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T eoria Matematica dei Giochi - PROV A IN ITINERE 14-12-2013 Cognome: Nome: Matricola: Exercise 1 Let ( N, v) be the cooperative game, defined by N = {1, 2, 3} such that v(i)=0 ,v (1, 2) = v(1, 3) = 5 ,v (2, 3) = 20 ,v (N ) = 20 . 1. Draw the core of this game; 2. find the Shapley value; 3. find the nucleolus of this game. Solution 1. x =( x1,x 2,x 3) is in the core i↵: 8 >>>>< >>>>: x1 + x2 + x3 = 20 x2 + x3 20 x1 + x2 5 x1 + x3 5 from those equations we get: x1 = 0, x2,x 3 2 [5, 15]. So the core of the game is C(v)= co{(0, 5, 15), (0, 15, 5)}. 2. The Shapley value is: 1 = 1 6 [5 0] + 1 6 [5 0] = 5 3 2 = 3 = (20 5 3 )/2= 55 6 since player 2 and 3 are symmetric. 3. If the core is non-empty, then the nucleolus belongs to the core. In this case, the nucleolus is ⌫(v)= (0, 10, 10). 1 Exercise 2 Given the zero sum game A = ✓36 5 5 a 2 27 a 21 4 ◆ 1. find the conservative values of the players for a 2 R; 2. find if there is any value of a 2 R such that there is an equilibrium in pure strategy; 3. find a 2 R such that both players have only one optimal strategy; 4. find the optimal strategies for players 1 and 2 if a = 6. Solution 1. If a  2, vI = vII = a;i f2 <a  21 4 , vI = 2, vII = a and if a> 21 4 , vI = 2, vII = 21 4 . 2. There is an equilibrium in pure strategy i↵ vI = vII , that is for a  2. 3. There is only one optimal strategy for both players 8a> 4. If a< 4 player 1 does not have a unique optimal strategy (see graphic interpretaion of Von Neuman’s theorem); if a = 4 the second player can play ( q1,q 2, 1 q1 q2) such that 13 q1 +5 q2 = 5, with 0  q1,q 2  1. 4. If a = 6, the second and third columns are dominated by a convex combination of the first and fourth columns. We can use the indi↵erence principle with the matrix ✓36 5 2 2 21 4 ◆ to get 36 5 p + 2(1 p)=2 p + 21…