1662537824 February 2019

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GAME THEORY 5 cfu February 4 2019 Surname: Name: Matricola: Exercise 1 Given the following game, where Pl3 chooses the matrix ( (3, 0, 7) (1 , 4, 2) (2, 1, 0) (3 , 4, 5) ) ( (40, 1, 9) (0 , 4, 4) (9, 1, 40) ( a, 4, 0) ) 1. Reduce the game by eliminating strictly dominated strategies; 2. Find all NE in pure strategies for every a∈ R; 3. Find all NE for every a̸= 0. Answer of exercise 1 1. The second column strictly dominates the first one in both matrices, thus for PL2 is strictly dominant to play second row. 2. From before, the game reduces (with Pl3 as column player) to ((1, 2) (0 , 4) (3, 5) ( a, 0) ) From this we see that (3, 5) is a N.E. outcome and (0, 4) is another NE iff a≤ 0. 3. If a> 0, the second row strictly dominates the first one, and thus (3, 5) is the unique NE. If a< 0, applying the indifference principle we get the fully mixed [( 5 7, 2 7 ), ( a a−2, −2 a−2 )]. Finally, if a = 0 , looking at the Best Reactions, it is seen that all equilibria are of the form [(p, 1−p), (0, 1)], with p≥ 5 7. 1 Exercise 2 Let N ={1, 2, 3} and let v be the following characteristic function of the game: v({1}) = 1 = v({2}) = v({3});v({1, 2}) = v({1, 3}) = 5;v({2, 3}) = 6,v (N ) = a 1. Find a such that the game is superadditive; 2. Find a such that the core is nonempty; 3. Draw the core for a = 50; 4. Find the Shapley value of the game for all a. Answer of exercise 2 1. a≥ 7; 2. a≥ 8 3. C(v) = co{(4, 1, 45), (1, 4, 45), (1, 45, 4), (4, 45, 1), (44, 5, 1), (44, 1, 5)} 4. σ(v) = ( 2a−2 6 , 2a+1 6 2a+1 6 ) Exercise 3 Pl1 can decide where to stay among nodes a,b,c, Pl2 among c,d,e. In every node a prize of 2 is located. Each player takes the prize in the nodes where she is closer than the other player, while they share in case they are equidistant: the connections are the following:…