博弈论 斯坦福game theory stanford week 7.1_

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title: 博弈论 斯坦福game theory stanford week 7-1
tags: note
notebook: 6- 英文课程-15-game theory
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博弈论 斯坦福game theory stanford week 7-1

1。第 1 个问题
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=3N=3 and v({1})=v({2})=v({3})=0v(1)=v(2)=v(3)=0, v({1,2})=v({2,3})=v({3,1})=0.8v(1,2)=v(2,3)=v(3,1)=0.8, v({1,2,3})=1v(1,2,3)=1.
Which allocation is in the core of this coalitional game?

a) (0,0,0);

b) (0.4, 0.4, 0);

c) (1/3, 1/3, 1/3);

d) The core is empty;

正确 
(d) is true.

By definition, the core of this game is formed by a triplet (x_1, x_2, x_3) \in R_{+}^3(x 
1
​    ,x 
2
​    ,x 
3
​    )∈R 
+
3
​     that satisfies:
x_{i}+x_{j}\geq0.8x 
i
​    +x 
j
​    ≥0.8 for i \neq ji≠j
x_1 + x_2 + x_3 \geq 1x 
1
​    +x 
2
​    +x 
3
​    ≥1
There is no triplet (x_1, x_2, x_3)(x 
1
​    ,x 
2
​    ,x 
3
​    ) that satisfies all inequalities. Then, the core is empty.

第 [1-9] 个问题

正确
1 / 1 分
2。第 2 个问题
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]

a) Each seller receives 1 and each buyer receives 0.

正确 
(a) is true.

By definition, the core of this game is formed by a vector of payoffs to buyers (b1b1 and b2b2) and to the seller (ss) (x_{b1}, x_{b2}, x_s) \in R_{+}^3(x 
b1
​    ,x 
b2
​    ,x 
s
​    )∈R 
+
3
​     that satisfies:
x_{b1}+x_{b2}\geq 0x 
b1
​    +x 
b2
​    ≥0;
x_{bi}+x_{s}\geq 1x 
bi
​    +x 
s
​    ≥1 for i =1,2i=1,2;
x_{b1}+x_{b2}+x_s\geq 1x 
b1
​    +x 
b2
​    +x 
s
​    ≥1;
and the feasibility constraint x_{b1}+x_{b2}+x_s \leq 1x 
b1
​    +x 
b2
​    +x 
s
​    ≤1.
It is easy to verify that allocation (a) is the only one that satisfies the set of inequalities.

b) Each seller receives 0 and each buyer receives 1.

未选择的是正确的 

c) Each seller receives 1/2 and each buyer receives 1/2.

未选择的是正确的 

第 [1-9] 个问题

正确
1 / 1 分
3。第 3 个问题
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]

a) Each seller receives 1 and each buyer receives 0.

正确 
All are in the core.

Again, the core of this game is formed by a vector of payoffs to buyers and sellers (x_{b1}, x_{b2}, x_{s1}, x_{s2}) \in R_{+}^4(x 
b1
​    ,x 
b2
​    ,x 
s1
​    ,x 
s2
​    )∈R 
+
4
​     that satisfies:
x_{b1}+x_{b2}\geq 0x 
b1
​    +x 
b2
​    ≥0;
x_{s1}+x_{s2}\geq 0x 
s1
​    +x 
s2
​    ≥0;
x_{bi}+x_{sj}\geq 1x 
bi
​    +x 
sj
​    ≥1 for i =1,2i=1,2 and j =1,2j=1,2;
x_{b1}+x_{b2}+x_{s1}+x_{s2}\geq 2x 
b1
​    +x 
b2
​    +x 
s1
​    +x 
s2
​    ≥2;
and the feasibility constraint x_{b1}+x_{b2}+x_{s1}+x_{s2} \leq 2x 
b1
​    +x 
b2
​    +x 
s1
​    +x 
s2
​    ≤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 \alphaα to all sellers and 1-\alpha1−α to all buyers (with \alpha \in [0,1]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.

b) Each seller receives 0 and each buyer receives 1.

正确 
All are in the core.

Again, the core of this game is formed by a vector of payoffs to buyers and sellers (x_{b1}, x_{b2}, x_{s1}, x_{s2}) \in R_{+}^4(x 
b1
​    ,x 
b2
​    ,x 
s1
​    ,x 
s2
​    )∈R 
+
4
​     that satisfies:
x_{b1}+x_{b2}\geq 0x 
b1
​    +x 
b2
​    ≥0;
x_{s1}+x_{s2}\geq 0x 
s1
​    +x 
s2
​    ≥0;
x_{bi}+x_{sj}\geq 1x 
bi
​    +x 
sj
​    ≥1 for i =1,2i=1,2 and j =1,2j=1,2;
x_{b1}+x_{b2}+x_{s1}+x_{s2}\geq 2x 
b1
​    +x 
b2
​    +x 
s1
​    +x 
s2
​    ≥2;
and the feasibility constraint x_{b1}+x_{b2}+x_{s1}+x_{s2} \leq 2x 
b1
​    +x 
b2
​    +x 
s1
​    +x 
s2
​    ≤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 \alphaα to all sellers and 1-\alpha1−α to all buyers (with \alpha \in [0,1]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.

c) Each seller receives 1/2 and each buyer receives 1/2.

正确 
All are in the core.

