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.
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- 2. Consider a finitely repeated game with a unique Nash equilibrium in each stage game. What is the subgame-perfect outcome for such a game?6) The following is a static game D 2,2 0,3 D 3,0 1,1 a) Convert this game into dynamic form game b) Find the Nash equilibrium and subgame perfect Nash equilibrium of this game. c) If you consider this game as dynamic then what kind of dynamic game is this.2. Assume a Hawk -Dove game with the following payoff matrix, where the first entry is Animal A's payoff, and the second entry is Animal B's payoff: Animal A Hawk Dove (rows)/Animal B (columns) (-5,-5) (0,10) (10,0) (4,4) Hawk Dove An animal that plays Hawk will always fight until it wins or is badly hurt. An animal that plays Dove makes a bold display but retreats if his opponent starts to fight. If two Dove animals meet they share. (a) Explain why there cannot be an equilibrium where all animals act as Doves. (b) Explore whether there are any Nash equilibria in pure strategies and explain which these are and why. (c) Derive a mixed strategy Nash equilibrium (MSNE). What is the proportion of Hawks and Doves? If the proportion of Hawks in the population of animals is greater than the mixed strategy equilibrium proportion you calculated, which strategy does better, Hawks of Doves? Explain your answer. (d) Draw the best response functions and show in the diagram all pure and mixed…
- 2. Assume a Hawk -Dove game with the following payoff matrix, where the first entry is Animal A's payoff, and the second entry is Animal B's payoff: Animal A Hawk Dove (rows)/Animal B (columns) (-5,-5) (0,10) (10,0) (4,4) Hawk Dove An animal that plays Hawk will always fight until it wins or is badly hurt. An animal that plays Dove makes a bold display but retreats if his opponent starts to fight. If two Dove animals meet they share. (a) Explain why there cannot be an equilibrium where all animals act as Doves. (b) Explore whether there are any Nash equilibria in pure strategies and explain which these are and why. (c) Derive a mixed strategy Nash equilibrium (MSNE). What is the proportion of Hawks and Doves? If the proportion of Hawks in the population of animals is greater than the mixed strategy equilibrium proportion you calculated, which strategy does better, Hawks of Doves? Explain your answer.1. Consider the following N-player game. Each agent has a choice of strategy A or B. The state variable is x, the proportion of agents choosing strategy A. For every agent, the utility of choosing strategy A is U(A) = 10 +2x and the utility of choosing strategy B is U(B) = 10+3x. What is the Nash equilibrium to this game? Explain.1. Consider the following normal form game: Player 2 Player 1 U (2,2) (1,6) Find the mixed-strategy Nash equilibrium, supposing that this is a simultaneous non- repeated game.
- (a) Consider the following bimatrix of a normal form game. Show that the set of strategies that survive elimination of weakly dominated strategies depends on the order in which the weakly dominated strategies are eliminated. Player 2 L R 1,1 0,0 M| 1,1 0,0 | 2,1 Player 1 2,1 (b) Find an example of a game where the unique Nash equilibrium in pure strategies would not survive the eliminination of weakly dominated strategies. (c) Find an example of a game where both players' strategies are weakly dominated in the Nash equilibrium.4. Assume a Hawk -Dove game with the following payoff matrix, where the first entry is Animal A's payoff and the second entry is Animal B's payoff: Animal A Hawk (rows)/Animal B (columns) Hawk Dove (-10,-10) (0,20) Dove (20,0) (8,8) An animal that plays Hawk will always fight until it wins or is badly hurt. An animal that plays Dove makes a bold display but retreats if his opponent starts to fight. If two Dove animals meet they share. (a) Explain why there cannot be an equilibrium where all animals act as Doves. (b) Explore whether there are any Nash equilibria in pure strategies and explain which these are and why they are equilibria.In a game theory payoff matrix. Your company (A) and a major competitor (B) havetwo potential strategies: to advertise or to not advertise during the Super Bowl. The payoffs in each cell represent the change in firm profits from advertising. How would you create payoffs in each cell such that the Nash equilibrium is that both firms advertise despite having a higher profit if neither firm advertised
- 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?1. Consider the following game in which two firms decide how much of a homogeneous good to produce. The annual profit payoffs for each firm are stated in the cell of the game matrix, and Firm A's payoffs appear first in the payoff pairs: Firm B - low output 300, 250 Firm B - high output 200, 100 Firm A - low output Firm A - high output 200, 75 75, 100 a. Draw a tree (extensive form) of this game.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.