## Transcribed Text

1
Marcia, Marcia, Marcia
Greg is deciding whether to ask Marcia out on date. However, Greg isnt sure whether Marcia likes
him, and he would rather not ask he expects to be rejected. Whether Marcia likes Greg private
information to her. Thus, her preferences regarding Greg constitute her type. Greg does not
have
any
private information. Assume that there is 25% chance that Marcia likes Greg. The Bayesian game
is
shown here. Should Greg ask Marcia? Find BNE that answers this question.
Nature
Marcia
Marcia doesn'
Greg
life Greg
Probability
.25
Probability 75
Greg
Invito
Do not
Do not
invite
invife
invite
Marcia
Marcia
4
4
Ves
No
Yes
No
Greg
10
8
2
Marcia
,
1
3
2 Auctions: First Price vs. Second Price
Fill in the blanks in the table below.
Bidder
Value
1st price bid
2nd price bid
A
92
B
95
C
31
D
75
E
88
a) Who wins the first price auction and what price do they pay?
b) Who wins the second price auction and what price do they pay?
3 NE in Auctions
Consider first-price sealed bid auction with risk-neutral bidders (we've been examining risk neutral
bidders all along). Each bidder has private value independently drawn from uniform distribution on
[0,1]. That is. for each bidder, all values between 0 and are equally likely. The complete strategy of
each bidder is bid function that will tell us. for any value v. what amount b(v) that bidder will choose
to bid.
It is proposed that the equilibrium bid function for is b(v) 0/2 for each of the two bidders.
That is. if we have two bidders. each should bid half her value.
a) (4 points) Suppose you're bidding against just one opponent whose value is univormly distributed on
10, and always bids half her value. What is the probability that you will win you bid b=.17 If you
bid b=.47 If you bid b=.6?
b) (4 points) Put together the answers topart a). What is the correct mathematica expression for p(win),
the probability that you win. as function of your bid b?
c) (4 points) Find an expression for the expected profit you make when your value is and you bid is
b. given that your opponent is bidding half her value. Remember that there are two cases: either you
win the auction or you lose the auction. You need to average the profit between these cases.
d) (4 points) What is the value of that maximizes your expected profit? This should bez function of
your value v.
e) (4 points) Use your results to argue that it NE for both biddres to follow the same bid function
4 Lemons
In the country of Autolandia, used cars all sell for the same price. In Autolandia, the quality of cars is
uniformly distributed with sellers valuing the cars from $0 to $1,000 and buyers valuing the cars
from
$100 to $1 100 (buyers value each individual car at $100 more than the sellers). For example consider
a market with 1,001 cars. There will be one worth $0 to seller/$100 to buyer, one worth $1 to a
seller/$101 to buyer, etc. up to one worth $1,000 to seller/$1,100 to a buyer.
a) (5 points) If neither the buyer nor the seller knows the quality of car. will any cars sell? If so. at
what price?
b) (5 points) If only the seller knows the quality of car. what will buyer think is the true value of a
car with price to the seller of that car? How much would buyer value the average car for sale at
prioe p? Give numerical example.
c) (5 points) In the version of the market where only the seller knows the true quality of the car, what
will be the equilibrium price?
d) (5 points) What percentage of the total number of cars get sold?
5 No such thing as a free lunch
A local charity has been given grant to serve free meals to the homeless in its community, but it is
worried that its program might be exploited by nearby college students who are always interested in a
free
meal. Both homeless person and college student receive payoff of 10 for free meal. The cost
of standing in line for the meal for homeless person and The for college student where m is the
number of minutes spent waiting in line. Assume that the staff of the charity cannot observe the true
type of people coming for free meals
a) (5 points) What the minimum wait time, m. that will achieve separation of types?
b) (5 points) After while. the charity finds that it can successfully identify and turn away college
students half fo the time. College students who are turned away receive no free meal and also incur
a cost of 5 for their time and embarrassment. Will the ability to partially identify college students
reduce or increase the answer in part a)? Explain.
6
To Stand or Not to Stand, that is the Question
In scene at the end of The Princess Bride (same movie as the poison scene from lesson L01-1) the
hero. Wesley confronts the evil Prince Humperdinck. This interaction can be modeled as the following
game. Wesley can either be strong or weak, as randomly picked by nature with equal probability. Wesley,
knowing if he strong or weak. can then choose to stand or continue lying on the bed. The
prince
observes this decision but does not know Wesley's type and chooses to fight or surrender. The prince can
beat weak Wesley (Wesley was "mostly' dead only few hours earlier) and can really beat up weak
Wesley who stays in bed. If weak Wesley stands and the prince fights, then the base payoffs are for
the prince and -1 for Wesley. If Wesley stays in bed. then the prince gets and Wesley still gets -1. If
Weslev is weak and the prince surrenders regardless of whether Wesley has stood or not the base payoffs
are for Wesley and 0 for the prince. strong Wesley, however, will destroy the prince and enjoy doing
it should the prince try to fight. It easier for Weslev to beat the Prince he stands. though, so in
that case he gets payoff of while the prince gets -2. If Wesley stays in bed and the Prince chooses
to
fight the payoffs are for Wesley and -1 for the prince. In the event that the prince chooses not
to
fight. Wesley gets if he stands and if he stays in bed while in either case the prince gets 0. The final
complication to the payoffs is that if Wesley is weak, standing costs him some extra amount c, while it
costs strong Wesley nothing.
a) (5 points) derive the extensive form of the game
b) (10 points) derive pooling BNE where q 0,5 represents the prince's initial belief that Wesley is
weak. Be sure to include what are the prince's beliefs about Wesley's type if he observes Wesley
in
bed or observes him standing. Find the range of values for that makes the belief valid.
c) (10 points) derive separating BNE for this game as well as the range of e over which it valid

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