1. (30 points) Suppose Wal-Mart and Target are playing a "War of
Attrition" game in various small towns. Each small town is only a big
enough market for one firm to be profitable, but initially both stores
operate in each town. Each store can either Stay or Leave. The
payoffs from each strategy are shown in the table below. Neither firm
intends to use a pure strategy; each will randomize the decision for
each town as to whether to stay or leave.
WAL-MART
Stay Leave
Stay -1, -2 4,0
TARGET
Leave 0, 5 0,0
(a) Consider one small town, Gnawbone, Indiana. In equilibrium, what
is Target's equilibrium expected payoff if it elects to stay in
Gnawbone? ANSWER. Zero, the same as if it left.
(b) What is Wal-Mart's equilibrium probability of staying in Gnawbone?
ANSWER. Wal-Mart's probability X of staying must make Target
indifferent between staying and leaving. Thus,
Payoff(Stay) = -X + 4 (1-X) = Payoff(Leave) = 0,
so 4 = 5X, and X = 4/5.
(c) What is Target's equilibrium probability of staying in Gnawbone?
ANSWER. Target's probability Y of staying must make Wal-Mart
indifferent between staying and leaving. Thus,
Payoff(Stay) = -2Y + 5(1-Y) = Payoff(Leave) = 0,
so 5 = 7Y, and X = 5/7.
2. (30 points) Two companies are trying to decide on the price of what
is an
acquisition for one of them and a deacquisition for the other. Ford
Motor Company has just decided to get out of the luxury car business
and sell that division of the company to General Motors. The value of
Ford is currently 30 billion dollars and that of GM is 80 billion
dollars. The value of the luxury car division is 2 billion dollars
for Ford and 4 billion dollars for GM, including all synergies, etc..
There are no other potential buyers who would value the division at
more than 0. In the first stage of bargaining, the two companies have agreed to be bound by the following procedure: In the first round, Ford moves first and offers a price; GM can accept or reject the offer. If GM rejects Ford's offer, then in the second round of bargaining it can counteroffer and Ford gets to accept or reject.
This process is to continue in a back and forth alternating fashion for a fixed number, N, of rounds. The two companies have not yet decided on N, however.
a. If the number of rounds is even, what will the price be? Why?
ANSWER. Working back from the end, if the number of rounds is even, then GM makes the last offer. It will offer a price of at least 2 billion, the minimum Ford would accept. If Ford would accept 2 billion, that is what will be offered; otherwise, GM will offer slightly more.
Nothing that happens in any previous round matters, since Ford can just wait till the last round and hold out for 4 billion.
b. If the number of rounds is odd, what will the price be? Why?
ANSWER. 4 billion dollars. Working back from the end, if the number of rounds is odd, then Ford makes the last offer. It will want to offer a price low enough that GM will accept. GM will be indifferent at a price of 4 billion dollars, so Ford will offer that price, or perhaps 4.01 billion to make GM strictly willing to accept.
Nothing that happens in any previous round matters, since Ford can just wait till the last round and hold out for 4 billion.
(c) What will happen when the companies try to agree on N? Have the companies been smart in taking the negotiations down this path?
ANSWER: The two companies will not be able to agree on N, since N decides which of them gets the entire surplus from the transaction. It was foolish of them to start down this path, because bargaining over N is even harder than bargaining over the price itself, since there is no room for compromise unless a company throws in extra cash to get its choice of N.
3. (20 points) Mr. Turner is thinking of entering the
garbage
collection business in a certain large city. Currently, Cutright
Enterprises has a monopoly, earning 40 million per year from the 40
routes the city offers up for bids. Turner thinks he can take away
as many routes as he wants from Cutright, at a profit of 1.5 million
per route for him. He is worried, however, that Cutright might resort
to assassination, killing him to regain their lost routes. He would
be willing to risk assassination for profit of 80 million dollars, and
assassination would cost Cutright 6 million dollars in expected
legal costs and possible prison sentences. a. How many routes should Turner try to take away from Cutright?
ANSWER. 5 routes. That will make Cutright unwilling to resort to assassination. If Turner takes away 5 routes, then Cutright's payoff is 5 million lower (5 times 40/40 million) than it would be otherwise, unless Cutright kills Turner, in which case it will be 6 million lower. Hence, Turner should take away 5 or 6 routes. 5 is safer, given that Cutright probably would enjoy killing Turner, expenses aside.
(For an article pertaining to the dangers of this business, see N.Y. CLEANING MOB FROM DIRTY BUSINESS OF HAULING TRASH. MERRILL GOOZNER. 12/15/1996. The Seattle Times. A14.)
b. Explain which of these strategies Turner's optimal strategy most resembles: Fat cat, lean-and-hungry-look, puppy dog, top dog.
ANSWER. This is a puppy dog strategy. Turner is playing a ``soft'' strategy of limited entry, so that Cutright will play ``soft'' also.
4. (20 points) Dixit and Nalebuff describe in Chapter 5 the
situation of the European Union in trying to decide whether to buy
European Airbus jets rather than American Boeings. Suppose there is
some chance that Boeings might turn out to be the better jet. What
advantage is there to the European Union from committing in advance
to buy at least some Airbus jets rather than waiting to decide later
when the quality of each company's product is clearer? Ignore the
obvious political reason: that the EU wants to make European jet
workers happy, and discuss only the advantages to the EU as a buyer of
jets. ANSWER. By committing itself now, the European Union encourages Airbus to continue producing jets. Not only does this increase the selection in the future, it also prevents Boeing from being a monopolist and charging high prices. If the EU waits, then maybe Airbus will exit the market, and there will be no future choice.