameters. Further,

Liang et al. [1] only account for fuzzy costs without fuzzy demand.

In this paper, we propose the Cournot game with fuzzy demand and cost, which to our knowledge has not appeared in the literature. We highlight two important drawbacks in previous studies: an unclear restriction of occurrence of a negative equilibrium quantity, and limited flexibility for modification of the ranking method in fuzzy modeling. We propose a fuzzy Cournot model with rigorous definitions to ensure a positive equilibrium quantity and with a flexible controlling mechanism for decision-makers. This paper describes a method solving for the equilibrium quantity of each competing firm in a duopoly market with fuzzy parameters that give several important managerial insights by examining the variation of the profit function in the fuzzy environment.

We utilize the weighted center of gravity (WCoG) which is proposed by Bender and Simonovic [17] to defuzzify the

profit function of each firm into a crisp value. Each firm desires to make decisions that optimize its particular objective profit functions. In the equilibrium quantity, no firm can be better off by a unilateral change in its solution. Mathematically the equilibrium quantity can be obtained by simultaneously solving the first-order condition of each firm’s objective profit

function. In addition, we utilize the variation index proposed by Lee and Li [18] to calculate the variation in the firms’ profit

functions. For simplicity, we assume that the demand function and cost function of the firms exist with the form of linearity

and fuzzy parameters. The linearity assumption is commonly used in the literature [11,13] and it assists in obtaining the

qualitative managerial insights with less analytical complexity. It also has the desirable properties for approaching the equilibrium quantities.

The remainder of this paper is organized as follows. In Section2, we introduce the concept and definitions. Section3

addresses the fuzzy Cournot model followed by the proposed method to solve for the equilibrium quantity of each firm in

the fuzzy context. Section4conducts the sensitivity analysis to investigate the effect of parameter perturbation on firms’

outcomes. Section5presents the conclusions and suggestions for future research.

2. Definitions

Zadeh [4] introduced fuzzy set theory to analyze and solve problems with sources of vagueness called fuzziness. The word

‘‘fuzziness’’ captures the properties of parameters when in reality decision-makers rarely have sharp boundaries and/or cannot precisely determine them. Below we briefly introduce the definitions and notations used in this paper.

Let X be a universal set. A fuzzy subset

e

A of X is defined by its membership function

µ

eA

:

X

→ [

0

,

1

]

. We denote by

e

Aα

= {

x

:

µ

(3)

µ

eA

(

x

) =

1. In addition, the fuzzy number,

e

A, is called a convex fuzzy set if

µ

eA

(λ

x

+

(

1

−

λ)

y

) ≥

min

(µ

eA

(

x

), µ

eA

(

y

))

for

λ ∈ [

0

,

1

]

. This paper assumes the universal set X is a real number system, i.e., X

= <

. A fuzzy number,

e

A, must satisfy:

(i)

e

A is a normal and convex fuzzy set,

(ii) its membership function,

µ

eA, is upper semi-continuous, and

(iii) the

α

-level set,

e

Aα, is bounded for each

α ∈ [

0

,

1

]

.

The fuzzy number,

e

A, is called a nonnegative fuzzy number if

µ

eA

(

x

) =

0 for all x

<

0, and a nonpositive fuzzy number

if

µ

eA

(

x

) =

0 for all x

>

0. It is obvious that

e

A

L

αand

e

AU_αare nonnegative real numbers for all

α ∈ [

0

,

1

]

if

e

A is a nonnegative

fuzzy number, and

e

AL_αand

e

A_αUare nonpositive real numbers for all

α ∈ [

0

,

1

]

if

e

A is a nonpositive fuzzy number.

2.1. Fuzzy arithmetic operation

Let ‘‘

’’ be any binary operation

⊕

or

⊗

between two fuzzy numbers

_e

a and

e

b. The membership function of

_e

a

e

b is

defined by

µ

eaeb

(

z

) =

sup x◦y=z

min

µ

ea

(

x

), µ

eb

(

y

)

using the extension principle in [4], where the operations

,

=

,

⊕

, and

⊗

correspond to the operations

◦

,

=

,

+

, and

×

. Thus

we review the following definition for derivation purposes.

Definition 1 ([19]). Let

e

a and

e

b be two fuzzy numbers. Then

e

a

⊕

e

b and

e

a

⊗

e

b are also fuzzy numbers. Further, we have

e

a

⊕

e

b

α

=

e

a L α

+e

bLα

,e

aUα

+

e

bU_α

and

e

a

⊗

e

b

α

=

min

_e

aL_α

·e

b_αL

,e

aU_α

·e

bU_α

,e

aL_α

· e

bU_α

,e

aU_α

· e

bL_α

,

max

_e

aL_α

· e

bL_α

,e

aU_α

·e

bU_α

,e

aL_α

·e

b_αU

,e

aU_α

·e

bL_α

.

Definition 1is useful when considering the linear model in this paper.

