Management Engineering - Game Theory

Full exam

GAME THEORY 5 cfu July 14Surname: Name: Matricola: These and only these papers must be returned for the evaluation of the exam. All answers MUST be justi ed by reporting the main calculations. No copies with corrections will be graded. Points:Ex. 1: 2+3+3+3+3Ex. 2: 3+3+3Ex. 3: 2+2+3Ex. 4:2+3+3.SOLVE THE EXERCISES PROVIDING SHORT EXPLANATIONS USING ONLY THESE PAPERS Exercise 1 Given the following bimatrix game, wherea; bare real parameters, 0 @(3 ;5) (2;2) (4;8) (2;1) (4;1) (4;5) (3; b) (a;9) (1;7)1 A; 1.Find the pure NE pro les for everya; b; 2. nda; bsuch that the game is a potential game; 3. nd the best reaction of player II to the strategy(12 ;14 ;14 ) of Player I; 4.Say if there is a NE pro le with Player I playing(12 ;14 ;14 ) ; 5.Say if there is a NE pro le such that Player I plays(p;0;1p)with0< p < > :(1 ;0;0);ifb >17 (0;0;1);ifb =P. 3.In order to nd the optimal mixed strategies it is sucient to solve the following linear system of inequalities:8 > > > > > > > > > < > > > > > > > > > :2 x 2 3x 4 0 2x 1+ x 3 0 x 2+ 4 x 4 0 3x 1 4x 3 0 x1; x 2; x 3; x 4 0 x1+ x 2+ x 3+ x 4= 1 which admits as solutions all the vectors of the form(0; p;0;1p)for somep2(35 ;45 ) . 4 Exercise 4 A University has apartments available on campus for students. The evaluations of the students are summarized in the following table:Apt. 1Apt. 2Apt. 3Apt 4 Alberto50404030 Beatrice7002010 Carla60704030 Emanuele20306040 So a40306050 The University decides to apply the VCG mechanism. 1.Who remains without apartment? 2.How much pays So a? 3.How much pays Emanuele? Answer of exercise 4 1.Because of eciency, apartments are assigned in the following way: 1 to Beatrice, 2 to Carla, 3 to Emanuele,4 to So a; 2.If So a is not present, her apartment is assigned to Alberto thus the loss in valuation of the other players dueto Beatrice is 30, so she will pay 30; 3.If Emanuele is not present, the assignment will be 1 to Beatrice 2 to Carla, 3 to So a, 4 to Alberto. The lossin total valuation without Emanuele is 40 (a loss for Alberto of 30, and for So a of 10), so he pays 40. 5