Proceedings of 8th Asia-Pacific Business Research Conference
9 - 10 February 2015, Hotel Istana, Kuala Lumpur, Malaysia, ISBN: 978-1-922069-71-9
Cournot Duopolies with Symmetric Information and Their
Risk-Taking Behaviors
Suyeol Ryu, Iltae Kim** and Yu-Seok Jeong***
This paper introduces the simple and standard Cournot duopoly model facing the
symmetric situation in which each firm has same information on the cost function of
its rival, respectively. We present two types of the firms’ risk-taking behaviors and
two classes of symmetric information. First, we consider that both firms are risk
neutral and have perfect information. In this case, we obtain the unique Cournot
equilibrium. Second, when both firms are risk neutral and have partial information,
that is, the set of information of each firm changes from perfect information to partial
information compared with case 1, the number of equilibrium changes from one to
four compared with case 1. Third, when both firms are risk averse and have perfect
information, that is, risk taking behaviors of firms change from risk neutral to risk
averse compared with case 1, the number of equilibrium changes from one to three
compared with case 1. Therefore, the information structure and risk taking
behaviors of both firms affect their optimal output strategies.
Keywords: Cournot equilibrium, symmetric information, risk neutrality, risk
aversion
JEL Classification: D81, L13
1. Introduction
There are a growing number of papers which concern the duopoly models with
information assuming that each firm is risk neutral and can share or exchange its
private information on market uncertainty with its rival. The line of research focuses
on the key factor of market uncertainty such as uncertain market demand or
unknown constant marginal costs of firms.
In case of uncertain market demand, Novshek and Sonnenschein (1982) analyzed
the incentives of firms to share their private information about unknown market
demand, and concluded that firms would not benefit from sharing their information.
Clarke (1983a, b) and Vives (1984) examined Novshek and Sonnenschein’s result
in Cournot duopoly model and confirmed it. Gal-Or (1985) also found that firms
would have higher profit by sharing their information on unknown market demand
rather by keeping it private. However, Li (1985) showed that firms producing
homogenous products would not be profitable when they exchange their
information on uncertain demand. Uncertainty made firms to be more profitable. On
the other hand, with respect to uncertain marginal costs of firms, Gal-Or (1986) and
Shapiro (1986) showed that firms producing substitute goods would benefit from
sharing their private information on their costs. Other important works concerning
information sharing on uncertain demand or unknown costs are included in Shapiro
(1986), Kirby (1988), Sakai and Yamato (1989), Sakai (1990, 1991), Hwang (1993),

