CUBO A Mathematical Journal

Vol.11, N
o
¯ 02, (55–84). May 2009

Cournot Models: Dynamics, Uncertainty and Learning

Ferenc Szidarovszky

Systems & Industrial Engineering Department,

The University of Arizona, Tucson,

Arizona, 85721-0020, USA

email: szidar@sie.arizona.edu

and

Vernon L. Smith and Steven Rassenti

Interdisciplinary Center for Economic Science,

George Mason University, 3330 Washington Boulevard

Arlington, VA 22201

emails: vsmith2@gmu.edu, srassent@gmu.edu

ABSTRACT

This chapter gives an overview of the recent developments in the theory of dynamic oligopolies

including some new results. We will discuss the Cournot classical model and its extensions

to product differentiation, multiproduct models, price adjusting oligopolies, labor managed and

rent seeking games. The dynamic process based on these models will be analyzed. From the

theoretical point of view we will investigate models with and without full information, with

partial cooperation among the firms, and under the assumption that the information about the

production levels of the rivals has time delay. We will also introduce and discuss special learning

procedures based on repeated price information. We will also briefly discuss investigations based

on laboratory experiments, in which more realistic cases can be examined.

RESUMEN

Damos una descripción general de los recientes desarrollos de la teoria de dinámica de oligopolios

incluyendo algunos resultados novos. Discutimos el modelo clásico de Cournot y sus extensiones



56 Ferenc Szidarovszky, Vernon L. Smith and Steven Rassenti CUBO
11, 2 (2009)

a la diferenciación producto, modelos multiproducto, ajuste de precios de oligopolios, labores

dirigidas y juegos de rent seeking. La dinámica de procesos basados en estos modelos será anal-

izada. Desde el punto de vista teórico investigaremos modelos con y sin información completa,

con cooperación parcial entre las firmas, y bajo la suposición que la información al respecto de los

niveles de producción de los rivales tiene un tiempo de retrazo. Introduzimos y discutimos proce-

sos especiales de aprendizaje basados en información de precios repetidos. También brevemente

discutimos investigaciones sobre experimentos de laboratórios, en los cuales casos mas realisticos

son examinados.

Key words and phrases: n-person game, oligopoly, dynamic systems, stability learning.

Math. Subj. Class.: 91A20, 91A80.

1 Introduction

Since the pioneering work of [1] oligopoly models are the most frequently discussed topics in the literature

of mathematical economics. They describe the interaction of manufacturers and service suppliers through

some market demand structure. Most of the authors consider this problem as a game in which the supplied

quantities (or selected prices) are the strategies and the payoff functions are defined as the profits of the

firms.

In a competitive environment the Nash equilibrium is the solution of the game, in which none of the firms

can increase its profit by changing its production level alone. In an N -person oligopoly there is no guarantee

in general for the existence of such equilibrium, and even if equilibrium exists, there is the possibility of the

existence of multiple equilibria ([5]).

Several versions of the Cournot model has been developed in the literature including oligopolies without

and with product differentiation, multiproduct oligopolies, labor managed and rent seeking models as well

as oligopsonies, in which the firms also compete on the factor market. A comprehensive summary of the

different model variants is presented in [6].

If the state of an oligopoly is a Nash equilibrium at a certain time, then no firm has the interest to move

away from the equilibrium, therefore the state will remain at the equilibrium for all future times. However in

a disequibilium state at least one of the firms is able to increase its profit by changing strategy, so the state

of the system will change. If the new state is an equilibrium, then it will remain the state of the system for

all future times. Otherwise another change of the state will occur, and so, a dynamic process will develop.

The dynamics of this process depend on the desires of the firms, and also on the accuracy and timing of the

available information. During this process the firms are able to monitor repeated information on the demand

structure and the actions of the competitors, which rises the possibility of some learning mechanisms.

The research on dynamic behavior of the firms can be done in two fundamentally different ways. The

mathematical models are always based on certain assumptions on the objectives of the firms, on the types

and analytical properties of the functions involved, and on the information structure. Those assumptions

very often differ from the economic reality. In more realistic cases it is very often impossible to obtain

nice mathematical results, so simulation is used. The applied simulation methodology still depends on the

basic assumptions of the model, it simple generates experimental results in the absence of analytical tools.



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Cournot Models: Dynamics, Uncertainty and Learning 57

Another methodology is based on actual laboratory experiments, when real people make the decisions under

computer generated environment, and by monitoring their repeated actions we are able to gain a detailed

understanding of their priorities, information usage, and decision making mechanism.

In this paper we will give a brief overviews of the most important problems arising in examining dynamic

oligopolies, and in addition we will offer some new results on this topic. This chapter is developed as

follows. Static oligopoly models will be first introduced in Session 2 and the existence and uniqueness of the

equilibrium will be discussed briefly. Dynamic models with full information on the demand structure will be

then examined, and Session 4 will be devoted to partial cooperation among the firms. Session 5 will deal with

the effect of uncertainty in the information on the demand structure. Information delays will be considered

in Session 6 when we will discuss how the stability of the equilibrium is lost through Hopf bifurcation giving

the possibility of the birth of limit cycles. Some special learning schemes will be introduced in Session 7.

The fundamentals of experimental economics will be outlined in Session 8. Finally, some conclusions will be

drawn.

2 The Cournot Model and Its Extensions

Consider an economy of N firms that produce the same product or offer the same service to a homogeneous

market. Let k = 1, 2, . . . , N denote the firms and xk the produced or offered quantity of firm k. The total

output of the industry is Q =
∑N

k=1 xk, and we assume that the market price f is a function of Q. If Ck(xk)

is the cost of firm k, then its profit is given as

ϕk(x1, . . . , xN ) = xkf (Q) − Ck(xk). (2.1)

Let Lk denote the capacity limit of firm k. Then an N -person noncooperative game is defined, in which

the firms are the players, interval [0, Lk] is the set of strategies of player k, and ϕk is its payoff function. This

model is known as the classical Cournot model, or the single-product quantity adjusting oligopoly without

product differentiation. In the literature of oligopoly theory it is usually assumed that functions f and Ck

(k = 1, 2, . . . , N ) are twice continuously differentiable and for all xk ∈ [0, Lk] and Q ∈ [0,
∑N

k=1 Lk],

(A) f
′
(Q) < 0;

(B) xkf
′′
(Q) + f

′
(Q) < 0;

(C) f
′
(Q) − C′′k (xk) < 0.

Notice that the payoff ϕk of player k does not depend explicitly on the outputs of each individual

competitor, it depends on only the output Qk =
∑

l 6=k xl of the rest of the industry. For each feasible value of

Qk we can easily determine the best strategy choice of firm k, which is called its best reply. Under assumptions

(A)-(C), function ϕk is strictly concave in xk with any fixed value of Qk, and simple differentiation shows

that



58 Ferenc Szidarovszky, Vernon L. Smith and Steven Rassenti CUBO
11, 2 (2009)

arg max
xk

{xkf (xk + Qk) − Ck(xk)} =















0, if f (Qk) − C
′
k(0) < 0;

Lk, if f (Qk + Lk) + Lkf
′
(Qk + Lk)

−C′k(Lk) > 0;

zk, otherwise.

(2.2)

where zk is the unique solution of the strictly monotonic equation

f (Qk + xk) + xkf
′
(Qk + xk) − C

′
k(xk) = 0 (2.3)

in [0, Lk]. Since the value of zk depends on Qk, we may write zk = Rk(Qk).

In our future analysis we will need the derivative of the best response function. In the neighborhood of an

interior optimum, Rk is differentiable as the consequence of the implicit function theorem. By differentiating

equation (2.3) implicitly with respect to Qk, we have

f
′ · (1 + R′k) + R

′
k · f

′
+ xkf

′′ · (1 + R′k) − C
′′
k · R

′
k = 0

implying that

R
′
k(Qk) = −

f
′
(Q) + xkf

′′
(Q)

2f ′(Q) + xkf
′′(Q) − C′′

k
(xk)

.

Assumptions (B) and (C) imply that

−1 < R′k(Qk) < 0.

Clearly a strategy vector x
∗

= (x
∗
1, . . . , x

∗
N ) is a Nash-equilibrium if and only if for all k,

(i) x
∗
k ∈ [0, Lk];

(ii) x
∗
k = Rk

(
∑

l 6=k x
∗
l

)

.

