# Game Theory Week 7 Quiz Answers

## Game Theory Week 7 Quiz Answers

In this article i am gone to share Coursera Course Game Theory Week 7 Quiz Answers with you..

### Week 7 Quiz Answers

Question 1)

Three players together can obtain 11 to share, any two players can obtain 0.80.8, and one player by herself can obtain zero.
Then, N=3 and v({1})=v({2})=v({3})=0, v({1,2})=v({2,3})=v({3,1})=0.8,v({1,2,3})=1.
Which allocation is in the core of this coalitional game?

• (0,0,0);
• (0.4, 0.4, 0);
• (1/3, 1/3, 1/3);
• The core is empty;

Question 2)
• There is a market for an indivisible good with B buyers and S sellers.
• Each seller has only one unit of the good and has a reservation price of 0.
• Each buyer wants to buy only one unit of the good and has a reservation price of 1.
• Thus v(C)=min(BC,SC) where BC and SC are the number of buyers and sellers in coalition C (and so, for instance, v(i)=0 for any single player, and v(i,j)=1 if i,j are a pair of a buyer and seller).
If the number of buyers and sellers is B=2B=2 and S=1S=1, respectively, which allocations are in the core? [There might be more than one]

• Each seller receives 0 and each buyer receives 1.
• Each seller receives 1/2 and each buyer receives 1/2.
• Each seller receives 1 and each buyer receives 0

Question 3)
• There is a market for an indivisible good with B buyers and S sellers.
• Each seller has only one unit of the good and has a reservation price of 0.
• Each buyer wants to buy only one unit of the good and has a reservation price of 1.
• Thus v(C)=min(BC,SC) where BC and SC are the number of buyers and sellers in coalition C (and so, for instance, v(i)=0 for any single player, and v(i,j)=1 if i,j are a pair of a buyer and seller).
Now assume that we increase the number of sellers so that B=2B=2 and S=2S=2. Which allocations are in the core? [There might be more than one]

• Each seller receives 1 and each buyer receives 0.
• Each seller receives 0 and each buyer receives 1.
• Each seller receives 1/2 and each buyer receives 1/2.

Question 4)
• The instructor of a class allows the students to collaborate and write up together a particular problem in the homework assignment.
• Points earned by a collaborating team are divided among the students in any way they agree on.
• There are exactly three students taking the course, all equally talented, and they need to decide which of them if any should collaborate.
• The problem is so hard that none of them working alone would score any points. Any two of them can score 4 points together. If all three collaborate, they can score 6 points.
Which allocation is in the core of this coalitional game?

• (0,0,0);
• (2, 2, 0);
• (2, 2, 2);
• The core is empty;

Question 5)
• The instructor of a class allows the students to collaborate and write up together a particular problem in the homework assignment.
• Points earned by a collaborating team are divided among the students in any way they agree on.
• There are exactly three students taking the course, all equally talented, and they need to decide which of them if any should collaborate.
• The problem is so hard that none of them working alone would score any points. Any two of them can score 4 points together. If all three collaborate, they can score 6 points.
What is the Shapley value of each player?

• phi=(0,0,0)ϕ=(0,0,0)
• phi=(2,0,2)ϕ=(2,0,2)
• phi=(1/3,1/3,1/3)ϕ=(1/3,1/3,1/3)
• phi=(2,2,2)ϕ=(2,2,2)

Question 6)
There is a single capitalist (cc) and a group of 2 workers (w1w1 and w2w2).

The production function is such that total output is 0 if the firm (coalition) is composed only of the capitalist or of the workers (a coalition between the capitalist and a worker is required to produce positive output).

The production function satisfies:
• F(c∪w1)=F(c∪w2)=3
• F(c∪w1∪w2)=4
Which allocations are in the core of this coalitional game? [There might be more than one]

• x_c=2xc​=2, x_{w1}=1xw1​=1, x_{w2}=1xw2​=1;
• x_c=4xc​=4, x_{w1}=0xw1​=0, x_{w2}=0xw2​=0;
• x_c=2.5xc​=2.5, x_{w1}=0.5xw1​=0.5, x_{w2}=1xw2​=1;

Question 7)
There is a single capitalist (cc) and a group of 2 workers (w1w1 and w2w2).

The production function is such that total output is 0 if the firm (coalition) is composed only of the capitalist or of the workers (a coalition between the capitalist and a worker is required to produce positive output).

The production function satisfies:
• F(c∪w1)=F(c∪w2)=3
• F(c∪w1∪w2)=4
What is the Shapley value of the capitalist?

• 3;
• 4;
• 7/3;
• 7;

Question 8)
There is a single capitalist (cc) and a group of 2 workers (w1w1 and w2w2).

The production function is such that total output is 0 if the firm (coalition) is composed only of the capitalist or of the workers (a coalition between the capitalist and a worker is required to produce positive output).

The production function satisfies:
• F(c∪w1)=F(c∪w2)=3
• F(c∪w1∪w2)=4
What is the Shapley value of each worker?
• 1;
• 5/6;
• 3/4;
• 1/2;

Question 9)
There is a single capitalist (cc) and a group of 2 workers (w1w1 and w2w2).

The production function is such that total output is 0 if the firm (coalition) is composed only of the capitalist or of the workers (a coalition between the capitalist and a worker is required to produce positive output).

The production function satisfies:
• F(c∪w1)=F(c∪w2)=3
• F(c∪w1∪w2)=4
True or False: If there was an additional 3^{rd}3rd worker that is completely useless (i.e., his marginal contribution is 0 in every coalition), then the sum of the Shapley Values of the capitalist and the first two workers will remain unchanged.

• True
• False