Department of Economics, Andong National University, Andong 760-749, Republic of Korea,
E-mail: syryu@andong.ac.kr
**
Corresponding author, Department of Economics, Chonnam National University, Gwangju
500-757, Republic of Korea, E-mail: kit2603@chonnam.ac.kr
***
Department of Economics, Chonnam National University, Gwangju 500-757, Republic of Korea,
E-mail: go21ctr@naver.com
Proceedings of 8th Asia-Pacific Business Research Conference
9 - 10 February 2015, Hotel Istana, Kuala Lumpur, Malaysia, ISBN: 978-1-922069-71-9
Jin (1996), Malueg and Tsutsui (1996, 1998), Kultti and Niinimaki (1998) and Doyle
and Snyder (1999).
This paper presents the simple and standard Cournot duopoly model in which
each firm has the same information set about the rival’s uncertain cost function, but
there is not information sharing or information exchange between firms. The main
purpose of this paper is twofold. First, we investigate how firms with same
risk-taking behaviors react to the rival’s strategy for change in the information on the
cost function. Second, we examine how firms with same information structure react
to the rival’s strategy for change in risk taking behaviors. In a traditional Cournot
duopoly game, each firm's optimal output depends on its own cost function as well
as the other firm's cost function. Thus, when the information on the cost function is
given, each firm has to use its beliefs about the other firm's cost function and the
beliefs can be summarized in its own subjective probability distribution.
In this paper, we introduce two types of the firms’ risk-taking behaviors and two
classes of symmetric information. First, when both firms are risk neutral and the set
of information of each firm changes from perfect information to partial information,
we compare these firms’ optimal strategies for the change in information set.
Second, when both firms have perfect information and risk taking behaviors of firms
change from risk neutral to risk averse, we compare these firms’ optimal strategies
for the change in their risk taking behaviors. It is well known that the expected utility
maximizing firm with risk aversion shows some different behavior from the expected
profit maximizing firm with risk neutrality in Sandmo (1971). We use the Sandmo
approach in order to analyze the change in risk taking behaviors of firms.
This paper is organized as follows. Section 2 introduces the basic and standard
duopoly model and analyzes three cases: (i) both firms are risk neutral and have
perfect information, (ii) both firms are risk neutral and have partial information and
(iii) both firms are risk averse and have perfect information. Finally, section 3
includes concluding remarks.
2. The Model and Analysis
We introduce a simple Cournot duopoly model in which two firms producing identical
products face a linear market demand. The inverse demand function is given by:
P  a q1 q 2 ,
where q i denotes the amount of output produced by firm i, the demand intercept is a and
the slope of market demand is 1 for simplicity. We also assume that firms have no fixed
costs and the marginal cost of firm i is constant and equal to c i ( c i  0 ). The profit function
of firm i is described as:
 i  (a  q i  q j  c i )q i , for i, j=1,2 i  j .
(1)
From the profit maximization problem of (1), when under perfect information, the cost
functions of firms are exactly known to each other, the standard Cournot equilibrium for
firm i under perfect information, q ic is :
q ic 
1
(a  2c i  c j ), for i, j = 1,2 i  j .
3
(2)
Proceedings of 8th Asia-Pacific Business Research Conference
9 - 10 February 2015, Hotel Istana, Kuala Lumpur, Malaysia, ISBN: 978-1-922069-71-9
We consider that each firm knows exactly its own constant marginal cost but the set of
information about the rival’s marginal cost available to each firm is different. That is, each
firm has its subjective probability distribution about the rival’s constant marginal cost.
Before proceeding, we define the random variables to distinguish from the non-random
variables given the set of information available to each firm in the following way:
q j i , c j i , and  i i , for i, j = 1,2 i  j ,
where i is the set of information available to firm i.
(1) Case 1: Both firms are risk neutral and have perfect information
We assume that both firms are risk neutral and have perfect information. Each firm
chooses its output to maximize its expected profit conditional on its private information.
Firm i’s expected profit function given its private information,  i is equal to:
E ( i i )  E(a  qi  q j  ci i )qi 


 a  q i  c i  E (q j i )q i , for i, j = 1,2 i  j ,
(3)
where E is the expectations operator.
We assume that each firm has its subjective probability distribution about the rival’s
constant marginal cost in the following way.
Assumption: Let firm j’s expectation about c i be written as E (c i  j )  c i  b i , when b i
is assumed to be a small (relative to c i ) non-systematic bias. This implies that firm i’s
expectation about firm j’s expectational bias, b i is zero: E (b i i )  0 . Then

 

E    E E (c  )        E E (c  )   c
E    E E (c  )        E (c  ) c  b .
E E (c i  j )i  E (c i  b i )i  c i  E (b i i )  c i
i
j
i
i
i
j
i
j
i
i
j
j
i
i
i
i
Now we consider the symmetric situation in which both firms have perfect information
about the constant marginal cost of the rival firm. That is,
E (c 1 2 )  c 1 and E (c 2 1)  c 2 .
From the first-order condition of (3), the best reaction function for each can be written as
Proceedings of 8th Asia-Pacific Business Research Conference
9 - 10 February 2015, Hotel Istana, Kuala Lumpur, Malaysia, ISBN: 978-1-922069-71-9


(4)


(5)
q1 
1
a  c 1  E (q 2 1)
2
q2 
1
a  c 2  E (q 1 2 ) .
2
This unique Cournot equilibrium can be varied, depending on each firm’s expectation based
on the information set of the other firm, E (q j i ). Thus, expected reaction function of one
firm based on the given information set of the other firm is shown as follows.
Expected reaction function of firm 1 based on the information set of firm 2 in round 0:
1
2



E (q 1 2 )  E  a  c 1  E (q 2 1) 2  .
(6)

Expected reaction function of firm 2 based on the information set of firm 1 in round 0:
1
2



E (q 2 1)  E  a  c 2  E (q 1 2 ) 1  .
(7)

Substituting (6) into (7), we get

1 

 2 
1
2

 



E (q 2 1)  E  a  c 2  E  a  c 1  E q 2 1 2  1  





 
 
 


 
 
 
1 1
1
1
1
   a  E c 2 1  E c 1 2 1  E q 2 1 2 1
2
4
4
2 4 
(8)


.

