CUBO A Mathematical Journal

Vol.11, N
o
¯ 02, (107–115). May 2009

A Multiobjective Model of Oligopolies under Uncertainty

Carl Chiarella

School of Finance and Economics,

University of Technology, Sydney, P.O. Box 123,

Broadway, NSW 2007, Australia

email: carl.chiarella@uts.edu.au

and

Ferenc Szidarovszky

Systems & Industrial Engineering Department,

The University of Arizona, Tucson,

Arizona, 85721-0020, USA

email: szidar@sie.arizona.edu

ABSTRACT

It is assumed that in an n-firm single-product oligopoly without product differentiation

the firms face an uncertain price function, which is considered random by the firms. At

each time period each firm simultaneously maximizes its expected profit and minimizes

the variance of the profit since it wants to receive as high as possible profit with the least

possible uncertainty. It is assumed that the best response of each firm is obtained by

the weighting method. We show the existence of a unique equilibrium, and investigate

the local stability of the equilibrium.

RESUMEN

Es asumido que en un oligopolio de n-firmas “single-product” sin diferenciación pro-

ducto firmas con función de precio variable, son consideradas randon por las firmas.



108 Carl Chiarella and Ferenc Szidarovszky CUBO
11, 2 (2009)

En todo peŕıodo de tiempo todo firma simultaneamente maximiza la utilidad esperada

y minimiza la variación de utilidad desde que estan quisen obtener la utilidad mas alta

posible con la menor incertidumbre posible. Es asumido que la mejor respuesta de toda

firma es obtenida por el metodo weighting. Mostramos la existencia de un equilibrio

único y investigamos la estabilidad local del equilibrio.

Key words and phrases: Uncertainty, n-person games, multiobjective optimization.

Math. Subj. Class.: 91A06, 90C29.

1 Introduction

The uncertainty in the inverse demand functions of oligopoly models has been previously examined

by many researches. Cyert and DeGroot (1971, 1973) investigated mainly duopolies. Kirman (1975,

1983) examined the case of differentiated products and linear demand functions and analysed how

the resulting equilibria depend on the way the firms misspecify and try to assess the demand

function. Gates et al. (1982) also examined linear demand functions and differentiated products.

Leonard and Nishimura (1999) assumed that the firms know the shape of the demand function but

misspecify its scale, and investigated the asymptotic behavior of the equilibrium under discrete

time scales. Chiarella and Szidarovszky (2001) have introduced the continuous counterpart of

the Leonard-Nishimura model and in addition to equilibrium and local stability analysis, the

destabilising effect of time delays, in obtaining and implementing information on the competitors’

output, was analysed. Bischi et al. (2004) consider the situation in which the firms’ reaction

functions are unimodal and analyse the various equilibria that may arise and their complicated

basins of attraction. All these earlier studies assumed that the firms maximized their misspecified

or expected payoffs at each time period depending on the type (deterministic or stochastic) of the

model being used.

In this paper we introduce a new approach. The uncertainty of the inverse demand function

is treated here also with a stochastic model, but we assume that at each time period each firm

maximize its expected profit and at the same time tends to reduce profit uncertainty by minimizing

its variance. That is, at each time period the firms face a “Pareto-game”, each with multiple payoffs

(see for example Szidarovszky et al., 1986). In our model, in each time period each firm uses a

multiobjective optimization approach to find its best response. Based on these best response

functions a dynamic process develops. The subject of this paper is the properties of this dynamic

process including equilibrium analysis and the investigation of the asymptotic behavior of the

equilibrium.



CUBO
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A Multiobjective Model of Oligopolies under Uncertainty 109

2 The Mathematical Model and Equilibrium Analysis

Consider an n-firm single-product oligopoly without product differentiation. Let qj be the output

of firm j, and Cj (qj ) the associated cost function. Assume that firm j believes that the true price

function is f (Q) + ηj , where Q =
∑n

j=1 qj is the total output of the industry and ηj is a random

variable such that

E(ηj ) = 0 and V ar(ηj ) = σ
2
j . (2.1)

Using πj to denote profit, the expected profit of firm j is given as

E(πj ) = qj f (Q) − Cj (qj ), (2.2)

and the variance of profit is

V ar(πj ) = q
2
j σ

2
j . (2.3)

Assume that firm j wants to maximize its expected profit and at the same time to minimize the

variance of the profit. That is, the firm tends to obtain as high a profit as possible with minimum

uncertainty. It is also assumed that firm j uses the weighting method (see for example, Szidarovszky

et al. 1986), therefore it maximizes a linear combination of the two objective functions, i.e.

max

[

E(πj ) −
αj

2
V ar(πj )

]

, (2.4)

where αj shows the relative importance of reducing uncertainty compared to the increase of the

expected profit.