It is well known (see for example, [5]) that under conditions (A)-(C) there is a unique Nash equilibrium.

Assume next that the firms produce different but related goods. Let x1, . . . , xN denote again the pro-

duced quantities and let fk(x1, . . . , xN ) denote the price of the product of firm k. Then the profit of this firm

is given as

ϕk(x1, . . . , xN ) = xkfk(x1, . . . , xN ) − Ck(xk). (2.4)

This model is known as a single product oligopoly with product differentiation.

In Bertrand (or price adjusting) oligopolies we consider a single product oligopoly with product differ-

entiation in which each firm selects its price. If P1, . . . , PN are the selected prices and dk(P1, . . . , PN ) is the

demand function of the product of firm k, then the profit of this firm can be obtained again by equation

(2.4), where for all k, fk = Pk and



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Cournot Models: Dynamics, Uncertainty and Learning 59

xk = dk(P1, . . . , PN ),

so the profit functions now depend on only the price selections.

Multiproduct oligopolies are obtained by assuming that the firms produce M different items. Let x
(m)
k

denote the output of firm k of product m, then the firm’s production can be characterized by its production

vector xk = (x
(1)
k

, . . . , x
(M)
k

). The vector Q =
∑N

k=1 xk shows the total production vectors of the industry. If

Ck(xk) is the production cost of firm k, then its profit is given as

ϕk(x1, . . . , xN ) = x
T
k f (Q) − Ck(xk). (2.5)

Here f (Q) =
(

f1(Q), . . . , fM (Q)
)

with fm(Q) (1 ≤ m ≤ M ) being the price function of product m.

Consider again the classical Cournot model (2.1) and let lk(xk) denote the labor force needed by firm k

to produce output xk. Then the profit per labor of this firm can be given as

Ψk(x1, . . . , xN ) =
ϕk(x1, . . . , xN )

lk(xk)
=

xkfk(Q) − Ck(xk)

lk(xk)
. (2.6)

If the firms maximize profits per labor instead of their total profits, then the oligopoly is said to be

labor managed.

Let N denote the number of agents involved in rent seeking activity. Let xk be the expenditure of agent

k and fk(xk) its production function for lotteries, then the probability that agent k will win the rent is

Pk =
fk(xk)

∑N

l=1 fl(xl)
.

If the rent is normalized to 1, then the expected net rent of agent k is given as

Πk =
fk(xk)

∑N

l=1 fl(xl)
− xk. (2.7)

This model is known as a rent-seeking game. By introducing the new variables yk = fk(xk) and Ck = f
−1
k

it

is clear that the payoff function (2.7) has the new form

Πk =
yk

∑N

l=1 yl

− Ck(yk). (2.8)

Notice that function form (2.8) reduces to (2.1) by selecting f (Q) = 1/Q, so rent-seeking games are usually

considered as special oligopolies.

The equilibria of the above extensions can be defined in the same way as it has been presented for the

classical Cournot model. Existence and uniqueness results are represented in [6].



60 Ferenc Szidarovszky, Vernon L. Smith and Steven Rassenti CUBO
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3 Dynamic Models with Full Information

In this session we consider only the classical Cournot model, and assume that the firms know the true price

function and the simultaneous output values of all competitors. In this case for firm k, the best output choice

is Rk(Qk) as it was shown in Session 2.2. If xk is the current output of the firm and Rk(Qk) is its desired

output, then under continuous time scales the firm will change its output in the direction toward the desired

output, since it cannot ”jump” with the output value instantaneously. Therefore the output change of the

firms can be described by the system of ordinary differential equations

ẋk(t) = Kk (Rk(Qk(t)) − xk(t)) (k = 1, 2, . . . , N ) (3.1)

where Kk > 0 is a constant being called the speed of adjustment of firm k.

Here we assume that the firms know their best response functions, and instantaneous information is

available about the market price. If P (t) denotes the price at time period t, then

P (t) = f (xk(t) + Qk(t))

from which firm k is able to calculate the output if the rest of the industry:

Qk(t) = f
−1

(P (t)) − xk(t).

In this way the firms have all necessary information to proceed with output adjustments (3.1).

An output vector x
∗

= (x
∗
1, . . . , x

∗
N ) is a steady state of system (3.1) if and only if for all k,

x
∗
k = Rk





∑

l 6=k

x
∗
l



 ,

that is, when x
∗

is a Nash equilibrium.

Example 1 Assume linear price and cost functions:

f (Q) = B − AQ, Ck(xk) = akxk + bk (k = 1, 2, . . . , N ).

In this case equation (2.3) can be written as

B − A(Qk + xk) + xk(−A) − ak = 0

implying that

xk = −
Qk

2
+

B − ak
2A

,

so the best response of firm k is



CUBO
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Cournot Models: Dynamics, Uncertainty and Learning 61

Rk(Qk) = −
Qk

2
+

B − ak
2A

by assuming interior optimum. Therefore the dynamic system (3.1) can be rewritten as follows:

ẋk = Kk



−
1

2

∑

l 6=k

xl − xk +
B − ak

2A



 (3.2)

for k = 1, 2, . . . , N .

∇

Let now x
∗

= (x
∗
1, . . . , x

∗
N ) denote an interior equilibrium of the classical Cournot model. We can

examine the local asymptotic stability of this equilibrium with respect to the dynamic process (3.1). The

Jacobian J
C

of the system at the equilibrium has the special structure










−K1 K1r1 · · · K1r1

K2r2 −K2 · · · K2r2
.
.
.

.

.

.
. . .

.

.

.

KN rN KN rN · · · −KN










(3.3)

where rk = R
′
k(Q

∗
k) with Q

∗
k =

∑

l 6=k x
∗
l . We have seen earlier that −1 < rk < 0. The characteristic

polynomial of J
C

can be written as

ϕ(λ) = det(D + ab
T
− λI)

with D = diag(−K1(1 + r1), . . . , −KN (1 + rN )), a = (K1r1, . . . , KN rN )
T
, and b

T
= (1, . . . , 1). In obtaining

a closed form representation of ϕ(λ) we can use the well-known fact that with any N -element vectors a and

b, det(I + ab
T
) = 1 + a

T
b. Therefore

ϕ(λ) = det(D − λI) · det(I + (D − λI)−1abT )

=

N
∏

k=1

(−Kk(1 + rk) − λ) ·

[

1 +

N
∑

k=1

Kkrk

−Kk(1 + rk) − λ

]

. (3.4)

The main result of this session can be formulated as

Theorem 1 All eigenvalues of J C have negative real parts implying the local asymptotic stability of the

equilibrium.

Proof The roots of function (3.4) are λ = −Kk(1 + rk), which are all negative, and the roots of the

bracketed factor. This equation clearly can be rewritten as



62 Ferenc Szidarovszky, Vernon L. Smith and Steven Rassenti CUBO
11, 2 (2009)

s
∑

i=1

αi

βi − λ
+ 1 = 0 (3.5)

where αi < 0, βi < 0 and the βi values are different. This equation is equivalent to a polynomial equation of

degree s, so there are s real or complex roots. Let g(λ) denote the left hand side, then

lim
λ→±∞

g(λ) = 1, lim
λ→±βi

g(λ) = ±∞,

g
′
(λ) =

s
∑

i=1

αi

(βi − λ)2
< 0.

The graph of function g is shown in Figure 1. Clearly there is a root before β1 and one root between βi and

βi+1 for i = 1, 2, . . . , s − 1. Since βi < 0 for all i, we found s real negative roots. Hence all eigenvalues of J
C

are real and negative, which completes the proof.

2

1 2 1s s

g

1

Figure 1: The graph of function g.

Assume next that the time scales are discrete. If xk(t) denote the production level of firm k at time

period t, then its best choice with given xl(t) (l 6= k) values is Rk(Qk(t)), where Qk(t) =
∑

l 6=k xl(t). In many

industries the firms are unable to make large changes in their production levels during a single time period,

therefore they select levels in the direction toward their best choices. This dynamism can be conveniently

modelled as

xk(t + 1) = αkxk(t) + (1 − αk)Rk(Qk(t)) (3.6)

for k = 1, 2, . . . , N , where 0 ≤ αk < 1 is a given constant for all k. In order to examine the asymptotic

behavior of the equilibrium notice first that the Jacobian of this system at the equilibrium can be given as

follows:



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Cournot Models: Dynamics, Uncertainty and Learning 63

J
D

=










α1 (1 − α1)r1 · · · (1 − α1)r1

(1 − α2)r2 α2 · · · (1 − α2)r2
.
.
.

