Game Theory Assignment | Get Paper Help

It is supposed to an assignment that should be complete within 2 hours. There is a word limit of 12 pages but 6 pages are said to be able to complete all questions.

UNIVERSITY COLLEGE LONDON
EXAMINATION FOR INTERNAL STUDENTS
MODULE CODE : ECON0027
ASSESSMENT : ECON0027A6UB/ECON0027A6UA
PATTERN
MODULE NAME : Game Theory
LEVEL : Undergraduate

DATE
TIME : 12:00
This paper is suitable for candidates who attended classes for this
module in the following academic year(s):
Year

TURN OVER
SUMMER TERM 2020
24-HOUR ONLINE EXAMINATION
ECON0027: GAME THEORY
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ECON0027 1 TURN OVER
1. Sonic the hedgehog wants to purchase running shoes from Dr Robotnik. It is
commonly known that Sonic values the shoes at v and Dr Robotnik’s cost of
procuring the shoes is zero.
Dr Robotnik chooses a price and makes an oer to Sonic. Sonic can either accept
the oer or decline it. If Sonic accepts, he pays Dr. Robotnik’s price and gets his
shoes. If he declines the oer, the trade does not occur and the game ends.
(a) Formulate this situation as a game, dene the appropriate equilibrium notion
and solve for all the equilibria of this game.
(b) Suppose that in order for Dr. Robotnik to make his oer, Sonic has to travel
to his laboratory and meet him there. The cost of travel for Sonic is c > 0
(also commonly known). Find all equilibria of this game when c < v. Explain
what would change in your analysis if c v.
(c) Suppose there are a hundred clones of Dr. Robotnik, each with his own
laboratory, indexed by i 2 f1; 2; :::100g. Each clone can sell shoes to Sonic
(Sonic only needs one pair). Each Robotnik i secretly chooses a price in the
beginning of the game. After the prices are chosen, Sonic is randomly placed
in one of the hundred laboratories. At a laboratory, the price chosen by its
owner is revealed to Sonic. Sonic can either accept it, decline it and travel to
another (randomly chosen) laboratory, or decline the oer, give up the search
and end the game. If the oer is accepted, Sonic pays the price and gets the
shoes. If Sonic declines the oer and travels to another laboratory, he faces
a travel cost c. Solve for all pure strategy equilibria of this game. Explain
how competition between Robotniks aects Sonic’s equilibrium payo.
2. Two chefs are competing for a position at a restaurant called \Food for Thought”.
The value of the position to each chef is equal to v. The competition takes a form
of a contest in which one of the two chefs who bakes a bigger cake wins. In order
to bake a cake of size x > 0, chef i has to procure x kilograms of
our from the
restaurant at a price pi. The restaurant will charge a chef for the
our only if he
wins the contest. The price of
our pi for each chef i is drawn randomly from
a uniform distribution on [1; 2]. The prices for the two chefs are independent.
The chefs choose the sizes of their cakes simultaneously to maximize the expected
value of the position net of the expenses for the
our.
(a) Suppose the realized prices for the
our are publicly observed before the chefs
make their choices. Let p1 < p2 and suppose that in the event the chefs bake
the cakes of equal size, the chef who can source cheaper
our|i.e., chef 1|
wins. Find a Nash equilibrium of this game.
ECON0027 2 CONTINUED
(b) Suppose the chefs privately observe the realization of their prices: chef 1
observes only p1 and chef 2|only p2. Solve for a Bayes-Nash equilibrium of
the game.
(c) Continue assuming that the chefs privately observe the realization of their
prices. In addition, suppose that the restaurant charges chefs for the
our
independently of the contest outcome. Solve for a Bayes-Nash equilibrium
of the game.
3. Consider the following modied beer-and-quiche game
1 N 1
2
2
2
2
3,-1 x,0 y,1 7,0
5,-1 9,0 1,1 5,0
p
strong
1 p
weak
Q
B
Q
B
D N
d n
D N
d0 n0
(a) Suppose y = 3 and x = 7. Find all weak sequential equilibria of this game.
(b) Suppose y = 3. Find all values of x such that there is an equilibrium in which
both types of player 1 play B.
(c) Suppose x = 7. Find all values of y such that there is an equilibrium in which
dierent types of player 1 play dierent pure actions.
ECON0027 3 TURN OVER
4. Consider an outbreak of an infectious disease that is transmitted in a human-
to-human interaction. There is a society of N individuals. Each individual i
independently chooses a level of an economic activity ai 0. The economic
activity is valuable for the individual, but it facilitates the spread of the infection.
The payo of an individual i is
f

XN
j=1
aj
!
+ ai;
where the rst part is the expected cost of being infected which is increasing
in the total economic activity in the society
NP
i=1
ai, and the second part is the
individual benet of the economic activity. Assume that f is twice continuously
dierentiable, f0(x) > 0 and f00(x) > 0 for all x. Also, assume that f0(0) = 0
and lim
x!1
f0(x) = 1
(a) Formulate this situation as a game. Dene the equilibrium notion that is the
most appropriate. Find all symmetric equilibria in this game.
(b) Find the symmetric Pareto-ecient outcome in this game. In order to do
that, solve the social planner’s problem in which the social planner treats the
population symmetrically. Compare your answer to your ndings in question
4a. Explain the economic intuition behind this comparison.
(c) Suppose that there are two types of individuals: V (vulnerable) and I (im-
mune). The types dier by their valuation of the infection risk which is
represented by parameters V and I (V > 1 > I ). There are M individu-
als of type I and N M individuals of type V and the type of each individual
is publicly observable. The payo of agent i who is of type t 2 fV; Ig is
tf

XN
j=1
aj
!
+ ai:
Find an equilibrium in which all individuals within a given type make the same
choice (the choices may dier across types). Discuss the dierences between
this equilibrium and the one you found in question 4a.
ECON0027 4 END OF PAPER