Exercise 5 solution

Page 1

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? Solution The Prisoner Dilemma is described by a bimatrix like the following one: ( (5, 5) (0 , 7) (7, 0) (1 , 1) ) By eliminating strictly dominated strategies we find that the only Nash equilibrium is given by (5, 5). For this reason this is also the unique 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. Solution 1. The first matrix is strictly dominated by the second one (for the third player), then it will never be played at the equilibrium. 2. The Nash equilibria in pure strategies are (2, 4, 5) and (4, 2, 1) 3. One equilibrium in mixed strategies is ((1 2 , 1 2 ) , (1 2 , 1 2 ) , (0, 1) ) . Exercise 65. Given the bimatrix ((3, 3) (1 , 4) (4, 1) (0 , 0) ) , is ( 1 3 1 31 3 0 ) a correlated equilibrium? 1 Solution By writing the incentive constraints, we verify that ( 1 3 1 31 3 0 ) is indeed a correlated equilibrium for the game. 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. Solution 1.   1/2 0 0 0 0 0 0 0 1…