.

.

.
. . .

.

.

.

(1 − αN )rN (1 − αN )rN · · · αN










. (3.7)

We can rewrite this matrix similarly to the continuous case:

J
D

= D + ab
T

with

D = diag ((α1 − 1)r1 + α1, . . . , (αN − 1)rN + αN ) ,

a = ((1 − α1)r1, . . . , (1 − αN )rN )
T

, and b
T

= (1, . . . , 1).

Therefore the characteristic polynomial of J
D

is given as

ϕ(λ) = det(D + ab
T − λI) = det(D − λI) det(I + (D − λI)−1abT )

=

N
∏

k=1

((αk − 1)rk + αk − λ) ·

[

1 +

N
∑

k=1

(1 − αk)rk
(αk − 1)rk + αk − λ

]

. (3.8)

In this case Theorem 1 can be modified as follows:

Theorem 2 All eigenvalues of J D are inside the unit circle if and only if

N
∑

k=1

(1 − αk)rk
(αk − 1)rk + αk + 1

> −1. (3.9)

In this case the equilibrium is locally asymptotically stable. If the left hand side is smaller than −1, then

the equilibrium is unstable.

Proof The eigenvalues are λ = (αk − 1)rk + αk, which are all positive, and the roots of the bracketed

factor. First we show that the roots (αk − 1)rk + αk are inside the unit circle. Since they are positive it is

sufficient to show that

(αk − 1)rk + αk < 1,

which is obviously true, since it can be rewritten as

(αk − 1)(−rk − 1) > 0,

where both factors are negative. The other roots are the solutions of equation (3.5) where αi < 0 and

0 < βi < 1 for all i. The graph of function g is the same as it is shown in Figure 1 with the only difference



64 Ferenc Szidarovszky, Vernon L. Smith and Steven Rassenti CUBO
11, 2 (2009)

that all βi values are between 0 and 1. All roots between the pairs βi and βi+1 are inside the unit circle, and

the smallest root is also inside the unit circle if and only if g(−1) > 0. It is easy to see that this inequality

is equivalent to condition (3.9).

2

If all firms select the best response, then αk = 0 for all k, and in this case condition (3.9) can be rewritten

as

N
∑

k=1

rk

1 − rk
> −1, (3.10)

which certainly holds if all values rk are sufficiently close to zero. In the case of symmetric oligopolies the Rk

best responses are identical, so rk ≡ r. In this further special case relation (3.10) simplifies to the following:

N r

1 − r
> −1

which can be rewritten as

r >
−1

N − 1
. (3.11)

Example 2 Consider again the linear case given in the previous example. Since R′k(Qk) = −
1
2

for all k,

relation (3.11) holds only for N = 2, so the equilibrium is asymptotically stable for only duopolies with αk = 0

(k = 1, 2, . . . , N ).

In this case relation(3.9) simplifies as

N
∑

k=1

αk − 1

αk + 3
> −1 (3.12)

which certainly holds if the αk values are sufficiently close to 1.

∇

In this session we have focused on the classical Cournot model, however its extensions and all model

variants can be examined similarly to this case.

4 Models with Partial Cooperation

A usual way how the firms might increase their profits is to seek certain cooperation with the rivals. A

common way to model such ”partial” cooperation can be given as follows. Let ϕk denote the profit of firm k

(1 ≤ k ≤ N ), and let γkl denote a nonnegative constant for all k and l, which shows the cooperation level of



CUBO
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Cournot Models: Dynamics, Uncertainty and Learning 65

firm k toward firm l. Then firm k maximizes ϕk +
∑

l 6=k γklϕl that is, it takes a certain portion of the profits

of the competitors into account in addition to its own profit. Thus the payoff function of firm k becomes

Ψk(x1, . . . , xN ) = (xkf (Q) − Ck(xk)) +
∑

l 6=k

γkl(xlf (Q) − Cl(xl)) (4.1)

in the case of the classical Cournot model. For the sake of mathematical convenience assume that γkl ≡ γk

for all k and l, that is, each firm has identical cooperation levels toward its rivals. In this special case

Ψk(x1, . . . , xN ) = (xk + γkQk)f (Q) − Ck(xk) − γk
∑

l 6=k

Cl(xl). (4.2)

With given value of Qk, the best response of firm k can be obtained by simple differentiation. Assuming

interior optimum, then at the optimum

f (xk + Qk) + (xk + γkQk)f
′
(xk + Qk) − C

′
k(xk) = 0. (4.3)

The derivative of the left hand side with respect to xk is the following:

2f
′
(Q) + (xk + γkQk)f

′′
(Q) − C′′k (xk).

Assume that conditions (A) and (C) introduced in Session 2.2 are satisfied with a modified version of condition

(B):

(B’) (xk + γkQk)f
′′
(Q) + f

′
(Q) < 0

for all xk ∈ [0, Lk], Qk ∈ [0,
∑

l 6=k Ll] and Q = xk + Qk. Then Ψk is strictly concave in xk, so the payoff

maximizing xk value is unique. If it is interior, then it can be obtained as the unique solution of the monotonic

equation (4.3). Let Rk(Qk) denote the solution as before. By differentiating equation (4.3) implicitly with

respect to Qk it is easy to see that

R
′
k(Qk) = −

(1 + γk)f
′
(Q) + (xk + γkQk)f

′′
(Q)

2f ′(Q) + (xk + γkQk)f
′′(Q) − C′′

k
(xk)

. (4.4)

Clearly both the numerator and denominator are negative, so R
′
k(Qk) is always negative. In addition,

R
′
k(Qk) > −1 if the following stronger version of condition (C) is satisfied:

(C’) (1 − γk)f
′
(Q) − C′′k (xk) < 0

for all 0 ≤ xk ≤ Lk and 0 ≤ Q ≤
∑N

l=1 Ll.

Under conditions (A), (B’) and (C’) all results of the previous session remain true for this case with the

only difference that R
′
k(Qk) is now given by equation (4.4).



66 Ferenc Szidarovszky, Vernon L. Smith and Steven Rassenti CUBO
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5 Models with Uncertain Price Function

In this session we will examine again the classical Cournot model, however the methodology and the results

to be discussed here can be extended to other model variants.

Assume now that the firms do not know the true price function f , but they have certain estimates of it.

Let fk denote the estimated price function by firm k. Then this firm believes that its profit is

ϕk(x1, . . . , xN ) = xkf k(Q) − Ck(xk). (5.1)

Assume that conditions (A)-(C) are satisfied with f being replaced by f k. Then with all fixed Qk, firm k

has a believed profit maximizing output Rk(Qk), which is usually different than the ”true” best response of

the firm.

Consider first continuous time scales and assume that similarly to the full information case, each firm

adjusts its production level into the direction toward its believed best reply. Then we have a modified version

of system (3.1):

ẋk(t) = Kk

(

Rk

(

Qk(t)
)

− xk(t)
)

(k = 1, 2, . . . , N ) (5.2)

where Qk(t) is the estimate of Qk(t) by firm k. Notice that the true price function is not known by the firm,

so it cannot compute the true value of Qk. Instead the following method is used. The true market price,

which the firm observes is f (xk + Qk). On the other hand, firm k believes that it equals f k(xk + Qk), so in

fact, firm k solves the equation

f k(xk + Qk) = f (xk + Qk)

to get its estimate

Qk = (f
−1

k ◦ f )(xk + Qk) − xk. (5.3)

For the sake of simplicity introduce the function Gk = f
−1

k ◦ f . Then system (5.2) can be rewritten as

ẋk(t) = Kk

(

Rk (Gk (xk(t) + Qk(t)) − xk(t)) − xk(t)
)

(5.4)

for k = 1, 2, . . . , N . Notice that the steady state of this system (if exists) is usually different than the Nash

equilibrium, since Rk usually differs from the true best response function Rk. We can refer to this steady

state as the ”believed” equilibrium, which will be denoted as x
∗

= (x
∗
1, . . . , x

∗
N ). The Jacobian J

C
of the

system has a similar form to matrix (3.3):










K1 (r1(g1 − 1) − 1) K1r1g1 · · · K1r1g1

K2r2g2 K2 (r2(g2 − 1) − 1) · · · K2r2g2
.
.
.

