Econ 431: David Reiley
Due: Wednesday, 31 January 2007

Problem Set #2

Questions labeled as "DS" come from the end-of-chapter exercises in Dixit and Skeath. "Question 3.1" refers to question 1 at the end of chapter 3.

Remember that although you may discuss these questions with your classmates as much as you like, you are responsible for preparing your own independent writeup, in your own words. Merely copying someone else's work constitutes academic dishonesty, and will be punished.

1. (5 points) DS, question 2.4.
2. (10 points) DS, question 3.1.
3. (10 points) DS, question 3.3.
4. (5 points) DS, question 3.5.
5. (10 points) DS, question 3.7. The reasoning here will be somewhat similar to that in the game of Nim, which we played on the first day of class. Solving this game requires some creativity, so get started early. I recommend brainstorming and collaborating with your classmates.
6. (10 points) DS, question 3.8.
7. (10 points) DS, question 3.9.
8. (5 points) Consider the ultimatum bargaining game ("proposer/responder") that we played in class.
   1. Describe the rollback-equilibrium outcome, including the complete strategies for each of the two players. Recall the proper definition of a "strategy" from page 25 of your text, and recall that the proposer in the game had eleven different possible offers to choose from.
   2. Discuss why so many people in class played strategies other than the rollback equilibrium. What part of the theory needs to be changed, in order to fit the data better?
   3. (Optional - 5 extra-credit points) Can you think of another experiment you could perform in order to figure out whether your answer in (b) is correct? Describe your experiment, and how you could tell whether your proposed theory in (b) was correct.
9. (15 points) DS, question 4.2. See the bottom of page 86 and the bottom of page 112 for help in understanding the notation for a zero-sum game table.
10. (5 points) DS, question 4.4.
11. (5 points) DS, question 4.5.
12. (10 points) DS, question 4.6.
13. (10 points) DS, question 4.8.
14. (15 points) DS, question 4.10.
15. (15 points) Return to the mall-location game of question 3.8 in the text. The rules of this game specify that, when all three stores request space in Urban Mall, the two bigger (more prestigious) stores get the available spaces. The original version of the game also specifies that the firms move sequentially in requesting mall space.
    1. Suppose that the three firms make their location requests simultaneously. Draw the payoff table for this version of the game and find all of the Nash equilibria. Which one of these equilibria do you think is most likely to be played in practice? Explain.
       
       Now suppose that when all three stores simultaneously request Urban Mall, the two spaces are allocated by lottery, giving each store an equal chance of getting into Urban Mall. With such a system, each would have a two-thirds probability (or a 66.67% chance) of getting into Urban Mall when all three had requested space there.
       
    2. Draw the game table for this new version of the simultaneous-play mall-location game. Find all of the Nash equilibria of the game.Which one of these equilibria do you think is most likely to be played in practice? Explain.
    3. Compare and contrast the equilibria found in part (b) with the equilibria found in part (a). Do you get the same Nash equilibria? Why or why not?
16. (15 points) Three friends (Julie, Kristin, and Larissa) independently go shopping for dresses for their high-school prom. Upon reaching the store, each girl sees only three dresses worth considering: one black, one lavender, and one yellow. Each girl furthermore can tell that her two friends would consider the same set of three dresses, because all three have somewhat similar tastes.
    
    Each girl would prefer to have a unique dress, so a girl's utility is zero if she ends up purchasing the same dress as at least one of her friends. All three know that Julie strongly prefers black to both lavender and yellow, so she would get a utility of 3 if she were the only one wearing the black dress, and a utility of 1 if she were either the only one wearing the lavender dress or the only one wearing or the yellow dress. Similarly, all know that Kristin prefers lavender and secondarily prefers yellow, so her utility would be 3 for uniquely wearing lavender, 2 for uniquely wearing yellow, and 1 for uniquely wearing black. Finally, all know that Larissa prefers yellow and secondarily prefers black, so she would get 3 for uniquely wearing yellow, 2 for uniquely wearing black, and 1 for uniquely wearing lavender.
    
    1. (5 points) Provide the game table for this three-player game. To make your work easier to grade, please make Julie the row player, Kristin the column player, and Larissa the page player.
    2. (5 points) Identify any dominated strategies in this game, or explain why there are none.
    3. (5 points) What are the pure-strategy Nash equilibria to this game?
17. (10 points) Construct the payoff matrix for your own two-player game that satisfies the following requirements. First, each player should have three strategies. Second, the game should not have any dominant strategies. Third, the game should not be solvable using minimax. Fourth, the game should have exactly two pure-strategy Nash equilibria. Provide your game matrix, and then demonstrate that each of the above conditions are true.
18. (Optional - 10 extra-credit points) This is a modified version of DS, question 3.10. Consider the Survivor game tree illustrated in Figure 3.9. We might not have guessed exactly the values Rich estimated for the various probabilities, so let's generalize this tree by considering other possible values. In particular, suppose that the probability of winning the immunity challenge when Rich chooses Continue is x for Rich, y for Kelly, and 1-x-y for Rudy; similarly, the probability of winning when Rich gives up is z for Kelly and 1-z for Rudy. Further, suppose that Rich's chance of being picked by the jury is p if he has won immunity and has voted off Rudy; his chance of being picked is q if Kelly has won immunity and has voted off Rudy. Continue to assume that, if Rudy wins immunity, he keeps Rich with probability 1, and that Rudy wins the game with probability 1 if he ends up in the final two. Note that in the example of Figure 3.9, we had x=0.45, y=0.5, z=0.9, p=0.4, and q=0.6. (In general, the variables p and q need not sum to 1, though this happened to be true in Figure 3.9.)
    1. Find an algebraic formula, in terms of x, y, z, p, and q, for the probability that Rich wins the million dollars if he chooses Continue. (Note: Your formula might not contain all of these variables.)
    2. Find a similar algebraic formula for the probability that Rich wins the million dollars if he chooses Give Up. (Again, your formula might not contain all of these variables.)
    3. Use these results to find an algebraic inequality telling us under what circumstances Rich should choose Give Up.
    4. Suppose all the values are the same as in Figure 3.9 except for z. How high or low could z be, so that Rich would still prefer to Give Up? Explain intuitively why there are some values of z for which Rich is better off choosing Continue.
    5. Suppose all the values are the same as in Figure 3.9 except for p and q. Assume that since the jury is more likely to choose a "nice" person who doesn't vote Rudy off, we should have q>0.5>p. For what values of the ratio (p/q) should Rich choose Give Up? Explain intuitively why there are some values of p and q for which Rich is better off choosing Continue.