UNIVERSITY OF SOUTHERN DENMARK
FACULTY OF BUSINESS AND SOCIAL SCIENCES
Strategy and markets
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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).
Left
(2,1)
Ida
Up
Right
Peter
(-2,3)
(5,1)
Left
Down
Ida
Quick
(-1,1)
Right
Peter
Slow
(1,4)
a) How many decision nodes does the game consist of? How many terminal nodes does the
game consist of?
Decision nodes: 4
Terminal nodes: 5
b) Describe and list the strategies (complete plan of actions) of Peter and Ida.
Peter: UQ, US, DQ, DS
Ida: LL, LR, RL, RR
c) Find the Rollback equilibrium.
Rollback equilibrium: DRS
The NE is: (1,4)
Peters equilibrium strategies are 𝑆𝑝 = 𝐷𝑆
Idas equilibrium strategies are 𝑆𝑖 = 𝑅𝑅
d) Write the game in strategic form (game table).
Ida
LL
Peter
UQ
US
DQ
DS
LR
RL
2.1
2.1
5.1
5.1
2.1
2.1
-1.1
1.4
RR
-2.3
-2.3
5.1
5.1
-2.3
-2.3
-1.1
1.4
e) Find the Nash equilibria of the game and identify the subgame perfect Nash equilibria.
There is no dominant strategy, so we use best response
We find following NE in pure strategy:
𝑁𝐸 = [(𝐷𝑄, 𝐿𝐿)(𝐷𝑄, 𝑅𝐿)(𝐷𝑆, 𝑅𝑅)]
We know that the only subgameperfect equilibrium is (DS,RR) = (1.4)
Here both players use their best strategies to get the optimal payoff.
Exercise 2:
Consider a lottery with three possible events:
- with probability 0.4, you receive 200.
- with probability 0.1, you receive 400.
- With probability 0.5, you receive 500.
Assume the individual’s utility function is given by u(x)=x.
1) Compute the expected payoff of this lottery.
𝐸𝑃 = 0.4 ∗ 200 + 0.1 ∗ 400 + 0.5 ∗ 500
𝐸𝑃 = 370
The expected payoff is 370.
2) Compute the individual’s expected utility for the lottery?
𝑈(𝑥) = 0.4 ∗ √200 + 0.1 ∗ √400 + 0.5 ∗ √500
𝑈(𝑥) = 18.837
The expected utility is 18.837
3) Is the individual risk averse, risk neutral or risk lover? Explain.
There is 3 ways to find out.
1. plot in the function  The person is risk averse, because of the √𝑥 function. the graph is concav
2. We find the first derivative of the utility function u(x)=x.:
𝑢′ (𝑥) = 𝑥 −0.5
Since the power have a negative sign the person is risk averse
3. We can take a look at expected payoff and the expected utility:
Eu<Ep
18,84<370
Then the person is risk avers. If EP havde været mindre eller lig med 18,84 så ikke tale om risk
averse
4) For which price is the individual willing to buy this lottery?
We find the amount of money this person is willing to pay for this lottery ticked by putting his
utility gained from the possibly outcomes of lottery equal to his utility function and isolating.
√𝑥 = 18.837
𝑥 = 18.8372
𝑥 = 354.832
5) Explain what is the risk premium and compute it.
A risk premium, for a risk averse person, describes how much this person would give to get out of
the game. They have already said yes to buy, so you need to get rid of the risk of being in the game.
Risk premium is the number of money, which the expected payoff must be higher than for the
person to take the risk.

To find the RP(risk premium):
𝑅𝑃 = 𝐸𝑃 − 𝑊(𝐸𝑈)
𝑅𝑃 = 370 − 354.832
𝑅𝑃 = 15.17
Therefor the person is willing to pay 15.17 to get rid of the risk.
Exercise 3:
Consider a village where people either have blue hair or green hair, in equal proportion. When they
go fishing and meet someone with hair from a different color, they can choose between:
- A peaceful strategy, consisting in sharing their catch if the other person is also peaceful, and
giving the other person their catch if the other person is aggressive.