2.2. Triangular fuzzy number

In practice, the most commonly used fuzzy numbers are triangular fuzzy numbers because they are easy to handle

arithmetical operations and they have intuitive interpretations [20]. The triplet,

(

x

,

l

,

r

)

, denotes a triangular fuzzy number

with the apex x, left-spread l, and right-spread r. In other words, the lower and upper bounds of x are x

−

l and x

+

r

respectively. For notational simplicity, we let xU_(xL_{) denote the upper (lower) bound of x.}

2.3. Weighted center of gravity measure

Many ranking methods have been proposed (see [21,22]). The center of gravity (COG) method, also called the centroid

method [23], is widely used because of its straightforward geometrical meaning. However, the COG method is inappropriate

to distinguish two fuzzy sets that may have the same centroid, but greatly differ in the degree of fuzziness. In this case, the

WCoG method is more useful [17]:

WCoG

=

R

g

(

x

)µ(

x

)

k_dx

R

µ(

x

)

k_dx

,

(1)

where g

(

x

)

is the horizontal component of the area under scrutiny and

µ(

x

)

is the membership function value. The value of

k is a control parameter ranging from k

=

1 to k

= ∞

. In practice, decision-makers determine g

(

x

)

and the magnitude of k.

This paper applies the WCoG method to retrieve crisp values. In(1), there are two control parameters: g

(

x

)

and k; the final

selection of appropriate control parameters depends upon the decision-makers’ level of risk tolerance (see [17]).

2.4. Measurement of variation

Lee and Li [18] propose the use of generalized mean and the standard deviation, listed in(2)and(3), based on the

probability measure of fuzzy events to rank fuzzy numbers. This method ranks fuzzy numbers on the basis of the fuzzy mean and the spread of fuzzy numbers, and assumes two kinds of the probability distributions of fuzzy numbers: uniform distribution and proportional distribution. We adopt the uniform distribution for calculation simplicity in this paper. The

uniform distribution, U, with the spreads

|

e

A

|

of the fuzzy number has probability density function

(4)

The mean,

e

x

(

e

A

)

, and standard deviation,

e

σ(

e

A

)

, of fuzzy number

e

A are defined as

e

x

(

e

A

) =

Z

S(eA) x

µ

eA

(

x

)

dx

Z

S(eA)

µ

eA

(

x

)

dx

,

(2)

e

σ(

e

A

) =

Z

S(eA) x2

µ

eA

(

x

)

dx

Z

S(eA)

µ

eA

(

x

)

dx

−

_e

x

(

e

A

)

2

1/2 (3)

where S

(

e

A

)

is the support of fuzzy number

e

A. When fuzzy number

e

A is a triangular fuzzy number,(2)and(3)can be written as

e

x

(

e

A

) =

1

/

3 x L

₊

_x

₊

_xU

,

(4)

e

σ(

e

A

) =

1

/

18

xL

2

+

(

x

)

2

+

xU

2

−

xLx

−

xxU

−

xLxU

.

(5)

3. The fuzzy Cournot model

As we have noted it is almost impossible to find the exact economic assessment data for parameters’ estimation in the real world. Therefore, we apply the fuzzy set theory to solve the fuzzy Cournot game with the fuzzy demand function and fuzzy cost function.

3.1. The Cournot equilibrium quantity in a duopoly market

Let qidenote the production quantity of firm i

,

i

=

1

,

2. Consider the general fuzzy profit function of firm i,

e

π

i, as

e

π

i

=

e

p

(

Q

) ·

qi

−

TC

e

(

qi

)

(6)

where

_e

p

(

Q

)

is the fuzzy inverse demand function,TC

e

(

qi

)

is the fuzzy cost function of firm i and Q is the total quantity in the

duopoly market, namely Q

=

q1

+

q2.

The fuzziness of

e

p

(

Q

)

andTC

e

(

qi

)

result from the fuzzy parameters of the inverse demand function and the cost function.

From(6)it is clear that if

_e

p

(

Q

)

orTC

e

(

qi

)

is unbounded, then

e

π

iis unbounded. However, since unboundedness is irrational

in the real world, this paper assumes that both

e

p

(

Q

)

andTC

e

(

qi

)

are bounded. Thus,

e

π

iis bounded as well. The upper (lower)

bound of

e

π

idenoted by

π

U

i (

π

L

i) can be obtained by the extension principle. According to fuzzy arithmetic operations, the

profit function of firm i is a fuzzy number, which may cause difficulty in solving for each firm’s equilibrium quantity. In this

paper, we utilize(1)to defuzzify the profit function into a crisp value. The center of the fuzzy profit function can be stated as

WCoG

(e

π

i

) =

R

πiU πL i g

(π

i

)µ(π

i

)

kd

π

i

R

πiU πL i

µ(π

i

)

kd

π

i

.