.

.

.
. . .

.

.

.

KN rN gN KN rN gN · · · KN (rN (gN − 1) − 1)










(5.5)



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Cournot Models: Dynamics, Uncertainty and Learning 67

with gk = G
′
k(x

∗
k + Q

∗
k).

In the full information case, fk ≡ f for all k, therefore Gk is the identity function with gk = 1. In this

case J
C

reduces to matrix (3.3). Otherwise fk is only an estimate if f , and if this estimate is sufficiently

good, then Gk is close to the identity function with gk ≈ 1. The location of the eigenvalues of J
C

can be

similarly examined as it was shown in the full information case. Observe that

J
C

= D + ab
T

where in this case D = diag (−K1(1 + r1), . . . , −KN (1 + rN )), a = (K1r1g1, . . . , KN rN gN )
T

and b
T

=

(1, . . . , 1). The characteristic polynomial of this matrix is also similar to (3.4):

N
∏

k=1

(−Kk(1 + rk) − λ) ·

[

1 +

N
∑

k=1

Kkrkgk

−Kk(1 + rk) − λ

]

. (5.6)

Since f is strictly decreasing and we may assume that all estimates fk are also strictly decreasing (otherwise

the firms’ estimates are irrealistic), Gk is strictly increasing with nonnegative derivative gk. Therefore the

proof of Theorem 1 can be used without any changes to show that the ”believed” equilibrium is locally

asymptotically stable in this case as well.

Assume next that the time scales are discrete. In this case the discrete dynamic system (3.6) is modified

as

xk(t + 1) = αkxk(t) + (1 − αk)Rk (Gk (xk(t) + Qk(t)) − xk(t)) (5.7)

for k = 1, 2, . . . , N , where we used equation (5.3) again. The steady state of this system (if exists) is usually

different than the Nash equilibrium, so we might refer to the steady state as a ”believed” equilibrium similarly

to the continuous case.

The Jacobian J
C

of system (5.7) has the special structure










α1 + (1 − α1)r1(g1 − 1) (1 − α1)r1g1 · · · (1 − α1)r1g1

(1 − α2)r2g2 α2 + (1 − α2)r2(g2 − 1) · · · (1 − α2)r2g2
.
.
.

.

.

.
. . .

.

.

.

(1 − αN )rN gN (1 − αN )rN gN · · · αN + (1 − αN )rN (gN − 1)










(5.8)

where rk and gk are as before. In the full information case gk = 1, so J
C

reduces to matrix (3.7). Otherwise

fk is only an estimate of f , and if this estimate is sufficiently good, then Gk is close to the identity function with

gk ≈ 1. This matrix can also be rewritten as J
D

= D+ab
T

with D = diag ((α1 − 1)r1 + α1, . . . , (αN − 1)rN + αN ),

a = ((1 − α1)r1g1, . . . , (1 − αN )rN gN )
T

and b
T

= (1, . . . , 1). The characteristic polynomial of J
D

has a sim-

ilar form to the previously discussed cases:

N
∏

k=1

((αk − 1)rk + αk − λ) ·

[

1 +

N
∑

k=1

(1 − αk)rkgk
(αk − 1)rk + αk − λ

]

. (5.9)



68 Ferenc Szidarovszky, Vernon L. Smith and Steven Rassenti CUBO
11, 2 (2009)

It is easy to see that Theorem 2 remains valid in this case with the only difference that condition (3.9) has

to be modified as

N
∑

k=1

(1 − αk)rkgk
(αk − 1)rk + αk + 1

> −1. (5.10)

6 The Effect of Information Delay

In this session we will examine how delayed information affects the asymptotic behavior of the equilibrium.

For the sake of simplicity we will consider only the classical Cournot model.

Assume that at time period t firm k obtains a delayed information on the production level of the rest of

the industry, Qk(sk), where t− sk is the delay. If firm k uses this latest information to form its best response,

then the dynamic system becomes

ẋk(t) = Kk (Rk (Qk(sk)) − xk(t)) (k = 1, 2, . . . , N ). (6.1)

If the delay is known and it is denoted by dk(t), then sk = t − dk(t), and so equation (6.1) becomes a

difference-differential equation. However the delay is uncertain in real economies, therefore a convenient

modelling way is offered by considering it as a random variable and replacing the random right hand sides of

equation (6.1) by their expected values. In this way a Volterra-type integro-differential equation is obtained:

ẋk(t) = Kk

(∫ t

0

w(t − s, Tk, mk)Rk (Qk(s)) ds − xk(t)

)

. (6.2)

The weighting function w is defined as

w(t − s, T, m) =

{
1
T

e
− t−s

T , if m = 0;

1
m!

(
m
T

)m+1
(t − s)me−

m(t−s)

T , if m ≥ 1,
(6.3)

where T > 0 is a real and m ≥ 0 is an integer parameter. This weighting function has the following properties:

(a)
∫ ∞

0
w(s, T, m)ds = 1;

(b) If m = 0, then weights are exponentially decreasing with the largest weight given to the most

current data. If m ≥ 1, then the most current data has zero weight, the weight is increasing to a maximal

value at t − s = T , and decreases thereafter.

(c) With increasing value of m, the weighting function becomes more peaked around t − s = T , as

m → ∞ the weighting function converges to the Dirac delta function centered at t − s = T .

(d) If T → 0, then the weighting function converges to the Dirac delta function centered at zero.

Theorem 3 System (6.2) is equivalent to a system of ordinary differential equations by introducing additional

unknown functions.



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Cournot Models: Dynamics, Uncertainty and Learning 69

Proof Assume first that m = 0 and let P be any function of time. Introduce the new function

P0(t) =

∫ t

0

1

T
e
− t−s

T P (s)ds.

By simple differentiation,

Ṗ0(t) =
1

T
(P (t) − P0(t)) .

Assume next that m ≥ 1, and for all l = 0, 1, . . . , m introduce the functions

Pl(t) =

∫ t

0

1

l!

(
m

T

)l+1

(t − s)le−
m(t−s)

T P (s)ds.

Then by differentiation,

Ṗl(t) =
m

T
(Pl−1(t) − Pl(t))

and

Ṗ0(t) =
m

T
(P (t) − P0(t)) .

By selecting Pk(s) = Rk (Qk(s)), the integro-differential equation system is equivalent to the following

system of ordinary differential equations:

ẋk(t) = Kk (Pkm(t) − xk(t)) (1 ≤ k ≤ N )

Ṗkl(t) =
qk

Tk
(Pk,l−1(t) − Pkl(t)) (1 ≤ k ≤ N, 1 ≤ l ≤ mk)

Ṗk0(t) =
qk

Tk
(Pk(t) − Pk0(t)) (1 ≤ k ≤ N )

with

qk =

{

1, if mk = 0;

mk, if mk ≥ 1.
(6.4)

2

Linearizing equation (6.2) we have for all k,

ẋkδ(t) = Kk



rk

∫ t

0

w(t − s, Tk, mk) ·
∑

l 6=k

xlδ(s)ds − xkδ (t)



 (6.5)

where rk = R
′
k(Q

∗
k), and xkδ (t) is the deviation of xk(t) from its equilibrium level. As it is usual in the theory

of ordinary differential equations we look for the solution as xkδ (t) = vke
λt

. By substituting this form into

(6.5) and letting t → ∞ we have

(λ + Kk)vk − Kkrk

∫ t

0

w(s, Tk, mk)e
−λs

ds

∑

l 6=k

vl = 0



70 Ferenc Szidarovszky, Vernon L. Smith and Steven Rassenti CUBO
11, 2 (2009)

and by using simple integration and the definition of the gamma function this equation simplifies as

(λ + Kk)vk − Kkrk

(

1 +
λTk

qk

)−(mk+1) ∑

l 6=k

vl = 0. (6.6)

Introduce function

Ak(λ) = (λ + Kk)

(

1 +
λTk

qk

)mk+1

to see that equations (6.6) are equivalent to a determinantal equation

det










A1(λ) −K1r1 · · · −K1r1

−K2r2 A2(λ) · · · −K2r2
.
.
.

