Management Engineering - Game Theory

First partial exam

GAME THEORY 5 cfu November 5Surname: Name: Matricola: Exercise 1 Given the following bimatrix game, wherea; bare real parameters 0 @(3 ;2) (4;6) (4;10) (a;4) (2; b) (1;1) (4;3) (1;0) (2;0)1 A; 1. nd the Nash equilibria in pure strategies for dierent values ofa; b2R; 2. nd the best reaction of the rst player to the strategy(14 ; 0;34 ) of the second player 3. nd for which values ofaandbthere is a Nash equilibrium with support rst and third strategy for both players; 4.prove that fora= 5there is no NE pro le with full support. Answer of exercise 1 1.(4;10)for everya; b,(a;4)ifa4; b4,(4;3)ifa4. 2.utilities (multiplied by 4) from the rows: 15; a+ 3;10 . Thus8 > < > :(1 ;0;0);ifa 12 (p;1p;0);otherwise: 3.Setting respectively(p;0;1p)and(q;0;1q)for the strategies of the two players, the inequalities/equalities to be checked are: 3q+ 44q= 4q+ 22q > aq+ 1q for the rst player and2p+ 33p= 10p >6p providing the conditionsp=311 ; q =23 ; a < 92 4.fora= 5a convex combination of the rst and second rows strictly dominates the third one. This implies that no NE can exist with full support for the rst player 1 Exercise 2 Two players have three cards: the rst has one 1 one 2 and one 3, the second has one 1 one 3 and one 4. They must select a card. The rst player wins if the sum of the two cards is odd, otherwise the second wins. Setting 1 the utility of the victory, write the matrix of the (zero sum) game. Answer of exercise 2 0 @ 11 1 1 11 11 11 A: Exercise 3 Consider the following cooperative game(N ; v)in whichN=f1;2;3gand v(S) =X i2Si 2 ; write the characteristic function of the game, nd the Shapley value and the core.Answer of exercise 3 v(1) = 1; v(2) = 4; v(3) = 9; v(1;2) = 5; v(1;3) = 10; v(2;3) = 13; v(N) = 14: Since the game is additive, thenC(v) =(v) = (1;4;9). Exercise 4 Provide an example of a weighted majority game with 4 players such that the core of the game is nonempty and not a singleton. Answer of exercise 4 (At least) Two veto players are needed. An example is[9; 4;4;1;1]. Exercise 5 Draw a length three game with backward induction outcome(4;4)for the players and such that it is not ecient (i.e. there is another outcome where the sum of the payos of the players is greater). Answer of exercise 5a bA c CDdBI IIII I (4,4)(30,0)(1,2) (3,0)(0,2) 2