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Homework Feedback | Social and Economic Networks: Models and Analysis
Feedback — Problem Set 7
You submitted this homework on Mon 27 May 2013 3:44 AM IST (UTC +0530). You
got a score of 16.00 out of 16.00.
Question 1
Consider the following two games played on a network.
In both cases the players are sellers. A seller's neighbors in a the network are a list of those
other sellers whose actions affect the seller's profits.
a). Each player
i
chooses whether or not to set a high price (and then the market will determine
the quantity sold).
In particular, a i
a low price
Player
i
= 1
stands for charging a high price
pi = 5 ,
and
ai = 0
stands for charging
pi = 3 .
's profit as a function of her price and the prices of her neighbors in the network
j ∈ N (i)
is (100 − pi
b). Each player
i
+ ∑
j∈N (i)
pj /2)pi
.
chooses whether or not to produce a high quantity (and then the market
determines the price at which the player sells).
In particular, a i
= 1
stands for a high quantity q i
= 5,
and
ai = 0
stands for a low quantity
qi = 3.
Player
i
's profit as a function of her quantity and the quantities of her neighbors in the network
j ∈ N (i)
is (100 − q i
− ∑
j∈N (i)
q j /2) qi
.
According to the definition of strategic complements/substitutes, which statement is correct?
Your Answer
Score
Explanation
a): actions are strategic complements
b): actions are strategic complements
a): actions are strategic complements
b): actions are strategic substitutes
✔
2.00
a): actions are strategic substitutes
b): actions are strategic complements
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b): actions are strategic
complements
a): actions are strategic substitutes
b): actions are strategic substitutes
a): actions are neither strategic substitutes nor
complements;
b): actions are neither strategic substitutes nor complements
Total
2.00 /
2.00
Question Explanation
In a), actions are strategic complements as the relative benefit from setting a higher price is
higher, as more neighbors' prices are high;
In b), actions are strategic substitutes as the relative benefit from setting a higher quantity is
lower, as more neighbors' quantities are high.
Question 2
Consider the following game of complements with threshold 2 played on the network depicted in
the picture.
In particular, the players' utility functions are such that for player i:
ud (0, mN ) = 0
i
i
;
ud (1, mN ) = −1.5 + mN
i
i
i
;
In lecture 7.2 we have already seen three equilibria, as depicted in the picture.
There is one more pure-strategy equilibrium. How many nodes are choosing action 1 in that
equilibrium?
Your Answer
Score
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Explanation
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Homework Feedback | Social and Economic Networks: Models and Analysis
0
3
4
✔
2.00
7
11
Total
2.00 / 2.00
Question Explanation
Here is the other equilibrium. There are 4 nodes choosing action 1.
Question 3
Consider again a game of complements with threshold 2, now on a different network as depicted
in the picture.
Which of the following is the largest pure strategy equilibrium ("largest" in the sense of
having most players choosing 1)?
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Your Answer
Score
Explanation
a
b
c
✔
2.00
d
none of them
Total
2.00 / 2.00
Question Explanation
c is the largest pure strategy equilibrium. Notice that d is not an equilibrium. a and b are
equilibria but not the largest.
Question 4
Consider the following best shot public goods game on a directed network g
= {12, 23, 31} .
In particular, player i can borrow a textbook from j if she has a directed link (arrow) pointing at j,
and j buys the textbook.
Each player strictly prefers to buy the textbook (a
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= 1)
if she cannot borrow a textbook from
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Homework Feedback | Social and Economic Networks: Models and Analysis
anyone;
Otherwise, she strictly prefers to not buy (a
= 0)
the textbook if she can borrow a textbook
from her neighbor.
Which is an pure-strategy equilibrium of this game? The numbers in the circles are the actions of
the nodes
Your Answer
Score
Explanation
a
b
c
d
none of them
Total
✔
2.00
2.00 / 2.00
Question Explanation
None is an equilibrium. Actually this tells you that this game has NO pure-strategy equilibria.
