ECON 212
THE UNIVERSITY OF AUCKLAND
SEMESTER ONE, 2020
Campus: City
ECONOMICS
Game Theory
FINAL ASSESSMENT
(Designed to be completed in: THREE hours)
NOTE: There are FIVE questions. Answer ALL FIVE questions. Each question
is worth 20 marks. Please allocate your time appropriately. Good luck.
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ECON 212
Academic honesty declaration
By completing this assessment, I agree to the following declaration:
I understand the University expects all students to complete coursework with
integrity and honesty. I promise to complete all online assessment with the same
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practice or academic misconduct will be followed up and may result in disciplinary
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As a member of the University’s student body, I will complete this assessment
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• I declare that this assessment is my own work, except where acknowledged
appropriately (e.g., use of referencing).
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lecturer or tutor) in completing this assessment.
• I declare that this work has not been submitted for academic credit in another
University of Auckland course, or elsewhere.
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within the assessment period.
• I will not reproduce the content of this assessment anywhere in any form.
Submission Instructions
You answer should be submitted on CANVAS in the form af a single pdf file.
Please ensure that your answer clearly labels on each page the Question number
that you are answering on that page. Please start each Question on a new page.
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ECON 212
Question 1.
(20 marks)
This question deals with two player constant sum games.
(a) Consider the following two person zero sum game.
Player 2
P Q R S
0 4 3 4
2 7 2 5
8 6 1 2
7 9 0 3
A
B
Player 1
C
D
What is the maxmin pure strategy for Player 1 and what payoff can Player 1
guarantee using only pure strategies?
(3 marks)
(b) In the same game, what is the minmax pure strategy for Player 2 and what
payoff can Player 2 guarantee using only pure strategies?
(3 marks)
(c) In the same game, without doing any further calculations, what can you say
about the value of the game when mixed strategies are allowed?
(4 marks)
(d) Consider now the following two person zero sum game.
Player 1
T
B
A
8
2
Player 2
B C D
5 1 4
3 5 4
E
0
8
Find the optimal (mixed) strategy for Player 1 and the (mixed strategy) value
of this game.
(6 marks)
(e) Without further calculation what can you say about the optimal mixed strategy for Player 2?
(4 marks)
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ECON 212
Question 2.
(20 marks)
This question deals with the equilibria of two person games where each player has
two or three strategies.
(a) Consider the following two person game.
Player 2
L
R
T
0, 1 7, 4
B
2, 5 3, 2
Player 1
Graph the best reply correspondences for each player and mark on the graph
all the mixed strategy equilibria, including the ones that are equivalent to
pure strategy equilibria. Label your diagram completely, labelling the axes,
any relevant values on each axis, and clearly indicating which is the graph of
the best reply correspondence of Player 1 and which of Player 2. (10 marks)
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ECON 212
(b) Consider a two player game that has best replies as indicated in the following
diagram.
BR1
(0, 0, 1) ∼ R
(0, 31 , 23 )
M
( 45 , 0, 51 )
T
B
( 12 , 14 , 41 )
( 12 , 21 , 0)
(1, 0, 0) ∼ L
(0, 1, 0) ∼ C
Σ2
BR2
(0, 0, 1) ∼ B
( 14 , 0, 43 ) R (0, 14 , 43 )
C
( 34 , 0, 41 )
L
(1, 0, 0) ∼ T
(0, 34 , 14 )
(0, 1, 0) ∼ M
Σ1
Use the best reply diagrams to find all mixed strategy equilibria. Carefully
explain the process of how you found the equilibria and explain why the process
does find all equilibria. You are not required to copy the diagrams into your
answer unless you add some construction or labelling to them and refer to that
addition in your answer.
(10 marks)
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ECON 212
Question 3.
(20 marks)
Consider the following extensive form game. [Hint: Take care. While the “shape”
is the same as games you have seen before there may be some quite significant
differences.]
4, 1
1, 4
1, 4
4, 1
L
R
L
R
2
X
Y
1
2, 3
3, 2
W
E
1
T
B
1
(a) List all the pure strategies of each player.
(2 marks)
(b) Find all the subgames of the original game.
(2 marks)
(c) Find all the subgame perfect equilibria of this game, including those in which
there are mixed equilibria in the subgame(s).
(8 marks)
(d) Find the associated normal form of this game. If there are pure strategies
that are Kuhn-equivalent, keep only one representative pure strategy for each
set of equivalent strategies. Find all the pure strategy Nash equilibria of this
game and identify which are Kuhn equivalent to subgame perfect equilibria.
(8 marks)
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ECON 212
Question 4.
(20 marks)
Consider the following four player coalitional game.
v(123) = 10
v(12) = v(13) = 4
v(23) = 3
v(1) = v(2) = v(3) = 1.
(a) Are any players substitutes for each other? If so, which players?
(5 marks)
(b) What is the core of this game? Is there any allocation in the core in which
Player 1 gets 1? If so, give one such allocation. If not, explain why not.
(5 marks)
(c) What is the payoff to Player 1 in the Shapley value?
(5 marks)
(d) What is the payoff to each player in the Shapley value?
(5 marks)
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ECON 212
Question 5.
(20 marks)
(a) Consider the following Housing Assignment Game
1
2
3
4
5
6
7
8
h3
h2
h1
h6
h5
h4
h7
h8
h1
h6
h7
h2
h5
h8
h3
h4
h6
h1
h4
h2
h8
h3
h5
h7
h6
h1
h5
h2
h7
h3
h8
h4
h7
h2
h8
h4
h1
h5
h3
h6
h4
h8
h2
h6
h5
h3
h7
h1
h1
h4
h3
h8
h5
h2
h7
h6
h6
h4
h2
h7
h8
h3
h1
h5
Use Gale’s Top Trading Cycle algorithm to find the unique allocation in the
core, carefully explaining how the algorithm works and how you apply it to
find the answer.
(10 marks)
(b) Consider the following Marriage Market.
m1
m2
m3
m4
w1
w2
w3
w4
w2
w1
w4
w3
w1
w4
w3
w2
w1
w2
w3
w4
w4
w1
w2
w3
m1
m3
m2
m4
m3
m1
m4
m2
m1
m2
m3
m4
m1
m4
m3
m2
Use the Gale-Shapley deferred acceptance algorithm to find the stable matching most preferred by the women, carefully explaining how the algorithm works
and how you apply it to find the answer.
(10 marks)
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