This exam can take the full 2 hours.
1. (15 points) In the following game find all the (a) dominant strategies, (b) dominated strategies, and (c) pure-strategy Nash equilibria.
Column
Consumers Producers
Africa 6,1 2,45
Row
Europe 5,2 0,0
ANSWER. (a) Africa is dominant for Row. (b) Europe is dominated
for Row. (c) The pure-strategy Nash equilibria is
(Africa, Producers). 2. (30 points) Acme Autos is bargaining with Selco Steel on the price of steel that Acme buys. Currently the price is 10 dollars per ton. Selco's production cost is 3 dollars per ton.
For each dollar the price falls, Acme gains 1 million in profit, and Selco loses 1 million. If they do not agree immediately, Acme can run down its inventory of steel, but the inventory will run out in 4 weeks, after which Acme must give in and accept an offer of a price as high as 20 dollars per ton, the price offered by other steel companies. For each week of disagreement over prices, however, Selco loses 6 million dollars in profits. Acme starts by making an offer in the first week, then Selco offers, then Acme, and finally Selco.
(a) When will the two companies come to an agreement?
ANSWER. They will agree immediately. There is no asymmetric information, and delay is costly while teaching neither side anything about the other.
(b) What will the price be?
ANSWER. The initial price of $10 is not relevant to the ultimate price agreed upon. Just because that was the old price does not mean that the new price cannot move far away from it.
Selco makes the offer in week 4, Acme makes it in 3, Selco in 2, and Acme in 1. The price would be 20 in week 4, 14 in week 3 (because if Acme offers 14, Selco will accept it), 14 in week 2 (because if Selco offers 14, Acme will accept it, rather than wait for the same amount in week 3), and 8 in week 1. Thus, the price would be 8.
Suppose, alternatively, that Acme makes the offer in week 4, Selco makes it in 3, Acme in 2, and Selco in 1. Then the sequence of prices would be 20 offered by Acme in week 4 (because Selco knows that if it turns this down, Acme will have to accept the price of 20), 20 offered by Selco in week 3 (because Acme could go to an outside company at that price immediately if it wanted), 14 offered by Acme in week 2 (because it costs Selco 6 to wait), and 14 by Selco in week 1 (since Acme has no cost of waiting, and knows it can get 14 in week 2. Thus, the price would be 14.
(c) If Acme could precommit to a strategy and then not deviate from it, what strategy would it pick?
ANSWER. Acme would commit to as low a price as Selco would possibly accept-- 3 dollars per ton. Selco would then give in and agree immediately.
3. (20 points) In the movie, "A Beautiful Mind," John Nash gets
the idea for Nash equilibrium in a student hangout where he is sitting
with three buddies. Five women walk in, four brunettes and a stunning
blonde. Each of the four buddies starts forward to introduce himself
to the blonde. Nash stops them, though, saying, "If we all go for the
blonde, we will all be rejected and none of the brunettes will talk
to us afterwards because they will be offended. So let's go for the
brunettes." The next thing we see is the four buddies dancing with
the four brunettes and the blonde standing alone, looking unhappy.
Nash then goes to his professor to discuss his new idea of Nash
equilibrium and how individual pursuit of payoff maximization does not
lead to optimal results, contrary to Adam Smith.(a)Make up payoff numbers to go with the story for the choices of the four buddies who are the players in this game.
ANSWER. Ending up with the blonde has a payoff of 4, ending up with a brunette has a payoff of 1, and ending up alone is 0. If more than one buddy goes after a single woman, they both end up alone with the payoff of zero.
(b) Is the result in the story a Nash equilibrium?
ANSWER. No. Any one of the four buddies could deviate and increase his payoff by pursuing the blonde instead the brunette he chose.
(c) If the result in the story is Nash equilibrium, what is another Nash equilibrium? If not, what is a Nash equilibrium for this situation?
ANSWER. It would be a Nash equilibrium for Buddy 1 to go for the blonde and Buddies 2,3, and 4 to go for brunettes 2,3, and 4. This is an asymmetric Nash equilibrium, of course. Note that this is Pareto- superior to the result in the movie, since Buddy 1 would be better off and Buddies 2, 3, and 4 no worse off.
There is an equilibrium in mixed strategies also. Suppose each buddy goes for a different brunette with probability N and otherwise goes for the blonde. In equilibrium, each one must be indifferent between going for his brunette or for the blonde. Suppose the payoffs are 0 for ending up alone, 1 for ending up with a brunette, and B for ending up with the blonde. If more than one buddy goes for the blonde, then nobody ends up with her. So the only way a buddy can end up with the blonde is if the other 3 all choose not to approach her, which has probability N*N*N. In the mixed strategy equilibrium, his payoffs from his pure strategies are
Payoff (Approach Blonde)= N*N*N*B = Payoff (Approach Brunette)= 1.Solving this for N yields N being the cube root of 1/B. If, for example, B=8, then N = 1/2. Each buddy approaches the blonde with probability 1/2, expecting that when he does approach her, he has a 1/8 chance of being the only buddy to do so.
