#### Auction

Economics help

Suppose a seller runs a second-price, sealed-bid auction for a painting. There are two bidders with independent, private values. The seller does not know their precise valuations, but knows: (i) each bidder i has one of three values, vi = 2,4,8; and (ii) each of these values is equally likely (i.e., occurs with probability 1/3). When running the auction, if the two bids are tied (say, at x), the winner is chosen at random (and pays x). The seller has no value for the painting (i.e., his valuation is 0)
a) Assume both bidders use their dominant strategies. What is the seller’s expected revenue in this auction?
b) Now the seller decides to set a reserve price of r. This means that if the highest bid is at least r, then the painting will go to the highest bidder, and the winner will pay the maximum of r and the second-highest bid. Suppose the reserve price is set to r = 4. Assume both bidders use their dominant strategies. What is the seller's expected revenue in this auction?