Answer key.

EconS 424 - Strategy and Game Theory
Quiz #2 - April 1st, 2015
Instructions: You have 15 minutes to complete this exam. Please read the questions
carefully, answer them in a formal and concise manner, but include all your steps, this will
allow you to obtain partial credit. Good luck!!
Let us consider a Cournot oligopoly game where two …rms compete in quantities. Both …rms
have the same marginal costs, M C = $2, but they are asymmetrically informed about the
actual state of market demand. In particular, Firm 2 does not know what is the actual state
of demand, but knows that it is distributed with the following probability distribution
p(Q) =
20 Q with probability 23
8 Q with probability 13
On the other hand, …rm 1 knows the actual state of market demand, and …rm 2 knows that
…rm 1 knows this information (i.e., it is common knowledge among the players).
1. Let us …rst focus on Firm 1, the informed player in this game, as we usually do when
solving for the BNE of games of incomplete information.
(a) Find …rm 1’s best response function when the …rm operates in a high-demand
market. Denote it as q1H (q2 ).
(b) Find …rm 1’s best response function when the …rm operates in a low-demand
market. Denote it as q1L (q2 ).
2. Let us now turn to Firm 2, the uninformed player in this game.
(a) Write the expected pro…ts of this …rm, taking into account the above probabilities
of operating in a high or low-demand market.
(b) Find …rm 2’s best response function. Denote it as q2 (q1H ; q1L ). [Recall that its best
response function is only one, since …rm 2 does not know whether the market is
in high or low demand.]
3. Insert q1H (q2 ).from exercise 1(a) and q1L (q2 ).from 1(b) into q2 (q1H ; q1L ) from 2(b). Then
solve for q2 in order to …nd …rm 2’s equilibrium production level (this production should
be a number).
4. Insert the value of q2 you found in part (3) into the expression of q1H (q2 ).you obtained
in exercise 1(a) and in q1L (q2 ).from 1(b). Summarize your results (you have just found
the Bayesian Nash Equilibrium of this game of incomplete information!!).
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EconS 424 – Strategy and Game Theory
Quiz #2 - Answer Key
1) Firm 1 (informed firm):
1a) If high demand:
𝜋𝜋1𝐻𝐻 = (20 − 𝑞𝑞1𝐻𝐻 − 𝑞𝑞2 ) ∗ 𝑞𝑞1𝐻𝐻 − 2𝑞𝑞1𝐻𝐻
Taking FOCs with respect to 𝑞𝑞1𝐻𝐻 ,
20 − 2𝑞𝑞1𝐻𝐻 − 𝑞𝑞2 − 2 = 0 → 18 − 𝑞𝑞2 = 2𝑞𝑞1𝐻𝐻
Hence, solving for 𝑞𝑞1𝐻𝐻 , we obtain firm 1’s best response function when facing a high demand
1
𝐵𝐵𝐵𝐵𝐹𝐹1𝐻𝐻 → 𝑞𝑞1𝐻𝐻 (𝑞𝑞2 ) → 𝑞𝑞1𝐻𝐻 = 9 − 𝑞𝑞2
2
1b) If low demand:
𝜋𝜋1𝐿𝐿 = (8 − 𝑞𝑞1𝐿𝐿 − 𝑞𝑞2 )𝑞𝑞1𝐿𝐿 − 2𝑞𝑞1𝐿𝐿
Taking FOCs with respect to 𝑞𝑞1𝐿𝐿 ,
8 − 2𝑞𝑞1𝐿𝐿 − 𝑞𝑞2 − 2 = 0 → 6 − 𝑞𝑞2 = 2𝑞𝑞1𝐿𝐿
Hence, solving for 𝑞𝑞1𝐿𝐿 , we obtain firm 1’s best response function when facing a low demand
2) Firm 2 (uninformed firm):
1
𝐵𝐵𝐵𝐵𝐹𝐹1𝐿𝐿 → 𝑞𝑞1𝐿𝐿 (𝑞𝑞2 ) → 𝑞𝑞1𝐿𝐿 = 3 − 𝑞𝑞2
2
2a) Expected profit for Firm 2
2
3
1
3
E𝜋𝜋2 = �(20 − 𝑞𝑞1𝐻𝐻 − 𝑞𝑞2 )𝑞𝑞2 − 2𝑞𝑞2 � + �(8 − 𝑞𝑞1𝐿𝐿 − 𝑞𝑞2 )𝑞𝑞2 − 2𝑞𝑞2 �
2
1
= (20𝑞𝑞2 − 𝑞𝑞1𝐻𝐻 𝑞𝑞2 − 𝑞𝑞22 ) + (8𝑞𝑞2 − 𝑞𝑞1𝐿𝐿 𝑞𝑞2 − 𝑞𝑞22 ) − 2𝑞𝑞2
3
3
2b) Taking FOCs with respect to 𝑞𝑞2 ,
8 1
2
40 2 𝐻𝐻 4
− 𝑞𝑞1 − 𝑞𝑞2 + − 𝑞𝑞1𝐿𝐿 − 𝑞𝑞2 − 2 = 0
3
3 3
3
3 3
And solving for 𝑞𝑞2 , we obtain firm 2’s best response function.
2
1
𝐵𝐵𝐵𝐵𝐹𝐹2 → 𝑞𝑞2 (𝑞𝑞1𝐻𝐻 , 𝑞𝑞1𝐿𝐿 ) → 𝑞𝑞2 = 7 − 𝑞𝑞1𝐻𝐻 − 𝑞𝑞1𝐿𝐿
6
6
1
3) Plugging 𝑞𝑞1𝐻𝐻 and 𝑞𝑞1𝐿𝐿 into firm 2’s best response function, we find
1
1
1
1
𝑞𝑞2 = 7 − �9 − 𝑞𝑞2 � − �3 − 𝑞𝑞2 �
2
6
2
3
and solving for 𝑞𝑞2 , we obtain firm 2’s equilibrium output level,
𝑞𝑞2 = 4.66
4) We can now find firm 1’s equilibrium output level. First, plugging 𝑞𝑞2 =4.66 into 𝑞𝑞1𝐻𝐻 = 9 −
1
𝑞𝑞 , we obtain:
2 2
1
𝑞𝑞1𝐻𝐻 = 9 − (4.66) = 6.7
2
1
Similarly, plugging 𝑞𝑞2 =4.66 into 𝑞𝑞1𝐿𝐿 = 3 − 𝑞𝑞2 ,we obtain.
2
1
𝑞𝑞1𝐿𝐿 = 3 − (4.66) = 0.66
2
Summarizing, the BNE of this incomplete information Cournot game is:
(𝑞𝑞1𝐻𝐻 , 𝑞𝑞1𝐿𝐿 , 𝑞𝑞2 ) = (6.7, 0.66, 4.66)
2