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GAME THEORY 2017-2018 5 cfu 5 Nash and correlated equilibria Exercise 63. Find the Nash equilibria in pure strategies for the Prisoner Dilemma. Is there any correlated equilibrium? Exercise 64. Given the following game, in which player III chooses the matrix, I the row and II the column: ( (1, 0, 0) (7 , 3, 1) (1, 10, 1) (4 , 2, 0) ) , ( (2, 4, 5) (3 , 3, 2) (1, 1, 3) (4 , 2, 1) ) , 1. prove that in every correlated equilibrium player III never chooses the first matrix 2. find the Nash equilibria in pure strategies 3. find a Nash equilibrium in mixed strategies. Exercise 65. Given the bimatrix ((3, 3) (1 , 4) (4, 1) (0 , 0) ) , is ( 1 3 1 31 3 0 ) a correlated equilibrium? Exercise 66. Given the bimatrix   (2, 2) (0 , 3) (0 , 0) (1, 1) ( −1,−1) (1 , 1) (0, 0) (0 , 3) (2 , 2)   , 1. Is the probability distribution: play (first row, first column) and (third row, third column) with probability 1 2 each a correlated equilibrium? 2. Find, if there exists, a correlated equilibrium assigning positive probability to both the above strategy profiles. 1 Exercise 67. How many repetitions of the game:   (100, 100) (0 , 160) ( −5,−5) (160, 0) (1 , 1) ( −5,−5) (−5,−5) ( −5,−5) ( −5,−5)   , are necessary to make the players to gain at least 90 on average? Exercise 68. Consider the following bimatrix: A =   (10, 10) (0 , 11) (0 , 0) (11, 0) (1 , 1) (0 , 0) (0, 0) (0 , 0) ( −1, 1)   . Given ε > 0, find how many times the game should be played in order that there exists a Nash equilibrium providing at least 10− ε, on average, to both players. Exercise 69. Given the following bimatrix:   (4, 3) (0 , 1) (2 , 3) (a, 2) (2 , 3) (3 , 2) (1, 1) (1 , b) (2 , 0)   1. find the equilibria in pure strategies; 2. find the best reaction of the second player to the strategy ( 1 4 , 1 4 , 1 2 )…