Management Engineering - Game Theory

First partial exam

GAME THEORY 5 cfu November 9 2021Surname: Name: Matricola: Only papers distributed with the text of the exercise must be returned for the evaluation of the exam. Solution of the Exercise below the exercise and on the back. All answers MUST be justi ed by reporting the main calculations. No copies with wide corrections will be graded. Points: 2 for every questionExercise 1 Given the following two player game in strategic form:0 @(2 ;4) (5; a) (2;2) (3;3) (1;0) (0;4) (3;3) (2;0) (2;4)1 A whereais a real parameter, 1.Find the best reaction of Player 2 to the strategy(13 ;13 ;13 ) of Player 1; 2.Find the mixed strategies(q 1; q 2; 1q 1 q 2) for which the best reaction of Player 1 is the rst row; 3.Check, for every non negative reala, if there is a NEp with support rst third row for Player 1 and rst second column for Player 2; Answer of exercise 1 1.Since twice the expected utilities of the columns are respectively:10; a;10the best reaction is(0;1;0)ifa >10, (q;0;1q)fora > < > > :x 1+ x 3= 1 x2+ x 4= 1 x1+ x 4= 1 x2+ x 3= 1 and ndC(v). Answer of exercise 2 1.Clearly, the players are symmetric thus(v) = (12 ;12 ;12 ;12 ) ; 2.(12 ;12 ;12 ;12 ) is in the core since it is an imputation, and no coalition can object since the only coalition able to form two pairs, the grand coalition of course does not object to an imputation, while coalitions forming one pair of gloves need at least one right and one left glove, thus the coalition actually gets at least one; 3.Let(x 1; x 2; x 3; x 4) be a vector in the core. Observe that the following set of inequalities/equality must hold; 8 > > > > < > > > > :x 1+ x 3 1 x2+ x 4 1 x1+ x 4 1 x2+ x 3 1 x1+ x 2+ x 3+ x 4= 1 : Thus it follows, from the last equality, that the following system must hold:8 > > < > > :x 1+ x 3= 1 x2+ x 4= 1 x1+ x 4= 1 x2+ x 3= 1 On the other hand, clearly if(x 1; x 2; x 3; x 4) is a solution of the above linear system, it belongs to the core. It is mathematically simple to see that the system has in nitely many solutions (the rank of the matrix of the coecient is 3) and these are of the form(x; x;1x;1x). 2