In oligopoly theory it is usually assumed that the functions f and Cj (j = 1, 2, · · · , n) are

twice continuously differentiable, f is decreasing, Cj is increasing, furthermore

(a) f
′
(Q) + qj f

′′
(Q) ≤ 0,

(b) f
′
(Q) − C′′j (qj ) < 0,

for all j and nonnegative qj and Q.

Under conditions (a) and (b) the composite objective function (2.4) is strictly concave in qj

with fixed value of Qj =
∑

l 6=j ql. The derivative of the objective function (2.4) with respect to qj

can be given as

qj f
′
(Q) + f (Q) − C′j (qj ) − αj qj σ

2
j .

Notice that f
′ ≤ 0, C′ ≥ 0, f (Q) ≤ f (0), therefore with positive αj and σj , this derivative

converges to −∞ as qj → ∞ implying that there is a unique maximizing value, qj ≥ 0, with any

fixed Qj ≥ 0. The best response of firm j, Rj (Qj ), can be obtained in the following way. If

f (Qj ) − C
′
j (0) ≤ 0, (2.5)

then Rj (Qj) = 0, otherwise it is the unique positive solution of the equation

qj f
′
(qj + Qj) + f (qj + Qj ) − C

′
j (qj ) − αj qj σ

2
j = 0. (2.6)



110 Carl Chiarella and Ferenc Szidarovszky CUBO
11, 2 (2009)

Since f is decreasing, from condition (2.5) we see that if Rj (Qj ) = 0 then for all Q̄j > Qj ,

Rj (Q̄j ) = 0. Assume next that Rj (Qj) > 0. Then equation (2.6) is satisfied with qj = Rj (Qj ).

By implicit differentiation we have
1

R
′
j f

′
+ Rj f

′′
(1 + R

′
j ) + f

′
(1 + R

′
j ) − C

′′
j R

′
j − αj R

′
j σ

2
j = 0,

from which we have

R
′
j = −

f
′
+ Rj f

′′

2f ′ + Rj f
′′ − C′′j − αj σ

2
j

. (2.7)

Conditions (a) and (b) imply that

−1 < R′j ≤ 0. (2.8)

Therefore Rj is a decreasing function of Qj. We can rewrite equation (2.6) as

qj f
′
(qj + Qj ) + f (qj + Qj) − C

′
j (qj ) = αj qj σ

2
j . (2.9)

The left hand side strictly decreases in qj , therefore the solution qj = Rj (Qj) decreases if αj

and/or σj increases.

We can also consider qj as a function of the total output level of the industry, qj = qj (Q),

which can be defined as follows. If

f (Q) − C′j (0) ≤ 0, (2.10)

then qj (Q) = 0, otherwise it is the unique positive solution of the equation

qj f
′
(Q) + f (Q) − C′j (qj ) − αj qj σ

2
j = 0. (2.11)

With fixed values of Q, the left hand side is strictly decreasing in qj , it has a positive value

at qj = 0 and converges to −∞ as qj → ∞. Similarly to the previous case, condition (2.5) implies

that if qj (Q) = 0 then for all Q̄ > Q we have qj (Q̄) = 0. If qj (Q) > 0, then by letting q
′
j =

d
dQ

qj (Q)

we have

q
′
j f

′
+ qj f

′′
+ f

′ − C′′j q
′
j − αj q

′
j σ

2
j = 0,

implying that

q
′
j = −

f
′
+ qj f

′′

f ′ − C′′j − αj σ
2
j

≤ 0, (2.12)

so qj (Q) is decreasing in Q. We can rewrite equation (2.11) as

qj f
′
(Q) + f (Q) − C′j (qj ) = αj qj σ

2
j . (2.13)

The left hand side is decreasing in qj , therefore the solution qj (Q) decreases if αj and/or σj

increases.

