Week 1
Week 1 - In Video Quizzes?
Which of the following outcomes is Pareto-optimal? (click all that apply: There might be more than one, or none.) [a,b]
a) (3,3);
b) (1,4);
c) (1,1);
Problem set 1
Week 2
In-video quizzes
博弈論 斯坦福game theory Stanford week 2-0
習題
1浆劲。第 1 個問題
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
正確
(b) is true.
Ina mixed strategy equilibrium in this game both players must mixandso2must be indifferent betweenLeftandRight.Leftgives2an expected payoff:2p+0(1?p)Rightgives2an expected payoff:1p+3(1?p)Setting these two payoffs to be equal leadstop=3/4.
c) 1/2
d) 2/3
第 2 個問題
正確
1 / 1 分
2牡借。第 2 個問題
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.
正確 (a)?is true. In a mixed strategy equilibrium,1and2areeachindifferent betweenLeftandRight.For p: Left gives 2 an expected payoff:2pRightgives2an expected payoff:2(1?p)These two payoffs are equal, thus we have p=1/2. For q: setting the Left expected payoff equaltotheRightleadstoXq=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 個問題
正確
1 / 1 分
3。第 3 個問題
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.
正確 (d) is correct. In a mixed strategy equilibrium, worker1and2must be indifferent between applyingtofirm1and2. For a given p, worker2's indifference condition is given by2p+4(1?p)=6p+3(1?p). Similarly, for a given q, worker1's indifference condition is given by2q+4(1?q)=6q+3(1?q). Both conditions are satisfiedwhenp=q=1/5.
a) p=q=1/2;
第 4 個問題
正確
1 / 1 分
4憋沿。第 4 個問題
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;
正確 (a)istrue.Thereisnopurestrategy equilibrium, soina 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.Forq: indifferenceforthe king requires that5q+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 個問題
正確
1 / 1 分
5。第 5 個問題
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);
正確 (d)istrue.Check (a): Pirate inspects (XY, YZ, XZ)withprob (4/9,4/9,1/9); Y is inspected with prob 8/9 while X (orZ)isinspectedwithprob5/9; King preferstohideinXorZ, which contradicts the fact thatina mixed strategy equilibrium, the king should be indifferent. Similarly, you can verify that (b)and(c) arenotequilibriainthe same way.In(d), every placeischosenbykingandinspectedbypiratewithequal probabilityandthey are indifferent between all strategies.
