Econ 431: David Reiley
Due: Wednesday, 7 March 2007

Problem Set #5

  1. (10 points) DS, question 9.1.
  2. (10 points) DS, question 9.2.
  3. (15 points) DS, question 9.5.
  4. (25 points) DS, question 9.6.
  5. (40 points) DS, question 9.7.
  6. (30 points) Corporate lawsuits may sometimes be signaling games. Here is one example. In 2003, AT&T filed suit against eBay, alleging that its Billpoint and PayPal electronic-payment systems infringe on AT&T's 1994 patent on "mediation of transactions by a communications system." Let's consider this situation from the point in time when the suit was filed. In response to this suit, as in most patent-infringement suits, eBay can offer to settle with AT&T without going to court. If AT&T accepts eBay's settlement offer, there will be no trial. If AT&T rejects eBay's settlement offer, the outcome will be determined by the court.

    The amount of damages claimed by AT&T is not publicly available. Let's assume that AT&T is suing for $300 million. In addition, let's assume that if the case goes to trial, the two parties will incur court costs (paying lawyers and consultants) of $10 million each.

    Because eBay is actually in the business of processing electronic payments, we might think that eBay knows more than AT&T does about its probability of winning the trial. For simplicity, let’s assume that eBay knows for sure whether it will be found innocent (i) or guilty (g) of patent infringement. From AT&T's point of view, there is a 25% chance that eBay is guilty (g) and a 75% chance that eBay is innocent (i).

    Let’s also suppose that eBay has two possible actions: a generous settlement offer (G) of $200 million or a stingy settlement offer (S) of $20 million. If eBay offers a generous settlement, assume that AT&T will accept, thus avoiding a costly trial. If eBay offers a stingy settlement, then AT&T must decide whether to accept (A) and avoid a trial, or reject and take the case to court (C). In the trial, if eBay is guilty it must pay AT&T $300 million in addition to paying all the court costs. If eBay is found innocent it will pay AT&T nothing, and AT&T will pay all the court costs.
    1. (10 points) Write down the extensive-form game tree for this game. Be careful about information sets.
    2. (5 points) Which of the two players has an incentive to bluff in this game? What would bluffing consist of? Explain your reasoning.
    3. (15 points) Write down the strategic-form game matrix for this game. Find all of the Nash equilibria to this game. What are the expected payoffs to each player in equilibrium?
  7. (40 points) In class, we played a simplified version of poker, played by the professor against a student. We each ante, then I draw a card, then I bet or fold, and if I bet, you either call or fold. If I fold, you win my $1 ante. If you fold, I win your $1 ante. If I bet and you call, I win $2 from you ($1 ante plus $1 additional bet) if I draw a king, and lose $2 to you if I draw a queen. We computed the equilibrium to that game in class.

    Now let's consider a slightly more complicated version of that game. Suppose that I now use a deck with three types of cards: 4 kings, 4 queens, and 4 jacks. All rules remain the same as before, except for what happens when I bet and you call. When I bet and you call, I win $2 from you if I have a king, we "tie" and each get back our money if I have a queen, and I lose $2 to you if I have a jack.
    1. (10 points) Draw the game tree for this game. Label the two players as "professor" and "student." Be careful to label information sets correctly. Indicate the payoffs at each terminal node.
    2. (5 points) How many pure strategies does the professor have in this game? Explain your reasoning.
    3. (5 points) How many pure strategies does the student have in this game? Explain your reasoning.
    4. (5 points) Represent this game in strategic form. This should be a matrix of expected payoffs for each player, given a pair of strategies.
    5. (5 points) Find the unique pure-strategy Nash equilibrium to this game. Explain in English what the strategies are.
    6. (5 points) Would you call this a pooling equilibrium, a separating equilibrium, or a semi-separating equilibrium? Explain.
    7. (5 points) In equilibrium, what is the expected payoff to the professor of playing this game?
  8. (20 points) Suppose you are the owner of Perfecta Potato Corporation, looking for a manager to undertake a project on genetic engineering of square potatoes. The outcome of the project is uncertain, and the probability of success depends on the quality of the manager’s effort. If successful, the project will earn $1,000,000. The probability of success is 50% if the effort is of routine quality, but 80% if it is of high quality.

    Putting out high-quality effort entails a subjective cost to the manager. For example, she may have to think about the project night and day, and her family may suffer. Her opportunity cost of routine effort is $200,000, and for extra effort her opportunity cost is an additional $150,000. You cannot observe the quality of the effort she puts forth; all you can observe is whether the project is successful.