.

.

.
. . .

.

.

.

−KN rN −KN rN · · · AN (λ)










= 0. (6.7)

Notice that by introducing

D = diag (A1(λ) + K1r1, . . . , AN (λ) + KN rN ) , a = (−K1r1, . . . , −KN rN )
T

and b
T

= (1, . . . , 1) this equation can be rewritten as

det(D + ab
T
) = det(D) · det(I + D−1abT )

=

N
∏

k=1

(Ak(λ) + Kkrk) ·

[

1 −
N
∑

k=1

Kkrk

Ak(λ) + Kkrk

]

= 0.

First we prove that all roots of equation

Ak(λ) + Kkrk = 0 (6.8)

have negative real parts. Clearly λ 6= 0, and assume that with some root λ, Reλ ≥ 0. Then

|λ + Kk| > Kk and

∣
∣
∣
∣
1 +

λTk

qk

∣
∣
∣
∣
> 1

implying that

Kk > |Kkrk| = |Ak(λ)| > Kk

which is an obvious contradiction. As the asymptotic behavior of the equilibrium is concerned we have to

examine the locations of the solutions of equation



CUBO
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Cournot Models: Dynamics, Uncertainty and Learning 71

1 −
N
∑

k=1

Kkrk

Ak(λ) + Kkrk
= 0. (6.9)

Notice first that it is equivalent to a polynomial equation, so there are finitely many roots. In the general case

computer methods are needed to locate the roots, however in special cases analytic results can be obtained.

Assume now that the firms are identical and the equilibrium is symmetric. Then K1 = . . . = KN = K,

T1 = . . . = TN = T , m1 = . . . = mN = m, q1 = . . . = qN = q, and r1 = . . . = rN = r showing that equation

(6.9) reduces to the following:

(λ + K)

(

1 +
λT

q

)m+1

+ (1 − N )Kr = 0. (6.10)

Consider first the case of T = 0, which corresponds to models without information delay. In this case

(6.10) has a unique root λ < 0, so the equilibrium is locally asymptotically stable. Note that this symmetric

case is a special case of Theorem 1.

Assume next that T > 0 and m = 0. Then (6.10) becomes quadratic:

λ
2
T + λ(1 + KT ) + K (1 + (1 − N )r) = 0.

Since all coefficients are positive, all roots have negative real parts implying the local asymptotic stability of

the equilibrium (see for example, [8]).

Consider next the case of T > 0 and m = 1. In this case (6.10) is a cubic equation:

λ
3
T

2
+ λ

2
(2T + T

2
K) + λ(1 + 2KT ) + K (1 + (1 − N )r) = 0. (6.11)

All coefficients are positive and the Routh-Hurwitz criterion implies that all roots have negative real parts if

and only if

(2T + T
2
K)(1 + 2KT ) > T

2
K (1 + (1 − N )r) , (6.12)

which can be rewritten as

2T
2
K

2
+ T K (4 + r(N − 1)) + 2 > 0. (6.13)

The discriminant of the left hand side is

r(N − 1) [8 + r(N − 1)] ,

where r(N − 1) < 0. Therefore we have the following cases.

Case 1. If 8 + r(N − 1) > 0, then the left hand side of (6.13) has no real root, so it always holds.

Therefore the equilibrium is locally asymptotically stable.



72 Ferenc Szidarovszky, Vernon L. Smith and Steven Rassenti CUBO
11, 2 (2009)

Case 2. If 8 + r(N − 1) = 0, then there is a unique root

T K =
−4 − r(N − 1)

4
=

− (8 + r(N − 1)) + 4

4
= 1

and if T K 6= 1, then the equilibrium is locally asymptotically stable.

Case 3. If 8 + r(N − 1) < 0, then there are two real roots

(T K)
∗
1,2 =

−4 − r(N − 1) ±

√

(4 + r(N − 1))
2
− 16

4T 2
.

Notice that −4 − r(N − 1) = −(8 + r(N − 1)) + 4 > 0, so both roots are positive. Hence the equilibrium is

locally asymptotically stable if

T K < (T K)
∗
1 or T K > (T K)

∗
2,

where (T K)
∗
1 < (T K)

∗
2. The equilibrium is unstable if

(T K)
∗
1 < T K < (T K)

∗
2.

Figure 2 shows the stability region of the equilibrium.

1

8

N

0-1
r

KT

1

Figure 2: Stability region in the (r, KT ) space

From the above analysis we can draw the following interesting conclusions. If N < 9, then −8/(N − 1) < −1,

so Case 1. occurs regardless of the value of r, so the equilibrium is always locally asymptotically stable. That

is, we need at least 9 firms to have instability. If N = 9, then Case 1. occurs for r > −1 and Case 2. occurs

for r = −1. Assume next that N ≥ 10. If r > − 8
N−1

, then the equilibrium is always locally asymptotically

stable, if r = − 8
N−1

, then it occurs if KT 6= 1 (that is, always except a particular value of KT ), and if

r < − 8
N−1

, then KT has to be sufficiently small or sufficiently large to guarantee local asymptotical stability.

It is also interesting to note that the stability conditions depend on only the product KT and not on the

individual values of these parameters. It shows a certain compensation between the speed of adjustment and

average information delay.



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Cournot Models: Dynamics, Uncertainty and Learning 73

We also know from the above analysis that by crossing the critical values (T K)
∗
1 and (T K)

∗
2 with the

value of T K, stability is lost or gained. It is interesting to examine what happens at these critical values. We

will show that Hopf-bifurcation occurs (see for example, [4]) implying the possibility of limit cycles around the

equilibrium. At the critical values inequality (6.13) as well as (6.12) become equality implying that equation

(6.11) can be rewritten as

0 = λ
3
T

2
+ λ

2
(2T + T

2
K) + λ

T
2
K (1 + (1 − N )r)

2T + T 2K
+ K(1 + (1 − N )r)

=
(

λT
2
+ (2T + T

2
K)
)
(

λ
2

+
K(1 + (1 − N )r)

2T + T 2K

)

showing that one eigenvalue is negative, λ1 = −
2+T K

T
, and the other two are pure complex. Differentiating

equation (6.11) implicitly with respect to T we have

3λ
2
λ̇T

2
+ λ

3
2T + 2λλ̇(2T + T

2
K) + λ

2
(2 + 2T K) + λ̇(1 + 2T K) + λ2K = 0

implying that

λ̇ =
−2T λ3 − λ2(2 + 2T K) − 2Kλ

3λ2T 2 + 2λ(2T + T 2K) + (1 + 2T K)
. (6.14)

By letting

α
2

=
K(1 + (1 − N )r)

2T + T 2K

clearly

λ̇ =
2T α

3
i + α

2
(2 + 2T K) − 2Kαi

−3α2T 2 + 2αi(2T + T 2K) + (1 + 2T K)

with real part

Reλ̇ =
−2α4T 2(2 + 2T K) + 4α2T (T α2 − K)(2 + T K)

(−2α2T 2)2 + 4α2(2T + T 2K)2

where we used the simple fact that 1 + 2T K = T
2
α

2
. The numerator is

4α
2
T (α

2
T − T K2 − 2K).

The firs factor is positive, and the second factor can be rewritten as



74 Ferenc Szidarovszky, Vernon L. Smith and Steven Rassenti CUBO
11, 2 (2009)

K(1 + (1 − N )r)

2 + T K
− T K2 − 2K =

K

2 + T K
·
(

(1 + (1 − N )r) − (2 + T K)2
)

=
K

2 + T K

(

1 +
2(T K + 1)

2

T K
− (2 + T K)2

)

=
1

(2 + T K)T

(

−(T K)3 − 2(T K)2 + (T K) + 2
)

=
1 − (T K)2

T
6= 0 (6.15)

where we used the equality form of inequality (6.13). Since Reλ̇ 6= 0, there is the possibility of a limit cycle

around the equilibrium.

The discrete version of this model and its asymptotical properties can be discussed similarly.

