title: 博弈论 斯坦福game theory stanford week 2-0
tags: note
notebook: 6- 英文课程-15-game theory
---

博弈论 斯坦福game theory stanford week 2-0

习题

1。第 1 个问题

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?

a) 1/4

b) 3/4

正确
(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.

c) 1/2

d) 2/3

第 2 个问题

正确
1 / 1 分
2。第 2 个问题
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)?

a) p is the same, q decreases.

正确 
(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.

b) p increases, q increases.

c) p decreases, q decreases.

d) p is the same, q increases.

第 3 个问题

正确
1 / 1 分
3。第 3 个问题
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.

b) p=q=1/3;

c) p=q=1/4;

d) p=q=1/5.

正确 
(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.

a) p=q=1/2;

第 4 个问题

正确
1 / 1 分
4。第 4 个问题
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:

a) p=1/2, q=1/2;

正确 
(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.

b) p=4/9, q=2/5;

c) p=5/9, q=3/5;

d) p=2/5, q=4/9;

第 5 个问题

正确
1 / 1 分
5。第 5 个问题
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 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:

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);

正确 
(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.