# Game Theory and Social Sciences POLS 2125

Game Theory and Social Sciences POLS 2125

Writing Time: ONE DAY

You must attempt to answer all questions.

Section A.

Answer the following question (30 points, 10% total).

⦁ Consider the following simultaneous games. For each game, find the pure strategy equilibria (DO NOT find mixed strategy equilibria) and report. Mark the best responses.

7, 5 2, 5

4, 6 4, 2

Player 2 C D

Player 1 A

B

U

Player 1 C

D

Player 2

L M R

5, 2 4, 1 6, 3

6, 3 3, 4 4, 2

4, 0 7, 2 3, 1

A

Player 1 B

C

D

Player 2

W X Y Z

4, 7 4, 3 3, 5 2, 1

2, 4 3, 4 2, 3 4, 6

3, 4 5, 2 4, 3 2, 5

0, 3 4, 2 5, 4 1, 3

Player 1 A

B

Player 2

3, 4, 1 3, 3, 4 4, 2, 3

2, 1, 3 5, 2, 2 1, 1, 2

C D E

F

Player 1 A

B

Player 3

Player 2

C D E

2, 5, 2 2, 4, 5 7, 5, 4

1, 3, 4 3, 4, 1 2, 3, 2

G

Section B.

Answer the following question (30 points, 10% total).

⦁ Consider the following game. Show all work.

5, 4 1, 3

2, 3 4, 5

Player 2 C D

Player 1 A

B

⦁ Find all pure strategy Nash equilibria and report. Mark best responses. (5 points)

⦁ Find the mixed strategy Nash equilibrium. First set up the problem, writing out the utilities of Player 1 (given Player 2’s mixed strategy) and Player 2’s utilities (given Player 1’s strategy). (5 points)

⦁ Solve for the mixed strategy equilibrium and report it. (11 points)

⦁ Calculate the average utility (“value of the game”) for each pure strategy for each player in order to check if the players are actually indifferent. (4 points)

⦁ Graph the best response correspondences. (5 points)

Section C.

Answer the following question (25 points, 8% total).

⦁ Consider the following game. Show all work.

A

Player 1 B

C

D

Player 2

W X Y Z

5, 4 2, 1 3, 1 6, 3

3, 4 3, 7 1, 4 2, 8

3, 3 4, 5 4, 2 2, 4

4, 2 6, 3 2, 1 3, 1

⦁ In the entire game, are there any strategies that are strictly dominated by pure strategies? If so, identify the strategy and which strategy (or strategies) is/are strictly better. (4 points)

⦁ In the entire game, are there any strategies that are weakly dominated by pure strategies? If so, identify which strategy (or strategies) is/are weakly better. (4 points)

⦁ Perform iterative elimination of strictly dominated strategies. Cross out the eliminated strategies and write out the order in which the strategies are eliminated. What remains of the payoff matrix after IESDS? Create the reduced strategic form (i.e., recreate the payoff matrix with only the strategies that survive). What are the pure strategy Nash equilibria? (10 points)

⦁ Find the mixed strategy Nash equilibria (hint: calculate based on the reduced payoff matrix). (7 points)

Section D.

Answer the following question (30 points, 10% total).

⦁ Consider the following game. Show all work.

A

Player 1 B

C

Player 2

D E F

4, 2 2, 3 2, 3

2, 5 3, 4 5, 3

3, 4 5, 2 3, 5

⦁ Set up the problem, writing out the utilities of Player 1 (given Player 2’s mixed strategy) and Player 2’s utilities (given Player 1’s strategy). (6 points)

⦁ Solve for the equilibrium and report it. (18 points)

⦁ Calculate the average utility (“value of the game”) for each pure strategy for each player in order to check if the players are actually indifferent. (6 points)

Section E.

Answer the following question (20 points, 6% total).

⦁ Consider the following game. Show work as indicated in subparts.

⦁ How many information sets does each player have? Circle them on the figure. How many rows and columns would the strategic form for the entire game have? (Do not write out the possible strategies, there are too many.) (6 points)

⦁ Solve the game through backwards induction (do not solve using the strategic form of each subgame). Cross out eliminated strategies as you work backwards and indicate with arrows which actions each player takes at each decision. What are the final payoffs in equilibrium? (6 points)

⦁ Report the equilibrium (i.e., in standard format, Player 1’s complete equilibrium strategy, followed by Player 2’s). (8 points)

Section F.

Answer the following questions (45 points, 15% total).

⦁ Consider the game below. Show all work.

⦁ How many information sets do each player have? Circle them. How many subgames are there? Place a rectangle around each one in the figure. (7 points)

⦁ Create the strategic form for the entire game and find the pure strategy Nash equilibria. (13 points)

⦁ Find the pure strategy subgame perfect Nash equilibria by comparing the strategic forms for each subgame to the Nash equilibria from the entire game. (10 points)

⦁ Find any subgame perfect Nash equilibri(um/a) that involves mixed strategies, using the method of generalised backwards induction. (10 points)

⦁ Which equilibri(um/a) survive(s) Forward Induction? Explain/Demonstrate why. (5 points)

Section G.

