Exercise 4 solution

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GAME THEORY 2017-2018 5 cfu 4 Zero sum games Exercise 47. Consider the zero sum game in extensive form in Figure 1. Write the game in strategic form and solve it. d e f a b c I II -1 0 2 1 5 Figure 1: Exercise 47 Solution The game in strategic form is given by the following matrix:   2 1 5 −1 −1 −1 0 0 0   The conservative values of the two players are vI = vII = 1 , therefore there exists an equilibrium in pure strategies, that corresponds to the first row ans second column. This is the only optimal result, since this outcome can be derived by elimination of dominated strategies. Indeed, the second and third rows are (strictly) dominated by the first one. Then we can eliminate the two rows and further observe that the first and third column are dominated by the second one and we get the result. Exercise 48. Find if there are equilibria in pure strategies in the following matrices: A =   9 6 11 7 6 8 3 0 6 11 3 2 8 6 12 7 6 7 0 1 10 3 2 6   , B =   3 2 1 0 0 1 2 0 1 0 2 1 3 1 2 2   . 1 Solution Game A : Player I can obtain at least 6 playing the first row, 0 playing the second, 6 from the third and 0 from the fourth. The maximum is 6, then the conservative value of the first player is vI = 6 playing the first or the third row. Similarly the conservative value of the second player is vII = 6 playing the second or the fifth column. Therefore v = 6 is the value of the game and a12, a15, a32 and a35 are equilibria. Observe that, on the other hand, a23 and a46 are not equilibria. Game B : vI̸= vII , therefore there are no equilibria in pure strategies. Exercise 49. Given the zero sum game described by the following matrix: A =   3 6 5 5 2 4 1 0 3   find a pair of optimal strategies for the players and the value of the game. Solution First of all, notice that…