game theory课程week2 problem set

Question 1

Mixed Strategy Nash Equilibrium

1\ 2	Left	Right
Left	4,2	5,1
Right	6,0	3,3

Find a mixed strategy Nash equilibrium where player 1 randomizes over the pure strategy Left and Right with probability  p  for Left. What is  p ?

Your Answer		Score	Explanation
a) 1/4
b) 3/4	Correct	1.00
c) 1/2
d) 2/3
Total		1.00 / 1.00

Question Explanation

(b) is true.

• In a mixed strategy equilibrium in this game both players must mix and so 2 must be indifferent between Left and Right.
• Left gives 2 an expected payoff:  2p+0(1−p)
• Right gives 2 an expected payoff:  1p+3(1−p)
• Setting these two payoffs to be equal leads to  p=3/4 .

Question 2

Comparative Statics

1\ 2	Left	Right
Left	x,2	0,0
Right	0,0	2,2

In a mixed strategy Nash equilibrium where player 1 plays Left with probability  p  and player 2 plays Left with probability  q . How do  p  and  q  change as  X is increased ( X>1 )?

Your Answer		Score	Explanation
a)  p  is the same,  q  decreases.	Correct	1.00
b)  p  increases,  q  increases.
c)  p  decreases,  q  decreases.
d)  p  is the same,  q  increases.
Total		1.00 / 1.00

Question Explanation

(a) is true.

• In a mixed strategy equilibrium, 1 and 2 are each indifferent between Left and Right.
• For p:
  • Left gives 2 an expected payoff:  2p
  • Right gives 2 an expected payoff:  2(1−p)
  • These two payoffs are equal, thus we have  p=1/2 .
• For  q : setting the Left expected payoff equal to the Right leads to  Xq=2(1−q) , thus  q=2/(X+2) , which decreases in X.

Question 3

Employment

• There are 2 firms, each advertising an available job opening.
• Firms offer different wages: Firm 1 offers  w1=4  and 2 offers  w2=6 .
• There are two unemployed workers looking for jobs. They simultaneously apply to either of the firms.
  • If only one worker applies to a firm, then he/she gets the job
  • If both workers apply to the same firm, the firm hires a worker at random and the other worker remains unemployed (and receives a payoff of 0).

Find a mixed strategy Nash Equilibrium where  p  is the probability that worker 1 applies to firm 1 and  q  is the probability that worker 2 applies to firm 1.

Your Answer		Score	Explanation
d)  p=q=1/5 .	Correct	1.00
a)  p=q=1/2 ;
c)  p=q=1/4 ;
b)  p=q=1/3 ;
Total		1.00 / 1.00

Question Explanation

(d) is correct.

• In a mixed strategy equilibrium, worker 1 and 2 must be indifferent between applying to firm 1 and 2.
• For a given  p , worker 2's indifference condition is given by  2p+4(1−p)=6p+3(1−p) .
• Similarly, for a given  q , worker 1's indifference condition is given by  2q+4(1−q)=6q+3(1−q) .
• Both conditions are satisfied when  p=q=1/5 .

Question 4

Treasure

• A king is deciding where to hide his treasure, while a pirate is deciding where to look for the treasure.
• The payoff to the king from successfully hiding the treasure is 5 and from having it found is 2.
• The payoff to the pirate from finding the treasure is 9 and from not finding it is 4.
• The king can hide it in location X, Y or Z.

Suppose the pirate has two pure strategies: inspect both X and Y (they are close together), or just inspect Z (it is far away). Find a mixed strategy Nash equilibrium where  p  is the probability the treasure is hidden in X or Y and  1−p  that it is hidden in Z (treat the king as having two strategies) and  q  is the probability that the pirate inspects X and Y:

Your Answer		Score	Explanation
a)  p=1/2 ,  q=1/2 ;	Correct	1.00
b)  p=4/9 ,  q=2/5 ;
c)  p=5/9 ,  q=3/5 ;
d)  p=2/5 ,  q=4/9 ;
Total		1.00 / 1.00

Question Explanation

(a) is true.

• There is no pure strategy equilibrium, so in a mixed strategy equilibrium, both players are indifferent among their strategies.
• For p:
  • Inspecting X \& Y gives pirate a payoff:  9p+4(1−p)
  • Inspecting Z gives pirate a payoff:  4p+9(1−p)
  • These two payoffs are equal, thus we have  p=1/2 .
• For  q : indifference for the king requires that  5q+2(1−q)=2q+5(1−q) , thus  q=1/2 .

Question 5

Treasure

• A king is deciding where to hide his treasure, while a pirate is deciding where to look for the treasure.
• The payoff to the king from successfully hiding the treasure is 5 and from having it found is 2.
• The payoff to the pirate from finding the treasure is 9 and from not finding it is 4.
• The king can hide it in location X, Y or Z.

Suppose instead that the pirate can investigate any two locations, so has three pure strategies: inspect XY or YZ or XZ. Find a mixed strategy Nash equilibrium where the king mixes over three locations (X, Y, Z) and the pirate mixes over (XY, YZ, XZ). The following probabilities (king), (pirate) form an equilibrium:

Your Answer		Score	Explanation
a) (1/3, 1/3, 1/3), (4/9, 4/9, 1/9);
b) (4/9, 4/9, 1/9), (1/3, 1/3, 1/3);
c) (1/3, 1/3, 1/3), (2/5, 2/5, 1/5);
d) (1/3, 1/3, 1/3), (1/3, 1/3, 1/3);	Correct	1.00
Total		1.00 / 1.00

Question Explanation

(d) is true.

• Check (a):
  • Pirate inspects (XY, YZ, XZ) with prob (4/9, 4/9, 1/9);
  • Y is inspected with prob 8/9 while X (or Z) is inspected with prob 5/9;
  • King prefers to hide in X or Z, which contradicts the fact that in a mixed strategy equilibrium, king should be indifferent.
• Similarly, you can verify that (b) and (c) are not equilibria in the same way.
• In (d), every place is chosen by king and inspected by pirate with equal probability and they are indifferent between all strategies.