(7)

The equilibrium quantity of the conventional Cournot game is directly related to each firm’s profit function. Additionally, we imagine that the equilibrium quantity derived by our proposed model is related to the resulting profit function. As

expected, if the profit function of each firm is unbounded, the equilibrium quantity is also unbounded. Thus,(7)requires a

more rigorous definition so that the estimated profit function of each firm is bounded and the equilibrium quantity indeed exists.

Proposition 1. If

R

πiU

πL i

g

(π

i

)

d

π

iis bounded, the WCoG

(e

π

i

)

is bounded.

Proof. Let

µ(π

i

)

be the membership function of the fuzzy profit function. Thus, we observe that the possibility of

µ(π

i

)

is always less than or equal to one. The membership function is contained by a rectangle; the height of the rectangle is equal to one and the base of the rectangle extends between the upper bound of the fuzzy profit function and the lower bound of

the fuzzy profit function,

π

_iU

−

π

_iLrespectively. The shape of the membership function

µ(π

i

)

kdiffers when the value of k

varies, but

µ(π

i

)

kis still contained by a rectangle whose height and base are 1 and

π

iU

−

π

iL. Therefore, one can argue that

the integration of membership function

µ(π

i

)

khas the inequality

0

≤

Z

π_iU πL i

µ(π

i

)

kd

π

i

=

w ≤ π

iU

−

π

L i

,

(8)

where

w

is a constant. Thus,(7)can be rewritten as

WCoG

(e

π

_i

) =

1

/w ·

Z

π_iU

πL i

(5)

Because the maximum value of

µ(π

i

)

kis 1,(9)can be rearranged as WCoG

(e

π

_i

) =

1

/w ·

Z

π_iU πL i g

(π

_i

)µ(π

_i

)

kd

π

_i

≤

1

/w ·

Z

π_iU πL i g

(π

_i

)

d

π

_i

.

(10) Thus, if

R

π U i πL i

g

(π

i

)

d

π

iis bounded, WCoG

(e

π

i

)

is bounded as well.

We note that decision-makers who utilize(7)to defuzzify the fuzzy profit function, must check whether

R

π

U i

πL i

g

(π

i

)

d

π

i is bounded. The best response function, an important concept in game theory, returns the quantity that yields the optimal

profit for firm i given other players’ decisions about the quantity. In this paper, we use(7)to defuzzify the fuzzy profit

function into a crisp value. Thus, the best response functions of firm i can be obtained by optimizing each firm’s profit

functions with respect to each firm’s decision variable qiwhile assuming the competitors’ quantity qjas given. The resulting

best response functions are

Ri qj

=

arg max WCoG

(e

π

i

) ,

i

6=

j

,

i

,

j

=

1

,

2

.

(11)

We can obtain the equilibrium quantity of each firm by simultaneously solving the first-order condition being obtained by letting the partial derivative of each firm’s profit function equal zero. Now that we have crisp values, we can examine the variations of the profit function to provide more information for decision-makers. As discussed earlier, we can express

in(3)or(5)the profit variation resulting from the gap of the parameter’s spread of both the firm’s cost and market demand

functions. In addition, the optimistic and pessimistic cases can be realized by membership function

µ(π

_i

)

, which can be

derived by the extension principle in [4]. Moreover, we can adapt our proposed method to fit the different criteria of

decision-makers or markets by setting different controlling mechanisms of g

(

x

)

and k.

3.2. Linear model

Next, we develop the fuzzy Cournot game that includes the parameters with triangular fuzzy numbers, a linear inverse demand and cost functions. Initially, we introduce the conventional Cournot game and generalize it in the fuzzy environ-ment. Given the linear inverse demand function

p

(

Q

) =

a

−

bQ

,

0

≤

Q

≤

a

b (12)

where a, b

>

0 are given numbers and p

(

Q

)

is the unit price, which is a function of the market demand quantity Q . The total

cost function of firm i

,

i

=

1

,

2, denoted by TCi

(

qi

)

, is stated as

TCi

(

qi

) =

ci

+

diqi (13)

where cidenotes the production fixed cost of firm i and direpresents the production variable cost of firm i. Then the profit

function of firm i is given by

π

i

=

p

(

Q

) ·

qi

−

TCi

(

qi

) = (

a

−

bQ

)

qi

−

TCi

(

qi

).

(14)

Without loss of generality, we assume that all parameters are fuzzy numbers since crisp values can be treated as degenerated fuzzy numbers. In other words, the last two elements in the triplet of the fuzzy number are equal to zero for a

crisp value. Particularly in the linear model, we let fuzzy sets

e

p

(

Q

) =

e

a

−e

bQ ,TC

e

i

(

qi

) =e

ci

+e

diqi, and

e

π

i

=

e

p

(

Q

) ·

qi

−

TC

e

i

(

qi

)

where

e

p

(

Q

)

,TC

e

i

(

qi

)

, and

e

π

irepresent the fuzzy price, fuzzy cost function of firm i and fuzzy profit function of firm i

,

i

=

1

,

2,

respectively. Triangular fuzzy numbers are adopted in this paper because they are considered the most suitable for modeling

the market demand; see [24,25]. We assume that all parameters are nonnegative triangular fuzzy numbers; in other words,

all elements in the following triplets of fuzzy numbers are nonnegative.

e

a

=

(

a

,

la

,

ra

) ,

e

b

=

(

b

,

lb

,

rb

) ,

e

ci

=

ci

,

lci

,

rci

,

e

di

=

di

,

ldi

,

rdi

.