    We want to look for the optimal contract (base salary s plus bonus b) that you could offer to the manager, generating maximum profit. Assume the manager is risk neutral, so her objective i s to maximize her expected payment minus her effort cost. The following subproblems will help you arrive at the optimal contract.
     
    1. When you offer a contract, there are three things that could happen. First, the manager could accept your contract and put forth extra effort. Second, the manager could accept your contract and put forth regular effort. Third, the manager could choose not to work for you at all. Suppose that you offer the manager a contract that pays her $200,000 as base salary, plus a $150,000 bonus if the project is successful. Under this contract, which of the three choices would you expect the manager to make? Explain your reasoning.
    2. Let's see what happens if you want to get the manager to put forth regular effort. What constraint on s and b needs to be satisfied in order to get the manager to put forth even regular effort? (That is, what is the “particip ation constraint” for regular effort?)
    3. What is the simplest and cheapest contract you can offer to get the manager to put forth regular effort? That is, what values of s and b would you want to choose?
    4. How much expected profit would you make with the contract in (c)?
    5. Now let's see what happens if you want to get the manager to put forth extra effort. What constraint on s and b needs to be satisfied in order to get the manager to put forth extra instead of regular effort? (That is, what is the “incentive constraint” for extra effort?)
    6. What constraint on s and b needs to be satisfied in order to get the manager to put forth extra effort instead of not working at all? (That is, what is the “participation constraint” for extra effort?)
    7. What is the optimal contract for you to offer that gets the manager to put forth extra effort? That is, what is the cheapest way for you to simultaneously satisfy the two constraints in (e) and (f)?
    8. How much expected profit would you make with the contract in (g)?
    9. Compare your answers to (d) and (h). What is the optimal contract for you to offer to the manager?
  9. (20 points) Consider the second version of the Attacker-Defender game in the text, where the cost of sending a signal is only 4. That is, C=4, F=10, G=5, and p=0.25.
    1. (10 points) Represent this game instrategic form. You should be looking for a 4x4 game matrix. Remember to condition the Attacker's strategies on whether or not she sees a signal from the Defender. Here's an example of how to compute expected payoffs for the table. Suppose the Defender decides to signal if tough but not if weak, and suppose the Attacker decides not to invade if there's a signal but to invade if there's not signal. Then the expected payoff to the Defender is -4.75 and the expected payoff to the Attacker is 3.75.
    2. (5 points) Find all pure-strategy Nash equilibria to the game. Describe which are pooling and which are separating equilibria. If you're uncertain of your results, you might find it helpful to check your work against the last paragraph of Section 9.5.A and all of Section 9.B.
    3. (5 points) Verify that the semiseparating equilibrium found in Section 9.5.C of the text is actually an equilibrium to this strategic-form game. I recommend using the same approach you used in verifying the mixed-strategy Nash equilibria of problems 7.9 and 7.10.
  10. (Extra credit - 15 points) Let's change the example in section 9.3 of the text. Now suppose there are three intrinsic types of student ability. To employers, type-A students are worth $200,000 per year in salary, type-B students are worth per year $150,000 in salary, and type-C students are worth $100,000 per year in salary. For taking each challenging course, type-A students incur disutility worth $6,000 per year of salary, type-B students incur disutility worth $8,000 per year of salary, and type-C students incur disutility worth $12,000 per year of salary.
    1. Suppose employers want to screen students for the different jobs, using the number of tough courses taken by each student. Assume that nA is the number of tough courses required to earn a type-A salary, and nB is the number of tough courses required to earn a type-B salary. Assume that if a student takes zero tough courses, they can earn a type-C salary. Now instead of two incentive-compatibility constraints, there will be six incentive-compatibility constraints. What are these six constraints?
    2. What are the minimum numbers of courses that a type-A student and a type-B student can be required to take so that the three types choose to separate themselves fully?
  11. (Extra credit - 15 points) In the stripped-down poker game played in class between the professor and the student, what fraction of the deck would have to be kings in order for it to be a "fair" game? That is, what would have to be the proportion of kings in order to make the equilibrium payoff equal to zero for both players? Show your work.
  12. (Extra credit - 10 points) In addition to the pure-strategy equilibrium you found above, there are multiple mixed-strategy Nash equilibria to the king-queen-jack poker game of problem 7. What are all the mixed-strategy equilibria?