代写ECON322 GAME THEORY MOCK MID-TERM ACADEMIC YEAR 2024-25代做回归

ECON322 GAME THEORY

MOCK MID-TERM

ACADEMIC YEAR 2024-25

1.  Al and Bo play a public contribution game.  Each starts with $9.  Each can choose to con- tribute either $9 or $0 to a common pot. If at least one player contributes, the pot returns $16 to both players with probability n/2, where nis the number of contributors.

(a)  Suppose that Al and Bo both have preferences over monetary amounts x that are repre- sented by the utility function u(x) = x.

i. Write out the game in matrix form.	[6 marks]
ii. Do either of the players have any dominant strategies?	[3 marks]
iii. Identify any Nash equilibria to the game.	[3 marks]

(b)  Suppose that Al and Bo both have preferences over monetary amounts x that are repre- sented by the utility function u(x) = √x .

i. Write out the game in matrix form.	[7 marks]
ii. Do either of the players have any dominant strategies?	[3 marks]
iii. Identify any Nash equilibria to the game.	[3 marks]
2. Consider the following game, where x,y ∈ R:

Colin

d         e        f          g        h

3,2	1,2	2,3	3,y	2, 1
3,2	4,3	2,4	4,3	2,3
3,2	0, 1	x, 2	4, 1	1,0

(a)  Do there exist values of y such that Colin has a weakly dominant strategy?     [5 marks] (b)  Do there exist values of x such that Rohan has a strictly dominant strategy?    [5 marks]

(c)  Let x = 5 and y = 3. Solve the game for the Iterated Elimination of Strictly Dominated Strategies.                        [5 marks]

(d)  Let x = 5 and y = 3.  Solve the game for the smallest set of strategies that survive the Iterated Elimination of Weakly Dominated Strategies.

[Note: to obtain the smallest set of IEWDS, delete any weakly dominated strategies as early as possible.]           [5 marks]

(e)  Let x = 2 and y = 2. Identify any Nash equilibria.                                            [5 marks]

3.  Consider the following game between Di, Effie and Flo:

a b

e

c d

1, 1, 5	1,0,0
1, 1, 2	0,0,0

a b

f

c d

1, 1, 3	1, 0, 1
0, 2, 1	0, 1, 7

a b

g

c d

1, 1, 0	1,0,0
0,0,0	0, 1, 5

Di is the row player (her choices are a and b), Effie is the column player (her choices are c and d) and Flo is the matrix player (her choices are e, f , and g). In each cell, the payoffs are listed in the order Di, Effie, Flo.

(a)  Solve the game for the Iterated Elimination of Strictly Dominated Strategies. [6 marks]

(b)  Solve the game for the smallest set of strategies that survive the Iterated Elimination of Weakly Dominated Strategies.

[Hint: to obtain the smallest set of IEWDS, delete any weakly dominated strategies as early as possible.]        [6 marks]

(c)  Identify any weak dominant strategy equilibria.                                                 [6 marks]

(d)  Identify any Nash equilibria.                                                                               [7 marks]

4.  There are 4 students in a class and the possible grades are A, B, and C. The professor is lazy and instead of preparing a final exam tells the students:

“On the last day each of you should give me a written note, requesting a grade and your request can be either an A or a B.  If 2 or less students request an A, then I will give to each student the grade that he/she requested; otherwise I will give a C to everyone.”

Assuming that each student only cares about his/her own grade and prefers an A to a B and a B to a C (and, by transitivity, an A to a C), list all the Nash Equilibria of this game. Explain your reasoning.            [25 marks]