Old Final Exam – Econ 0200

1. For the following game:

   A      B

A 1,1 15,4

B 4,15 7,7

a) (2 points) Which outcomes are Pareto optimal?

b) (4 points) List all NE in pure strategies

c) (1 point) If this fits the definition of one of our seven named games, name it. Otherwise, write “none”.

d) (2 points) When testing whether or not A is ESS, what is A’s fitness and B’s fitness?

e) (1 point) Is A ESS?

f) (2 points) When testing whether or not B is ESS, what is A’s fitness and B’s fitness?

g) (1 point) Is B ESS?

h) (4 points) What is the static midpoint in an evolutionary setting? Write “none” if none exists.

i) (1 point) If a static midpoint exists, is it stable? If none exists write “none”.

j) (1 point) List all dominant strategies for this game. If no strategy is dominant, write “none”.

k) (4 points) draw the evolutionary graph for this game

l) (4 points) draw the best response graph for this game

2. For the following game:

    A      B

A 12,12 5,12

B 12,5 16,16

a) (2 points) Which outcomes are Pareto optimal?

b) (4 points) List all NE in pure strategies

c) (1 point) If this fits the definition of one of our seven named games, name it. Otherwise, write “none”.

d) (2 points) When testing whether or not A is ESS, what is A’s fitness and B’s fitness?

e) (1 point) Is A ESS?

f) (2 points) When testing whether or not B is ESS, what is A’s fitness and B’s fitness?

g) (1 point) Is B ESS?

h) (4 points) What is the static midpoint in an evolutionary setting? Write “none” if none exists.

i) (1 point) If a static midpoint exists, is it stable? If none exists write “none”.

j) (1 point) List all dominant strategies for this game. If no strategy is dominant, write “none”.

k) (4 points) draw the evolutionary graph for this game

l) (4 points) draw the best response graph for this game

3. What is the present value of each of the following future payments?

a) (1 point) $1,000,000 received in 9 years if the interest rate is 7%?

b) (1 point) $100,000 received in 5 years if the interest rate is 6%?

c) (1 point) $200,000 received in 12 years if the interest rate is 2%?

d) (1 point) $500,000 received in 40 years if the interest rate is 11%?

e) (1 point) $1,000,000 received in 7 years if the interest rate is 60%?

4. For the following game:

   A    B

A 7,7 26,6

B 6,26 19,19

a) (2 points) Which outcomes are Pareto optimal?

b) (4 points) List all NE in pure strategies

c) (1 point) If this fits the definition of one of our seven named games, name it. Otherwise, write “none”.

d) (4 points) When starting from a population mix of 50-50, what is the population mix 1 generation later?

e) (1 point) Is A ESS?

f) (1 point) Is B ESS?

g) (2 points) In an infinitely repeated setting, what interest rate condition would be required to sustain cooperation if both players were to play grim trigger?

h) (2 points) In an infinitely repeated setting, what interest rate condition would be required to sustain cooperation if both players were to play tit-for-tat?

i) (2 points) In an indefinitely repeated game, if the probability of playing subsequent rounds is 40% after each round concludes, which strategies sustain cooperation – Grim Trigger, Tit-for-tat, both, or neither?

j) (4 points) draw the evolutionary graph for this game

k) (4 points) draw the best response graph for this game

5. For the following game:

   A           B

A 200,200 12,210

B 210,12 50,50

a) (2 points) In an infinitely repeated setting, what interest rate condition would be required to sustain cooperation if both players were to play grim trigger?

b) (2 points) In an infinitely repeated setting, what interest rate condition would be required to sustain cooperation if both players were to play tit-for-tat?

c) (2 points) In an indefinitely repeated game, if the probability of playing subsequent rounds is 7% after each round concludes, which strategies sustain cooperation – Grim Trigger, Tit-for-tat, both, or neither?