- An aggressive strategy, consisting in taking the other person’s catch if she is peaceful, and
fighting if the other person is also aggressive.
In case of a fight, all fish is gone, and the two person end up injuring each other. This game is
modeled as follows:
Blue hair player
Peaceful
Aggressive
Green hair player
Peaceful
Aggressive
0.5,0.5
0,1
1,0
-1,-1
a) Find the Nash equilibria in pure strategies.
There are no dominant strategies, so we use best response.
We find following 2 NE:
𝑁𝐸 = [(𝐴𝑔𝑟𝑒𝑠𝑠𝑖𝑣𝑒, 𝑃𝑒𝑎𝑐𝑒𝑓𝑢𝑙)(𝑃𝑒𝑎𝑐𝑒𝑓𝑢𝑙, 𝐴𝑔𝑟𝑒𝑠𝑠𝑖𝑣𝑒)]
b) If the Blue hair player chooses to be peaceful with probability p, what is the Green hair
players’s expected payoff from being Peaceful, and from being aggressive?
First, we write expected payoff for Green hair. Both when he chooses P and A:
𝔼𝐺𝐻 (𝑃) = 𝑝 · 0.5 + (1 − 𝑝) · 0
𝔼𝐺𝐻 (𝐴) = 𝑝 · 1 + (1 − 𝑝) · (−1)
Now, we need to find the point, where 𝔼𝐺𝐻 (𝑃) = 𝔼𝐺𝐻 (𝐴) because here Green hair is indifferent
between choosing P or A:
𝑝 · 0.5 + (1 − 𝑝) · 0 = 𝑝 · 1 + (1 − 𝑝) · (−1)
𝑝=
1
1.5
We now insert the p in our Expected payoff functions for Green hair
Peaceful:
𝔼𝐺𝐻 (𝑃) = 𝑝 · 0.5 + (1 − 𝑝) · 0
𝔼𝐺𝐻 (𝑃) =
1
1
∗ 0.5 + (1 −
)∗0
1.5
1.5
𝔼𝐺𝐻 (𝑃) = 0.3333333
Aggressive:
𝔼𝐺𝐻 (𝐴) = 𝑝 · 1 + (1 − 𝑝) · (−1)
𝔼𝐺𝐻 (𝐴) =
1
1
· 1 + (1 −
) · (−1)
1.5
1.5
𝔼𝐺𝐻 (𝐴) = 0.3333333
c) If the Green hair player chooses to be peaceful with probability q, what is the Blue hair
players’s expected payoff from being Peaceful, and from being aggressive?
Because it’s a symmetric game, they will have the same payoff
𝔼𝐵𝐻 (𝑃) = 0.333333
𝔼𝐵𝐻 (𝐴) = 0.333333
d) Find the Nash equilibrium in mixed strategies.
For Green Hair.
1
= 0.6666666
1.5
1
1−𝑝 = 1−
= 0.3333333
1.5
𝑝=
For Blue Hair
1
= 0.6666666
1.5
1
1−𝑞 =1−
= 0.33333333
1.5
𝑞=
Can be written in several ways:
2 1 2 1
𝑁𝐸 = [( , ) ( , )]
3 3 3 3
𝑁𝐸 = [(𝑃, 1 − 𝑃)(𝑃, 1 − 𝑃)]
1
1
1
1
𝑁𝐸 = [( , 1 −
)( ,1 −
)]
1.5
1.5 1.5
1.5
e) Compute the average (expected) payoff for blue hair players and for green hair players at
this equilibrium.
For Green Hair:
𝔼𝑮𝑯 =
1
1
· 𝟎. 𝟓 + (𝟏 −
) · 𝟎 = 𝟎. 𝟑𝟑𝟑𝟑𝟑𝟑
1.5
1.5
𝔼𝑩𝑯 =
1
1
· 𝟎. 𝟓 + (𝟏 −
) · 𝟎 = 𝟎. 𝟑𝟑𝟑𝟑𝟑𝟑
1.5
1.5
For Blue Hair
f) Compute the average payoff if all players always choose the peaceful strategy. Compare to
the equilibrium outcome.
