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2 questions refer to the games above. (Fun fact, as an aside: the payoffs reflect the frequencies of the letter pairs formed by the names of the strategies in the English language, per 10,000 words.)
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- 7) What is a repeated game? Why are repeated games useful to consider? What is the difference between finitely repeated games and infinitely repeated games? Provide an example of a political situation that would be usefully modeled with a repeated game.2 questions refer to the games below. (Fun fact, as an aside: the payoffs reflect the frequencies of the letter pairs formed by the names of the strategies in the English language, per 10,000 words.)2- Consider the following game. Player 2 Player 1 U 12, 2 | 3, 9 5, 8 4, 2 D (a) Find all the Nash equilibria, pure and mixed. (b) Suppose that the payoff of the column player u:(D, L) is reduced from 8 to 6, but all other payoffs remain the same. Again, find all the pure- and mixed-strategy Nash equilibria. (c) Compare the mixed-strategy equilibria in parts (a) and (b). Did this worsening in one of player 2's payoffs change player 2's equilibrium mixed strategy? Did it change player l's? Give some intuition.
- Player 1 Cooperate (C) Defect (D) If the game has a dominant strategy, what is it? There is none. If the game has a Nash equilibrium in pure strategies, what is it? There is none. Cooperate (C) 3,3 8,0 Cooperate (C) is a dominant strategy for both players. Defect (D) is a dominant strategy for both players. Cooperate (C) is a dominant strategy for 1, and Defect (D) is a dominant strategy for 2. C, C is the only Nash equilibrium. D, D is the only Nash equilibrium. C, C and D, D are both Nash equilibria. Player 2 Defect (D) 0,8 1,1Imagine that two firms in two different countries want to bring a new product to market. Due to economies of scale, if both firms do this, they will both lose £50 million. But if only one firm does this, it will gain £300 million. (a) What is the best strategy for firm A, if firm B has not yet entered the market, and why? (b) Illustrate this with a game theory diagram, showing appropriate payouts. (c) What is the welfare-maximising strategy for a government, and why?Problem 2. Consider the partnership-game we discussed in Lecture 3 (pages 81-87 of the textbook). Now change the setup of the game so that player 1 chooses x = [0, 4], and after observing the choice of x, player 2 chooses y ≤ [0, 4]. The payoffs are the same as before. (a) Find all SPNE (subgame perfect Nash equilibria) in pure strategies. (b) Can you find a Nash equilibrium, with player 1 choosing x = 1, that is not subgame perfect? Explain.
- Question: There are 500 people who need to go to work at the same time. There are 3 roads possible : AE, BE and C. 1) What's the Nash equilibrium TA = 10 + (0.01)NA T, = 9+ (0.02)N, B T; = 10 + (0.01)Ng Tc = 20 + (0.01)N.Player 1 Cooperate (C) Defect (D) Cooperate (C) 3,3 8,0 Player 2 Defect (D) 0,8 1,1 In general, a combination of strategies is a Nash equilibrium if ... Every player is choosing a best response against the other players' strategies. Every player has a positive payoff. The players maximize the sum of their payoffs. The players choose identical strategies. If the game is repeated, which cooperative actions could benefit both players? O Both players choose C. Player 1 chooses C, Player 2 chooses D. O Player 1 chooses D, Player 2 chooses C. Both players choose D.the Consider a simultaneous-move integer game between two players: Marilyn and Noah. Each of player is required to announce a positive integer between 1 to 4. In other words, a player can announce 1, 2, 3, or 4. Two players announce their integers simultaneously. Notice that this game is different from the games we learned in class in that each player has four actions to take. The payoffs of the players in the game are specified as follows: (1) when the two announced integers are different, whoever reports the lower number pays $1 to the other player, so that the loser of the game has payoff -1 and the winner of the game has payoff 1; (2) when the two players announce the same integer, their payoffs are both O. What is Marilyn's maximin strategy? 01 02 03 O
- 3. Consider the following game: K Out 2 2 1 9 Player 1 Player 1 X In A Y Player 2 0 2 B 5 5 a) Does the game have perfect information? Why/why not? b) Circle the subgames. c) Find all backward induction solutions. d) Below are written five strategy profiles. One of them is a subgame perfect equilibrium; circle it! One of them is not a Nash equilibrium; write "X" across it! ((Out, X), A) ((Out, Y), A) ((In, Y),B) ((Out, Y), B) ((In, X),B)Define a new game as a modified version of the game in Problem 1, which we will call the Matching Two Pennies game: each player choose heads (H) or tails (T) for two pennies. In this game, Player 1 wins if both coins show the same combination of heads and tails, while Player 2 wins if they do not. (a) Write the strategy sets S1 and S2 for this game. (b) Give the payoffs for each player in this Matching Two Pennies game, by defining T;(81, 82) for each player and for each strategy pair, either as a list or in matrix form.Two politicians from the same party decide whether to support or oppose the gov- ernment's environmental policy. Politician A prefers that they both support, but Politician B prefers that they both oppose. Both politicians agree that the party should have a coherent position and the worst outcome is that one supports and the other opposes the policy. (a) Formulate this situation as a strategic game - specify the players, actions, and payoffs (you can set-up a table of the actions and payoffs). (b) What are the Nash equilibria of this game? Explain your answer.