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哪位大侠 帮忙做一下 下面的题 万分感谢
5. In this question you are asked to consider the seller’s auction or the reverse auction.
In the reverse auction there is one buyer and multiple sellers of an object. The sellers
bid by providing an asking price for the object. The seller who wins the auction is the
seller with the lowest asking price. Suppose there are two sellers trying to sell
identical used cars. A seller’s valuation of his car is Vi with i = 1, 2. The buyer’s
valuation of a car is VB. The buyer only values one car. Assume for all parts of the
question that VB > V1 and VB > V2, so that it is efficient for the buyer to purchase the
car from either seller. We only have trade between a buyer and a seller and not
between sellers. Asking prices by sellers are given by Pi with i = 1,2. Consider the
following cases.
(a) Suppose both sellers know their own valuation of the car and the other seller’s
valuation of the car. Also assume V1 > V2. List all the possible bid combinations
for the two sellers that are pure strategy Nash equilibria in the first price sealed
bid reverse auction. In the first price sealed bid reverse auction each seller submits
a selling price in a sealed envelope. Seller i wins the auction if Pi < Pj with i, j = 1,
2 and trade occurs at a price equal to the minimum of Pi or VB. If P1 = P2 then
player 2 wins and sells his car since he has the lower valuation of the car.(10)
(b) Again, suppose both sellers know their own valuation of the car and the other
seller’s valuation of the car. Also assume V1 > V2. List all the possible bid
combinations for the two sellers that are pure strategy Nash equilibria in the
second price sealed bid reverse auction. In the second price sealed bid reverse
auction each seller submits a selling price in a sealed envelope. Seller i wins the
auction if Pi < Pj with i, j = 1, 2 and trade occurs at a price equal to the minimum
of Pj (ie. the other seller’s bid) or VB. If P1 = P2 then player 2 wins since he has
the lower valuation of the car.(10)
(c) Suppose each seller only knows his own valuation of the car but does not know
the other seller’s valuation. Each seller knows that other seller’s valuation is
uniformly distributed on [0,1]. Further, each seller knows the buyer’s valuation
and VB > 1. You may assume that each seller sets an asking price between 0 and
1. Suppose we have a first price sealed bid reverse auction. If P1 = P2 then a
random draw with equal probability for each seller occurs to determine which
seller is the winner. Both sellers are risk neutral and only care about expected
payoffs. Find the pure strategy Bayesian Nash equilibrium where each seller sets
an asking price for his car that is a linear function of his valuation.(20)
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