𝔼𝑨𝒗𝒆𝒓𝒂𝒈𝒆 = 0.5 ∗ 0.5 + (1 − 0.5) ∗ 0.5
𝔼𝑨𝒗𝒆𝒓𝒂𝒈𝒆 = 𝟎. 𝟓
Here the players will have a payoff of 0,5 when they cooperate, which is higher payoff than 0,333
when they don’t cooperate.
g) What type of game is this?
It’s a chicken game.
Exercise 4:
Consider the two player version of the Rock-Paper-Scissors game where each player chooses
simultaneously to show (with his hand) either a rock, a paper or a scissor. Rock beats scissor,
scissor beats paper, paper beats rock. The payoffs are as follows:
- The loser pays 1 to the winner if the winner won using paper or rock.
- The loser pays 3 to the winner if the winner won using scissor.
- Both get 0 in case of a tie.
a) Write the game table for this simultaneous move game.
Player 2
Player 1
Rock
Paper
Scissor
Rock
0,0
1,-1
-1.1
Paper
-1.1
0,0
3,-3
Scissor
1,-1
-3.3
0,0
b) Find the Nash equilibrium in pure strategies.
There is no dominant strategi, so we use best response. There is no Nash equilibrium in this game
using pure strategy. It’s a zero-sum game.
c) If p_R is the probability of player 1 choosing Rock, and p_P is the probability of player 1
choosing Paper, what is player 1’s probability of choosing Scissor?
The probabilty of choosing scissor would be following:
𝑝𝑠 = (1 − 𝑝𝑅 − 𝑝𝑃 )
d) Compute the expected payoffs of player 2 playing Rock, Paper and Scissors when player 1
chooses Rock with probability p_R and Paper with probability p_P.
Remember:
Player 2
Player 1
Rock p_R
Paper p_P
Scissor (1-p_R-p_P)
Rock q_R
0,0
1,-1
-1.1
Paper q_P
-1.1
0,0
3,-3
Scissor (1-q_R-q_P)
1,-1
-3.3
0,0
Start by finding the expected payoffs of player 2 for all her three strategies:
𝐸𝑝2 (𝑟) = 𝑃𝑅 · 0 + 𝑃𝑃 · (−1) + (1 − 𝑃𝑅 − 𝑃𝑃 ) · 1 = 1 − 2𝑃𝑃 − 1𝑃𝑅
𝐸𝑝2 (𝑝) = 𝑃𝑅 · 1 + 𝑃𝑃 · 0 + (1 − 𝑃𝑅 − 𝑃𝑃 ) · (−3) = 3𝑃𝑃 + 4𝑃𝑅 − 3
𝐸𝑝2 (𝑆) = 𝑃𝑅 · (−1) + 𝑃𝑃 · 3 + (1 − 𝑅𝑃 + 𝑃𝑃 ) · 0 = 3𝑃𝑃 − 1𝑃𝑅
We want to find 𝑃𝑅 , so we:
𝐸𝑝2 (𝑠) = 𝐸𝑝2 (𝑝)
3𝑃𝑃 − 𝑃𝑅 = 3𝑃𝑃 + 4𝑃𝑅 − 3
3 = 5𝑃𝑅
𝑃𝑅 =
3
5
Now, we want to find 𝑃𝑃 , so we:
𝐸𝑝2 (𝑅) = 𝑃𝑝2 (𝑃)
1 − 2𝑃𝑃 − 1𝑃𝑅 = 3𝑃𝑃 + 4𝑃𝑅 − 3
We insert 𝑃𝑅 and isolate for 𝑃𝑃 :
1 − 2𝑃𝑃 − 1 ∗
3
3
= 3𝑃𝑃 + 4 ∗ − 3
5
5
3/5 = 0.6
1 − 2𝑃𝑃 − 1 ∗ 0.6 = 3𝑃𝑃 + 4 ∗ 0.6 − 3
1 − 2𝑃𝑃 − 0.6 = 3𝑃𝑃 + 2.4 − 3
1 − 0.6 − 2.4 + 3 = 5𝑃𝑃
1 = 5𝑃𝑃
𝑃𝑃 =
1
5
Now, we insert 𝑃𝑅 and 𝑃𝑃 in the following:
𝑃𝑆 = 1 − 𝑃𝑅 − 𝑃𝑃 = 1 −
3 1 1
− =
5 5 5
So there are 60% (3/5=0.6) probabillity that Player 2 plays rock, 20% (1/5=0.2) probabillity of
player 2 plays Paper and 20% (1/5=0.2) probabillity that player 2 plays scissor.
