1. 2 firms are considering whether to operate in a new market. Call them 1 and 2
ID: 1227151 • Letter: 1
Question
1. 2 firms are considering whether to operate in a new market. Call them 1 and 2 respectively
a. Assume that they make the decision whether to enter or not at the same time. They payoffs are given in the below matrix. Find all the pure strategy NE for this game.
b. Find all mixed strategies NE for this game or explain why no such strategies exist.
c. Now consider another markets in which these 2 firms compete. In this second market, demand is given by P(Q) = 20 2Q, and costs are c1(q1) = 4 + 4q1 and c2(q2) = 2 + 2q2. If the firms set quantity at the same time, what quantity do they each sell and what is the market price?
d. What are the profits for both firms in this second market?
PLEASE show and explain your work
Enter Don't Enter -10,-10 40,0 Don't 0,20 0,0Explanation / Answer
a) there are 2 NE strategies - (enter, don't) and (don't, enter)
If 1 decides to enter, it is best for 2 not to enter and if 1 decides not to enter, its best for 2 to enter and vice-versa hence we get 2 NE.
b) 1 would choose mixed strategy such that 2 is indifferent in entering and not entering
1 would choose p such that 2's payoff from enter=2's papyoff from don't enter
p(-10)+(1-p)(20)= p(0)+(1-p)(0)
p=2/3
similarly 2 would choose q such that 1's payoff from enter=1's papyoff from don't enter
q(-10)+(1-q)(40)=q(0)+(1-q)(0)
-10q+40-40q=0
q=4/5
In the mixed strategy Nash equilibrium 1 chooses enter with probability 2/3 and Don’t Enter with probability 1/3 and 2 chooses enter with probability 4/5 and Don’t Enter with probability 1/5
c) P=20-2Q
profit of 1, P1=q1(20-2(q1+q2))-4-4q1
dP1/dq1=0
4q1+2q2=16
profit of 2, P2=q2(20-2(q1+q2))-2-2q2
dP2/dq2=0
2q1+4q2=18
q1=7/3
q2=10/3
P=20-2(17/3)
P=26/3
d. profit of 1, P1= (7/3)(20-2(17/3))-4-4(7/3)
P1=6.89
P2=(10/3)(20-2(17/3))-2-2(17/3)
P2=15.56
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