Question 5
Questions 5 and 6 are based on the network depicted in the following picture.
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There are three groups, green, blue and yellow, in the network.
What is the cohesiveness of each of the three groups?
Your Answer
Score
Explanation
Green group 0.5; Blue group 0.5; Yellow group 0.5.
Green group 0.6; Blue group 0.25; Yellow group 0.5.
Green group 0.5; Blue group 0.67; Yellow group 0.75.
Green group 0.6; Blue group 0.5; Yellow group 0.75.
Total
✔
2.00
2.00 / 2.00
Question Explanation
By the definition of cohesiveness, we know that the correct answer should be:
Green group 3/5=0.6; Blue group 2/4=0.5; Yellow group 3/4=0.75.
Question 6
Again consider the following network:
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Consider a coordination game on this network, such that:
In each step, any player that was choosing action 0 changes to action 1 if and only if a fraction
of at least 0.6 of his or her neighbors have already chosen action 1;
Once a player chooses action 1, she keeps choosing it;
The process stops if in the last step there were no players changing from action 0 to action 1.
Start from 2 nodes (seeds) choosing action 1, and the others all choosing action 0:
Is that possible to have contagion to the whole network, i.e. all players in the network choose
action 1 eventually?
Your Answer
Score
Explanation
Yes. If the 2 seeds choosing 1 are both Yellow OR both
Green.
Yes. If the 2 seeds choosing 1 are both Blue.
Yes. There exists one such contagion process starting from
one Green seed and one Blue seed.
No. Contagion to the whole network never happens no matter
✔
2.00
which two seeds are picked.
Total
2.00 /
2.00
Question Explanation
Contagion to the whole network never happens no matter which two seeds are picked.
From question 5 we know that the cohesiveness of a mono-color group is at least 0.5 > 1-0.6.
This implies that a group never gets contagion if all its nodes were playing 0.
Therefore, there is at least one group in which all nodes stay with 0 eventually.
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Question 7
Consider a local public good game played on a ring network with 5 players, in which each player
chooses an action
(xi + ∑
j∈ N i (g)
xi ∈ [0, 10] ,
0.5
xj )
and such that player i's payoff is
− 0.5 xi .
Which of the following is a distributed and unstable equilibrium (as defined in lecture) ?
Your Answer
Score
Explanation
a
b
✔
2.00
c
d
none of them
Total
2.00 / 2.00
Question Explanation
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b is the correct answer.
Firstly, (xi
+ ∑
j∈ N i (g)
0.5
xi )
− 0.5 xi
implies x∗
= 1.
Therefore b and d are equilibria. In particular, d is a specialized and stable equilibrium, and b
is a distributed and unstable equilibrium.
Question 8
The picture depicts the Florentine Marriages (1430's) network we have seen in several lectures
[based on Padgett and Ansell’s (1993) Data, one isolated node dropped].
Consider the favor exchange model discussed in lecture with parameters δ
c = 1,
and
Thus, δp(v
= 0.5 , v = 4,
p = 0.3.
− c)/(1 − δ)
< c < 2δp(v
− c)/(1 − δ) .
For these parameters, nodes with just
one link cannot be expected to perform any favors in any equilibrium of the repeated favor
exchange game. So, they can be ignored. Similarly nodes with just one link in what remains
cannot be expected to perform any favors in any equilibrium, and you can iterate on this
process. Nodes that have two or more links can be induced to exchange favors in some
equilibria of the game.
What is the maximum number of nodes that can be induced to exchange favors in some
equilibrium of this game for these parameters?
Your Answer
Score
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Explanation
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Homework Feedback | Social and Economic Networks: Models and Analysis
4
8
10
✔
2.00
11
All the nodes
Total
2.00 / 2.00
Question Explanation
The four nodes with just one link cannot be counted upon. This also means that the Salvati
cannot be counted upon. The remaining nodes form a component in which each node has at
least two links. They can support favor exchange.
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