𝑃𝑅 = 0.6
𝑃𝑃 = 0.2
𝑃𝑆 = 0.2
e) Find the Nash equilibrium in mixed strategies.
It’s a symmetric gametable, therefore the Nash Equalibrium is the same for both:
3 1 1
3 1 1
𝑁𝐸 = [( , , ) , ( , , )]
5 5 5
5 5 5
Exercise 5:
An economy has two types of jobs, Good and Bad, and two types of workers, Qualified and
Unqualified. The population consists of 60% Qualified and 40% Unqualified. In a Bad job, either
type of worker produces an amount worth 10. In a Good job, a Qualified worker produces an
amount worth 100, and an Unqualified worker produces 0. There is enough demand for workers
that for each type of job, companies must pay what they expect the appointee to produce.
Companies must hire each worker without observing his type and pay him before knowing his
actual output. But Qualified workers can signal their qualification by getting educated. For a
Qualified worker, the cost of getting educated to level n is n2/2, whereas for an Unqualified worker,
it is n2. These costs are measured in the same units as output, and n must be an integer.
a) What is the minimum level of n that will achieve separation?
Price for each education:
High skilled: 𝑛2 /2
Low skilled: 𝑛2
To achieve the minimum level of education to stop low skilled workers from taking the high skilled
jobs, will be found, when the payment for education is bigger, than the expected payoff from jobs.
We find it when it cannot be profitable for the lowskilled workers to take education by solving the
following inequality:
2
10 ≥ 100−n
𝑛2 ≥ 100 − 10
𝑛2 ≥ 90
𝑛 ≥ √90
𝑛 = 9.4868
Therefore, if it is required to take at least 10 educations to get a high skilled job, it is no longer
worth it. We test, if it is worth it for the high skilled workers to take 8 educations:
10 ≥ 100 − 𝑛2 /2
𝑛2 /2 ≥ 90
𝑛2 ≥ 90 ∗ 2
𝑛 ≥ √180
𝑛 ≥ 13.4164
As we can see, the high skilled workers would take up to 13 educations, before it would no longer
be worth it. Therefore, we will only have the separation if
10<n<13
b) Now suppose the signal is made unavailable. Which kind of jobs will be filled by which
kinds of workers and at what wages? Who will gain and who will lose from this change?
We have asymmetry of information. Because the lowskilled workers know, that they are
lowskilled, and the highskilled workers know, that they are high skilled, but the companies don’t
know who are high or low skilled. This is a problem because the lowskilled workers will lie, and
tell that they are high skilled (more salary - they are expected to produce 100).
𝐸𝑃𝐺𝑜𝑜𝑑 = 100 ∗ 0.6 + 0 ∗ 0.4
Expected productivity Good:
𝐸𝑃𝐺𝑜𝑜𝑑 = 100 ∗ 0.6 + 0 ∗ 0.4
𝐸𝑃𝐺𝑜𝑜𝑑 = 60
Expected productivity Bad:
𝐸𝑃𝐵𝑎𝑑 = 10 ∗ 0.6 + 10 ∗ 0.4
𝐸𝑃𝐵𝑎𝑑 = 10
The companies is willing to pay 60 for a good job and 10 for a bad job
Since every worker will have 60 over 10, and therefore the good job will be filled fastest.
The qualified workers will now have a profit of 100, and the unqualified workers will now have
2
profit of 10. Qualified workers will have the cost of n^2 for education, and example: 10 =100 and
therefore, it is the qualified workers who pays for the education/seperation system
Both the high skilled workers and the companies have gained in complete information, because the
high skilled workers has increased their expected salary from 40 to 100, and the companies
increased their total production.
The low skilled lost, because they went from an expected payoff of 40 to 10.