Final_2023

Game Theory 45100 – Spring 2023: Final Exam The exam will last two hours and will be graded out of 75 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 [8 out of 75] 1 2 E F G H A 10 , 7 3 , 8 6 , 2 5 , 3 B 8 , 5 5 , 7 8 , 9 7 , 3 C 9 , 3 7 , 6 5 , 7 3 , 5 D 8 , 2 2 , 5 3 , 4 2 , 4 a.) Which pure strategies are dominated by a pure strategy (for Player 1 and for Player 2)? b.) Perform the iterated removal of dominated strategies. What strategies survive? c.) Using your answer to part (b) or otherwise, identify all the Nash equilibria of the game. d.) Use your answer to part (b) to identify what players with common knowledge of rationality must do in this game. Question 2 [6 points out of 75] 1 2 L C R U 5 , 9 2 , 1 1 , 4 M 3 , 2 2 , 1 3 , 8 D 1 , 2 2 , 6 3 , 5 Suppose that θ 1 = (0 . 1 , 0 . 7 , 0 . 2). a.) What is Player 2’s expected payoff from choosing L? Page 1 of 4

b.) What is Player 2’s best response? c.) Find the value of u 2 ( L, U ) that would make Player 2 indifferent between L and C. Suppose that θ 2 = (0 . 6 , 0 . 1 , 0 . 3). d.) What is Player 1’s expected payoff from choosing M? e.) What is Player 1’s best response? f.) Find the value of u 1 ( U, L ) that would make Player 1 indifferent between U and M. Question 3 [11 points out of 75] 1 2 (3 , 4) C (2 , 6) D A 2 (9 , 3) E (1 , 5) F B a.) * Does Player 1 have a dominant strategy? b.) Find the subgame-perfect Nash equilibrium. c.) * Show that any Nash equilibrium of this game is subgame-perfect. d.) * Now suppose that Player 1 gets a payoff of x < 2 if Player 1 chooses B and Player 2 then chooses E. Find the two Nash equilibria of this game. Which one of the two is a subgame-perfect Nash equilibrium? Page 2 of 4

Question 6 [12 points out of 75] Consider the following game: 1 2 L R U 10 , 10 20 , 5 D 5 , 20 0 , 0 a.) Find the pure-strategy Nash equilibrium. Is it a strict Nash equilibrium? b.) * Prove that there is no mixed-strategy Nash equilibrium. c.) Which strategy profiles are Pareto efficient? d.) * Now suppose that this game is repeated three times. Find the subgame-perfect Nash equilibrium. Question 7 [14 points out of 75] Consider again the game from Question 6. Now assume that the game is repeated an indefinite number of times, with a continuation probability δ ∈ (0 , 1) in every period, and consider the fol- lowing ‘alternating’ strategies. Alternating strategy for Player 1: Choose U in odd periods (periods 1, 3, and so on) and choose D in even periods (periods 2, 4, and so on), as long as both players have followed their alternating strategy in every previous period; otherwise choose U. Alternating strategy for Player 2: Choose R in odd periods (periods 1, 3, and so on) and choose L in even periods (periods 2, 4, and so on), as long as both players have followed their alternating strategy in every previous period; otherwise choose L. For parts (a), (b) and (c), assume that Player 1 follows her alternating strategy. a.) * Starting from an odd period (e.g., from the first period), show that Player 2’s expected payoff from following her alternating strategy is 5+20 δ 1 − δ 2 . Hint: Use the fact that 1 + δ 2 + δ 4 + ... = 1 1 − δ 2 . b.) * Starting from an odd period (e.g., from the first period), show that Player 2’s expected payoff from deviating from her alternating strategy is 10+10 δ 1 − δ 2 . c.) * Explain why Player 2 has no incentive to deviate from her alternating strategy in even periods or following a deviation by either player. d.) * Find the values of δ for which the alternating strategies form a subgame-perfect Nash equilibrium. e.) Explain intuitively why the alternating strategies are not a subgame-perfect Nash equilibrium when the continuation probability is low. Page 4 of 4