(15)

ByDefinition 1, when

_e

a and

e

b are triangular fuzzy numbers, we have

e

p

(

Q

) = (

a

−

bQ

,

la

+

rbQ

,

ra

+

lbQ

) .

(16)

To derive the lower bound of price, we must substitute the lower bound of a, namely a

−

la, and the upper bound of b, namely

b

+

rb, into(12). Also, the upper bound of price can by derived by substituting the upper bound of a, namely a

+

ra, and the

lower bound of b, namely b

−

lb, into(12). Similarly,TC

e

_i

(

q_i

)

and

_e

π

_iare given by

e

TCi

(

qi

) =

TCi

,

lci

+

ldiqi

,

rci

+

r_diqi

,

e

π

i

=

π

i

, (

la

+

rbQ

)

qi

+

rci

+

r_diqi

, (

ra

+

lbQ

)

qi

+

lci

+

ldiqi

.

(17)

(6)

The controlling mechanism should be clearly defined and followsProposition 1before applying(1)to defuzzify the fuzzy

profit function into a crisp value. For illustrative purposes, this paper assumes k

=

1 and g

(

x

) =

x. Simply put, it is the same

as the centroid method and it weights all of the values with different possibility to form a single value to represent a fuzzy

number. Since each firm’s fuzzy profit function is a triangular fuzzy number, WCoG

(e

π

i

)

can be easily calculated as

WCoG

(e

π

_i

) =

R

∞ −∞

π

i

µ

eπid

π

i

R

∞ −∞

µ

eπid

π

i

=

1 3

π

L i

+

π

i

+

π

iU

.

(18)

Thus, the partial derivative of WCoG

(e

π

i

)

can be stated as

∂

WCoG

(e

π

i

)

∂

qi

=

a

−

2bqi

−

bqj

−

di

+

1

3

[

(

ra

−

la

) +

2

(

lb

−

rb

)

qi

+

(

lb

−

rb

)

qj

+

(

ldi

−

rdi

)],

i

,

j

=

1

,

2

,

i

6=

j

.

(19)

By letting(19)equal zero, the first-order condition of firm i, i

=

1

,

2, can be obtained. The equilibrium quantity of firm i

follows by simultaneously solving the first-order conditions of firms 1 and 2. Thus, we have qi

=

a

+

dj

−

2di

−

1₃

(

la

−

ra

) +

2₃ ldi

−

rdi

−

1 3 ldj

−

rdj

[

3b

−

(

lb

−

rb

)]

,

i

,

j

=

1

,

2

,

i

6=

j

.

(20)

The denominator of(20), 3b

−

(

lb

−

rb

)

, can be decomposed into three terms

3b

−

(

lb

−

rb

) = (

b

−

lb

) +

b

+

(

b

+

rb

).

(21)

Because b

−

lbis nonnegative,(21)is obviously greater than zero. To ensure a nonnegative equilibrium quantity of firm i,

we impose the condition such that qi

≥

0, i

=

1

,

2. From this condition followsAssumption 1.

Assumption 1. a

+

(

dj

−

2di

) −

1₃

(

la

−

ra

) +

2₃

(

ldi

−

rdi

) −

1

3

(

ldj

−

rdj

) ≥

0.

In addition,(20)is the same as the equilibrium quantity of the conventional Cournot game when the spreads of all

parameters are equal to zero. Next, we calculate the variance of each firm’s profit by the Lee and Li [18] index and we

suppose the distribution of the fuzzy profit function is the uniform distribution. The standard deviation of the fuzzy profit function can be defined as

σ(e

π

i

) =

Z

S(eπi)

(π

2 i

µ

eπid

π

i

)

Z

S(eπi)

(µ

eπid

π

i

)

2

₋

₍

_WCoG

_(e

_π

i

))

2

1/2

.

(22)

By substituting(18)into(22), when the fuzzy profit function is a triangular fuzzy number,(22)can be rewritten as

e

σ (e

π

i

) =

1

/

18

π

iL

2

+

(π

_i

)

2

+

π

_iU

2

−

π

_iL

π

_iU

−

π

_i

π

_iU

−

π

_iL

π

_i

,

(23)

where

π

_iU,

π

i, and

π

iLcan be obtained by(17). Using the solution procedure proposed in this section, we now have both the

equilibrium quantity and the standard deviation of the fuzzy profit function of firms 1 and 2. 3.2.1. Uniqueness and existence of equilibrium quantity

The equilibrium quantity(20)is a solution to a system of two first-order conditions. Non-existence of an equilibrium

quantity is a potential outcome depending on the value of the parameters. In the linear model proposed in this paper, the equilibrium quantity does exist under the condition we discuss later and there is a simple way to show that the equilibrium quantity is unique.

