ECON 44115 Durham University Programming Questions

EXAMINATION PAPERexamination session
year
exam code
May/June
2022
ECON44115-WE01
title
GAME THEORY
Release date/time:
13 May 2022: 09:30 hours (UK time)
Lastest submission date/time:
14 May 2022: 09:30 hours (UK time)
Form of exam:
Take home exam
Duration:
2 hours
Word limit:
3 000 words
Additional material provided:
None
Expected form of submission:
PDF of handwritten work
Your uploaded file should be named with
your anonymous ID and the exam code, e.g.,
Z0123456 ECON44115-WE01
Submission method:
Blackboard/Turnitin
Instructions to candidates:
Answer TWO questions.
Each question has equal weight. Each part of each
question has equal weight unless otherwise stated.
page number
exam code
ECON44115-WE01
2 of 3
1. Consider two firms, each of which has one job opening. The firms offer different wages.
Firm 1 offers w1 and firm 2 offers w2 where 0 < 0.5w1 < w2 < 2w1 . Two workers want these jobs, but each worker can apply to only one of the two firms. The workers simultaneously decide which firm they will apply to. If each of the two workers applies to a different firm, both get hired. If both apply to the same firm, each has a 50% chance of getting the job, and a 50% chance of unemployment (with a wage of zero). Solve for all of the Nash equilibria of this game. The normal form is shown below. In each cell, worker 1’s (expected) payoff is shown first. Worker 2 Worker 1 apply to firm 1 apply to firm 2 apply to firm 1 0.5w1 , 0.5w1 w1 , w2 apply to firm 2 w2 , w1 0.5w2 , 0.5w2 2. Consider the following simultaneous move stage game. In each cell, player 1’s payoff is shown first. Player 2 Player 1 L C R T 3,1 0,0 5,0 M 2,1 1,2 3,1 B 1,2 0,1 4,4 This game is played twice, without discounting of the second stage payoffs. Both players observe the outcome of the first stage prior to making their second stage choices. Determine whether or not (4, 4) can be the first stage payoffs from a pure strategy subgame perfect Nash equilibrium. Explain your answer carefully. 3. Two firms engage in Cournot competition in an industry with inverse demand curve P = a − Q where a is a parameter and Q = q1 + q2 . The total cost function for firm i is Ci (qi ) = cqi . The value of the demand parameter a is uncertain: a = aH with probability θ and a = aL with probability 1 − θ, where aH > aL > c > 0.
continued
page number
exam code
ECON44115-WE01
3 of 3
The information structure is as follows. Firm 1 knows whether a equals aH or aL ,
while firm 2 does not. The firms choose their quantities qi > 0 simultaneously. The
structure of the game is common knowledge to both firms. Solve for the Bayesian
Nash equilibrium of this game, and state any parameter restrictions that are needed
to ensure all quantities are strictly positive in equilibrium.
4. A signaling game is illustrated below in which nature selects type t1 or t2 with equal
probability. The sender observes nature’s selection and sends the signal L or R. The
receiver observes the signal, but does not observe nature’s selection, and chooses u
or d. At each terminal node, the sender’s payoff is shown first. Provide a complete
description of all pure strategy perfect Bayesian equilibria of this game.
1, 1 s
@
@
u
@
@
@s
L
sender
t1
s
R
@s
0, 0
s
1, 0
@s
1, 2
s
@
@
@
d @
s
prob = 0.5
s nature
receiver
0, 0 s
@
@
2, 2
u
d
2, 0
s
receiver
prob = 0.5
u
u
@
@
@s
s
L
d
0, 1
t2
sender
R
s
END OF EXAMINATION
s
@
@
@
d @