In many situations, a seller of an item entertains bids from a number of buyers but allows one buyer to obtain the good by matching the highest competing offer. Consider the owner of a house who must sell the property immediately. These are only two potential buyers, one of whom currently rents the house. The owner agrees to solicit a final price from the outside buyer and allow the renter to match it if he wishes. All the parties believe that the buyers’ values for the house are independent of each other and that, for each buyer, values between $200,000 and $260,000 are equally likely.
a. How should the renter decide whether or not to match the outside offer?
b. What offer should the outside buyer make if her value is $240,000? More generally, what offer should she make as a function of her value, v?
c. What is the seller’s expected revenue? Confirm that it is lower than under either the English or sealed-bid auctions.