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.
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 first-price auction in which ties are broken by a coin flip. prepare down the payoff matrix, and find all the pure-strategy Nash equilibria.
(2) Conisider that the unit is sold via second-price auction in which ties are broken by a coin flip. prepare down the payoff matrix. What is the dominant-strategy equilibrium of this auction game?
(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. prepare down the payoff matrix, and find all the pure-strategy Nash equilibria (if any).