By making use of E (c 2 1)  c 2 , E c 1 2 2  c 1 and E q 2 1 2 1  E q 2 1 , we
have


1
1
1
1
1


E q 2 1    a  c 2  c 1  E q 2 1 .
2
4
4
2 4 
(9)
Substituting (7) into (9), we get expected reaction function of firm 2 as


1
1


1
1
1
1  1
1



E q 2 1    a  c 2  c 1  E  a  c 2  E (q 1 2 ) 1  .
2
4
4  2
2 4 

That is,
1
1
1
1
1

 
E q 2 1     a    c 2  c 1  E q 1 2 1 .
4
8
2 4 8 
2 8 
We call this equation expected reaction function of firm 2 based on the information set of
firm 1 in round 1.
In round 2,
Proceedings of 8th Asia-Pacific Business Research Conference
9 - 10 February 2015, Hotel Istana, Kuala Lumpur, Malaysia, ISBN: 978-1-922069-71-9


1
1
1
1
1 
1
1 
1
a    
c 2
E q 2 1     

 2 4 8 16 32 
 2 8 32 

 
1 1 
1
c 1 
  
E q 1 2 1 .

32
 4 16 
Hence, we have sequence of expected reaction function of firm 2 based on the information
set of firm 1.
In round n, we get
2 n 1
n
1 1 1

1 1
1
1
1 1
1
1  1  



E q 2 1 
  

      
a 
 
       c 2
 2 4 8 16 32

 2 8 32
2 2 
2 4  






n 1
n 1
1
1
1 1 
1 1
 
       c 1    E q 1 2 1 .
 4 16
4 4  
8 4 



 
Thus, in the limit, we have


E q 2 1 
1
2
1
a  c 2  c1 .
3
3
3
(10)
Similarly, we have expected reaction function of firm 1 based on the information set of
firm 2


E q 1 2 
1
2
1
a  c1  c 2 .
3
3
3
(11)
Inserting (10) and (11) into (4) and (5), finally we get the unique Cournot equilibrium as:
q 1c 
1
(a  2c 1  c 2 )
3
q 2c 
1
(a  2c 2  c 1).
3
These best reaction curves of firm 1 and 2 according to the round are shown graphically in
Figure 1.
Proceedings of 8th Asia-Pacific Business Research Conference
9 - 10 February 2015, Hotel Istana, Kuala Lumpur, Malaysia, ISBN: 978-1-922069-71-9
𝑞2
Firm 1’s best reaction curve in round 0
Firm 1’s best reaction curves in the limit
Firm 2’s best reaction curve in the limit
𝑞2𝑐
𝑞1𝑐
𝑞1
Firm 2’s best reaction curve in round 0
Fig.1. Cournot equilibrium under perfect information
(2) Case 2: Both firms are risk neutral and have partial information
Next, we investigate the case in which both firms are risk neutral and have partial
information about the constant marginal cost of the rival firm. That is,
E (c 1 2 )  c 1  b 1 and E (c 2 1)  c 2  b 2


 
 

By making use of E (c 2 1)  c 2  b 2 , E c 1 2 2  c 1 and E q 2 1 2 1 


E q 2 1 , from (8) we have


1
1


E q 2 1    a  c 2  b 2   c 1  E q 2 1 .
2
4
4
2 4 
1
1
1
As the round goes on, in the limit, we get


E q 2 1 
1
2
1
a  c 2  b 2   c 1 .
3
3
3
(12)
Similarly, we have expected reaction function of firm 1 based on the information set of firm


2, E q 1 2 .


