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Determine the Nash equilibria
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- Consider the following voting game. There are three players, 1, 2 and 3. And there are three alternatives: A, B and C. Players vote simultaneously for an alternative. Abstaining is not allowed. Thus, the strategy space for each player is {A, B, C}. The alternative with the most votes wins. If no alternative receives a majority, then alternative A is selected. Denote ui(d) the utility obtained by player i if alternave d {A, B, C} is selected. The payoff functions are, u1 (A) = u2 (B) = u3 (C) = 2 u1 (B) = u2 (C) = u3 (A) = 1 u1 (C) = u2 (A) = u3 (B) = 0 a. Let us denote by (i, j, k) a profile of pure strategies where player 1’s strategy is (to vote for) i, player 2’s strategy is j and player 3’s strategy is k. Show that the pure strategy profiles (A,A,A) and (A,B,A) are both Nash equilibria. b. Is (A,A,B) a Nash equilibrium? Comment.Two friends, Khalid and Mahmood, are going to a watch a world cup football match. They play a simple game in which they hold out one or two fingers to decide who will pay for the other's ticket. Khalid wins if the fingers held out add up to an even number; Mahmood wins if the fingers held out add up to an odd number. The price of the ticket is 25 OMR. Construct a payoff matrix for the game. Is there a unique Nash equilibrium in this game? Which strategy should a player use to maximize her chances of winning the game?What is the secure strategy for player B in the game presented in Table
- This activity has 4 theorical and practical exercises for which you will have to describe the representation of each proposed game and its solutions.1. Consider the following game in strategic or normal form.A2 B2 C2A1 1,0 1,2 -2,1B1 6,2 0,3 2,3C1 2,2 -2,1 2,3▸ Use the iterative elimination of strictly dominated strategies to reduce the game as much as possible.▸ What is the set of rationalizable strategies for each player? Microeconomics (code ECO2023)Actividades 2© MIU City University Miami ▸ What is/are Nash equilibrium(s) in this game?▸ Explain the differences between cooperative and non- cooperative games. What are the fundamental hypotheses about the behavior of the persons in Game theory?2. Two hunters may choose, individually and simultaneously, to go deer hunting, as the deer can provide abundant and appetizing food, or to go hare hunting, also appetizing, but scarce. Deer hunting represents a challenge and requires mutual coordination. Hunting a…Consider the following two-player game.First, player 1 selects a number x≥0. Player 2 observes x. Then, simultaneously andindependently, player 1 selects a number y1 and player 2 selects a number y2, at which pointthe game ends.Player 1’s payoff is: u1(x; y1) = −3y21 + 6y1y2 −13x2 + 8xPlayer 2’s payoff is: u2(y2) = 6y1y2 −6y22 + 12xy2Draw the game tree of this game and identify its Subgame Perfect Nash Equilibrium.Paramter y = 0 What is the highest payoff any player can receive in any subgame perfect Nashequilibrium of the repeated game?
- Consider the following game in strategic or normal form. A2 B2 C2 A1 1,0 1,2 -2,1 B1 6,2 0,3 2,3 C1 2,2 -2,1 2,3 Use the iterative elimination of strictly dominated strategies to reduce the game as much as possible. What is the set of rationalizable strategies for each player? What is/are Nash equilibrium(s) in this game?Consider the following representation of a Normal form game. actions w a (45,22) (10,38) (42,13) (10,7) (p,28) (15,40) (q,10) (44,10) (20,22) (14,31) (27,13) (12,8) d. (20,41) (9,48) (28,24) (18,32) Here each cell in the table represents an ordered pair. First element is payoff of the first player and second element is payoff of the second player. The letters a, b, c, d, x, y, z, w represent the actions. Write down the table in your answer script too. Now answer the following questions: 1. What is the distinction between strictly dominant strategy and weakly dominant strategy? Is it reasonable for a player to play a strictly dominated strategy? Explain why. 2. What are the minimum values for p and q that will make ba strategy that strictly dominates all other strategies for player 1, assuming both p and q are natural numbers? 3. Does player 2 have any strictly dominated pure strategy? If yes, which pure strategy dominates that strategy? If the submit button is off it is beacuse the due…Consider the following game 1\2 Y Z A 10,3 3,9 B 8,5 6,1 Suppose Player 2 holds the following belief about Player 1: θ1 (A,B) = (9/10,1/10) What is the expected payoff from playing ‘Y’ ? What is the expected payoff from playing ‘Z’ ? Based on these beliefs, player 2 should respond by playing _____
- Country A is rich and country B is poor. Consequently, residents of country B want to immigrate to country A. The trouble is that there is no legal way to do. So they attempt to cross the border Illegally. There are 3 border crossings at which a resident of B may attempt to cross the border. An attempt is successful if there is no border guard at the crossing, and unsuccessful if there is a guard at the crossing. Country A has only 1 border guard. Model the game in which a resident of country B attempts to enter country A at one of the 3 border crossings, and the government of country A chooses to post its border guard at one of the 3 border crossings. In this game, each player has 3 pure strategies. A) What are the ultimate outcomes in this game? B) What are the player's Bernoulli Payoffs? C) Find a mixed strategy equilibrium of the game.You are given the payoff matrix below. B1 B2 B3 A1 1 Q 6 A2 P 5 10 A3 6 2 3 a. Determine the range of values of P and Q in order for Player A to choose strategy A2 and PlayerB to choose strategy B2.b. Solve the problem in the perspective of the opponent. Are the ranges the same?c. What are the values of the game for (a) and (b)?A game involves two players: player A and player B. Player A has three strategies a1, a2 and a3 while player B has three strategies b1, b2 and b3. Player B b1 b2 b3 a1 -40,30 70,20 -10,120 Player A a2 40,60 80,80 60,20 a3 -30,40 -50,110 150, -70 Assuming that this is a one-time game, answer the following questions: Is there any dominant strategy for each player? What is the secure strategy of each player. What is the Nash equilibrium of the game?