4. (35 points) Smith is arranging for a friendly takeover of
Brown's hospital company. They have worked out the basic details,
including the price, and they can fall back on that. Now, however,
Smith is considering adding a
proposal that he not buy Brown's laboratory subsidiary. He knows
that the value of the lab is 20 million dollars to Brown, but
only 10 million dollars to himself. So he is thinking of offering to
not buy the laboratory, in exchange for dropping the price in the
merger deal by 15 million dollars. It would cost Smith 1 million
dollars in legal fees to put together the details of this new
``Superior'' proposal. Alternatively, Smith can offer a ``sneaky'' modification to the contract, in which he buys only the best parts of the laboratory. Then, the proposal still drops the price by 15 million dollars but leaves Brown with only 7 million dollars in lab value and Smith with the 10 million dollars in lab value that he would get in the original deal.
Brown does not know which proposal is being offered. At a cost of 3 million dollars, Brown can hire accounting consultants and discover the truth as to what the fine print in the contract modification implies. Or, he can just accept or reject the proposal without learning anything more.
(a) Explain why there is no equilibrium in pure strategies in which Smith offers the Superior proposal and Brown accepts his proposal.
ANSWER. We must ask whether either player would unilaterally deviate from such an equilibrium. Brown is indeed willing to accept Smith's proposal if it is the Superior one. Smith, however, would deviate and offer the Sneaky proposal instead, given that Brown will accept whatever is proposed.
The answer above is enough for full credit. One might note further, however, that in this proposed equilibrium, Brown would never spend the $3 million to discover the truth about the proposal, because in equilibrium he expects it to be the Superior one anyway.
(b) There are two equilibria, one in mixed strategies, and one in which Smith does not offer the new proposal. Lay out the strategies for each player in the equilibrium in which Smith does not offer any new proposal. A strategy must say what action a player would take in any circumstance he might happen to observe, so it must say what Jones's response would be if a proposal were made.
ANSWER. The equilibrium strategy for Smith is not to offer either the sneaky or the superior proposal. The equilibrium strategy for Brown is to reject any proposal that is made, without hiring consultants. Neither player can benefit by deviating. If Smith goes to the expense of offering a proposal, it will be turned down, and he will have wasted the expense. If, out of equilibrium, Smith does make a proposal, Brown has nothing to gain by accepting it or hiring the consultants if he has enough belief that if Smith did deviate Smith would offer the sneak proposal.
It is not enough to say that the equilibrium strategy is for Smith not to offer any new proposal. That leaves open the question of what Brown's strategy would be in response to a proposal that was made.
This equilibrium is like the equilibrium in the signalling game in which signalling fails to work. Recall that if employers do not think good workers will be any more likely than bad workers to get diplomas, employers will not offer higher wages for workers with diplomas. Therefore, in that equilibrium, no worker will get a diploma. Signalling is not expected to work, so it doesn't.
The outcome here is simple-- Smith makes no proposal-- but that is not the entire equilibrium, which must also specify what Brown's strategy is. This is like an entry deterrence game with credible threats, in which the equilibrium outcome might be that the new firm fails to enter, but the interesting part of the equilibrium is the credible threat that keeps it out.
(c) Calculate the mixed strategy equilibrium. With what probability S does Smith offer the Superior modification, and with what probability B does Brown accept the proposal without hiring any consultants?
ANSWER. In another equilibrium, Smith offers a modification, which might be sneaky and might be superior. Brown either accepts without hiring consultants, or hires them and accepts or rejects based on the consultant report. To find the mixing probabilities, equate the payoffs from the pure strategies:
First, let us equate the expected payoffs of Smith from his two pure strategies. We will say his payoff is 0 under the original deal, and calculate all payoffs relative to that.
If Smith offers the Superior contract he will get a payoff of 4. 5 of that is from the contract's difference between the price of 15 and the value of 10 to him, and we subtract 1 for the cost of the Superior proposal. This is so whether Brown audits the proposal or not. If Smith uses the Sneaky proposal, then he either gets the price reduction of 15 (if Brown accepts outright) or gets 0 (if Brown hires the consultants and rejects the modification).
Profit(Smith, Superior) = B(4) + (1-B) (4) = Profit(Smith, Sneaky) = B(15) + (1-B) (0).
This yields 4 B + 4 - 4B = 15B, so 4=15B, and B = 4/15.
Now let's equate Brown's payoffs from his two actions of Accepting outright and Hiring Consultants. If Brown accepts the proposal, his payoff is 5, equal to 20 minus 15, if it is Superior. If it is Sneaky his payoff is -8, because he pays 15 for something worth 7 to him. If Brown hires the consultants his payoff is 5 minus 3 if it is Superior, because he signs the deal but has to pay the consultants; but 0-3 if it is Sneaky, because then he refuses the deal but has to pay the consultants 3.
Profit(Brown, Accept) = S(5) + (1-S) (-8) = Profit(Brown, Hire Consultants) = S( 5-3) + (1-S) (0-3).
This yields 5S -8 +8S = 2S -3 +3S, so 8S = 5, and S = 5/8.