EconS 491: Strategy and Game Theory Oligopoly games with ...

Firm 2. Let us now analyze Firm 2 (the uninformed player in this game). First note that its
pro…ts must be expressed in expected terms, since …rm 2 does not know whether market demand
is high or low.
Rearranging,
P rofits 2 = 1
(10 q
H
2 1 q 2 )q 2 1q 2 + 1
(5 q
L
| {z } 2 1 q 2 )q 2 1q 2
| {z }
demand is high
demand is low
P rofits 2 = 1 2
i
h10q 2 q1 H q 2 (q 2 ) 2 q 2 + 1 i
h5q 2 q1 L q 2 (q 2 ) 2 q 2
2
Di¤erentiating with respect to q 2 , we can obtain …rm 2’s best response function, BRF 2 (q L 1 ; qH 1 ):
Note that we do not have to di¤erentiate for the case of low and high demand, since …rm 2 does
not observe such information). In particular,
Rearraging,
And solving for q 2 ,
q 2 q L 1 ; q H 1
1
10 q
H
2 1 2q 2 1 + 1
5 q
L
2 1 2q 2 1 = 0
=
13 q L 1 q H 1
4
13 q H 1 4q 2 q L 1 = 0
= 3:25 0:25 q1 L + q1
H
(BRF 2 q1 L; qH 1 )
After …nding the best response functions for both types of Firm 1 and for the unique type of
Firm 2 we are ready to plug the …rst two BRFs into the latter. Speci…cally,
0
1
h
q 2 = 3:25 0:25 B 2
@
And solving for q 2 , we …nd q 2 = 2:167.
q 2
i
| {z 2 }
q1
L
h
+
i
C
2 A
q 2
4:5
| {z }
q1
H
With this information, it is easy to …nd the particular level of production for …rm 1 when
experiencing low market demand,
q L 1 (q 2 ) = 2
q 2
2 = 2 2:167
= 0:916
2
As well as the level of production for …rm 1 when experiencing high market demand,
q H 1 (q 2 ) = 4:5
q 2
2 = 4:5 2:167
= 3:4167
2
Therefore, the Bayesian Nash equilibrium (BNE) of this oligopoly game with incomplete
information about market demand prescribes the following production levels
q H 1 ; q L 1 ; q 2
= (3:416; 0:916; 2:167)
2