ECON 223, Final - Exam
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ECON 223, Final - Exam
Friday, 20th December, 2019.
INSTRUCTIONS:
1. The test is two hours long (10:00 am to 12:00 pm) .
2. Write your name and student number on the test answer booklet and hand them in at the end of the test .
3. This question booklet consists of 8 pages including cover sheet . Please check and ask the examiner for a fresh one in case you have missed any page .
4. The test is worth 60 marks:
PART A: 1 question worth 8 marks . PART A is worth 8 marks . PART B: 1 question worth 7 marks . PART B is worth 7 marks . PART C: 5 questions worth 9 marks each . PART C is worth 45 marks
5. Answer ALL questions . Answer every question of each part and their respective sub- questions . Write your answers clearly on the answer booklet provided .
PART A
NOTE: Answer ALL the sub questions of the first question in PART A . Each sub question is worth 1 mark, for 8 marks in total . Write your answers clearly on the answer booklet provided .
1. Define the following:
a . Subgame Perfect Nash Equilibrium (SPNE)
b . Backward Induction
c . Preempition Game
d . Attrition
e . Subtree
f. Regular Subtree
g . Subgame
h . Grim-Trigger Strategy
(Each of these carries 1 mark, for 8 marks in total)
PART B
NOTE: Answer ALL the sub questions of the first question in PART B . Marks for
each sub question is in the brackets . This constitutes 7 marks for PART B Write
your answers clearly on the answer booklet provided .
1. Answer the following:
a . Write the differences between pure strategy and mixed strategy? (2 marks)
b . What are the properties of a Nash Equilibrium in mixed strategy? (3 marks)
c . What is randomised version of the game? (2 marks)
PART C
NOTE: Answer ALL questions . Each question is worth 9 marks . PART C is worth 45
marks . Write your answers clearly on the answer booklet provided .
For all questions, remember to show your working.
1. a . Consider the two-player game below, where Player 1 chooses the row and Player 2 chooses the column . Find all of the mixed-strategy Nash equilibria . (3 marks)
|
Left |
Right |
Top |
1,2 |
0,2 |
Bottom |
1,0 |
3,4 |
b . Consider the two-player game below, where Player 1 chooses the row and Player 2 chooses the column . Assume that players are allowed to randomize .
i) Derive players’ best-reply functions and show them on the graph . (3 marks)
ii) Find all of the mixed strategy Nash equilibria . (3 marks)
|
x |
y |
a |
3,3 |
4,2 |
b |
6,3 |
2,6 |
c |
5,3 |
3,2 |
2. Consider the game represented in the table below, where Player 1 chooses the row and Player 2 chooses the column .
|
Swerve |
Don’t Swerve |
Swerve |
0,0 |
- 1,1 |
Don’t Swerve |
T,- 1 |
-2,-2 |
a. Find all of the pure strategy Nash equilibrium profiles for this game if T < 0. (1 mark)
b. If T > 0, there is a mixed strategy Nash equilibrium strategy profile that is not a pure strategy Nash equilibrium . Find it and find the payoffs to each player in this equilibrium . (2 marks)
c. In a mixed strategy Nash equilibrium with T = 2, which player is more likely to swerve? If T = 2, which player gets the higher expected payoff in equilibrium? Which player’s equilibrium mixed strategy depends on T . (3 marks)
d. Is there anything paradoxical about the results in question a and b? If so, what?
(3 marks)
3. Consider the following figures and find their SPNE . For this particular question, you need not show the working. But the strategypairs of the answer (SPNE) must be in proper order.
a . (2 marks)
b . (2 marks)
c . Find the payoffs of those alphabets at the bottom of the tree, such that, it matches the answer - as shown by the arrow mark down the game tree . (5 marks)
4. It is the week before the Yule Ball Dance, and Victor and Ron are each contemplating whether to ask Hermione . As portrayed in figure. Victor moves first by deciding whether or not to approach Hermione . (Keep in mind that asking a girl to a dance is more frightening than a rogue bludger) . If he gets up the gumption to invite her, then Hermione decides whether or not to accept the invitation and go with Victor . After Victor (and possibly Hermione) have acted, Ron decides whether to conquer his case of nerves (perhaps Harry can trick him by making him think he’s drunk Felix Felicis) and finally tell Hermione how he feels about her (and also invite her to the dance) . However, note that his information set is such that he doesn’t know what has happened between Victor and Hermione . Ron doesn’t know whether Victor asked Hermione and, if Victor did, whether Hermione accepted . If Ron does invite Hermione and she is not going with Victor—either because Victor didn’t ask, or he did and she declined—then Hermione has to decide whether to accept Ron’s invitation . At those decision nodes for Hermione, she knows where she is in the game, since she is fully informed about what has transpired . The payoffs are specified so that Hermione would prefer to go to the dance with Ron rather than with Victor . Both Ron and Victor would like to go with Hermione, but both would rather not ask if she is unable or unwilling to accept .
a . Use the concept of subgame perfect Nash equilibrium to find out what will happen .
(4 marks)
b . For the above question on the Yule Ball Dance, now assume that, before he decides whether to ask Hermione, Ron observes whether or not Victor asked her . However, if Victor does invite Hermione, Ron does not know her answer to Victor when he decides whether to invite her himself. Write down the extensive form game and find all subgame perfect Nash equilibria. (5marks)
5. An embezzler wants to hide some stolen money . An inspector is looking for the stolen money . There are two places that the embezzler can put the money . One place is difficult to access and one is easy to access . The inspector only has time to look in one of the two places . It is more costly to hide the money in the difficult place than in the easy place and also more costly for the inspector to look in the difficult place than in the easy case . The payoffs are as follows:
• If the embezzler hides the money in the difficult place and the inspector looks in the difficult place, the payoff is 0 for the embezzler and 2 for the inspector .
• If the embezzler hides the money in the difficult place and the inspector looks in the easy place, the payoff is 2 for the embezzler and 1 for the inspector .
• If the embezzler hides the money in the easy place and the inspector looks in the difficult place, the payoff is 3 for the embezzler and 0 for the inspector .
• If the embezzler hides the money in the easy place and the inspector looks in the easy place, the payoff is 1 for the embezzler and 3 for the inspector .
Answer the following based on the above situation:
a . If the inspector believes that the embezzler randomizes in choosing his hiding place and hides the money in the hard place with probability 2/3, the inspector will maximize his expected payoff by looking in the hard place with probability 2/3 . This statement is TRUE or FALSE? Justify your answer.
b . Find a Nash equilibrium in mixed strategies for this game .
(3 marks)
(2 marks)
c . In Nash equilibrium: What is the expected payoff for the embezzler? What is the expected payoff for the inspector? What is the probability that the inspector finds the money? (4 marks)
2022-11-30