Answer the following question. (40 points, 13% total)

⦁ Consider the following game of incomplete information, where there is a probability of that player 2 is Type I, and probability of being Type II. Show all of your work.

Player 1 A

B

Player 2 C D

3, 2 3, 4

4, 4 0, 3

Type I

Player 1 A

B

Player 2 C D

2, 3 3, 2

4, 3 1, 3

Type II

⦁ What are the possible strategies for player 1 and 2? For strategies that condition by type, write a short statement indicating what the order of actions in the strategy signifies. (5 points)

⦁ Create the normal form of the game and find the pure strategy equilibri(um/a). Are there any strictly or weakly dominated strategies? (20 points)

⦁ Find a mixed strategy equilibrium where both players mix between two or more strategies (10 points)

⦁ Find a mixed strategy equilibrium where one player plays a pure strategy. (5 points)

Section H.

Answer the following questions. (50 points, 16% total)

⦁ Consider the following sequential game. The first move is one of chance. Player 1 observes the move by Nature, but Player 2 does not.

⦁ Find the perfect Bayesian equilibria for this game and the beliefs that support them. Since all of Player 1’s information sets are singletons, the only relevant set of beliefs are Player 2’s, labelled in the figure (, ); report the probabilities for these nodes when updating beliefs in your answers. When possible, update beliefs using Bayes Rule, which is (for example)

⦁ Does this game have any separating perfect Bayesian equilibria? Show your analysis, and, if there is such an equilibrium, report it. (16 points)

⦁ Does this game have any pooling perfect Bayesian equilibria? Show your analysis, and, if there is such an equilibrium, report it. (22 points)

⦁ Find a mixed strategy semi-separating PBE where at least one type of Player 1 mixes between A and B, and Player 2 mixes in response to either A and/or B. Show your work. (12 points)

Note, you may answer parts (a) and (b) by either method: (1) create the normal form, find the Nash equilibria, and check those for PBE, or (2) check directly each combination of separating and pooling strategies for PBE. I would strongly suggest method (2) (method (1) is useful as a check though). For part (c), either approach can be used (each are equally complicated).

Section I.

Answer ONE of the following questions. (30 points, 10% total)

⦁ Consider an alternating offer bargaining model to divide $1 between two players over five rounds. Player 1 makes an offer to Player 2 in rounds 1, 3 and 5, Player 2 makes an offer to Player 1 in rounds 2 and 4. Instead of a common discount factor , player 1’s payoffs are discounted by factor after each round, and player 2’s payoff is discounted by a factor of . Show all of your work.

⦁ Derive what proposals are made in every round, and report payoffs each player receives if an agreement is made on the proposal. (15 points)

⦁ Report the payoffs that occurs in equilibrium, and in what round the players arrive at an agreement. (5 points)

⦁ Report the equilibrium (i.e., identify what each player proposes and accepts in each round in standard format). (10 points)

⦁ Consider the following stage game played a finite or infinite number of periods with common time discount factor δ for both players. Show all of your work.

Player 2

D E F

11, 10 3, 12 5, 9

14, 5 3, 6 8, 8

4, 4 4, 4 3, 3

A

Player 1 B

C

Recall the following formula

⦁ First, consider the repetition of this stage game twice. Propose a punishment strategy (stating it clearly) such that (A,D) is played in the first round. For what value of is this a Nash equilibrium? Report the equilibrium. (15 points)

⦁ Now consider the repetition of this game infinitely often. Find a Nash equilibrium using Grim Trigger strategies that results, in equilibrium, with (A,D) occurring every period. What value of is necessary for cooperation to occur? (15 points)

Section J.

BONUS QUESTIONS. (maximum of 21 points)

⦁ Return to Section I. You may solve the problem you did NOT chose in Section I (questions 9 and 10) here, for a maximum of 6 extra credit points (2%). Only attempt if you have finished the rest of the exam and have enough time. Note, partial credit will be more limited than in Section I.

⦁ Consider the following game from Section A (problem 1, third game). Find all mixed strategy equilibria, for a maximum of 15 extra credit points (5%). Only attempt if you have finished the rest of the exam and have enough time. Note, this is a challenging problem. Partial credit is limited to cases where a clear understanding of the problem is demonstrated through the work and the subsequent results. I.e., just setting up the problem and starting will not earn bonus points.

Player 2

W X Y Z

4, 7 4, 3 3, 5 2, 1

2, 4 3, 4 2, 3 4, 6

3, 4 5, 2 4, 3 2, 5

0, 3 4, 2 5, 4 1, 3

A

Player 1 B

C

D

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