22 - ECON 3600 Problem Set 2(from the Yale Open Course in...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
ECON 3600 Problem Set 2 (from the Yale Open Course in Game Theory Problem Sets) Due Wednesday, January 19. 1. Recall the voting game we discussed in class. There are two candidates, each of whom chooses a position from the set S i = {1, 2, 3,..., 10}. The voters are equally distributed across these ten positions. So, 10% of the voters are at each position. 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, u 1 (8, 8) = 50 and u 1 (7, 8) = 70. (Where u 1 (7, 8) means Player 1's payoff when Player 1 plays 7, Player 2 plays 8.) [Hint: in answering this question, you do not need to write out the full payoff matrices!] (a) In class, we showed that strategy 2 strictly dominates strategy 1. In fact, other strategies strictly dominate strategy 1. Find all the strategies that strictly dominate strategy 1. Explain your answer. [Hint: try some guesses and see if they work.] Solution: Strategies 2, 3, 4, 5, 6, and 7 strictly dominate 1. Strategies 8, 9, and 10 do not. (e.g. u 1 (1, 7) = 35 > u 1 (8, 7) = 30). (b) Suppose now that there are three candidates. Thus, for example, u 1 (8, 8, 8) = 33.3 and u 1 (7, 9, 9) = 73.3. (Where u 1 (7, 9, 9) means Player 1's payoff when Player 1 plays 7, Player 2 plays 9 and Player 2 plays 9.) Is strategy 1 dominated, strictly or weakly, by strategy 2? How about by strategy 3? Explain. Suppose we delete strategies 1 and 10. That is, we rule out the possibility of any candidate choosing either 1 or 10, although there are still voters at those positions. Is strategy 2 dominated, strictly or weakly, by any other (pure) strategy s i in the reduced game? Explain. Solution: Strategy 1 is now weakly dominated by strategy 2 and strategy 1 is weakly dominated by strategy 3. For example: u 1 (1, 2, 3) = 10 = u 1 (2, 2, 3) = 10, u 1 (1, 2, 4) = 10 = u 1 (3, 2, 4) = 10 If 1 and 10 are no longer options, then 2 is not dominated by any other pure strategy. For example, u 1 (2, 3, 4) = 20 > u 1 (3, 3, 4) = 15 and the payoffs for playing 2 will be strictly larger for any strategies 4 through 8 when it is "surrounded" by the other two players.
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern