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Assignment 4 Soln

The document discusses game theory concepts including subgames, proper subgames, Nash equilibrium, and sequential Nash equilibrium using examples. It also provides an example of a Cournot duopoly model to derive the reaction functions and equilibrium quantities, price, and profits.

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Prathibha Vikram
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22 views2 pages

Assignment 4 Soln

The document discusses game theory concepts including subgames, proper subgames, Nash equilibrium, and sequential Nash equilibrium using examples. It also provides an example of a Cournot duopoly model to derive the reaction functions and equilibrium quantities, price, and profits.

Uploaded by

Prathibha Vikram
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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1.

A Subgame is an extensive form game


○ That begins at some decision node, say A, with a singleton information set.
○ Includes all the successor nodes[decision and terminal nodes] of A.
○ If includes one node from an I-set, then include all the nodes of that I-set.

Here we can see only 1 singleton information set exists containing the whole game hence
Subgame =1

2. Proper Subgames = Subgames -1 = 1-1 = 0


(i.e. excluding the whole game.)
3. P1 has strategies as - {L,C,R} -3
4. P2 has strategies as- {Y,N} - 2
5.

Y N

L 2,1 0,0

C 1,1 0,0

R 0,2 0,0
As we can see from the game table (L,Y) is the Nash Equilibrium.
6. Here SPNE is actually the NE which is (L,Y) i.e. onle 1 SPNE.
7. SPNE is (L,Y)

8.

FHJ FHK FIJ FIK GHJ GHK GIJ GIK

C 3,0 3,0 3,0 3,0 1,6 1,6 1,6 1,6

D 1,1 1,1 2,1 2,1 1,1 1,1 2,1 2,1

E 2,2 3,5 2,2 3,5 2,2 3,5 2,2 3,5

As we can see from the table P2 has 8 strategies.


9. Using backward induction, we can see that
- P2 will choose K if P1 chose E.
- P2 will choose G if P1 chose C.
- P2 can choose either H/ I if P1 chose D.
P1 knows what P2 will choose so P1 will choose E because this maximizes his/her
payoff. Hence 2 SPNE - { (E,GHK), (E,GIK) }
10. NE - { (D,GIJ), (E,FHK), (E,FIK), (E,GHK), (E,GIK)} - 5

11. From the lectures, just put the values.


12. From the lectures, just put the values.

13.
P = 40- (qa+qb)
CA(qA) = 20qA
CB(qB) = qB2

In a Cournot duopoly, the reaction function of Firm A identifies its optimal response to any
quantity produced by Firm B. The optimal quantity is the one that maximizes ΠA,
Firm A’s profit, where
ΠA = P(Q)qA − C(qA )
= (40 − qA − qB ) qA − 20qA
The first-order condition for profit maximization is:
∂ΠA/∂qA = 40 − 2qA − qB − 20 = 0
The reaction function of firm B identifies the quantity qB that maximizes firm’s B profits helding
constant qA . Firm’s B profit is given by:
ΠB = P(Q)qB − C(qB )
= (40 − qA − qB ) qB − qB2
The first order condition is:
∂ΠB /∂qB = 40 − 2qB − qA − 2qB= 0

On solving these two equations -


The equilibrium quantities are qA = 40/7 ; qB = 60/7
Q = 100 /7
14. And the equilibrium price is
P = 40- 100/7 = 180/7
15. And firms’ profits are:
ΠA = 180/ 7 qA − 20qA = 1600 /49
ΠB = 180/ 7 qB − qB2 = 7200/49

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