Econ 431: David Reiley
Due: Wednesday, 18 April 2007

Problem Set #8

  1. (20 points) Two friends intend to meet at a coffeeshop. There are two choices: Starbucks (a national chain), and Espresso Art (a local hangout). Unfortunately, the friends don’t have cell phones, so they can’t confirm where it was that they are supposed to meet. Each friend prefers meeting at Café Paraiso to meeting at Starbucks, but they also prefer meeting to a failure to meet. The payoff matrix is as follows:

    Starbucks
    Espresso Art
    Starbucks
    1,1
    0,0
    Espresso Art
    0,0
    3,3

    1. What are all the Nash equilibria to this game? Make sure to consider equilibria in both pure and mixed strategies.
    2. Now consider an evolutionary rather than a rational/strategic approach to this game. Suppose that there is a large population of players of two types. Proportion s of the population is type S, which is hard-wired to go to Starbucks every time the game is played. Proportion (1-s) of the population is type P, which is hard-wired to go to Espresso Art every time. What is the fitness of the S type?
    3. What is the fitness of the P type?
    4. Graph both fitness curves, with the proportion s on the horizontal axis.
    5. What are the evolutionary equilibria to this game? Which of them are stable, and which are unstable? Explain your reasoning.
  2. (20 points) DS, question 13.2.
  3. (25 points) DS, question 13.3.
  4. (20 points) (This is a replacement for DS, Problem 13.4.) In Section 7.A, we considered testing for ESS in the thrice-repeated restaurant-pricing prisoners' dilemma. Figure 13.13 shows the fitness table for three different types: A (always defect), T (tit-for-tat), and N (never defect). To check your understanding, let's expand the problem to a four-period repeated prisoners' dilemma.
    1. Using the original one-period game as shown in Figure 13.1, construct a 3x3 fitness table for the four-period prisoner's dilemma, analogous to what we did for the three-period PD in Figure 13.13.
    2. Explain completely, on the basis of this fitness table, why an all-A population cannot be invaded by either N-type or T-Type mutants.
    3. Explain also why an all-N-type population can be invaded by both type-A and type-T mutants.
    4. Explain finally why an all-T-type population cannot be invaded by type-A mutants but can be invaded by mutants that are type N.
  5. (20 points) (This is a replacement for DS, Problem 13.7.) Refer to the previous problem. Suppose that we add a fourth possible type (type S) to the population. This type cooperates on the first play and defects on the second play of each two-period game, no matter what the opponent does.
    1. Draw the four-by-four fitness table for this game.
    2. If the population initially has some of each of the four types, show that the types N, T, and S will die out in that order.
    3. Show that, after a strategy (N, T, or S) has died out, as in part (b), if a small proportion of mutants of that type reappear in the future, they cannot successfully invade the population at that time.
    4. Assuming there are only four possible hard-wired types, what is the ESS in this game?
  6. (15 points) DS, question 13.8.
  7. (20 points) Consider an evolutionary game with two species of individuals: Bakers and Cutlers. Each time a Baker meets a Cutler, they play the following game. The Baker chooses the total prize to be either $10 or $100. The Cutler chooses how to divide the prize chosen by the Baker: the Cutler can choose either a 50:50 split, or a 90:10 split in the Cutler’s own favor. The Cutler moves first, and the Baker moves second.

    There are two types of Cutlers in the population: type F chooses a fair (50:50) split, while type G chooses a greedy (90:10) split. There are also two types of Bakers: type S simply chooses the large prize ($100) no matter what the Cutler has done, while type T chooses the large prize($100) if the Cutler chooses a 50:50 split, but the small prize ($10) if the Cutler chooses a 90:10 split.

    Let f be the proportion of type F in the Cutler population, so that (1-f) represents the proportion of type G. Let s be the proportion of type S in the Baker population, so that (1-s) represents the proportion of type T.
    1. Find the fitness of each of the four types of players.
    2. Use this information to sketch a two-dimensional graph (s versus f) displaying the population dynamics.
    3. Describe the evolutionary equilibria to this game, and indicate which ones are stable.
  8. (20 points) DS, question 13.9. I recommend that you follow the same procedure (parts a through c) as in the preceding problem. Refer to the game in Figure 13.8. There is a typo in the first printing of the book; the problem should have x be equal to the proportion of S types among the population of row players.
  9. (20 points) Analyze an evolutionary version of the tennis-point game (Figure 4.15). Regard servers and receivers as separate species, each of which has two phenotypes (DL or CC).
    1. Construct a figure analogous to Figure 13.11.
    2. What are the evolutionary equilibria to this game?
    3. Which one of the equilibria is most stable?
    4. If you started at the evolutionary equilibrium you described in (c) and perturbed it by introducing some mutant DL servers, describe qualitatively how you would expect things to change dynamically.
  10. (20 points) DS, question 14.1.
  11. (20 points) DS, question 14.2.
  12. (50 points) DS, questions 14.4 or 14.5. Choose just one of the six scenarios to analyze. You may find it helpful to know that the UA Library has electronic databases of newspaper and magazine articles available online. The "optional" mathematical model is worth only 5 out of the 50 points.
  13. (Extra credit, 20 points) DS, question 13.10.
  14. (Extra credit, 25 points) DS, questions 14.4 or 14.5. If you chose to analyze a success above, for the extra credit you can choose to analyze a failure as well. If you chose to analyze a failure, then here you can choose to analyze a success.