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ADVANCED MICROECONOMICS I HOMEWORK 4
发布时间：2023-01-05
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ADVANCED MICROECONOMICS I
HOMEWORK 4
Exercise 1. Consider the following game
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|
L |
C |
R |
|
T |
2,0 |
1,1 |
4,2 |
|
M |
3,4 |
1,2 |
2,3 |
|
B |
1,3 |
0,2 |
3,0 |
1. What strategies survive iterated elimination of strictly dominated strategies?
2. Can you still get the same result if some payoffs are affected by a common factor as follows
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|
L |
C |
R |
|
T |
2,0 |
1-x,1 |
4,2 |
|
M |
3,4 |
1,2-2x |
2,3 |
|
B |
1+2x,3 |
0,2 |
3,0 |
(a) Determine all the values of x (the range of x) such that B is still strictly dominated strategy.
(b) Determine all the values of x such that you still get the same result as in the original game (with the precise payoffs) after iterated elimination of strictly dominated strategies.
Exercise 2 (The odd couple). Thomas and Frank share an apartment. They have different views on cleanliness and, hence, on whether or not they would be willing to put in the hours of work necessary to clean the apartment. Suppose that it takes at least 12 hours of work (per week) to keep the apartment clean, 9 hours to make it livable, and anything less than 9 leaves the apartment filthy. Suppose that each person can devote either 3, 6 or 9 hours to cleaning.
Suppose Thomas and Frank agree that a livable apartment is worth 2 on the utility index. They disagree on the value of a clean apartment-Thomas thinks it is worth 10 utility units, while Frank thinks it is only worth 5. They also disagree on the unpleasantness of a filthy apartment-Thomas thinks it is worth - 10 utility units, while Frank thinks it is only worth -5. Each person’s payoff is the utility from the apartment minus the number of hours worked: for example, a clean apartment on which each person has worked 6 hours gives Thomas a payoff of 4, while it gives Frank a payoff of - 1.
1. Give the normal-form representation of the game.
2. Find the game’s outcome by applying "iterated elimination of weakly dominated strategies“ .
3. Is the outcome from (2) the unique Nash equilibrium of this game?
Exercise 3. Find all Nash equilibria in the following games:
2
R S
|
0, 0 |
1,- 1 |
- 1,1 |
|
- 1,1 |
0,0 |
1,- 1 |
|
1,- 1 |
- 1,1 |
0,0 |
Table 1: The rock, scissors, paper game
Owner
I NI
|
0, -h |
w, -w |
|
|
w-g,v-w |
Table 2: Inspection game
Exercise 4. Please show that in G + }I; {Si , ui(
1 (, if iterated elimination of strictly dominated strategies eliminates all but the strategy vector s* + }s1(*) , . . . , sn(*)(, then s* is the unique NE of the game.
Exercise 5. Two people use the following procedure to share two desirable identical indivisible objects. One of them (player 1) proposes an allocation among the following possibilities }2, 0(, }1, 1(, or }0, 2(, which the other player (player 2) either accepts or rejects. In the event of rejection, neither person receives either of the objects. Each person cares only about the number of objects he obtains.
1. Please Write down the extensive form of the game.
2. Please write down the strategic-form of the game (as a bimatrix)
3. Find pure-strategy subgame-perfect equilibria of this game.
4. Find subgame perfect Nash equilibria.
Exercise 6. Consider a market with three firms and the inverse demand function p}X ( + 1 - X , where X + x1 ) x2 ) x3 and xi}i + 1, 2, 3( is the quantity produced by firm i. Each firm has a constant marginal cost of production, c, where 1 > c > 0, and no fixed cost. The firms choose their quantities as follows:
1. First firm 1 chooses its quantity x1 and announces it to its two opponents. Then firm 2 chooses its quantity x2 and announces it to firm 3. Finally, firm 3 chooses its quantity x3 .
2. First firm 1 chooses its quantity x1 and announces it to its two opponents. Then firms 2 and 3 choose simultaneously their quantities x2 and x3 .
3. First firms 1 and 2 choose simultaneously their quantities x1 and x2: Then firm 3 chooses its quantity x3 without knowing the decisions of the two other firms.
4. First firms 1 and 2 choose simultaneously their quantities x1 and x2 and announce them to firm 3. Then firm 3 chooses its quantity x3 . What are the subgame-perfect outcomes of these games?
