ECON451 Midterm 2 2021Spring PDF

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ECON451 Midterm 2 2021Spring...

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Game Theory 45100 – Spring 2021: Midterm Exam 2 The exam will last one hour and will be graded out of 45 points. No electronic devices are allowed, except for non-graphing calculators. No notes or textbooks are allowed. For the parts that have an asterisk (*), please give a detailed answer that shows all the steps of reasoning that you used to come to your answer. Question 1 of 5. [8 points out of 45] Consider the following simultaneous-move game played by Player 1 (row player) and Player 2 (column player). 2

F

1

G 4

3

A

2

1

5

1 2

5

7 4

4

4 5

3

2

D

2 1

6

C

2 3

3

B

H

1

1 6

a.) Find all pure-strategy Nash equilibria of the above game. * Which of the pure-strategy Nash equilibria involves indifference for one of the players? Would you expect that player to ever choose the strategy that they play in this equilibrium (explain)? c.) * Prove that there is a Nash equilibrium in which Player 2 chooses F, while Player 1 chooses B with probability 12 and chooses C with probability 21 . Question 2 of 5. [12 points out of 45] 2

L

1 U D

R 2

x 2

1 4

1 3

0

a.) When x = 1 in the game shown above, find all pure-strategy Nash equilibria. b.) * When x = 1, prove that there is no mixed-strategy Nash equilibrium. c.) When x = 3, find all Nash equilibria (in pure strategies or mixed strategies). d.) Suppose that x > 2 is a known parameter of the above game. Find the mixed-strategy Nash equilibrium as a function of x. (Hint: only Player 1’s mixing probabilities depend on x). Give an intuitive explanation for why, in the mixed-strategy Nash equilibrium from part (d), Player 1’s probability of choosing U falls in x.

Question 3 of 5. [8 points out of 45] Tom and Jerry work together on a project. Each individual chooses to exert effort (E) or not (N). The project succeeds if at least one individual exerts effort. The cost of effort is a known parameter x ∈ (0, 10). Jerry

E

Tom E N

N 10

10 − x 10 − x

10 − x 0

10 − x 10

0

a.) Find all pure-strategy Nash equilibria of this game. b.) Find the symmetric mixed-strategy Nash equilibrium. That is, find the equilibrium probability p∗ with which each individual puts in effort as a function of x. c.) * What happens to p∗ as x tends to 0? Explain intuitively. Question 4 of 5. [9 points out of 45] In a Cournot duopoly, Firm A chooses quantity qA and Firm B chooses quantity qB . The market price is p = 40 − qA − qB . Firm A’s cost of production is 20qA , while Firm B’s cost of production is q 2B (that is, Firm A has a linear cost of production, while Firm B has a convex cost of production). a.) * Calculate each firm’s best-response function. b.) * By substituting one best-response function into the other, show that in the Cournot-Nash equilibrium of this game, q ∗A = 40/7 and qB∗ = 60/7. c.) Calculate the profits for each firm in the equilibrium. Question 5 of 5. [8 points out of 45] Alice and Bob are in an Italian restaurant and simultaneously make demands over a four-slice pizza. Alice demands sA ∈ {0, 1, 2, 3, 4} slices and Bob demands sB ∈ {0, 1, 2, 3, 4} slices. If sA + sB ≤ 4, they each get the number of slices they demand (and the remaining slices get thrown away). If sA + sB > 4, they cannot agree how to share the pizza, and neither Alice nor Bob eats any pizza. Alice and Bob each like consuming pizza slices (the more the better). a.) * What are the pure-strategy Nash equilibria of this game? b.) Suppose now that Alice dislikes inequality and so never asks for more than half the pizza: her strategy set is now SA = {0, 1, 2}. What are the pure-strategy Nash equilibria? c.) Finally, suppose that both Alice and Bob dislike inequality, so SA = SB = {0, 1, 2}. Find all pure-strategy Nash equilibria....