ECON 3600 Assignment 4

1. (4 points)
   An executive has been given two stock options as a bonus. Each option gives the executive the right (but not the obligation) to purchase one share of the company’s stock for $50, as long as she does it before the close of the stock market tomorrow afternoon. When the executive exercises either option, she must immediately sell the stock bought from the company at the market price in effect at the time. The stock price today is $55, so if she exercises either option today she is guaranteed a profit of $5 per option exercised. The stock price tomorrow will be either $45 or $65 with equal probability. This means that if she waits until tomorrow and the stock price rises to $65, she can exercise any remaining options for a profit of $15 per option. On the other hand, if the stock price falls to $45, then exercising either option results in a loss
   of $5 per option. Today the executive can (i) exercise both options, (ii) exercise one option today and wait until tomorrow to decide about the second one, and (iii) exercise neither option today and wait until tomorrow to decide what to do about both. Tomorrow the executive must either exercise any remaining option(s) or let them expire unused.
2. (a) Draw a game tree for this game between the executive and Nature; find the subgame perfect equilibrium. Remember to label the action taken along each branch, who moves at each decision node (the executive or Nature), and the total change in the executive’s wealth at each terminal node. Assume that there are no brokerage commissions or taxes. The executive is risk-neutral and her actual payoff (utility) at
   each terminal node equals the change in total wealth. The executive maximizes her expected utility.
   (b) Suppose that the executive knows today what the firm’s stock price will be tomorrow (she has "insider information"). Draw a second game tree that reflects these changes (now Nature moves first) and find the executive’s optimal strategy. Could the executive’s actions today and tomorrow be used to show that she had insider information at the time she acted? (compare the optimal strategies in (a) and (b)).
3. (1 point)
   Consider a population of 10,000 persons in which 100 persons (1%) have a certain genetic defect and 9,900 do not. Suppose that everyone takes a test. For persons with the defect, the test will be (correctly) positive with probability 99%. Of the remaining 9,900 without the defect, 2% will receive a false positive result. Given this information, what is the probability of the genetic defect in a random person with a positive test
   result?
4. (3 points)
   Two people are involved in a dispute. Person 2 does not know whether person 1 is strong or weak. It is common knowledge that Prstrong  0. 5. Person 1 is fully informed. Each person can either fight or yield. Each person receives a payoff of 0 if she yields (regardless of the other person’s action) and a payoff of 1 if she fights and her opponent yields; if both people fight, then their respective payoffs are (1,-1) if person 1 is strong and (-1,1) if person 1 is weak. Formulate this interaction as a
   Bayesian game and find its Nash equilibrium (equilibria).
5. (3 points)
   Two players simultaneously and independently have to decide how much to contribute to a public good. If player 1 contributes x1 and player 2 contributes x2, then the value of the public good is 2x1  x2  x1x2, which they each receive. Assume that x1 and x2 are positive numbers. Player 1 must pay a cost x1 2 of contributing. Player 2 pays the cost tx2
6. The number t is private information to player 2. It is common
   knowledge that t equals 2 with probability 2/3 and equals 3 with probability 1/3. For each player, the payoff is the difference between the value of the public good and the player’s cost of contributing. Find the Bayesian Nash equilibrium of this game; report expected payoffs.
7. (5 points)
   Consider a game with two players, a host and a guest. The host moves first; he decides whether to invite the guest or not. If the host does not invite the guest, each player’s payoff is 0. If the host invites the guest and the invitation is accepted, the host’s payoff is 1; if rejected, the host’s payoff is -1. The host may be good or bad (boring); the host knows his own type. The guest has to decide whether to accept or reject the invitation, without knowing the host’s type. If the host is good (bad), the
   guest’s payoff from accepting the invitation is 1 (-1). If the guest rejects the invitation, the guest’s payoff is 0. In the absence of any other information, the guest assumes that the probability of the host being good is p. This is common knowledge.
   (a) Draw a game tree for this game.
   (b) What range of the values of p will support a pooling equilibrium, in which both types of hosts invite guests and the guests accept? Describe carefully the strategies for both players and the guests’ beliefs.
   (c) Does there exist a separating equilibrium, in which only good hosts invite guests and such invitations are always accepted? Explain carefully why or why not.
8. (4 points)
   Consider a game with two players, an Attacker and a Defender. The Defender may be Tough or Weak. The Attacker has to decide whether to Invade, without knowing the Defender’s type. It is common knowledge that the probability of the Defender being Tough is 0.25. The Defender moves first. His two available actions are to signal that he is tough (action S) or not to signal (NS). Signaling is costly.
   (a) Does there exist a separating equilibrium, in which the Defender chooses S if Tough and NS if Weak? If yes, describe carefully the strategies for both players and the Attacker’s beliefs. If no, explain why.
   (b) Does there exist a pooling equilibrium, in which both types of the Defender choose NS? If yes, describe carefully the strategies for both players and the Attacker’s beliefs. If no, explain why.