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Q1 (3pt. There are 100 people who want to sell used cars and 100 people who want to buy a used car. Assume they are risk-neutral. 80 cars are 'plum'...

Q1 (3pt.): There are 100 people who want to sell used cars and 100 people who
want to buy a used car. Assume they are risk-neutral. 80 cars are ‘plum’ and 20
cars are ‘lemon.’ The owners of a plum is willing to part with it for \$2000. The
owners of a lemon is willing to part with it for \$900. The buyers are willing to pay
\$2400 for a plum and \$1000 for a lemon.
When the buyers cannot verify at the point of purchase whether a given car is a
plum or a lemon, they have to accept a common price p. Determine the possible
range of p, and the numbers of each type traded.
Q2 (17pt.): A single unit of a good is to be sold via an auction. There are two
bidders, A and B. They are assumed to be risk-neutral. The seller knows that
there are five possible values of willingness to pay, \$100, \$200, \$300, \$400 and
\$500. Bids are also restricted to those values.
Suppose A’s willingness to pay is \$400 whereas B’s willingness to pay is \$200.
(1) Consider that the unit is sold via rst-price auction in which ties are broken
by a coin flip.
Write down the payoff matrix, and find all the pure-strategy Nash equilibria. (6pt.)
(2) Conisider that the unit is sold via second-price auction in which ties are
broken by a coin flip.
Write down the payoff matrix. What is the dominant-strategy equilibrium of this
auction game? (6pt.)
(3) (innovative) Conisider that the unit is sold via all-pay auction in which ties are
broken by a coin flip. In all-pay auction, each bidder has to pay her bid, regardless
of winning or losing.
Write down the payoff matrix, and find all the pure-strategy Nash equilibria (if
any). (5pt.)

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