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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 di erent 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 payo s 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