1We use the notation R′
j

= d
dQj

Rj (Qj ).



CUBO
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A Multiobjective Model of Oligopolies under Uncertainty 111

The total industry output Q at the equilibrium is the solution of

n
∑

j=1

qj (Q) − Q = 0. (2.14)

At Q = 0, all qj (Q) ≥ 0, so at Q = 0 the left hand side is nonnegative. Since

n
∑

j=1

qj (Q) ≤

n
∑

j=1

qj (0),

it follows that the left hand side converges to −∞ as Q → ∞, furthermore it is strictly decreasing

in Q. Consequently there is a unique nonnegative solution of equation (2.14) proving the existence

and the uniqueness of the equilibrium. In summary, we have the following result.

Theorem 1 Under conditions (a) and (b) there is a unique equilibrium of the modified n-

person oligopoly with payoff functions (2.4).

The unique equilibrium of theorem 1 is usually different from the Cournot-Nash equilib-

rium of the n-firm oligopoly under the assumption that all firms know the true price function

f . Let Q
∗
(α, σ) denote the total industry output at the equilibrium with given parameters

α = (α1, · · · , αn) and σ = (σ1, · · · , σn). We will prove the following result:

Theorem 2 The value of Q∗(α, σ) decreases if any αj or σj increases.

Proof: Assume that αj < ᾱj with all other αi and all σi values unchanged. Let qi and q̄i

denote the corresponding equilibrium outputs and let Q =
∑

i qi and Q̄ =
∑

i q̄i. Contrary to the

assertion assume that Q < Q̄. From the monotonicity of the functions qi(Q) we have for all i 6= j,

qi(Q) ≥ qi(Q̄) = q̄i(Q̄), (2.15)

since αi and σi do not change. However

qj (Q) ≥ q̄j (Q) ≥ q̄j (Q̄), (2.16)

where we have used the monotonicity of the function qj (Q) in αj . Hence

Q =

n
∑

i=1

qi(Q) ≥

n
∑

i=1

q̄i(Q̄) = Q̄, (2.17)

which is an obvious contradiction.

The assertion of theorem 2 can be reformulated as follows: If any firm increases its weight αj

of uncertainty and, or assumes larger level σ
2
j of uncertainty of the price function, then the total

industry output decreases at the equilibrium.

3 Dynamic Models and Local Stability Analysis

We recall from the previous section that Rj (Qj) denotes the best response of firm j. In this section

we consider dynamic processes with the firms’ adjustment of output based on their best responses.



112 Carl Chiarella and Ferenc Szidarovszky CUBO
11, 2 (2009)

Considering continuous time scales first and assuming that each firm adjusts its output in the

direction toward its best response, we obtain the system of ordinary differential equations

q̇j = Kj (Rj (Qj ) − qj ), (j = 1, 2, · · · , n), (3.1)

where Kj is a sign preserving function, i.e.

Kj (∆)







= 0, if ∆ = 0,

> 0, if ∆ > 0,

< 0, if ∆ < 0.

Theorem 3 The equilibrium is locally asymptotically stable under conditions (a) and (b) and

by assuming that K′j (0) > 0 for all j.

Proof: The Jacobian of system (3.1) at the equilibrium has the special form,

J C =








−K′1 K
′
1R

′
1 · · · K

′
1R

′
1

K
′
2R

′
2 −K

′
2 · · · K

′
2R

′
2

.

.

.
.
.
.

.

.

.

K
′
nR

′
n K

′
nR

′
n · · · −K

′
n








, (3.2)

where K
′
j = K

′
j(0), and R

′
j is the derivative of Rj at the equilibrium.

The Jacobian (3.2) may be represented as

J C = K + a · 1
T
, (3.3)

with

K = diag(−K′1(1 + R
′
1), · · · , −K

′
n(1 + R

′
n)), 1

T
= (1, · · · , 1)

and

a = (K
′
1R

′
1, K

′
2R

′
2, · · · , K

′
nR

′
n)

T
.

