1. (20 points) Find all of the Nash equilibria for the following game.
Table 1: A Takeover Game
Target
Hard Medium Soft
Hard -3, -3 -1,0 4,0
Raider: Medium 0, 0 2,2 3,1
Soft 0,0 2,4 3,3
Payoffs to: (Raider, Target).
ANSWER. The three equilibria are in pure strategies: (Hard, Soft), (Medium, Medium), and (Soft, Medium). I gave full credit if you found those, even if you ignored the mixed strategy equilibria. There are two mixed strategy equilibria.
(1) ( Raider: Hard. Target: Mix between Medium and Soft.) Notice that for Target, Hard is a dominated strategy. That means it will not be part of any mixed strategy equilibrium. Next, notice that Soft is weakly dominated. Thus, if Raider ever plays anything but Hard, Target will want strongly to play Medium. What if Raider plays Hard? Then Target would be willing to mix between Medium and Soft. If Target plays Medium with a probability of $\gamma$, Raider's payoff from Hard is $(-1) \gamma + 4 (1-\gamma)$, whereas his payoff from Medium or Soft is $(2) \gamma + 3 (1-\gamma)$. Equating these yields $\gamma^* = $. 25. If $\gamma$ is no bigger than .25, we have a Nash equilibrium.
(2) ( Raider: Mix between Medium and Soft. Target: Medium. ) How about if Target plays Medium and Raider mixes? Raider would only want to mix between Medium and Soft. But that would generate a Nash equilibrium, for any mixing probability, since Raider gets 2 no matter what, and Target prefers Medium no matter what the mixing probability may be.
2. (20 points) Elena and Mary are the two leading figure skaters in the world. Each must choose during her training what her routine is going to look like. She cannot change her mind later and try to alter any details of her routine. Elena goes first in the Olympics, and Mary goes next. Each has five minutes for her performance. The judges will rate the routines on three dimensions: beauty, how high they jump, and whether they stumble after they jump. A skater who stumbles is sure to lose, and if both Elena and Mary stumble, one of the ten lesser skaters will win, though those ten skaters have no chance otherwise.
Elena and Mary are exactly equal in the beauty of their routines, and both of them know this, but they are not equal in their jumping ability. Whoever jumps higher without stumbling will definitely win. Elena's probability of stumbling is P(h) , where h is the height of the jump, and P is increasing smoothly and continuously in h . (In calculus terms, P' and P'' both exist, and P' is positive) Mary's probability is (P(h) - .1) --- that is, it is 10 percent less for equal heights.
Let us define as h=0 the maximum height that the lesser skaters can achieve, and assume that P(0) = 0.
(a) Show that it cannot be an equilibrium for both Mary and Elena to choose the same value for h (Call them M and E ).
ANSWER. Suppose they did choose the same value, so E= B . Then Mary could deviate and choose a slightly higher h , and she would only slightly increase her probability of stumbling, but would win for sure if neither stumbled.
Notice that this is a simultaneous-move game. The two skaters do not skate simultaneously, but they must make their decisions prior to the Olympics, during training, so they effectively are making simultaneous choices.
This question proved surprisingly difficult. It is a good example of a simple proof. Another good exercise would be to do a careful mathematical answer to this question,using the definition of continuity and laying out the payoff functions.
(b) Show for any pair of values (M,E) that it cannot be an equilibrium for Mary and Elena to choose those values.
ANSWER. Part (a) showed that a pair would not be an equilibrium if M=E , so suppose M not equal to E . Then whoever had the higher value of h would want to reduce her value until it was infinitesimally higher than that of the other skater. But if the values became very close, then whoever had the lower value would want to increase her value to become the highest, for the reasons in part (a). Thus, there is no pure strategy equilibrium.
Both (a) and (b) show the power of the seemingly simple idea of Nash equilibrium. Start with a hypothesized equilibrium, and test for whether any player wants to deviate unilaterally. Do that, and the answer flows out naturally. Start by trying to think what each player ought to do as an optimal strategy and you get tied up in knots.
(c) Describe the optimal strategies to the best of your ability. (Do not get hung up on trying to answer this question; my expectations are not high here.)
ANSWER. The equilibrium must be in mixed strategies. Suppose each skater mixes between all the values of h between 0 and some maximal value Hbar which is the same for both skaters. The mixing must be all the way down to zero because if it started from some lower bound L greater than 0 , then there is no point in assigning any probability to L , since it is sure to lose but increases your chances of stumbling over h=0 . The maximal value must be the same for both players because there is no point in having a maximal value greater than the other skater-- that just increases the chance of stumbling.
The mixing would be over the interval between the two bounds, because if there were a hole in the support for mixing, the same reasoning as in the last paragraph would imply that there is no reason to put probability on the height which is the upper bound of the hole-- you are just as likely to win with a height just infinitesimally greater than the lower bound of the hole.
The mixing probabilties would be different for the two skaters because their skills are different, I conjecture, and Mary would have a greater probability of winning. But my conjecture might be wrong, since the marginal incentives for the two players are the same-- the better skater's stumbling probability is different by a constant, not by a multiple.
(d) What is a business analogy? Find some situation in business or economics that could use this same model.