Proposition 2. If the lower bound of parameter b is greater than zero, namely b

−

lb

>

0, there exists a unique equilibrium

quantity of firms 1 and 2.

Proof. By letting(19)equal zero, the equilibrium quantities of firms 1 and 2 are obtained by simultaneously solving their

two first-order conditions. Obviously,(19)is a linear system when the demand and cost functions are linear. We can solve

a linear system by Cramer’s Rule (see [26]).

Let∆be a 2

×

2 matrix defined as

∆

=

2B B B 2B

,

where B

=

b

−

1 3

(

lb

−

rb

).

(24)

The determinant of∆, denoted by det

(

∆

)

, is 4B2

₋

_B2

₌

_3B2_{. We can decompose entry B into three terms as}

B

=

1 3

(

b

−

lb

) +

1 3b

+

1 3

(

b

+

rb

).

(25)

The first term in(25),

(

b

−

lb

)

, is the lower bound of parameter b. If the lower bound of parameter b is greater than zero, we

have B

>

0. By Cramer’s Rule, we show that the equilibrium quantity of firms 1 and 2 is unique if and only if det

(

∆

) 6=

0.

(7)

3.2.2. Illustrative examples

The solution procedure that we have proposed is to find the equilibrium quantity in a duopoly market. To demonstrate the proposed method, we consider the Cournot game with the following problem setups.

Example 1. Suppose p

(

Q

) =

12

−

Q , 0

≤

Q

≤

12, then a

=

12

,

b

=

1 and cost function of firms 1 and 2, TC1

(

q1

) =

2

+

4q1,

TC2

(

q2

) =

1

+

5q2in the crisp case. The equilibrium quantity of the conventional Cournot game can be obtained by

substi-tuting zero spreads in our model. Hence, the equilibrium quantity of firm 1 is 3 units (q∗₁

=

3), and the equilibrium quantity

of firm 2 is 2 units

(

q∗₂

=

2

)

.

Example 2. We consider the case with fuzzy parameters. Let

e

a

=

(

12

,

1

,

2

)

,

e

b

=

(

1

,

1

,

1

)

,

e

c1

=

(

2

,

1

,

1

)

,

e

d1

=

(

4

,

1

,

2

)

,

e

c2

=

(

1

,

1

,

1

)

, and

e

d2

=

(

5

,

2

,

1

)

. We can obtain the equilibrium quantity of firms 1 and 2 as

q1

=

12

+

(

5

−

8

) −

1₃

(

1

−

2

) +

2₃

(

1

−

2

) −

1₃

(

2

−

1

)

3

×

1

=

25 9

,

q2

=

12

+

(

4

−

10

) −

1₃

(

1

−

2

) −

1₃

(

1

−

2

) +

2₃

(

2

−

1

)

3

×

1

=

22 9

.

Given the equilibrium quantity of each firm, we calculate the standard deviation of the fuzzy profit function of each firm as

e

σ (e

π

i

) =

1

/

18

π

L i

2

+

(π

_i

)

2

+

π

_iU

2

−

π

_iL

π

_iU

−

π

_i

π

_iU

−

π

_iL

π

_i

,

i

=

1

,

2 (26) where

π

L i

=

((

a

−

la

) − (

b

+

rb

)

Q

)

qi

−

ci

+

rci

−

di

+

rdi

qi

π

i

=

(

a

−

bQ

)

qi

−

ci

−

diqi

π

U i

=

((

a

+

ra

) − (

b

−

lb

)

Q

)

qi

−

ci

−

lci

−

di

−

ldi

qi i

=

1

,

2

.

Substituting the qi, Q , and parameters into(26), the standard deviation of the profit of firms 1 and 2 is

e

σ (e

π

1

) =

94

.

72

,

e

σ (e

π

2

) =

74

.

52

.

4. Sensitivity analysis

The goal of sensitivity analysis is to investigate the effect of parameter perturbations on the resulting outcome such as the equilibrium quantity and firms’ profits so that we gain several managerial insights from the proposed model. As we can

see,(20)indicates that the terms, la

−

raand ldj

−

rdj, have negative coefficients. Parameters laand racan be interpreted

as the left- and right-spreads of fuzzy parameter

e

a in the demand function. An increase in la

−

ra(the difference between

left- and right-spreads in a) would result in a decrease in each firm’s equilibrium quantity. Similarly, from(20), an increase

in ldj

−

rdj(the difference between left- and right-spreads of the competitor’s production variable cost dj) would lead to a

decrease in the equilibrium quantity of the firm as well. In(20), the term, ldi

−

rdi, is with a positive coefficient. As a result,

an increase in ldi

−

rdi(the difference between the left- and right-spreads of its own production variable cost di) would

lead to an increase in the equilibrium quantity. It is interesting to note that the coefficient magnitude of ldi

−

rdiis greater

than that of ldj

−

rdj; in other words, the equilibrium quantity decreases in ldj

−

rdj, but increases in ldi

−

rdiat a faster rate.

