Tutorial 1 Game Theory Matthew Robson University of

Tutorial 1: Game Theory Matthew Robson University of York Microeconomics 2 1

Question 1 What implication can you draw from the Boeing-Airbus game? Choose the best answer from the following: a) b) c) d) e) Trigger strategy can lead to favourable outcome. Such a game has infinitely many Nash equilibria. Nash equilibrium may not necessarily be efficient. Using a punishment strategy can lead to win-win outcome. Making a credible commitment can bring advantage for your play of the game. Airbus Boein g Jumbo No Jumbo -50, 50 90, -10 No -20, 100 0, 0 2

Question 1 a) Trigger strategy can lead to favourable outcome. • Trigger strategy is used in infinite horizon games (i. e. Tit-for-Tat or Grim), so not particularly useful. b) Such a game has infinitely many Nash equilibria. • This game has only two: (Jumbo, No) and (No, Jumbo). c) Nash equilibrium may not necessarily be efficient. • Both Nash equilibria are efficient. d) Using a punishment strategy can lead to win-win outcome. • The game is simultaneous, but if it wasn’t the punishment wouldn’t be credible. e) Making a credible commitment can bring advantage for your play of the game. • Yes. If one firm made a credible commitment, then the other would not make the jumbo jet. This is a coordination game, similar to the Chicken Game. 3

Question 2 • Bill Not Confess -1, -1 -9, 0 Confess 0, -9 -6, -6 Alan 4

Question 2 • If both players adopt a trigger strategy, then in order for the strategy profile (Not Confess, Not Confess) in all periods to be a Nash equilibrium: a) b) c) d) e) r should be greater than -1. r should be no greater than 3. r should be no greater than 4. r should be no greater than 5. None of the above. 5

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Question 3 • Consider the following two player (Alice and Bob) game, where each player has two strategies. Bob Left Right Up 2, 2 8, 4 Down 3, 9 5, 9 Alice 9

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Question 4 • Consider the following two-player (Alice and Bob) game, where each player has two strategies. Bob Left Right Up 3, 5 9, 5 Down 2, 8 5, 9 Alice 13

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Question 5 • The Stag Hunt game is based on a story told by Jean Jacques Rousseau in his book Discourses on the Origin and Foundation of Inequality Among Men (1754). The story goes something like this: “Two hunters set out to kill a stag. One has agreed to drive the stag through the forest, and the other to post at a place where the stag must pass. If both faithfully perform their assigned stag-hunting tasks, they will surely kill the stag and each will get an equal share of this large animal. During the course of the hunt, each hunter has an opportunity to abandon the stag hunt and to pursue a hare. If a hunter pursues the hare instead of the stag he is certain to catch the hare and the stag is certain to escape. Each hunter would rather share half of a stag than have a hare to himself. ” The matrix below shows payoffs in a stag hunt game. If both hunters hunt stag, each gets a payoff of 4. If both hunt hare, each gets 3. If one hunts stag and the other hunts hare, the stag hunter gets 0 and the hare hunter gets 3. 16

Question 5 a) b) c) d) If you are sure that the other hunter will hunt stag, what is the best thing for you to do? If you are sure that the other hunter will hunt hare, what is the best thing for you to do? Does either hunter have a dominant strategy in this game? No. If so, what is it? If not explain why not. This game has two pure strategy Nash equilibria. What are they? Hunter B Hunter A Hunt Stag Hunt Hare Hunt Stag 4, 4 0, 3 Hunt Hare 3, 0 3, 3 17

Question 5 a) b) c) d) If you are sure that the other hunter will hunt stag, what is the best thing for you to do? Hunt Stag If you are sure that the other hunter will hunt hare, what is the best thing for you to do? Hunt Hare Does either hunter have a dominant strategy in this game? No. If so, what is it? If not explain why not. No, the best response depends on other player This game has two pure strategy Nash equilibria. What are they? Both Hunter B Hare or Both Stag Hunt Stag Hare Hunter A Hunt Stag 4, 4 0, 3 Hunt Hare 3, 0 3, 3 18

Question 5 • Hunter B Hunter A Hunt Stag Hunt Hare Hunt Stag 4, 4 0, 3 Hunt Hare 3, 0 3, 3 19

Question 5 • Hunter B Hunter A Hunt Stag Hunt Hare Hunt Stag 4, 4 0, 3 Hunt Hare 3, 0 3, 3 20

Question 6 • State two main assumptions of game theory, and explain why these assumptions are crucial for the application of game theory to microeconomic analysis. • Rationality: The player has well-defined preferences over the set of possible outcomes and implements the best available strategy to optimize her outcome according to her preferences. • Complete information: The structure of the game and the payoff functions of the players are commonly known by all players. • These assumptions make it possible to analyse games in the way we analyse them. If players weren’t rational it would be much harder to analyse and predict players’ choices in a game. Without complete information, players would not be able to respond optimally to the choice that they expect their opponent to take. They would not be able to predict their opponents’ choices because they would not know their opponents’ payoffs. • Rationality is also consistent with our usual assumption on how agents behave in microeconomic theory, and thus makes Game Theory more easily applicable to the 21 analysis of economic problems.

Question 7 • Construct a game (a concrete numerical example) such that it has a unique Nash equilibrium, and moreover this Nash equilibrium is a mixed strategy equilibrium. • Matching pennies • Unique Nash Equilibrium: Mixed Strategy (0. 5, 0. 5) for both Bob and Alice Bob H T H 1, -1 -1, 1 T -1, 1 1, -1 Alice 22