ANSWER. One business analogy is to research in a patent race. Two firms are competing to make a discovery first and patent it. If a company moves too quickly, though, it makes a mistake in its research, and fails to make the discovery at all, because it chases down a blind alley.
For the analogy to be valid, it has to parallel the original model in a number of dimensions. Ask yourself the following questions about your analogy. Are there two players? Is the situation a tournament, where only one can win? What is ``stumbling'' in the situation? What is ``height'' in the situation? Are the decisions simultaneous?
It is very important for anyone--undergraduate, MBA, or PhD student -- to develop the skill of answering questions like (d).
3. (30 points) (Renegotiation) A bank must decide whether to offer Joe an auto loan or not. The loan would be for $11,000, and the bank must decide on a total amount L of principal and interest that he must repay at the end of the year. If Joe accepts the loan, he can buy a car worth $12,000 to him. He then decides whether to work hard, earning $15,000, or loaf, earning $8,000. Joe would pay up to $5,000 to be able to loaf. Joe has no other assets. If he decides not to repay the loan, he loses the car, and the bank collects $7,000 by reselling the car. If Joe keeps the car, it retains its value of $12,000 to him even if it is a year old. (You do not need to worry about discount rates to answer the problem)
(a) Draw a game tree for this situation, including payoffs for each player. Assume that if Joe does not repay the full amount of the loan, the bank will repossess the car.
ANSWER. Bank: Offer a loan at rate L, or don't offer one. Joe: Accept or reject the offer. Joe: Loaf or work. Joe: Repay or lose the car to the bank.
If the Bank does not offer a loan, or if Joe turns down the loan, its payoff is 0 and Joe's is 15 if he works or 13 (8+5) if he loafs. If Joe works and repays, the Bank gets L-11 and Joe gets 15+12 -L; if he does not repay, the Bank gets 7-11 and Joe gets 15. If Joe loafs and repays, the Bank gets L-11 and Joe gets 8+5 +12 -L; if he does not repay, the Bank gets 7-11 and Joe gets 8+5.
(b) What is the equilibrium?
ANSWER. The Bank will make the loan with L=12, Joe will accept, work hard, and repay.
(c) Suppose that if Joe decides not to repay the full amount of the loan, he can offer the bank some smaller amount S not to foreclose on it and repossess the car. What would that amount S be? How does the outcome change if both Joe and the bank know that this kind of renegotiation is possible?
ANSWER. The Bank will not make the loan . It knows that if it did, Joe will accept, loaf, and offer S=7, which the bank would accept, yielding the bank a payoff of 11-7. The possibility of renegotiation ends up hurting both Joe and the Bank, because it gives Joe bad incentives.
This question is similar to an example in the 2nd edition of Games and Information.
4. (20 points) (Auctions) Two companies, Ace and Basic, are considering taking over a third company, Crown. Each of them separately estimates the value of Crown, and each knows it either estimates exactly correctly, or 50 million too high, or 50 million too low. Each must decide how much to bid. Crown has set up the rules to be that each bidder submits one sealed bid. The winner is the company that bids the highest, but it pays the amount that the other company bid, so this is a ``second-price auction.''
(a) Suppose that the value of Crown to Ace is completely independent of its value to Basic. If Ace's estimate is 400 million, how much should it bid? Explain.
ANSWER. Ace should bid 400 million. That is the expected value of Crown to Ace, a value independent of Basic's valuation, since this is a private value auction. Winning the auction or otherwise learning about Basic's valuation conveys no information to Ace about the value of Crown.
(b) Suppose that the value of Crown to Ace is identical to its value to Basic, but Ace and Basic do independent measurements of the value. If Ace's estimate is 400 million, how much should it bid? Explain.
ANSWER. This is a common value auction, so the Winner's Curse is a worry. Ace should bid something between 350 and 400 million. If Ace and Basic pooled their information, they would better know the value of Crown. If one had a valuation of 350 and the other a valuation of 450, for example, they would conclude the true value was 400. If Ace bids 400, it might happen that Basic had an estimate of 350 and bid that, and 350 was the true value, in which case Ace regrets having won. Thus, Ace needs to be cautious and reduce its bid.
5. (10 points) Bigco and the United Widget Workers union are about to negotiate a new labor contract, when Bigco calls a press conference to announce that it has made a deal with its bank, Megabank. The deal says that Megabank will lend Bigco 100 million, and Bigco must repay that amount plus interest at 10 percent, the market rate at the time, if the new UWW contract is for 15/hour or less, but Bigco must repay the loan with 100 million plus 200,000 stock shares, half of its equity, if the new UWW contract is for more than 15/hour.
(a) Why would Bigco and Megabank agree to this contract?
ANSWER. If the contract is executed, then in equilibrium the wage is 15 dollars an hour or less, and Megabank earns the market return on its loan, so it is willing to make the loan. Bigco likes the contract because it gives it bargaining leverage with the UWW, who know that Bigco, having signed that contract, cannot afford to pay generous wages.
(b) Why don't Bigco and Megabank keep this contract secret?
ANSWER. If the contract were secret, Bigco would get no bargaining leverage with the union. Hence, Bigco will insist on its being public--something to which Megabank will readily agree.