$13\mathrm{.}3$ 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, theta$\_($i$)$in$\{0,1,2\},$ each value occurring with an equal probability

of $(1)/(3).$ 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"

of r $>=0$ 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

Chapter $13$ Auctions and Competitive Bidding

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?