Exercise 3 solution

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GAME THEORY 2017-2018 5 cfu 3 Games in strategic form and Nash equilibria Exercise 33. Find the values of (a, b) such that the following game has an outcome determined by elimination of strictly dominated strategies.   (3, 3) ( a, b) (4 , 2) (2, 2) (6 , 3) (3 , 1) (5, 4) (7 , 8) (6 , 1)   Solution The second row is always dominated by the third one and the last column is always dominated by the first one. Then if b≥ 3 the second column dominates the first one and there is an equilibrium in pure dominated strategies iff a̸= 7. On the other hand, if a < 7 the last row dominates the first one and then (7,8) is the outcome determined by the elimination of strictly dominated strategies. Exercise 34. Find the equilibria in pure strategies of the following game   (5, 4) (1 , 6) (0 , 3) (5 , 1) (5, 3) (3 , 2) (1 , 0) (4 , 3) (2, 5) (4 , 0) (1 , 5) (2 , 1)   Solution The equilibria in pure strategies are (5, 3) and (1, 5). Exercise 35. Consider the following game in strategic form: ( (1, 2) (1 , 3) (0, 2) ( a, b) ) Find the Nash equilibria in pure strategies for different values of a and b. What are the conditions on a and b for the game to be derived from a game in extensive form with perfect information? Solution The Nash equilibria in pure strategies are (1, 3) if a≤ 1 and (a, b) if a≥ 1, b≥ 2. If (a, b) = (0 , 2) the game is described by the tree in Figure 1a; if (a, b) = (1 , 3) the game is described by the tree in Figure 1b. Exercise 36. Find the Nash equilibria for the Battle of Sexes. 1 I II (0, 2) (1, 2) (1, 3) (a) Exercise 35 II I (1, 3) (1, 2) (0, 2) (b) Exercise 35 Solution The Battle of Sexes is described by a bimatrix like the following one: ( (10, 0) ( −5,−5) (−10,−10) (0 , 10) ) there are no dominated strategies. The Nash equilibria in pure strategies are…