斯坦福大学英属哥伦比亚大学 博弈论课程week7 problem set及其答案

Feedback — Problem Set 7

Question 1

Core

• Three players together can obtain 1 to share, any two players can obtain 0.8, 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?

Question Explanation

(d) is true.

• By definition, the core of this game is formed by a triplet (x1,x2,x3)∈R3+ that satisfies:
  • xi+xj≥0.8 for i≠j
  • x1+x2+x3≥1
  • There is no triplet (x1,x2,x3) that satisfies all inequalities. Then, the core is empty.

Question 2

Buyers and Sellers

• 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=2 and S=1, respectively, which of the following allocations are in the core? 
a) Each seller receives 1 and each buyer receives 0. 
b) Each seller receives 0 and each buyer receives 1. 
c) Each seller receives 1/2 and each buyer receives 1/2.

Question Explanation

(a) is true.

• By definition, the core of this game is formed by a vector of payoffs to buyers (b1 and b2) and to the seller (s) (xb1,xb2,xs)∈R3+ that satisfies:
  • xb1+xb2≥0;
  • xbi+xs≥1 for i=1,2;
  • xb1+xb2+xs≥1;
  • and the feasibility constraint xb1+xb2+xs≤1.
• It is easy to verify that allocation (a) is the only one that satisfies the set of inequalities.

Question 3

Buyers and Sellers

• 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 competition among sellers increases, so that B=2 and S=2. Which of the following allocations are in the core?
a) Each seller receives 1 and each buyer receives 0. 
b) Each seller receives 0 and each buyer receives 1. 
c) Each seller receives 1/2 and each buyer receives 1/2.

Question Explanation

All are in the core.

• Again, the core of this game is formed by a vector of payoffs to buyers and sellers (xb1,xb2,xs1,xs2)∈R4+ that satisfies:
  • xb1+xb2≥0;
  • xs1+xs2≥0;
  • xbi+xsj≥1 for i=1,2 and j=1,2;
  • xb1+xb2+xs1+xs2≥2;
  • and the feasibility constraint xb1+xb2+xs1+xs2≤2.
• It is easy to verify that allocations (a), (b) and (c) satisfy the set of inequalities.
• In fact, any split of the surplus that gives α to all sellers and 1−α to all buyers (with α∈[0,1]) is in the core. That is, any split of the surplus is possible; the only restriction imposed by the increase in competition (i.e., increase in the number of sellers) is that all pairs must receive the same share of the surplus.

Question 4

Core and Shapley Value

• 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?

Question Explanation

(c) is true.

• By definition, the core of this game is formed by a vector of payoffs to each student (x1,x2,x3)∈R3+ that satisfies:
  • xi+xj≥4 for i≠j
  • x1+x2+x3≥6
  • (2,2,2) is the only option that satisfies these inequalities. Then, it belongs to the core.

Question 5

Core and Shapley Value

• 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?

Question Explanation

(d) is true.

• Use the definition of the Shapley Value to compute its value for each player.
• Another way to find the Shapley Value is to remember that:
  • by the axiom of symmetry, all agents should receive the same payoff.
  • the Shapley value divides the payoff to the grand coalition completely
  • Then, all agents will have a Shapley value of 6/3=2.

Question 6

Production

• There is a single capitalist (c) and a group of 2 workers (w1 and w2).
• 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 of the following allocations are in the core of this coalitional game? 
a) xc=2, xw1=1, xw2=1; 
b) xc=2.5, xw1=0.5, xw2=1; 
c) xc=4, xw1=0, xw2=0;

Question Explanation

All the three allocations are in the core

• It can be shown more generally that for any given number n of workers and any increasing and concave production function f, the core of this coalitional game is defined by:
  • xwi≤f(c∪w1…∪wn)−f(c∪w1…∪w(n−1))
  • xc+∑ni=1xw1≤f(c∪w1∪…∪wn)
• Intuitively, the first equation requires each worker to receive less than the marginal product of the nth worker. If this condition would not hold for worker i, then the rest of the workers and the capitalist could abandon him and get a higher value for the new coalition.
• The second condition is a feasibility condition (the sum of payoffs of the grand coalition is not greater than the resources available).

Question 7

Production

• There is a single capitalist (c) and a group of 2 workers (w1 and w2).
• 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?

Question Explanation

(c) is true.

• Use the definition of the Shapley Value to compute its value for the capitalist.

Question 8

Production

• There is a single capitalist (c) and a group of 2 workers (w1 and w2).
• 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?

Question Explanation

(b) is true.

• Use the definition of the Shapley Value to compute its value for each worker.
• Another way to find the Shapley Value is to remember that:
  • by the axiom of symmetry, all workers should receive the same payoff
  • the Shapley value divides the payoff to the grand coalition completely
  • Then, all agents will have a Shapley value of (F(c∪w1∪w2)−7/3)/2=(4−7/3)/2=5/6.

Question 9

Production

• There is a single capitalist (c) and a group of 2 workers (w1 and w2).
• 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 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.

Question Explanation

(a) is correct.

• The Shapley Value satisfies the Dummy player Axiom:
  • if i is a dummy player, then he/she must have a Shapley Value of 0
• Since the 3rd worker is a Dummy player (check the definition), his/her Shapley Value must be 0.
• Thus, the statement is true because the Shapley Value divides the payoff of the grand coalition completely.