Game theory

Two firms play a Cournot duopoly stage-game repeatedly T times, observing past actions at every stage and discounting payoffs with a factor de (0, 1) per period. In the stage game, each firm i 1,2 simultaneously produces a quantity qi (0, 15), and payoffs are given by vi(21,92) = (60 - 91 - 92)91 and v2(91, 92) = (60 - 91 - 92)92. (a) [2 pointsFind the Nash equilibrium of the stage game. Would both firms prefer playing (41, 42) = (14, 16) instead? (b) [3 points) Suppose the stage game is repeated T 10 times. For which values of 8 does this game have a subgame perfect equilibrium in which the firms play (91, 92) = (14, 16) in all 10 stages on the equilibrium path? (c) [5 points) Suppose the stage game is repeated infinitely many times (T = 0). What is the lowest value of 8 for which the infinitely repeated game has a subgame perfect equilibrium in which (q1, 42) = (14, 16) is played in every stage? = Two firms play a Cournot duopoly stage-game repeatedly T times, observing past actions at every stage and discounting payoffs with a factor de (0, 1) per period. In the stage game, each firm i 1,2 simultaneously produces a quantity qi (0, 15), and payoffs are given by vi(21,92) = (60 - 91 - 92)91 and v2(91, 92) = (60 - 91 - 92)92. (a) [2 pointsFind the Nash equilibrium of the stage game. Would both firms prefer playing (41, 42) = (14, 16) instead? (b) [3 points) Suppose the stage game is repeated T 10 times. For which values of 8 does this game have a subgame perfect equilibrium in which the firms play (91, 92) = (14, 16) in all 10 stages on the equilibrium path? (c) [5 points) Suppose the stage game is repeated infinitely many times (T = 0). What is the lowest value of 8 for which the infinitely repeated game has a subgame perfect equilibrium in which (q1, 42) = (14, 16) is played in every stage? =