E q 1 2 
1
2
1
a  c 1  b 1   c 2 .
3
3
3
(13)
Inserting (12) and (13) into (4) and (5), finally we get the Cournot equilibrium with partial
information as:
q 1* 
1
a  2c 1  (c 2  b 2 )
3
Proceedings of 8th Asia-Pacific Business Research Conference
9 - 10 February 2015, Hotel Istana, Kuala Lumpur, Malaysia, ISBN: 978-1-922069-71-9
1
a  2c 2  (c 1  b1).
3
q 2* 
Let the upper and lower bound of the bias, b 2 (b 1) be denoted by b 2u (b 1u ) and b 2 (b 1 )
respectively, then expected reaction function of firm 2 (or firm 1) based on the information
set of firm 1 (or firm 2) can be written as:








q 1 * 
1
1
a  2c 1  (c 2  b 2 ) and q 1u *  a  2c 1  (c 2  b 2u )
3
3
q 2 * 
1
1
a  2c 2  (c 1  b 1 ) and q 2u *  a  2c 2  (c 1  b 1u ) .
3
3
Now, comparing with case 1 with perfect information, we have four Cournot equilibria
such as (q 1*,q 2 * ), (q 1*,q 2u * ), (q 1u *,q 2* ) and (q 1u *,q 2u * ). That is, if the set of information of
the firm changes from perfect information to partial information, the number of equilibrium
changes. Figure 2 illustrates this case.
Firm 1’s best reaction curve in case 2
𝑞2
Firm 1’s best reaction curve in case 1
Firm 2’s best reaction curve in case 2
𝑞2𝑢∗
𝑞2𝑐
Firm 2’s best reaction curve in case 1
𝑞2𝑙∗
𝑞1𝑙∗
𝑞1𝑐
𝑞1𝑢∗
𝑞1
Fig.2. Cournot duopoly equilibria in case 2 compared with case 1
(3) Case 3: Both firms are risk averse and have perfect information
Now we assume that both firms are is risk averse ( u ( )  0 and u ( )  0 ) and each firm
has perfect information about the constant marginal cost of the rival firm.
The objective function of the expected utility maximizing firm 1 is:




E u 1(1 1)  E u 1 (a  q 1  q 2  c 1 1)q 1 .
(14)
The first- and second-condition can be written as:



E u 1(1 1)a  2q 1  c 1  (q 2 1)  0
(15)
Proceedings of 8th Asia-Pacific Business Research Conference
9 - 10 February 2015, Hotel Istana, Kuala Lumpur, Malaysia, ISBN: 978-1-922069-71-9
and




2
E u 1(1 1)a  2q 1  c 1  (q 2 1)  2u 1(1 1)  0 .
(16)
The second-order condition is always satisfied because, for the risk averse firm, u 1()  0
and u 1()  0 .




From the first-order condition (15), E u 1(1 1)(a  2q 1  c 1)  E u 1(1 1)(q 2 1) .


Subtract E u 1(1 1)E (q 2 1) from both sides, then:

 



E u 1(1 1)a  2q 1  c 1  E (q 2 1)  E u 1(1 1)q 2 1  E (q 2 1)
.
(17)
From the symmetric uncertainty of expected utility maximization in Sandmo (1971, p. 67),
we propose that:


 1 1  (a  q 1  c 1  q 2 1)q 1 and E (1 1)  a  q 1  c 1  E (q 2 1)q 1 .


Thus,  1 1  E ( 1 1)   q 2 1  E (q 2 1)q 1 .
Clearly, it follows that:


u 1(1 1)  u 1 E (1 1) , if q 2 1  E (q 2 1).
Now the following inequality holds for all q2 1 :

 


u 1(1 1)q 2 1  E (q 2 1)  u 1 E (1 1) q 2 1  E (q 2 1) .
Taking expectations on both sides:


  


 )  0 .
E u 1(1 1)q 2 1  E (q 2 1)  E u 1 E (1 1) q 2 1  E (q 2 1)


 u 1 E ( 1 1) E (q 2 1) E (q 2
From (17) and (18),

1
(18)


E u 1(1 1)a  2q 1  c 1  E (q 2 1)



 a  2q 1  c 1  E (q 2 1)E u 1(1 1)  0 .
(19)
This implies that, for optimal decision, the following inequality must be satisfied:
q1 


1
a  c 1  E (q 2 1) .
2
(20)
which is the best reaction function of firm 1. Similarly, we obtain the best reaction of firm 2
as:
Proceedings of 8th Asia-Pacific Business Research Conference
9 - 10 February 2015, Hotel Istana, Kuala Lumpur, Malaysia, ISBN: 978-1-922069-71-9
q2 