7 Learning in Oligopoly Models

For the sake of simplicity we will discuss the classical Cournot model again and assume linear cost and

price functions. Therefore assume that the cost function of firm k is Ck(xk) = αkxk + βk, and the true

price function is f (Q) = B − AQ, where Q =
∑N

k=1 xk as before. Assume that the firms have only limited

knowledge on the price function, and during the dynamic process they repeatedly update their beliefs of the

price function giving rise of a learning process. In this session we will examine three cases.

Case 1. Assume that the firms know the value of Q, where the price becomes zero. In this case firm k

believes that the price function is fk(Q) = εk

(
B
A
− Q

)

, but does not know that εk = A is the true value.

Case 2. If the firms know only the slope of the price function, then firm k believes that the price function

is fk(Q) = εk − AQ, but does not know that εk = B is the true value.

Case 3. If the firms know the price at Q = 0 but they are uncertain about the slope, then firm k believes

that the price function is fk(Q) = B − εkQ, but does not know the true value εk = A.

As we will see, the dynamic learning processes will be significantly different in the above cases.

Starting with Case 1. we examine the game first from the viewpoint of firm k. It believes that the profit

of each firm (including itself) is

ϕ
(k)
l

(x1, . . . , xN ) = xlεk

(

B

A
− Q

)

− (αlxl + βl) (l = 1, 2, . . . N ). (7.1)

By assuming interior optima, the believed best response of firm l is

xl =
B

A
−

αl

εk
− Q. (7.2)

By adding these equations

Q =
N B

A
−

1

εk

N
∑

l=1

αl − N Q



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Cournot Models: Dynamics, Uncertainty and Learning 75

implying that firm k believes that at the equilibrium the total production of the industry is

Q
(k)

=
1

N + 1

(

N B

A
−

∑N

l=1 αl

εk

)

and the corresponding equilibrium price is

fk(Q
(k)

) = εk

(

B

A
− Q(k)

)

=
1

N + 1

(

Bεk

A
+

N
∑

l=1

αl

)

. (7.3)

Firm k also produces the corresponding believed equilibrium level

xk =
B

A
−

αk

εk
− Q(k) =

B

(N + 1)A
−

αk

εk
+

∑N

l=1 αl

εk(N + 1)
. (7.4)

Therefore in reality, the total production level of the industry becomes

Q =

N
∑

k=1

xk =
N B

(N + 1)A
−

N
∑

k=1

αk

εk
+

(
N
∑

l=1

αl

)(
N
∑

k=1

1

εk

)

1

N + 1
(7.5)

with the corresponding equilibrium price

P = B − AQ =
B

N + 1
+ A

N
∑

k=1

αk

εk
−

A

N + 1

(
N
∑

l=1

αl

)(
N
∑

k=1

1

εk

)

. (7.6)

The actual price is usually different than the believed prices by the firms. For firm k, the discrepancy between

the actual and believed price is

D
(k)

= P − fk(Q
(k)

) =
B

N + 1

(

1 −
εk

A

)

+ A

N
∑

k=1

αk

εk
−

1

N + 1

(
N
∑

l=1

αl

)[

A

N
∑

k=1

1

εk
+ 1

]

. (7.7)

Based on this price discrepancy firm k thinks as follows. If D
(k)

= 0, then the believed price equals the actual

price, so the believed price is considered correct. If D
(k)

> 0, then the believed price is too low, so firm k

wants to increase its estimate on the price function by increasing εk. If D
(k)

< 0, then its price estimate was

too high, so the firm wants to decrease it by decreasing the value of εk. By assuming continuous time scales

this adjustment process can be modelled as

ε̇k = KkD
(k)

(k = 1, 2, . . . , N ), (7.8)

and in the discrete case as

εk(t + 1) = εk(t) + KkD
(k)

(k = 1, 2, . . . , N ). (7.9)

Here Kk > 0 is the speed of adjustment of firm k.



76 Ferenc Szidarovszky, Vernon L. Smith and Steven Rassenti CUBO
11, 2 (2009)

First we prove that systems (7.8) and (7.9) have only one steady state εk = A (k = 1, 2, . . . , N ) which

is the full information case. Notice first that D
(k)

= 0 for all k may occur only if the εk values are identical.

Let ε denote this common value, then

0 =
B

N + 1

(

1 −
ε

A

)

+
A

ε

N
∑

k=1

αk −
1

N + 1

(
N
∑

k=1

αk

)
[

AN

ε
+ 1

]

=
B

N + 1

(

1 −
ε

A

)

+

(
N
∑

k=1

αk

)
(

A

ε
− 1

)

1

N + 1
.

If ε > A, then both terms are negative; if ε < A, then both are positive, and if ε = A, then they are zero.

Hence the only steady state is εk = A for all k.

In order to analyze the asymptotical stability of systems (7.8) and (7.9) notice first that

∂D
(k)

∂εk
= −

B

(N + 1)A
+

A

ε
2
k

(

−αk +
1

N + 1

N
∑

l=1

αl

)

and for l 6= k,

∂D
(k)

∂εl
=

A

ε
2
l

(

−αl +
1

N + 1

N
∑

k=1

αk

)

.

For the sake of simple notation introduce the variable

γk =
1

N + 1

N
∑

l=1

αl − αk. (7.10)

The Jacobian of the continuous system (7.8) has the form

J
C

=











K1

(

− B
(N +1)A

+
Aγ1
ε2
1

)
K1Aγ1

ε2
1

· · ·
K1Aγ1

ε2
1

K2Aγ2
ε2
2

K2

(

− B
(N +1)A

+
Aγ2
ε2
2

)

· · · K2Aγ2
ε2
2

.

.

.
.
.
.

. . .
.
.
.

KN AγN
ε2

N

KN AγN
ε2

N

· · · KN

(

− B
(N +1)A

+
AγN
ε2

N

)











(7.11)

= D + ab
T

with

D = diag

(

−K1B

(N + 1)A
, . . . ,

−KN B

(N + 1)A

)

, a =

(

K1Aγ1

ε
2
1

, . . . ,
KN AγN

ε
2
N

)T

and b
T

= (1, . . . , 1). Therefore the characteristic polynomial of this matrix is the following:



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Cournot Models: Dynamics, Uncertainty and Learning 77

ϕ(λ) =

N
∏

k=1

(

−KkB

(N + 1)A
− λ

)


1 +

N
∑

k=1

KkAγk
ε2

k

−KkB
(N +1)A

− λ



 . (7.12)

The eigenvalues are λ =
−KkB

(N +1)A
< 0 and the solutions of equation

N
∑

k=1

KkAγk
ε2

k

− KkB
(N +1)A

− λ
+ 1 = 0

which has the same form as equation (3.5) by assuming that γk ≤ 0 for all k. Notice that this condition holds

if the marginal costs αk are close to each other. Then by repeating the proof of Theorem 1 we can show that

all eigenvalues have negative real parts implying the local asymptotical stability of the equilibrium.

The Jacobian of the discrete system (7.9) is similarly J
D

= I + J
C

, which has the same structure as J
C
,

but the identity matrix has to be added to the diagonal matrix D. Therefore its characteristic polynomial

has the form

N
∏

k=1

(

1 −
KkB

(N + 1)A
− λ

)


1 +

N
∑

k=1

KkAγk
ε2

k

1 − KkB
(N +1)A

− λ



 . (7.13)

By repeating the proof of Theorem 2 we can show that all eigenvalue of J
D

are inside the unit circle at the

equilibrium εk = A if γk ≤ 0 and
KkB

(N +1)A
< 2 for all k, furthermore

N
∑

k=1

Kkγk(N + 1)

2(N + 1)A − KkB
> −1. (7.14)

In Case 2. we assume that firm k believes that the price function is fk(Q) = εk − AQ, and the firms

learn about the value of εk. Then firm k believes that the profit of any firm l (including itself) is

ϕ
(k)
l

(x1, . . . xN ) = xl(εk − AQ) − (αlxl + βl) (7.15)

so the believed best response of firm l is

xl =
εk − αl

A
− Q, (7.16)

the total output of the industry is believed to be

Q
(k)

=
N εk −

∑N
l=1 αl

(N + 1)A
(7.17)

with price

fk(Q
(k)

) =
εk +

∑N
l=1 αl

N + 1
. (7.18)



78 Ferenc Szidarovszky, Vernon L. Smith and Steven Rassenti CUBO
11, 2 (2009)

Based on this belief firm k produces the amount

xk =
εk − αk

A
− Q(k) =

εk − (N + 1)αk +
∑N

l=1 αl

(N + 1)A
. (7.19)

In reality however each firm thinks in the same way but believes in its own εl value in the price function, so

the actual total production level of the industry becomes

Q =

N
∑

k=1

xk =
1

(N + 1)A

(
N
∑

k=1

εk −

N
∑

l=1

αl

)

(7.20)

with actual equilibrium price

P = B − AQ = B −
1

N + 1

(
N
∑

k=1

εk −

N
∑

l=1

αl

)

. (7.21)

Based on the discrepancy

D
(k)

= P − fk(Q
(k)

) =
1

N + 1

(

(N + 1)B −

N
∑

l=1

εl − εk

)

(7.22)

the dynamic process become similar to (7.8) and (7.9) with the only difference that in this case D
(k)

is given

in equation (7.22).

