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Game Theory 25-9-2015 Surname: Name: Matricola: Exercise 1 (8p o i n t s) Given the bimatrix game: 0 @ (2, 4) (3 , 2) (2 , 1) (1, 5) (0 , 0) (4 , 6) (1, 4) ( a, 1) ( b, 3) 1 A , 1. find the Nash equilibria in pure strategies for di ↵erent values of a, b 2 R; 2. find all the Nash equilibria for b = 1; 3. for b=1, find a correlated equilibrium of the form 0 @ x 0 y 00 z 000 1 A , with x, y, z > 0. Solution 1. In pure strategies (2 , 4) for all a, b and (4 , 6) if b  4. 2. The second column is dominated by the firts one, then the third row is dominated by the first one. There are the equilibria in pure strategies we found before and the equilibrium in mixed strategies {( 1 4 , 3 4 , 0), ( 2 3 , 0, 1 3 )}. 3. All the correlated equilibria of that form are given by x + y + z =1 ,x 2y, z 3y. So for instance one correlated equilibrium is 0 @ 1/301 /6 00 1 /2 000 1 A , 43 Exercise 2 ( 7 points (5 cfu) / 8 points (8 cfu) ) Consider the following zero sum game in extensive form: c d e a c d e b I II II 1 x 3 5 3 1 1. Write the game in strategic form; 2. find the conservative values of the two players for di ↵erent values of x; 3. (5 cfu) find the optimal strategies of the players if x> 2. (8 cfu) find the optimal strategies of the players for di ↵erent values of x> 1. Solution 1. The game in strategic form is given by the following matrix: ✓ 1 x 3 531 ◆ 2. The conservative values of the two players are, respectively, vI = 1 and vII = 3 for all x. 3. (5 cfu) if x> 2, the second column is dominated by a combination of the first and third column and the optimal strategies of the players are given by ( 2 3 , 1 3 ), ( 1 3 , 0, 2 3 ) . (8 cfu) if 1 <x< 2, the first column is dominated and the optimal strategies of the players are given by (p, 1 p), (0,q , 1 q) ,w h e r ep = q = 2 (x 5) .I f x…