6. Suppose a very large number of players are trying to guess a target number T, where the target is equal to 3/4 of the average plus 10. Players can pick any number from -100 to 100.

a) (2 points) What is the Nash Equilibrium? (All players guess what number?)

b) (3 points) What are level 1 through level 3 rationalizable ranges?Old Final Exam – Econ 0200

7. For the following game:

    L    R

U 9,4 13,3

D 4,6 13,7

a) (2 points) Which outcomes are Pareto optimal?

b) (4 points) List all NE in pure strategies

c) (1 point) If this fits the definition of one of our seven named games, name it. Otherwise, write “none”.

d) (4 points) What is the mixed strategy equilibrium? If none, write “none”. If more than one, write “many”.

e) (4 points) draw the best response graph for this game

8. For the following game:

    L    R

U 12,7 3,1

D 2,2 9,13

a) (2 points) Which outcomes are Pareto optimal?

b) (4 points) List all NE in pure strategies

c) (1 point) If this fits the definition of one of our seven named games, name it. Otherwise, write “none”.

d) (4 points) What is the mixed strategy equilibrium? If none, write “none”. If more than one, write “many”.

e) (4 points) draw the best response graph for this game

9. For the following game:

a) (4 points) List all NE in pure strategies

b) (4 points) List all SPNE in pure strategies

10. For the following game:

a) (4 points) List all NE in pure strategies

b) (4 points) List all SPNE in pure strategies

c) (2 points) List all Pareto optimal outcomes

11. Suppose we have 2 firms in an industry. Demand is represented by the following equation: P = 750 – Qt and costs for firm 1 are 238/unit + 10,000 fixed cost, while costs for firm 2 are 174/unit + 20,000 fixed cost.

a) (3 points) What is the NE in the pure Bertrand model? State price, quantity and profit for each.

b) (4 points) What is each firm’s best response function under Cournot?

c) (3 points) What is each firm’s equilibrium quantity, price and profit under Cournot?

d) (3 points) What are the level 1 through level 3 rationalizable strategies for each firm under Cournot?

12. Suppose we have 3 identical firms in an industry. Demand is represented by the following equation: P = 125 – (1/100) Qt and each firm’s cost = 29 per unit + 20,000 fixed cost.

a) (1 point) What is the NE in the pure Bertrand model? State the price that each firm charges.

b) (2 points) What is firm 1’s best response function under Cournot?

c) (3 points) What is firm 1’s equilibrium quantity, price and profit under Cournot?

d) (1 point) If there were only 2 firms in the Cournot model, what is firm 1’s quantity?

e) (1 points) If there were only 1 firm in the BERTRAND model, what is that firm’s price?

f) (2 points) EXTRA CREDIT: What is the largest number of firms that can profitably coexist in this industry under the Cournot model?

g) (2 points) EXTRA CREDIT: What is the largest number of firms that can profitably coexist in this industry if the firms collude?

13. Suppose there are 400 voters, and 5 candidates: Steve Jobs, Bill Gates, Elon Musk, Jeff Bezos and Mark Zuckerberg,

87 voters prefer: Zuckerberg > Gates > Jobs > Bezos > Musk

30 voters prefer: Gates > Zuckerberg > Jobs > Bezos > Musk

73 voters prefer: Bezos > Zuckerberg > Gates > Jobs > Musk

129 voters prefer: Musk > Jobs > Gates > Bezos > Zuckerberg

16 voters prefer: Gates > Bezos > Jobs > Zuckerberg > Musk

65 voters prefer: Jobs > Bezos > Gates > Zuckerberg > Musk

For each candidate, please fill in their votes – if a candidate is not involved in the relevant round, please leave it blank (so for part b, you may only have vote totals for 2 candidates and 3 should be blank).

(3 points each)

a) Who wins a plurality vote, and what are the vote totals?

b) If you follow the instant runoff method (if no one reaches 50%, the top two from the plurality vote have a runoff), who wins, and what is the vote total for each?

c) With ranked-choice voting, eliminating the lowest and redistributing, who wins and what is the vote in the round that they win?

d) For a Borda count method, who wins and what are each candidates totals? (use 4/3/2/1/0 points for first/second/third/fourth/fifth)

e) Who is the Condorcet tournament winner, if any? If there is no winner, say “none”. What is a possible vote total in the final tournament round?

f) If using Donald Saari’s preferred method: Borda count, top two go head-to-head in a run-off, who wins and what is the vote total in that runoff election?

g) If the addition of a sixth candidate, Richard Branson, changes the Borda count relative ranking of Musk vs. Bezos, which axiom of Arrow’s Impossibility Theorem is violated?

14. (4 points) EXTRA CREDIT: Write down a non-trivial 2x2 normal form. game where the best response graph is two overlapping solid squares.