The characteristic polynomial of J C is given by

det(K + a · 1T − λI) = det(K − λI)det(I + (K − λI)−1a · 1T )

=Π
n
i=1(−K

′
i(1 + R

′
i) − λi)

[

1 +

n
∑

i=1

K
′
iR

′
i

−K′i(1 + R
′
i) − λ

]

. (3.4)

Notice that relation (2.8) implies that 1 + R
′
j > 0 for all j, so all roots of the first product are

real and negative. It is therefore sufficient to show that all roots of the equation

n
∑

i=1

K
′
iR

′
i

−K′i(1 + R
′
i) − λ

= −1, (3.5)

are also real and negative. We might assume that the denominators are different, otherwise the

sum of terms with identical denominators can be represented as a single term where the numerators



CUBO
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A Multiobjective Model of Oligopolies under Uncertainty 113

are added. Equation (3.5) is equivalent to a polynomial equation of degree n, so there are n real

or complex roots. Let g(λ) denote the left hand side, then clearly

lim
λ→±∞

g(λ) = 0, lim
λ→−K′

i
(1+R′

i
)±0

g(λ) = ±∞,

and

g
′
(λ) =

n
∑

i=1

K
′
iR

′
i

(−K′i(1 + R
′
i) − λ)

2
< 0,

so g is strictly decreasing locally. The graph of the function g is shown in figure 1. There are n

negative poles at λ = −K′j(1 + R
′
j ) (j = 1, 2, · · · , n), there is a root between each pair of consecu-

tive poles and there is an additional root before the first pole. We have demonstrated that there

are n real negative roots, consequently all roots are real and negative.

g(λ)

λ

−1
b

b b b b b

−K
′

1
(1 + R

′

1
)

−K
′

2
(1 + R

′

2
)

−K
′

n−2
(1 + R

′

n−2
)

−K
′

n−1
(1 + R

′

n−1
)

−K
′

n(1 + R
′

n)
. . .

Figure 1: Graph of the function g(λ)

Considering discrete time scales next, we assume again that the firms adjust their outputs

into the direction toward their best responses, and so the outputs adjust according to

qj (t + 1) = qj (t) + Kj (Rj (Qj ) − qj ), (j = 1, 2, · · · , n), (3.6)

where Kj is a sign-preserving function for all j.



114 Carl Chiarella and Ferenc Szidarovszky CUBO
11, 2 (2009)

Theorem 4 Assume that conditions (a) and (b) hold, furthermore K′j(0) > 0 for all j. The

equilibrium is locally asymptotically stable if

K
′
j(0) <

2

1 + R
′
j (Q

∗
j )

, (3.7)

for all j, where q∗ = (q∗j ) is the equilibrium and Q
∗
j =

∑

i6=j q
∗
i , and

n
∑

j=1

K
′
j(0)R

′
j (Q

∗
j )

2 − K′j(0)(1 + R
′
j (Q

∗
j ))

> −1. (3.8)

If any of these conditions is violated with strict inequality, then the equilibrium is unstable.

Proof: The Jacobian of system (3.6) at the equilibrium can be written as

J D = I + J C , (3.9)

where I is the n × n identity matrix. All eigenvalues of J D are inside the unit circle if and only if

all eigenvalues of J C are inside the interval (−2, 0). From the proof of theorem 3 we know that all

eigenvalues of J C are negative, so the eigenvalues are larger than −2 if and only if

−K′j(1 + R
′
j ) > −2, (3.10)

for all j, and

g(−2) > −1. (3.11)

Notice that inequality (3.10) can be rewritten as (3.7), and inequality (3.11) can also be

rewritten as (3.8).

If either (3.10) or (3.11) is violated with strict inequality, then at least one eigenvalue of J C

becomes less than −2, so at least one eigenvalue of J D is below −1, demonstrating the instability

of the equilibrium in this case.

Notice that all denominators of inequality (3.7) are positive because of the relation (2.8).

Condition (3.7) implies that all denominators on the left hand side of (3.8) are also positive.

Therefore condition (3.8) can be interpreted as stating that all derivatives K
′
j(0) must be sufficiently

small in order to guarantee the local asymptotical stability of the equilibrium.

Received: April 22, 2008. Revised: May 16, 2008.



CUBO
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A Multiobjective Model of Oligopolies under Uncertainty 115

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