1
a  c 2  E (q 1 2 ) .
2
(21)
In the present model, each firm does in fact react to expected reaction function of the other
firm based on its own information set. Let rij be firm i’s response for the change in firm j’s
expected output level based on its own information set. rij is equal to and less than  (1/2)
under perfect information in (20) and (21). This means that if firm i based on its own
information set anticipates reduce expected output of firm j to reduce by 1 unit, firm 1
increases its output by less than and equal to 1/2 unit. That is, this makes firm i’s reaction
curve to move inward by k i( 0) under perfect information. Let k i be the amount of output
which is reduced under perfect information by firm i. Therefore, the best reaction functions
of both firms in (20) and (21) can be written as:




q1 
1
a  c 1  E (q 2 1)  k1
2
q2 
1
a  c 2  E (q 1 2 )  k 2 .
2
(22)
(23)
From (22), expected reaction function of firm 1 based on the information set of firm 2 in
round 0:
1
2



E (q 1 2 )  E  a  c 1  E (q 2 1) 2  k1  .

(24)
From (23), expected reaction function of firm 2 based on the information set of firm 1 in
round 0:
1
2



E (q 2 1)  E  a  c 2  E (q 1 2 ) 1  k 2  .

(25)
By calculating as in case 1, we get the best reaction of each firm, respectively as:
1
2
1
2
4
2
4
a  c 1  c 2  k 2  k1  q 1c  k 2  k1
3
3
3
3
3
3
3
1
2
1
2
4
2
4
q~ 2*  a  c 2  c 1  k1  k 2  q 2c  k1  k 2 .
3
3
3
3
3
3
3
c
c
where q 1 and q 2 are the Cournot equilibrium in case 1.
q~1* 
Let
kj
ki
be the relative size of the amount of output from both firms which is reduced
under risk aversion. The values of k ’s measure the difference of firms’ reaction between
risk taking behaviors.
Proceedings of 8th Asia-Pacific Business Research Conference
9 - 10 February 2015, Hotel Istana, Kuala Lumpur, Malaysia, ISBN: 978-1-922069-71-9
Comparing to case 1 where both firms are risk neutral and have perfect information, we
get three possible Cournot equilibria for the different values of the ratio of k. Because they
have the same information set but show different risk taking behaviors, they have the
different strategies from case 1. Figure 3 illustrates this case.
Firm 1’s best reaction curve in case 3
𝑞2
Firm 1’s best reaction curve in case 1
Firm 2’s best reaction curve in case 3
a
𝑞2∗
𝑞2𝑐
c
b
𝑞2∗
𝑞1∗
𝑞1𝑐
Firm 2’s best reaction curve in case 3
𝑞1∗
𝑞1
Fig.3. Cournot duopoly equilibria in case 3 compared with case 1
(i) If 0 
k2 1
 , then q~1*  q 1c and q~ 2*  q 2c (a in figure 3).
k1 2
(ii) If
1 k2

 2 , then q~1*  q 1c and q~ 2*  q 2c (b in figure 3).
2 k1
(iii) If
k2
 2 , then q~1*  q 1c and q~ 2*  q 2c (c in figure 3).
k1
3. Concluding Remarks
This paper presents the Cournot duopolists facing the symmetric situation where both
firms have the same information structure and the same risk taking behaviors. First, we
consider that both firms are risk neutral and perfect information. In this case, the equilibrium
strategies are same as the standard Cournot ones. Second, when both firms are risk neutral
and have partial information, that is, the set of information of each firm changes from
perfect information to partial information compared with case 1, the number of equilibrium
changes from one to four compared with case 1. Third, when both firms are risk averse and
have perfect information, that is, risk taking behaviors of firms change from risk neutral to
risk averse compared with case 1, the number of equilibrium changes from one to three
compared with case 1. Therefore, the information structure and risk taking behaviors of both
firms affect their optimal output strategies.
Proceedings of 8th Asia-Pacific Business Research Conference
9 - 10 February 2015, Hotel Istana, Kuala Lumpur, Malaysia, ISBN: 978-1-922069-71-9
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