Bargaining and Cooperation (50 points) Many strategic interactions have two distinct phases: (1) bargaining over how to share the benefits of cooperation followed by (2) a repeated interaction in which the actors must continuously decide whether to abide by the terms or terminate the agreement. For example, the Organization of the Petroleum Exporting Countries must decide how to divide the revenue of sustaining a cartel, and then each coun-try must decide whether to exceed its quota. Alternatively, a criminal organization may try to extort a company to pay a "protection" fee in exchange for promising not to disrupt that company's computer network. Initially, they must agree to a bribe size; afterward, the company must decide whether to pay that amount in any given period, while the group must decide whether to hack the servers anyway. To model this type of interaction, consider the following game. In the first phase, player 1 offers an amount xe [0, 1]. Player 2 accepts or rejects. If she rejects, the game ends, and both players receive 0. If she accepts, the game moves into a second phase. The players play the following game over an infinite horizon: Cooperate Defect Cooperate 1-,2 0,1 Defect 0.0 If they both cooperate, they share the terms of the deal for the period. Note that this deal is a function of x, which the players decided upon in the first stage of the game. However, if one cooperates and the and a payoff of O for both players. Let the respective players have discount factors O1 (0, 1) and 2 (0, 1). a. Find the range of x values that induce player 1 to cooperate in the second phase of the game. Assume that both players are trying to use grim trigger strategies. b. Find the range of x values that induce player 2 to cooperate in the second phase of the game. Again, assume that both players are trying to use grim trigger strategies. c. What is the range of x values make grim trigger strategies an equilibrium? Your answer should be a function of O1 and 72. Is this range guaranteed to exist? If so, prove it. If not, what is the constraint on 1 and 2 that generates existence? d. Suppose that a mutually acceptable x value exists. (This may either be because it is guaranteed to exist or because we are narrowing our focus to values of 1 and on that allow this to be possible.) Find the subgame perfect equilibrium to the full game. Assume that (1) player 2 accepts when indifferent in the bargaining phase and (2) the players play their grim trigger strategies in the second stage. e. In standard bargaining environments (without the enforcement phase analyzed here), a player's share of the deal is increasing in his or her discount factor-i.e., as 0; increases, i's share of the deal becomes larger. Does this remain true when enforcement is necessary? Bargaining and Cooperation (50 points) Many strategic interactions have two distinct phases: (1) bargaining over how to share the benefits of cooperation followed by (2) a repeated interaction in which the actors must continuously decide whether to abide by the terms or terminate the agreement. For example, the Organization of the Petroleum Exporting Countries must decide how to divide the revenue of sustaining a cartel, and then each coun-try must decide whether to exceed its quota. Alternatively, a criminal organization may try to extort a company to pay a "protection" fee in exchange for promising not to disrupt that company's computer network. Initially, they must agree to a bribe size; afterward, the company must decide whether to pay that amount in any given period, while the group must decide whether to hack the servers anyway. To model this type of interaction, consider the following game. In the first phase, player 1 offers an amount xe [0, 1]. Player 2 accepts or rejects. If she rejects, the game ends, and both players receive 0. If she accepts, the game moves into a second phase. The players play the following game over an infinite horizon: Cooperate Defect Cooperate 1-,2 0,1 Defect 0.0 If they both cooperate, they share the terms of the deal for the period. Note that this deal is a function of x, which the players decided upon in the first stage of the game. However, if one cooperates and the and a payoff of O for both players. Let the respective players have discount factors O1 (0, 1) and 2 (0, 1). a. Find the range of x values that induce player 1 to cooperate in the second phase of the game. Assume that both players are trying to use grim trigger strategies. b. Find the range of x values that induce player 2 to cooperate in the second phase of the game. Again, assume that both players are trying to use grim trigger strategies. c. What is the range of x values make grim trigger strategies an equilibrium? Your answer should be a function of O1 and 72. Is this range guaranteed to exist? If so, prove it. If not, what is the constraint on 1 and 2 that generates existence? d. Suppose that a mutually acceptable x value exists. (This may either be because it is guaranteed to exist or because we are narrowing our focus to values of 1 and on that allow this to be possible.) Find the subgame perfect equilibrium to the full game. Assume that (1) player 2 accepts when indifferent in the bargaining phase and (2) the players play their grim trigger strategies in the second stage. e. In standard bargaining environments (without the enforcement phase analyzed here), a player's share of the deal is increasing in his or her discount factor-i.e., as 0; increases, i's share of the deal becomes larger. Does this remain true when enforcement is necessary