Problem Set 2 | Quizerry

a) 1/4

b) 3/4

c) 1/2

d) 2/3

$p$ is the same, $q$ decreases.

$p$ increases, $q$ increases.

$p$ decreases, $q$ decreases.

$p$ is the same, $q$ increases.

There are 2 firms, each advertising an available job opening.
Firms offer different wages: Firm 1 offers $w_1=4$ and 2 offers $w_2=6$ .
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.

1 point

$p = q = 1 / 3$ ;

$p = q = 1 / 5$ .

$p = q = 1 / 4$ ;

$p = q = 1 / 2$ ;

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:

1 point

$p = 1 / 2$ , $q = 1 / 2$ ;

$p = 4 / 9$ , $q = 2 / 5$ ;

$p = 5 / 9$ , $q = 3 / 5$ ;

$p = 2 / 5$ , $q = 4 / 9$ ;

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:

1 point

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);

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

1 \ 2	Left	Right
Left	$X$ ,2	0,0
Right	0,0	2,2

Problem Set 2 >> Game Theory

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