Problem Set 1
Problem Set 1
Ben Polak, Econ 159a/MGT522a. Four Questions due September 19, 2007.
1. Strictly and Weakly Dominated Strategies? What is the de nition of a strictly dominated strategy? What is the de nition of a weakly dominated strategy? Give an example of a two-player game matrix where one player has three strategies, one of which is strictly dominated; and the other player has three strategies, one of which is weakly (but not strictly) dominated. Indicate the dominated strategies. 2. Iterative Deletion of (weakly) Dominated two-player game 2 l c T 1; 1 0; 1 1 M 1; 0 2; 2 D 1; 3 3; 1 Strategies Consider the following r 3; 1 1; 3 2; 2
(a) Are there any strictly dominated strategies? Are there any weakly dominated strategies? If so, explain what dominates what and how. (b) After deleting any strictly or weakly dominated strategies, are there any strictly or weakly dominated strategies in the `reduced' game? If so, explain what dominates what and how. What is left? (c) Go back to your argument for deleting in the rst `round' and recall what dominated what and how. Compare this with what was deleted in the `second' round. Comment on how this might make you a bit cautious when iteratively deleting weakly dominated strategies? 3. Hotelling's Location Game. Recall the voting game we discussed in class. There are two candidates, each of whom chooses a position from the set Si := f1; 2; : : : ; 10g. The voters are equally distributed across these ten positions. Voters vote for the candidate whose position is closest to theirs. If the two candidates are equidistant from a given position, the voters at that position split their votes equally. The aim of the candidates is to maximize their percentage of the total vote. Thus, for example,
u1 (8; 8) = 50 and u1 (7; 8) = 70. [Hint: in answering this question, you do not need to write out the full payo matrices!] (a) In class, we showed that strategy 2 strictly dominates strategy 1. In fact, other strategies strictly
Ben Polak, Econ 159a/MGT522a. Four Questions due September 19, 2007.
1. Strictly and Weakly Dominated Strategies? What is the de nition of a strictly dominated strategy? What is the de nition of a weakly dominated strategy? Give an example of a two-player game matrix where one player has three strategies, one of which is strictly dominated; and the other player has three strategies, one of which is weakly (but not strictly) dominated. Indicate the dominated strategies. 2. Iterative Deletion of (weakly) Dominated two-player game 2 l c T 1; 1 0; 1 1 M 1; 0 2; 2 D 1; 3 3; 1 Strategies Consider the following r 3; 1 1; 3 2; 2
(a) Are there any strictly dominated strategies? Are there any weakly dominated strategies? If so, explain what dominates what and how. (b) After deleting any strictly or weakly dominated strategies, are there any strictly or weakly dominated strategies in the `reduced' game? If so, explain what dominates what and how. What is left? (c) Go back to your argument for deleting in the rst `round' and recall what dominated what and how. Compare this with what was deleted in the `second' round. Comment on how this might make you a bit cautious when iteratively deleting weakly dominated strategies? 3. Hotelling's Location Game. Recall the voting game we discussed in class. There are two candidates, each of whom chooses a position from the set Si := f1; 2; : : : ; 10g. The voters are equally distributed across these ten positions. Voters vote for the candidate whose position is closest to theirs. If the two candidates are equidistant from a given position, the voters at that position split their votes equally. The aim of the candidates is to maximize their percentage of the total vote. Thus, for example,
u1 (8; 8) = 50 and u1 (7; 8) = 70. [Hint: in answering this question, you do not need to write out the full payo matrices!] (a) In class, we showed that strategy 2 strictly dominates strategy 1. In fact, other strategies strictly