UNIVERSITY OF SOUTHERN DENMARK
FACULTY OF BUSINESS AND SOCIAL SCIENCES
Strategy and markets
1.
2.
3.
4.
5.
Assigner: Rafael Treibich
Hand-out:
Hand-in:
All pages, incl. the front page, should include the following: Exam no.
All pages must be numbered.
Course description:
Duration: 3 hours.
Place: The test takes place at the university. The test is carried out using a personal computer to be
able to access the university's wireless network.
Internet access: Forbidden.
Handing out the assignment: Physical in the examination room.
Submission of answer: Through SDUassignment in the course page in Blackboard.
Scope: No limitation.
Aids: Aids allowed but internet access forbidden. It is not allowed to communicate with others.
Exercise 1:
Consider the following sequential-move games (the payoff of Peter is on the left, the payoff of Ida
is on the right).
(4,1)
Left
(-2,-3)
East
Ida
Up
Right
Peter
Peter
(2,4)
(-3,5)
Soft
Down
West
Ida
(-1,1)
Quick
Hard
Peter
Slow
(4,2)
a) How many decision nodes does the game consist of? How many terminal nodes does the
game consist of?
DECISION NODES=5 TERMINAL NODES=6
b) Describe and list the strategies (complete plan of actions) of Peter and Ida.
IDA 2^2=4 LS,LH,RS,RH
PETER2^3=8; UEQ, UES,UWQ,UWS, DEQ, DES, DWQ,DWS
c) Find the Rollback equilibrium (equilibrium path, strategies and payoffs).
NE(2,4)
IDA.RS
PETERUP AND US
d) Write the game in strategic form (game table).
PETER
UEQ
UES
UWQ
UWS
DEQ
DWQ
DES
DWS
LS
4,1
4,1
4,1
4,1
-3, 5
-3, 5
-3, 5
-3, 5
IDA
LH
4,1
4,1
4,1
4,1
-1, 1
-1,1
4,2
4,2
RS
-2, -3
-2, -3
2,4
2,4
-3,5
-3,5
-3,5
-3,5
RH
-2, -3
-2, -3
2,4
2,4
-1,1
-1,1
4,2
4,2
DW
e) Find the Nash equilibria of the game and identify the subgame perfect Nash equilibria.
Consider now the following game in extensive form:
(2,1)
Left
Ida
Up
(5,-1)
Right
Peter
(1,3)
Left
Down
Ida
Right
(2,2)
f) What is the rectangle around Ida’s two decision nodes? How do you interpret it?
IDAS2DECISIONAS AND IS AN INFORMATION SET
g) Is this game simultaneous or sequential? Write the game in strategic form (game table) and
find all the Nash equilibria in pure strategies.
SIMULTANEOUS BECAUSE IDA DOES NOT KNOW PETERS MOVES
PETER
L,L
2,1
1,3
UP
DOWN
L,R
2,1
2,2
NE = (U, L,L) (U,LR)
Exercise 2:
Ida considers buying a lottery ticket with two possible outcomes:
- with probability1/3, she wins 100.
- with probability 2/3, she wins 400.
Assume Ida’s utility function is given by u_I(x)=ln(1+2x).
a) Compute the expected payoff of this lottery.
1
2
EPAYOFF=3 ∗ 100 + 3 ∗ 400 = 300
RL
5,-1
1,3
RR
5,-1
2,2
b) Compute Ida’s expected utility f or the lottery.
𝐸𝑈 =
1
2
∗ ln(201) + ∗ 𝑙𝑛 (801) = 6,22502
3
3
c) Is Ida risk averse, risk neutral or risk lover?
We will do the 1º and 2º derivate. Ln(1+2x)
2
1 + 2𝑥
Second
−
4
(1 + 2𝑥)2
Risk averse
d) What is Ida’s certainty equivalent for the lottery?
6,22502=ln(1+2x)
X=253,11
e) What is Ida’s risk premium?
Rp=300 − 253,11 = 46,89
Peter also considers buying a ticket for the same lottery but his utility function is u_P(x)= x.
f) Are either Peter or Ida willing to buy the lottery ticket for a price of 260?
peter willignes to pay 6,2250.2^2= 38,751
both of them
g) Who is more risk averse: Peter or Ida?
Have the sme
Exercise 3:
In (american) football the offense can either run the ball or pass the ball, whereas the defense can
either anticipate (and prepare for) a run or anticipate (and prepare for) a pass. The objective for the
offense team is to move forward so as to bring the ball in the end zone, in which case it scores a
touchdown. The objective for the defense team is to prevent the offense team from moving forward.
Assume that the expected gains (in meters) for the two teams on any given play are as follows:
Defense
Offense
Run
Pass
Anticipate Run
-3,3
7,-7
Anticipate Pass
4,-4
-3,3
a) Show that this simultaneous game has no pure-strategy Nash equilibrium.
Defense
Anticipate Run Q
Anticipate Pass 1-Q
RunP
-3,3
4,-4
Offense
Pass 1-P
7,-7
-3,3
b) Find the unique mixed-strategy Nash equilibrium to this game.
DEFENSE
𝐸𝐷𝑅 = 3𝑃 + (1 − 𝑃) ∗ −7
𝐸𝐷𝑃 = −4𝑃 + (1 − 𝑃) ∗ 3
10𝑃 − 7 = −7𝑃 + 3
10
7
𝑃=
1−𝑃 =
17
17
𝐸𝑂𝑅 = −3𝑄 + (1 − 𝑄) ∗ 4
𝐸𝑂𝑃 = 7𝑄 + (1 − 𝑄) ∗ −3
17𝑄 = 7
7
10
𝑄=
1−𝑄 =
17
17
NE =(10/17, 7/17), (7/17, 10/17)
c) Explain why the mixture used by the offense is different from the mixture used by the
defense.
OFFENSE WOULD CHOOSE TO PLAY PASS WHEREAS DEFENCE WPULD LIKE
TO RUN
d) How many meters is the offense expected to gain per down in equilibrium? (EXPECTED
PAYOFF)
EP=1,1176
Exercise 4:
Assume there are two type of mechanics, Competent and Incompetent. The population of mechanics
consists of 40% Competent and 60% Incompetent. Mechanics can either work in licensed garages or
as free lance. Competent mechanics generate a revenue of 100 in a licensed garage, while incompetent
mechanics only generate a revenue of 40. Competent mechanics generate a revenue of 65 in free
lance, while incompetent mechanics only generate a revenue of 30 in free-lance. There is enough
demand for mechanics such that licensed garages must pay them what they expect the mechanics to
produce. However, licensed garages must hire each mechanic without observing his type and pay him
before knowing his actual output.
a) At equilibrium, which type of mechanics end up working for the licensed garages and at
which salary? Is this equilibrium efficient or not? How do you call this particular
phenomena?
Mechanics can now signal their qualification by getting a certification, but for the incompetent types
certification takes extra time and effort. Competent mechanics have to spend C months preparing for
the certification exam; incompetent ones take three times as long, 3C. Assume the payoff of each
mechanic is equal to (x) – t, where x is his yearly salary and t is the number of months spent working
for the certification. The licensed garages assumes anyone with a certification is competent and pays
them 100.
b) What minimum level of C will achieve separation between the two types?
c) Who gains and who loses compared to the initial situation?