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]
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