However, there are other factors affecting the change in the equilibrium quantity rather than the differences between the

left- and right-spreads. From(20), decision-makers cannot examine how a total spread of a fuzzy parameter could affect the

equilibrium quantity; for example, it is not obvious how the change in la

+

raaffects qifor firm i. The next section describes

our study of parameter centers in sensitivity analysis. 4.1. Effect of parameter centers

As discussed earlier, fuzzy parameters can be defuzzified into a crisp value representing the center of the associated fuzzy parameter. We now look at the impacts of parameter centers.

Proposition 3. The equilibrium quantity of each firm is only affected by the centers of the parameters. Proof. Considering the center of triangular fuzzy parameter a, it is trivial having the center of a in(27).

ac

=

1

(8)

Table 2

Partial derivatives of outcomes with respect to different centers.

ac _bc _dc i dcj Q 2/3bc ₋₃₍_2ac₋_dc j−dci)/(3bc)2 −1/3bc −1/3bc qi 1/3bc ₋₃₍_ac₊_dc j−2d c i)/(3b c₎2 ₋₂_/_3bc ₁_/_3bc WCoG(eπi) 2(ac+dc j−2dci) 9bc −3(2ac−dc j−dci)(ac+dcj+dci) 27(bc₎2 −4(ac+dc j−2dci) 3bc 2ac+2dj−2di 9bc

Similarly, the centers of the other fuzzy parameters can be written as bc

=

1 3

(

b

−

lb

+

b

+

b

+

rb

) ,

dc_i

=

1 3 di

−

ldi

+

di

+

di

+

rdi

,

dc_j

=

1 3 dj

−

ldj

+

dj

+

dj

+

rdj

.

(28)

Substituting(27)into(20),(20)can be rewritten as

qi

=

ac

₊

_dc j

−

2dci

3bc

.

(29)

Although both(20)and(29)represent the equilibrium quantity of each firm,(29)is much more convenient for illustrative

purposes in sensitivity analysis and it completes the proof.

All possible perturbation in parameters can be represented in the right-hand side of(27)or(28)and translated into the

change in the center of the associated fuzzy parameter. FromProposition 3, the equilibrium quantity is a function of the

center of fuzzy parameters ac_{, b}c_{, d}c

i, and dcj as shown in(29). Therefore, the change in the equilibrium quantity due to any

perturbation in fuzzy parameters can be explored by(29). Suppose that the right-spread of fuzzy parameter a, namely ra,

increases. It is clear that acwould be larger and it is with a positive coefficient in(29). As the result, the equilibrium quantity

increases in ac_.

From(29), the total market demand is written as

Q

=

q1

+

q2

=

2ac

₋

_dc

i

−

dcj

3bc

.

(30)

We can now conduct the sensitivity analysis of the total market demand and equilibrium quantity of firms. Taking the partial

derivative of(29)and(30)with respect to ac, we have

∂

qi

∂

ac

=

1 3bc

,

(31)

∂

Q

∂

ac

=

2 3bc

.

(32)

Similarly, we also take the partial derivatives of(29)and(30)with respect to other centers of fuzzy parameters. Following

Assumption 1, the resulting equilibrium quantity is positive, and implies that the numerator of(29), ac

₊

_dc

j

−

2dci, is positive

as well. The total quantity of the market demand is positive simply because of the positive quantities of both individual firms;

therefore, the numerator of(30), 2ac

−

dc_j

−

dc_i, is positive.Table 2summarizes these results (for example,_∂∂_aQc

=

_3b2c is listed

in the upper-left cell inTable 2). FromTable 2, we have the following statements:

•

the total market demand, Q , increases in ac, but decreases in bc, dc_i, and dc_j, and

•

the equilibrium quantity of firm i, qi, increases in acand dcj, but decreases in bcand d

c i.

Note that the equilibrium quantity of firm i increases in dc_j, but the equilibrium quantity of firm j decreases in dc_j at a

faster rate so that an increase in dc

jwould lead to a decrease in Q .

Next, we consider the weighted center of the fuzzy profit function, WCoG

(e

π

i

)

, as perturbations of the center of fuzzy

parameters occur. After some algebraic manipulations,(18)can be rewritten as

WCoG

(e

π

i

) =

ac

−

bcQ

qi

−

cic

+

d c iqi

.