Again, the core of this game is formed by a vector of payoffs to buyers and sellers (x_{b1}, x_{b2}, x_{s1}, x_{s2}) \in R_{+}^4(x 
b1
​    ,x 
b2
​    ,x 
s1
​    ,x 
s2
​    )∈R 
+
4
​     that satisfies:
x_{b1}+x_{b2}\geq 0x 
b1
​    +x 
b2
​    ≥0;
x_{s1}+x_{s2}\geq 0x 
s1
​    +x 
s2
​    ≥0;
x_{bi}+x_{sj}\geq 1x 
bi
​    +x 
sj
​    ≥1 for i =1,2i=1,2 and j =1,2j=1,2;
x_{b1}+x_{b2}+x_{s1}+x_{s2}\geq 2x 
b1
​    +x 
b2
​    +x 
s1
​    +x 
s2
​    ≥2;
and the feasibility constraint x_{b1}+x_{b2}+x_{s1}+x_{s2} \leq 2x 
b1
​    +x 
b2
​    +x 
s1
​    +x 
s2
​    ≤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 \alphaα to all sellers and 1-\alpha1−α to all buyers (with \alpha \in [0,1]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.

第 [1-9] 个问题

正确
1 / 1 分
4。第 4 个问题
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?

d) The core is empty;

a) (0,0,0);

b) (2, 2, 0);

c) (2, 2, 2);

正确 
(c) is true.

By definition, the core of this game is formed by a vector of payoffs to each student (x_1, x_2, x_3) \in R_{+}^3(x 
1
​    ,x 
2
​    ,x 
3
​    )∈R 
+
3
​     that satisfies:
x_{i}+x_{j}\geq4x 
i
​    +x 
j
​    ≥4 for i \neq ji≠j
x_1 + x_2 + x_3 \geq 6x 
1
​    +x 
2
​    +x 
3
​    ≥6
(2,2,2) is the only option that satisfies these inequalities. Then, it belongs to the core.

第 [1-9] 个问题

正确
1 / 1 分
5。第 5 个问题
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?

a) \phi=(0,0,0)ϕ=(0,0,0)

b) \phi=(2,0,2)ϕ=(2,0,2)

c) \phi=(1/3,1/3,1/3)ϕ=(1/3,1/3,1/3)

d) \phi=(2,2,2)ϕ=(2,2,2)

正确 
(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=26/3=2.

第 [1-9] 个问题

正确
1 / 1 分
6。第 6 个问题
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]

a) x_c=2x
c
​ =2, x_{w1}=1x
w1
​ =1, x_{w2}=1x
w2
​ =1;

正确 
(d) is true.

It is easy to verify that allocations (a), (b) and (c) satisfy the definition of the core.
It can be shown more generally that for any given number nn of workers and any increasing and concave production function ff, the core of this coalitional game is defined by:
x_{wi}\leq f(c \cup w1 \ldots \cup wn) - f(c \cup w1 \ldots \cup w(n-1))x 
wi
​    ≤f(c∪w1…∪wn)−f(c∪w1…∪w(n−1))
x_c + \sum_{i=1}^{n} x_{w1} \leq f(c \cup w1 \cup \ldots \cup wn)x 
c
​    +∑ 
i=1
n
​    x 
w1
​    ≤f(c∪w1∪…∪wn)
Intuitively, the first equation requires each worker to receive less than the marginal product of the n^{th}n 
th
worker. If this condition would not hold for worker ii, 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).

b) x_c=2.5x
c
​ =2.5, x_{w1}=0.5x
w1
​ =0.5, x_{w2}=1x
w2
​ =1;

正确 
(d) is true.

It is easy to verify that allocations (a), (b) and (c) satisfy the definition of the core.
It can be shown more generally that for any given number nn of workers and any increasing and concave production function ff, the core of this coalitional game is defined by:
x_{wi}\leq f(c \cup w1 \ldots \cup wn) - f(c \cup w1 \ldots \cup w(n-1))x 
wi
​    ≤f(c∪w1…∪wn)−f(c∪w1…∪w(n−1))
x_c + \sum_{i=1}^{n} x_{w1} \leq f(c \cup w1 \cup \ldots \cup wn)x 
c
​    +∑ 
i=1
n
​    x 
w1
​    ≤f(c∪w1∪…∪wn)
Intuitively, the first equation requires each worker to receive less than the marginal product of the n^{th}n 
th
worker. If this condition would not hold for worker ii, 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).

c) x_c=4x
c
​ =4, x_{w1}=0x
w1
​ =0, x_{w2}=0x
w2
​ =0;

正确 
(d) is true.

It is easy to verify that allocations (a), (b) and (c) satisfy the definition of the core.
It can be shown more generally that for any given number nn of workers and any increasing and concave production function ff, the core of this coalitional game is defined by:
x_{wi}\leq f(c \cup w1 \ldots \cup wn) - f(c \cup w1 \ldots \cup w(n-1))x 
wi
​    ≤f(c∪w1…∪wn)−f(c∪w1…∪w(n−1))
x_c + \sum_{i=1}^{n} x_{w1} \leq f(c \cup w1 \cup \ldots \cup wn)x 
c
​    +∑ 
i=1
n
​    x 
w1
​    ≤f(c∪w1∪…∪wn)
Intuitively, the first equation requires each worker to receive less than the marginal product of the n^{th}n 
th
worker. If this condition would not hold for worker ii, 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).

第 [1-9] 个问题

正确
1 / 1 分
7。第 7 个问题
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?

a) 3;

b) 4;

c) 7/3;

正确 
(c) is true.

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

d) 7;

第 [1-9] 个问题

正确
1 / 1 分
8。第 8 个问题
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?

a) 1;

b) 5/6;

正确 
(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. (Where 7/3 is the Shapley value of the capitalist)

c) 3/4;

d) 1/2;

第 [1-9] 个问题

正确
1 / 1 分
9。第 9 个问题
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}3
rd
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.

a) True;

正确 
(a) is correct.

The Shapley Value satisfies the Dummy player Axiom:
if ii is a dummy player, then he/she must have a Shapley Value of 0
Since the 3^{rd}3 
rd
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.

b) False;