Similarly to the previous case we can prove that there is a unique steady state εk = B for all k, which

corresponds to the full information case. Clearly D
(k)

= 0 for all k, if the εk values are identical. Let ε

denote this common value, then (N + 1)B − N ε − ε = 0 implying that ε = B.

Notice that systems (7.8) and (7.9) are both linear in this case, so local asymptotical stability implies

global asymptotical stability. The coefficient matrix in the continuous case is

J
C

=
1

N + 1










−2K1 −K1 · · · −K1

−K2 −2K2 · · · −K2
.
.
.

.

.

.
. . .

.

.

.

−KN −KN · · · −2KN










(7.23)

and in the discrete case

J
D

= I + J
C

.

Similarly to Theorems 1 and 2 we can easily prove that the continuous system is always asymptotically stable

and the discrete system is asymptotically stable if and only if for all k, Kk < 2(N + 1), and

N
∑

k=1

Kk

2(N + 1) − Kk
< 1.



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Cournot Models: Dynamics, Uncertainty and Learning 79

We turn our attention next to Case 3, when the firms learn about the slope of the price function. In

this case we assume the firm k believes that the price function is fk(Q) = B − εkQ, the profit of firm l

(l = 1, 2, . . . , N ) is believed by firm k to be

ϕ
(k)
l

(x1, . . . , xN ) = xl(B − εkQ) − (αlxl + βl), (7.24)

so the best believed output choice is

xl =
B − αl

εk
− Q,

and the believed total production of the industry is

Q
(k)

=
N B −

∑N

l=1 αl

(N + 1)εk
. (7.25)

The believed equilibrium price,

fk(Q
(k)

) =
B +

∑N
l=1 αl

N + 1
, (7.26)

is the same for all firms. Firm k also believes that its equilibrium output is

xk =
B − (N + 1)αk +

∑N

l=1 αl

(N + 1)εk
. (7.27)

Therefore in reality the total production of the industry becomes

Q =

N
∑

k=1

xk =
1

N + 1

((

B +

N
∑

l=1

αl

)
N
∑

k=1

1

εk
− (N + 1)

N
∑

k=1

αk

εk

)

(7.28)

with actual equilibrium price

P = B − AQ = B −
A

N + 1

((

B +

N
∑

l=1

αl

)
N
∑

k=1

1

εk
− (N + 1)

N
∑

k=1

αk

εk

)

. (7.29)

Based on the discrepancy between the actual and believed price

D
(k)

=
1

N + 1

(

N B − A

(

B +

N
∑

l=1

αl

)
N
∑

k=1

1

εk
+ A(N + 1)

N
∑

k=1

αk

εk
−

N
∑

l=1

αl

)

(7.30)

the dynamic process is similar to the previous cases (7.8) and (7.9). Note that D
(k)

is the same for all firms,

and therefore a set of εk (k = 1, 2, . . . , N ) values is a steady state of the system if and only if

B

(

N − A

N
∑

k=1

l

εk

)

+ (N + 1)A

N
∑

k=1

αk

εk
−

(
N
∑

l=1

αl

)(

A

N
∑

k=1

1

εk
+ 1

)

= 0.



80 Ferenc Szidarovszky, Vernon L. Smith and Steven Rassenti CUBO
11, 2 (2009)

Since this is a single equation for N unknowns with a feasible solution εk = A, there are infinitely many

steady states. That is, there is the possibility that all firms believe in wrong price functions but none of them

notices it since believed and actual prices still coincide. In this case no learning is possible in this way.

In the above discussed cases we always assumed that the firms use instantaneous information about

prices, however in reality price informations are always delayed. The effect of information lag in the learning

process can be similarly examined to the cases being demonstrated in Session 2.6.

8 Laboratory Experiments

In November 2002 the Nobel Committee awarded Vernon Smith the prize in Economics for a body of work

spanning a half-century that demonstrated that controlled laboratory experiments could be used to study

economic behavior, exactly as experiments in the hard sciences study physical phenomena. Smith summa-

rized his methodological breakthrough in a highly referenced 1982 paper, ”Microeconomic Systems as an

Experimental Science.” Today, economic experiments are widely undertaken to explore three main avenues

of research: to inform economic theory, to test-bed newly designed institutions under stressful environmental

conditions, and to understand how brain activity leads to economic behavior. As economic theorists we

should be very concerned with rescuing our theories from the doldrums of mathematical curiosity by sub-

jecting them to the rigors of laboratory scrutiny. This chapter has so far presented a review and some new

developments in oligopoly theory that are enmeshed with an abundant theoretical literature in this area, but

have we not discussed how to assess whether those results relate to what people really do. We can greatly

increase the value of our theories to the society that invests in them by becoming concerned with how people

really organize themselves to form, sustain, and adapt rules of order in order to generate beneficial outcomes

for themselves.

Each laboratory that conducts experiments with human subjects approaches its research with different

auxiliary hypotheses but uses the common analytic framework that has three main components: the envi-

ronment, the institution, and the behavior of the human subjects. The environment includes the subjects’

preferences or incentives for achieving various allocations in the exchange system, subjects’ productive ca-

pabilities and system technical constraints on achieving those allocations, and knowledge about the initial

conditions and allocation in the exchange system in which subjects will participate. The experimenter controls

the environment using induced values, a mapping of outcomes to different monetary earnings, and carefully

worded instructions. For example, in a simple oligopoly environment the experimenter may privately inform

each experimental subject i through computerized instructions that he will be playing the role of a producer

of a fictitious good, and that he will be able to, in any given period during the upcoming experiment, produce

up to ui units of the good at a cost of $ci per unit, and that if he can sell those units he will earn a cash

profit, paid to him by the experimenter, which will be the difference between the market price, p, he sells

each unit for, and his cost of production.

The institution consists of a set of rules that completely specify (1) what messages subjects are allowed

to send and when they can send them, (2) how these messages are translated into reallocations of environ-

mental conditions, and (3) what feedback the subject receives about the messages that were sent and the

reallocations they produced. The experimenter typically controls for the institution with a computer network

that instantiates these rules. For example, in one simple oligopoly environment a ’producer’ may only be

able to make only one offer of a given total quantity that he is willing to sell at a particular per unit price in



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Cournot Models: Dynamics, Uncertainty and Learning 81

a given period, and the institution may gather those offers, order them from lowest to highest price, and sell

as many of them as possible at a uniform price to buyers whose bids to buy have been ordered from highest

to lowest. The institution in this simple example may decide not to reveal publicly what offers and bids are

submitted or what was the volume traded in the period.

The behavior is what the experimenter can then observe in the form of messages that are actually sent

by subjects as they participate in the experiment. By controlling the environment, E, and the institution,

I, and creating experimental ’treatments’ which can selectively alter either of them, the experimenter can

estimate the behavioral response function bi(E, I) for each subject, i. Institutions where there is repeated

interaction among subjects precipitate learning and the behavioral response functions themselves become

functions of time, b
t
i(E, I), that depend upon the sequence of information delivered to the subject by the

institution and the perceived reallocations in the environment.