(33)

Again, we take partial derivative of(33)with respect to ac_{and we have}

∂

WCoG

(e

π

i

)

∂

ac

=

qi

+

a c

∂

qi

∂

ac

−

b c_Q

∂

qi

∂

ac

−

b c

∂

Q

∂

acqi

−

d c i

∂

qi

∂

ac

.

(34)

(9)

Using the results of the first two rows inTable 2,(34)can be rewritten as

∂

WCoG

(e

π

i

)

∂

ac

=

qi

+

a c 1 3bc

−

1 3Q

−

2 3qi

−

d c i 1 3bc

=

a c 1 3bc

−

1 3qj

−

d c i 1 3bc

=

2 3qi

.

(35)

Similarly, we can also take the partial derivative of(33)with respect to the other centers of fuzzy parameters. From earlier

discussion, we know ac

₊

_dc

j

−

2dci

>

0 and 2ac

−

dcj

−

dci

>

0. Trivially, 2ac

+

2dcj

−

2dci is positive since 2ac

+

2dcj

−

2dci

is greater than ac

₊

_dc

j

−

2d

c

i. Therefore, the sign of the last row inTable 2can be determined by these inequalities. We

summarize these results indicating the change rate in WCoG

(e

π

i

)

with respect to the centers of fuzzy parameters. From the

last row inTable 2, we have the following statement:

•

the weighted center of the fuzzy profit function increases in acand dc_j, but decreases in bcand dc_i.

4.2. Analysis of variation of the fuzzy profit function

It is interesting to discuss the effect of the perturbation of fuzzy parameter centers on the standard deviation of firms’

profits. Taking the partial derivative of(23)with respect to ac, we have

∂σ (e

π

i

)

∂

ac

=

WCoG

(e

π

i

)

2 3qi

−

1 6

∂ π

L i

· π

U i

+

π

L i

· π

i

+

π

i

· π

iU

∂

ac

.

(36)

Our purpose is to show the mathematical relation between ac _and

_σ(e

_π

i

)

. Considering the second term of (36),

∂(πL i·π U i+π L i·πi+πi·πiU)

∂ac , it can be decomposed into three terms of

∂(π

iL

· π

iU

)/∂

ac,

∂(π

iL

· π

i

)/∂

acand

∂(π

i

· π

iU

)/∂

ac. The first

term can be further simplified as

∂ π

L i

· π

iU

∂

ac

=

∂π

L i

∂

ac

π

U i

+

∂π

U i

∂

ac

π

L i

=

(

3qi

+

b∗ 5 3d c i

−

1 3d c j

+

aL

−

d U i 3bc

)

π

U i

+

(

3qi

+

b∗∗ 5 3d c i

−

1 3d c j

+

(

a

+

ra

) −

di

−

ldi

3bc

)

π

L i

=

3qi

π

iU

+

θπ

U i

+

3qi

π

iL

+

δπ

L i

,

(37) where b∗

=

b

+

rb bc

,

b∗∗

=

b

−

lb bc

θ =

b ∗ 5 3d c i

−

13d c j

+

(

a

−

la

) −

di

+

rdi

3bc

,

δ =

b ∗∗ 5 3d c i

−

1 3d c j

+

(

a

+

ra

) −

di

−

ldi

3bc

.

Similarly, the second and third terms are

∂ π

L i

· π

i

∂

ac

=

∂π

L i

∂

ac

π

i

+

∂π

i

∂

ac

π

L i

=

(

3qi

+

b∗ 5₃dc_i

−

1 3d c j

+

(

a

−

la

) −

di

+

rdi

3bc

)

π

i

+

(

3qi

+

b∗∗∗ 5₃dc_i

−

1 3d c j

+

a

−

di 3bc

)

π

L i

=

3qi

π

i

+

θπ

i

+

3qi

π

iL

+

ηπ

L i

,

∂ π

i

· π

iU

∂

ac

=

∂π

i

∂

ac

π

U i

+

∂π

U i

∂

ac

π

i

=

(

3qi

+

b∗∗∗ 5 3d c i

−

13d c j

+

a

−

di 3bc

)

π

U i

+

(

3qi

+

b∗∗ 5 3d c i

−

13d c j

+

(

a

+

ra

) −

di

−

ldi

3bc

)

π

i

=

3qi

π

iU

+

ηπ

iU

+

3qi

π

i

+

δπ

i (38) where b∗∗∗

=

b bc

,

η =

b ∗∗∗ 5 3d c i

−

1 3d c j

+

a

−

di 3bc

.

