Management Engineering - Game Theory

Full exam

GAME THEORY 5 cfu February 3 2020Surname: Name: Matricola: Exercise 1 Given the following bimatrix game, wherea; bare real parameters 0 @(3 ;2) (4;6) (4;4) (a;3) (3;0) (10;0) (20;0) (0;0) (4; b)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(13 ;13 ;13 ) of the second player; 3.fora= 25say if there is a NE pro le such that the rst player plays the third row with positive probability; 4. nd for which values ofaandbthere is a Nash equilibrium such that the rst player plays only the third row. Answer of exercise 1 1.NE outcomes:(4;6)for alla; b,(20;0)ifb0; a20,(a;3)ifa20and allb; 2.Evaluating the utilities form the three rows we get8 > < > :(0 ;0;1);ifa 11 (0; p;1p)otherwise 3.Such a NE pro le cannot exist since the last row is strictly dominated by the second one; 4.It must beb0; suppose player two plays(q;1q;0). Then it must be 3q+ 44q20q; aq+ 33q20q: this impliesa20; qmaxf421 ; 323 ag : 1 Exercise 2 Let(N ; v)be the following TU game:N=f1; : : : ; ng,v(S) = (s+ 1)!(whereSis a nonempty coalition and s=jSj). 1.Find the Shapley value for everyn; 2.Find the core whenn= 3. Answer of exercise 2 1.All players are symmetric thus the Shapley value is the vector(( n+1)!n ; : : : ;( n+1)!n ) ; 2.The core of the game is cof(2;4;18);(4;2;18);(18;2;4);(18;4;2);(2;18;4);(4;18;2)g. Exercise 3 From a pile of cards the two players can take either 3 or 4 cards. The player that cannot move looses.1.Tell who is the winner if there are 10 cards and how she wins; 2.Find all P and N positions. Answer of exercise 3 1.the rst player, she takes 3 cards leaving the second with 7: he can go to 3 or 4, then the rst go to 0; 2.The P positions are 0,1,2 (mod 7), the other are winning positions. The starting string is PPPNNNN. Exercise 4 Given the following zero sum game:40 20 0 0 13 24 1.Find the values of the game in pure and mixed strategies; 2.Find all optimal strategies for the players. Answer of exercise 4 1.In pure strategiesv I= 0 ,v I I= 20 , the value in mixed strategies is15; 2.(38 ;58 ) for the rst player, for the second(38 ; 0;58 g . 2 Exercise 5 Consider the game below:c EFdA c GHdBI IIII II (x,y) (0,5)(4,7)(1,2) (3,0)(0,2) 1.Write the strategic form of the game (complete); 2.Find all(x; y)such that(AE G; c)is a NE pro le. Answer of exercise 5 1.The strategies for the rst player are AEG,AEH,AFG,AFH,BEG,BEH,BFG,BFH, for the second c and d. Hereis the strategic form of the game:0 B B B B B B B B B B @( x; y) (0;5) (x; y) (0;5) (x; y) (4;7) (x; y) (4;7 (1;2) (3;0) (1;2) (0;2) (1;2) (3;0) (1;2) (0;2)1 C C C C C C C C C C A 2.. Then the conditions arex1; y5. 3