The experimental economics laboratory typically uses economic theory to predict the form of the behav-

ioral functions bi(·, ·), and thus predicts the environmental reallocations that will be produced as the subjects

interact. It becomes possible to compare predicted outcomes to actual outcomes and predicted behavioral

functions to estimated behavioral functions to ask how well does the theory perform in the laboratory? Con-

sider the following example of a two-player game that has been extensively studied in the laboratory. In

this case the messages are very simple. Subject One must move first and decide whether to stop the game

immediately, in which case he receives a payment of $10 and Subject Two also receives a payment of $10, or

pass the decision on to Subject Two. If Subject One chooses to pass, Subject Two now must choose between

an allocation which pays Subject One $0 and himself $40, or an allocation which pays Subject One $ 15 and

himself $25. Using game theory, Nash equilibrium predicts that when this game is played once by anony-

mous traders who have complete information, Subject 2 would always choose the (0, 40), and that Subject

1, realizing what Subject 2 would do, will always to stop the game immediately for a ($10, $10) allocation.

In fact, when this experiment is run typically only 50% of the Subject 1’s elect to stop, and, conditional on

passing to Subject 2, 75% of the Subject 2’s choose the allocation ($15, $25). Thus in the typical population

the expected payoff of Subject 1 if he passes is .75 × 15 = 11.25, greater than 10. This simple experiment

demonstrates the failure of a theory that does not fully account for the evolved tendency of human subjects

to divine opportunity through understanding the history and intentions of those with whom they interact. A

greater irony reveals itself in this scenario when we relax the Nash assumptions by repeating the game and

giving subjects incomplete information in the form of their only their own payoffs: then the Nash prediction

is robust!

Fouraker and Siegel [3] conducted the first extensive experimental study of basic oligopoly theory more

than 40 years ago. They hypothesized that although the original quantity adjusting Cournot model does not

directly discuss the information conditions of the agents, those conditions as well as the number of agents

might have important consequences during repeated economic interactions. They were right: In Cournot

quantity adjusting experiments where subjects either knew (complete info.) or didn’t know (incomplete info.)

their rivals payoff function, the distribution of outcomes suffered more variability under complete information,

while the Cournot prediction was more robust with less information. There are many experiments in various

other environments that verify that when provided information subjects attempt to use it in not always

an entirely predictable manner. Further analysis provided by the Fouraker/Siegel data showed that both

rivalistic (the tendency to increase your output when you observe others are producing more than you) and

cooperative (the tendency to decrease your output when you observe others are producing more than you)

signaling behaviors are more prevalent when information is complete. This result was most prevalent in



82 Ferenc Szidarovszky, Vernon L. Smith and Steven Rassenti CUBO
11, 2 (2009)

duopolies and transferred strongly to a Bertrand price-setting environment.

Since the original oligopoly experiments, there has been an exponential growth of published oligopoly

theory that deals with a multitude of mathematical nuances and assumptions, but comparatively few experi-

mental tests that reign in the relevance of all those theories to human behavior. An interesting experimental

study executed by [2] points out an environmental condition, low profitability under competition, that may

confound the predictions of extant theory in a price setting scenario. They create a simple single product

multi-period environment where 5 independent producers have identical cost structures such that they can

each produce 100 units at cost c per unit, 100 units at cost c + d per unit, and 100 additional units at cost

c + d + e per unit. The demand function is linear and represented by more than a hundreds robot buyers who

simply reveal their value for consuming the product during the experiment. The demand function intersects

the aggregate supply function at 1200 units, where the marginal cost of production is c + d + e. At the

competitive price the sellers’ share of surplus is an order of magnitude less than the buyers’. Each period

during the experiment the oligopolists must post a single price at which they are willing to sell their product

and the infinitesimal buyers queue up electronically and buy in order of lowest price available. Buyers are

rationed uniformly amongst producers tied in price. A competitive Nash equilibrium exists that has each

producer offering his 300 units at price c + d + e and earning 100d + 200e, since a higher price would exclude

the producer completely and a lower price would produce a loss on his higher cost units. But this is not

what the average group of oligopolists does in this environment. They engage in an offering strategy that

creates price cycles that tend to rise rapidly and fall slowly. The highest price producer (perhaps 2 or 3)

is always excluded in a given period, but the remaining active sellers are rewarded with supra-competitive

profits. These subjects, under much more complicated environmental and institutional conditions than in

the simple two person game discussed above, and without direct communication, create a continuous tension

between competitive (gradual undercutting) and cooperative (raise prices) behavior that serves them well in

the long run.

Our experimental philosophy has been shaped largely by our interpretation of Friedrich Von Hayek’s

view that economics is the study of the co-emergent order of the brain and our social institutions. This

philosophical framework leads us to look for ecologically rational explanations of both individual and insti-

tutional behavior. In particular we assume that individuals are adapted to specific functional needs that

arose over evolutionary time, but that are now expressed by our modern brains through interactions with our

current institutions and environments. However, as our brains learn to cope with their modern scenario they

encounter cognitive opportunity costs that we attempt to overcome by the development of institutions that

can perform the requisite computations and reallocations cheaply and efficiently. Economic experiments are

crucial in testing the theory used to prescribe institutional innovation and in test-bedding the prescription

before its implementation.

9 Conclusions

In this paper dynamic oligopoly models were examined. After introducing the classical Cournot model

and its extensions, the stability of single product oligopoly was discussed with the assumption that full

and instantaneous information was available to all firms concerning the market price. The equilibrium is

always locally asymptotically stable with continuous time scales, however in order to preserve stability in

the discrete case we have to assume that either the derivatives of the best response functions of all firms are



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Cournot Models: Dynamics, Uncertainty and Learning 83

sufficiently small, or the firms do not change much their output levels. Then we assumed that there was

partial cooperation among the firms when each firm took a certain portion of the profits of the rivals into

account in its payoff function. Under realistic conditions we could obtain the same stability results as in the

previous case. The firms are usually uncertain in the market as they use only an estimate of the price function

in their decisions on the best choices on production levels. We could show that similar stability conditions

hold for this case as well, however the production level might converge to a steady state which differs from

the Nash equilibrium. We also investigated the effect of information delay in market price, and showed that

stability can be lost. We have derived conditions for stability and instability of the equilibrium and showed

that at the critical value of the bifurcation parameter Hopf bifurcation occurs giving the possibility of the

birth of limit cycles. Three particular learning processes were then introduced and examined when the firms

had only limited information on the linear price function. It was interesting to see that the number of steady

states (that is, the possibility of learning) and the asymptotic properties of the learning process were different

for different types of uncertainty.

All theoretical results discussed in these sessions were based on certain specific assumptions on cost

and market demand structure as well as on particular assumptions on the behavior of the decision makers.

Such special conditions are not always satisfied in real economies, and in addition, the decision making

managers do not think and decide always as we expect them to do. The most appropriate methodology in

examining human decision making and economic processes without special conditions is based on laboratory

experiments with realistic environment. Experimental economy is this very important procedure in which

the actual decisions of the participants are repeated, observed and analyzed and hence we are able to gain

the right insight into the minds of the decision making humans.

Received: March 14, 2008. Revised: May 15, 2008.

References

[1] Cournot, A., Recherches sur les Principes Mathèmatiques de la Thèorie de Richesses, Hachette, Paris

(English translation (1960): Researches into the Mathematical Principles of the Theory of Wealth. Kelley,

New York.), 1838.

[2] Durham, Y., McCabe, K., Olson, M., Rassenti, S., and Smith, V., Oligopoly Competition in

Fixed Cost Environments, International Journal of Industrial Organization, Fall, 2003.

[3] Fouraker, L.E., and Siegel, S., Bargaining Behavior, McGraw-Hill Book Company/New York, 1963.

[4] Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamic Systems and Bifurcations of

Vector Fields, Springer-Verlag, Berlin/Heidelberg/New York, 1983.

[5] Okuguchi, K., Expectations and Stability in Oligopoly Models, Springer-Verlag, Berlin/Heidelberg/New

York, 1976.

[6] Okuguchi, K. and Szidarovszky, F., The Theory of Oligopoly with Multi-Product Firms, Springer-

Verlag, Berlin/Heidelberg/New York (2nd edition), 1999.



84 Ferenc Szidarovszky, Vernon L. Smith and Steven Rassenti CUBO
11, 2 (2009)

[7] Smith, V.L., Microeconomic Systems as an Experimental Science, American Economic Review, Decem-

ber, 1982.

[8] Szidarovszky, F. and Bahill, T.A., Linear System Theory, CRC Press, Boca Raton/London (2nd

edition), 1998.


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