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- 5) Mixed strategy Nash equilibrium Consider a mixed strategy Nash equilibrium of the following coordination game: Player 2 Player 1 A B a 5.5 6.-2 b -2,6 1,1 a) In the above game, explain in words what condition player 1's probability p of playing strategy A must satisfy to induce player 2 to mix strategies between a and b in equilibrium. b) Solve for the mixed strategy Nash equilibrium where player 1 chooses A with probability P and player 2 chooses a with probability p, for 0 < p, P < 1.7 In bargaining theory, sometimes an individual’s patience level can affect his or her bargaining power. Consider a simple repeated game in which Person 1 makes an offer to purchase a good that Person 2 is selling; Person 2 can either accept this offer or reject it and make a counteroffer. This game continues until one player accepts the other’s offer. Explain how incorporating the rate of time preference (ββ) into such a repeated game might alter the outcome.4. David plays in a game where his strategy (set of possible choices) is denoted by x, which can be any integer between 1 and 7, against Goliath, whose strategy is denoted by y, which can be any integer between 1 and 7. David's and Goliath's utility functions are the same and given by: U(x, y) = (x-2)(y-2). Based on this information, determine whether each of the following statements are true or false. (a) The combination (x, y) = (2, 2) is a Nash equilibrium. (b) This game has no dominant strategy (a choice of integer that gives the highest payoff regardless of what the opponent chooses). (c) This game has only one dominated strategy (a choice of integer that would never be a best response to the opponent's choice of integer).
- 26. Consider the following game. There are ten churches in town: nine are Catholic and one is Eastern Orthodox. All churches conduct their services at 10am on Sunday morning. You and your friend, without communication, agree to meet at a church service. If you fail to meet at the same church, then you are sad. What is this game's focal point? Remember that you cannot communicate about this. (A) You meet at any of the Catholic churches. (B) You meet at any of the churches – denomination does not matter. (C) You meet at the Orthodox church. (D) There is no focal point. 27. Which of the following is the best example of Skin in the Game? (A) A politician retires to serve on the board of governors of a prestigious think tank. (B) A wealthy man donates his savings to charity. (C) A general leads his troops into battle. (D) Billie Eilish creates another cringe music video. 28. Public choice theory makes assumptions about the motives of politicians, firms, voters, and bureaucrats. Which of…(Demand Game) Two players can divide ten dollars. Find the Nash equilibria of the following two games. (a) The players simultaneously submit demands, numbers in {0, 1, ... ,10}. If the sum of the demands is at most 10, each player gets what she demands. Otherwise both get 0. (b) As in the previous part, except that if the sum of the demands exceeds 10, then (i) if the demands differ then the player who demands less gets her demand and the other player gets the rest, and (ii) if the demands are the same then each player gets 5. (Hint- write the best response functions).FOOP 6. (a) For the following extensive-form game: i. Identify the pure and mixed strategy Nash Equilibria. ii. Is the set of pure and mixed strategy Subgame Perfect Nash equilibria of the game different from the set of equilibria identified in part (a)? Explain (a couple of sentences should suffice). (3,1) A D B C (-2,-2) (2,5) D (0,7)
- 4. Game theory: An example of the prisoner's dilemma in the real world is when two competitors are battling it out in the marketplace. Often, many sectors of the economy have two main rivals. There can be rivalry such as between du and Etisalat in Telecommunications services, Coca-Cola and Pepsi-Cola in soft drinks. Suppose du plans to cut its price. Etisalat will likely follow suit to retain its market share. This may end up with low profits for both companies. A price drop by either company may thus be construed as defecting since it breaks an implicit agreement to keep prices high and maximize profits. Thus, if du drops its price but Etisalat continues to keep prices high, du is defecting, while Etisalat is cooperating (by sticking to the spirit of the implicit agreement). In this scenario, du may win market share and earn incremental profits by selling more. Assume that the incremental profits that accrue to du and Etisalat are as follows: o If both keep prices high (Cooperate),…10) Consider a repeated game in which the following one-shot game between two players is repeated "infinitely." Denote by "d" the discount factor. (Assume both players have the same discount factor.) Player 2 Cooperate Defect Player 1 Cooperate 5, 4 0, 5 Defect 6, 0 1, 1 The two players try to support cooperation by using "grim trigger strategy" as discussed in class. Answer YES or NO to each of the following three questions. (a) Can they support cooperation if the common discount factor, d, is equal to 0.23? (b) Can they support cooperation if the common discount factor, d, is equal to 0.4? (c) Can they support cooperation if the common discount factor, d, is equal to 0.15?6. Battle of the Networks 1\2 %share Sitcom Sitcom 48;52 Sport 42;58 Nature 55:45 58;42 Ans: s; = (0; 0.85; 0.14); V, = 55.43 s; = (0.43; 0.57;0); V½ = 44.57 40;60 63;37 60;40 Sport Nature 56:44 52;48 Find Nash Equilibrium.
- U9. Consider a revised version of the game from Exercise S9: PROFESSOR PLUM Revolver Knife Wrench Conservatory 1,3 2,-2 0,6 MRS. PEACOCK Ballroom 3,2 1,4 5,0 (a) Graph the expected payoffs from each of Professor Plum's strategies as a function of Mrs. Peacock's p-mix. (b) Which strategies will Professor Plum use in his equilibrium mixture? Why? (c) What is the mixed-strategy Nash equilibrium of this game?1. Each of two players simultaneously announces either Rock, Paper, or Scissors. Paper beats Rock, Rock beats Scissors, and Scissors beats Paper. The player who names the winning object receives $1 from her opponent. If both players make the same choice, no payment is made. Each players's preferences are represented by by the expected amount of money she receives. a) Formulate the situation as a strategic game and find all its mixed strategy equilibria. Make sure that you consider all different supports. b) Find all the mixed strategy equilibria of the modified game in which player 2 is prohibited from announcing Rock.8) Consider the following game: There are 10 players. Each player is asked to write a number between 0 and 100. The player whose choice is closest to half of the average is the winner. Describe the game and discuss what would you expect to be the average number if the game were played in class with your colleagues. Do you expect them to play the Nash equilibrium strategy? Explain.