Examen 28 Marzo 2017, preguntas y respuestas PDF

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Universitat Pompeu Fabra Introduction to Game Theory Final exam March 28, 2017 Group: Surname: NIA: Name: Read the instructions carefully before touching the exam! a) The exam sheet cannot be touched until permission is given. Touching the exam sheet before the instructor allows it will be recorded ...

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Universitat Pompeu Fabra Introduction to Game Theory Final exam March 28, 2017

Group:

Surname:

NIA:

Name:

Read the instructions carefully before touching the exam! a) The exam sheet cannot be touched until permission is given. Touching the exam sheet before the instructor allows it will be recorded as an attempt to cheat. All students must start the exam at exactly the same time. b) The very first thing you have to do when the exam starts is to fill out your name, group and NIA in the appropriate place. c) All the sheets of the exam should be kept stapled together. The only additional piece of paper allowed will be the optical sheet that will be distributed at the end of the exam. d) There is only one correct answer out of the five proposed answers. For each question you mark the answer that you believe is correct, if you do not know which answer is correct then you can also not mark any of the answers. Each question counts the same number of points. Four incorrect answers are compensated by one correct answer. Be sure you understand from the instructions how to change your answer (erasing is not an option). e) Solely and exclusively you are allowed to pose a question in case you believe there is an error in the exam. In such case you should directly contact professor and not the invigilators. f) You must remain seated until the end of the exam, whose length will be 115 minutes. The optical sheet will be distributed 5 minutes prior the end. By then you must have decided your answers. g) Avoid at any time gestures that can be interpreted as attempts to cheat: include holding a hand before your eyes, looking around the room, talking, writing things on a sheet and pushing it near someone else or helping someone else look at your sheet or calculations.

Problem 1 Consider three firms from Girona. Their profits depend on the weather conditions according to the following table: Profits according to weather

Sun

Rain

Travel agency Umbrella producer

4 0

0 4

Sun glasses producer

6

1

The probability distributions of the three firms are perfectly correlated. The probability of rain is p = 1/2. Mr. Escuer is the owner of the travel agency. Due to the high risk of his business, he is considering different investment √ alternatives. His preferences with respect to risk are represented by the utility function u(x) = x Question 1 What is the minimum price at which Mr. Escuer would be willing to sell his travel agency? a)  0 b)  × 1 c) 

3 2

d)  2 e)  3 Question 2 What is the maximum price that Mr. Escuer would be willing to pay to buy the umbrella producer? a)  0 b)  1 c) 

3 2

d)  2 e)  × 3 Question 3 If Mr. Escuer could buy either the umbrella producer or the sun glasess producer for 1 monetary unit, what would be best for Mr. Escuer? a)  buy the sun glasses producer for 1 monetary unit b)  × buy the umbrella producer for 1 monetary unit c)  do nothing d)  there is a tie between doing nothing and buying the umbrella producer e)  there is a tie between buying the sun glasses producer and the umbrella producer

2

Problem 2 An investor has an initial wealth ω = 1. If he invests all of it in a risky asset he can obtain 3 with probability 12 and 0 otherwise. He is also considering the possibility of investing only a certain proportion α ∈ [0, 1] of his wealth in the risky asset. The utility function representing his preferences is given by u(x) = ln(x + 1) Question 4 If α is the proportion of the wealth invested in the risky asset, what is the optimal value of α? a)  α = 0 b)  α =

1 4

c)  α =

1 3

d)  α =

3 5

e)  × α=

1 2

Problem 3 Consider the following game in normal form 2 1\ a m

e 2,2 3,1

c 0,2 1,0

d 1,0 3,1

b

0,0

0,4

2,1

where the first payoff is that of player 1. Question 5 Mark player 1’s dominant strategies a)  Only a b)  × Only m c)  Only b d)  All e)  There is no dominant strategy Question 6 Which are player 2’s weakly dominated strategies? a)  Only e b)  Only c c)  Only d d)  All e)  × Player 2 does not have a weakly dominated strategy

3

Question 7 Select player 2’s best reply to player 1’s mixed strategy (p1 , p2 , p3 ) =   a)  13 , 13 , 13

1

3

, 13 , 13



b)  × (0, 1, 0)   c)  12 , 0, 21   d)  14 , 12 , 14 e)  (1, 0, 0) Question 8 Which of the following is a mixed strategy Nash equilibrium profile?     a)  13 , 31 , 13 , 13 , 31 , 13     b)  14 , 21 , 14 , 13 , 31 , 13     c)  14 , 21 , 14 , 12 , 0, 21    d)  (1, 0, 0)) , 13 , 13 , 13    e)  × (0, 1, 0) , 23 , 0, 31

Problem 4 Consider the following normal form game where the first payment is that of player 1 1\

2

a b

e

c

d

3,3 0,1

5,1 6,1

1,2 2,4

Question 9 Which is the secure mixed strategy of player 1? a)  p = 1 b)  p =

1 4

c)  p =

1 3

d)  × p=

1 2

e)  p =

3 4

Question 10 Which is the secure mixed strategy of player 2?   a)  (q1 , q2 , q3 ) = 13 , 13 , 13 b)  (q1 , q2 , q3 ) = (0, 0, 1)   c)  (q1 , q2 , q3 ) = 13 , 16 , 12   d)  (q1 , q2 , q3 ) = 14 , 0, 43   e)  × (q1 , q2 , q3 ) = 12 , 0, 12

4

Problem 5 Consider the following extensive form game

m b

0, 0

d

2, 2

e

1, 3

d

3, 1

B

a

A

e

1, 4

B

Question 11 Which of the following is a subgame perfect Nash equilibrium profile? a)  (a, d ) b)  (a, (d, d)) c)  × (a, (d, e)) d)  (b, (d, e)) e)  (m) Represent this game in a normal form and respond to the following questions: Question 12 How many Nash equilibria in pure strategies does this game have? a)  5 b)  1 c)  2 d)  × 3 e)  4 Question 13 This game has an equilibrium achieved by the iterated elimination of weakly dominated strategies (IEWDS) with payoffs a)  × (2, 2) b)  (1, 4) c)  (3, 1) d)  (1, 3) e)  There is no IEWDS equilibrium

5

Question 14 Which of the following strategic profiles is a Nash equilibrium that is not subgame perfect? a)  (m, (e, d)) b)  (a, (d, d)) c)  (a, (d, e)) d)  × (b, (e, e)) e)  (a, d ) Question 15 Suppose that the original game is played simultaneously, so that it becomes an imperfect information game. No player can see what the other is doing. Which statement is correct about the new game? a)  All Nash equilibria in pure strategies are strict. b)  × There is a Nash equilibrium that is Pareto optimal c)  There is only one Nash equilibrium in pure strategies. d)  There is no pure strategy Nash equilibrium. e)  There is a IEDS equilibrium.

Problem 6 Consider the following game of two players. Each player chooses a real number between 0 and 4, without knowing the other player’s choice. In other words, player 1 chooses a real number x such that x ∈ [0, 4] and player 2 chooses a real number y ∈ [0, 4]. The payoff functions are Π1 (x, y) = x(4 + y − x) Π2 (x, y) =

y(4 − y)

Question 16 Which is the best reply of player 1 if player 2 chooses y = 0? Select the interval containing the correct value. a)  0 ≤ x < b) 

1 2

1 2

≤x...