Reserve Prices (Easier):Consider a seller who must sell a single private value good. There are two potential buyers, each with a valuation that can take on one of three values, θi ∈ {0, 1, 2}, each value occurring with an equal probability of 1/8. The players' values are independently drawn. The seller will offer the good using a second-price sealed-bid auction, but he can set a "reserve price" ofr ≥ 0 that modifies the rules of the auction as follows.

If both bids are below r then neither bidder obtains the good and it is destroyed. If both bids are at or above r then the regular auction rules prevail. If only one bid is at or above r then that bidder obtains the good and pays r to the seller.

a. Is it still a weakly dominant strategy for each player to bid his valuation when r > 0?

b. What is the expected revenue of the seller when r = 0 (no reserve price)?

c. What is the expected revenue of the seller when r = 1?

d. What explains the difference between your answers to (b) and (c)?

e. What is the optimal reserve price r for the seller, and what can you conclude about the value of reserve prices?