ECOS3012 Midterm Exam S1, 2021
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
ECOS3012 Midterm Exam S1, 2021
Total: 32 points
1. (3 points) Find all pure and mixed Nash equilibria of the following game.
|
Left |
Right |
Up |
3, 4 |
9, 5 |
Down |
6, 2 |
2, 1 |
2. (4 points) Find all pure and mixed Nash equilibria of the following game.
|
L |
R |
U |
5, 4 |
2, 8 |
M |
3, 10 |
3, 12 |
D |
2, 7 |
6, 1 |
3. (8 points) A game with positive externality: n people are choosing the frequency of mask- wearing.
Suppose that there are n (version 1: n = 10, version 2: n = 5) residents in a small community. Each person chooses how often they wear facial masks in public. Let xi ∈ [0, 1] denote the frequency that resident i wears a mask. xi = 0 means that resident i never wears a mask in public; xi = 1 means that resident i always wears a mask in public.
All residents think it is uncomfortable to wear a mask. This discomfort is described by an increasing “total cost” function for mask-wearing:
C (xi ) = 2xi(2)
All residents also agree that masks protect them. The benefit of this protection that resident i gets depends on (i) how often i wears a mask themselves, and (ii) how often the other residents wear masks. In particular, for i = 1, 2, 3, ..., n if resident i chooses to wear a mask with frequency xi , then the total benefit for resident i is equal to
Bi (x1 , x2 , ..., xn ) = (x1 + x2 + x3 + . . . + xn )1/2
All residents want to maximise their total net benefit, i.e., total benefit - total cost.
(a) (3 points) Suppose that in the socially optimal, symmetric scenario, every player chooses frequency x. Calculate the value of x.
(b) (2 points) If resident 1 through n - 1 are playing the socially optimal strategy x, what is resident n’s best response?
(c) (3 points) What is the Nash equilibrium strategy for each resident?
(d) (1 point) Is the NE higher, lower, or the same compared to the SO?
4. (3 points)
(a) How many subgames does this game have?
(b) How many pure strategies does player 2 have?
(c) How many pure strategies does player 3 have?
5. (5 points)
(a) Find the subgame perfect equilibrium and correctly write down the full equilibrium strategies. (2 points)
(b) (3 points) Is there a pure-strategy Nash equilibrium in which:
Player 3 plays Y in the equilibrium outcome.
If so, please write down the full equilibrium strategies for all three players that yield this outcome. If not, please explain why it is impossible.
6. (8 points) The following game is repeated infinitely many times. Suppose that the discount factor δ can be arbitrarily close to one (but not equal to 1).
|
Left |
Right |
Up |
4, 3 |
10, 6 |
Down |
7, 1 |
3, 0 |
(a) (2 points) What is the highest average payoff that player 1 can get in any subgame perfect equilibrium?
(b) (2 points) What is the highest average payoff that player 2 can get in any subgame perfect equilibrium?
(c) (2 points) What is the infimum (greatest lower bound) of the average payoff that player
1 can get in any subgame perfect equilibrium?
(d) (2 points) What is the infimum (greatest lower bound) of the average payoff that player
2 can get in any subgame perfect equilibrium?
2022-04-13