(10)

Substituting(37)and(38)into(36), we obtain

∂σ (e

π

i

)

∂

ac

= −

7 3WCoG

(e

π

i

)

qi

−

1 6

(θ + η) π

U i

+

(θ + δ) π

i

+

(δ + η) π

iL

where WCoG

(e

π

i

)

, qi,

π

iU,

π

i, and

π

iLcan be found in(26). Similarly, taking the partial derivative of(23)with respect to dci,

dc

j, and bc, respectively, we have

∂σ (e

π

i

)

∂

dc i

=

WCoG

(e

π

i

)

5 3qi

−

1 6

θ

0

₊

_η

0

π

U i

+

θ

0

₊

_δ

0

π

i

+

δ

0

+

η

0

π

L i

,

∂σ (e

π

i

)

∂

dc j

=

WCoG

(e

π

i

)

2ac

₊

_2dc j

−

2ci 9bc

−

1 6

θ

00

₊

_η

00

π

U i

+

θ

00

₊

_δ

00

π

i

+

δ

00

+

η

00

π

L i

,

∂σ (e

π

i

)

∂

bc

=

WCoG

(e

π

i

)

−

3 2ac

₋

_dc j

−

dci

ac

₊

_dc j

+

dci

27

(

bc

)

2

−

1 6

∂ π

L i

· π

iU

+

π

iL

· π

i

+

π

i

· π

iU

∂

bc (39) where

θ

0

₌

1 3b ∗ _5ac

₋

_dc j

−

4dci

−

2

(

a

−

la

) +

2 di

+

rdi

3bc

,

δ

0

₌

1 3b ∗∗ 5ac

−

dc_j

−

4dc_i

−

2

(

a

+

ra

) +

2 di

−

ldi

3bc

,

η

0

₌

1 3b ∗∗∗ _5ac

₋

_dc j

−

4dci

−

2a

+

2di 3bc

,

θ

00

₌

b ∗ _ac

₋

_dc i

+

(

a

−

la

) −

di

+

rdi

3bc

,

δ

00

₌

b ∗∗ ac

−

dc_i

+

(

a

+

ra

) −

di

−

ldi

3bc

,

η

00

₌

b ∗∗∗ _ac

₋

_dc i

+

a

−

di 3bc

,

∂ π

L i

· π

iU

∂

bc

=

b∗

−

3

Qqi

+

1 bcqi

dU i 3bc

−

π

L i

π

U i

+

b∗∗

−

3

Qqi

+

1 bcqi

dL i 3bc

−

π

U i

π

L i

,

∂ π

i

· π

iU

∂

bc

=

b∗∗∗

−

3

Qqi

+

1 bcqi

dU i 3bc

−

π

i

π

U i

+

b∗∗

−

3

Qqi

+

1 bcqi

di 3bc

−

π

U i

π

i

,

∂ π

L i

· π

i

∂

bc

=

b∗

−

3

Qqi

+

1 bcqi

dU_i 3bc

−

π

L i

π

i

+

b∗∗∗

−

3

Qqi

+

1 bcqi

dU_i 3bc

−

π

i

π

L i

.

Hence,(38)and(39)indicate the change rate in the standard deviation with respect to fuzzy parameter centers. One can

easily conduct sensitivity analysis of the firm’s profit variation by examining the above results. 5. Conclusion

The real world is with many uncertainties in the decision-making environment so that the conventional Cournot model is inadequate to explain or describe the real situation. Many stochastic models have been developed to handle uncertainty in game-theoretical models where the uncertain parameters are typically modeled by probability distributions. However, probability distributions may not be available in practice or may be difficult to estimate from limited or absent data points. The fuzzy set theory is an appropriate tool to model such a situation when uncertain parameters cannot be described in probability distributions. In addition, the fuzzy set theory provides a mathematical approach to model intrinsic vagueness and the imprecision of human cognitive processes.

In this paper, we propose the fuzzy Cournot game for resolving fuzziness aspect of demand and cost uncertainty. We present a solving procedure for the equilibrium quantity of a general fuzzy Cournot model and indicate the condition to ensure the existence and uniqueness of the equilibrium quantity. For simplicity, we assume that the demand and cost functions of firms exist with the form of linearity and fuzzy parameters. The linearity assumption is commonly used in the literature and it helps us obtain the qualitative managerial insights without computation complexity. The equilibrium quantities of firms can be obtained by our proposed method. We further investigate the standard deviation of the fuzzy profit function of each firm to provide decision-makers with more information about the variations in profits.

In addition, we conduct sensitivity analysis to examine the effect of parameter perturbation on firms’ outcomes including the equilibrium quantity, total market demand and weighted center of the fuzzy profit function. It is worth mentioning that

(11)

the resulting equilibrium quantity of each firm varies with its own parameter at a faster rate than with its competitor’s parameters. A second finding indicates that the center of parameter plays an important role in sensitivity analysis and it dominates how the equilibrium quantity varies due to a perturbation of fuzzy parameters. Further research may include the issue of fuzzy parameters with probability distributions, and our proposed procedure requires additional refinement for this series of research questions.

Acknowledgement

This research was supported in part by the National Science Council of Taiwan under Grant NSC96-2221-E-009-085. References

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The Cournot game under a fuzzy decision environment

Computers and Mathematics with Applications