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Game Theory (MID-TERM) 28-11-2015 Surname: Name: Matricola: Exercise 1 (3+3+3p o i n t s ) Consider the following zero-sum game: 0 @ 3243 b 546 2 a 31 1 A 1. find the conservative values of the players and say for which values of a and b there are equilibria in pure strategies; 2. find all the optimal strategies of the players for a< 2, b = 1; 3. find all the optimal strategies of the players for a< 2, b = 3. Exercise 2 (3+3+3p o i n t s ) Let ( N, v) be the TU-game defined as follows: N = {1,...,n } and for A ✓ N : v(A)= ⇢ 1i f |A| is odd |A| otherwise. 1. Say for which n the game is superadditive; 2. Evaluate the Shapley value for every n; 3. Say for which n the core of the game is nonempty and explain why. 1 Exercise 3 (3+3+3p o i n t s ) 1. Consider the following game in strategic form: 0 @ (3, 0) (2 , 1) (2, 4) (2 , 0) (0, 0) (2 , 2) 1 A . Is the pair of strategies ((1 , 0, 0), ( 1 3 , 2 3 ) a Nash equilibrium? And (( 1 2 , 0, 1 2 ), (0, 1))? 2. Consider the Nim game with starting position (9 , 7, 11, 10). Which player has a winning strategy? How many winning moves does he have when he plays for the first time? Enumerate them. 3. Consider the following game in extensive form. x y a z w b I II II (3,1) (0,1) (2,0) (1,2) (a) Enumerate the strategies of the two players. (b) Solve the game using backward induction and say which are the optimal (pure) strategies for the two players. First Question (4p o i n t s) The core of a TU-game. Second Question (4p o i n t s) The Bouton theorem. 2