Consider an auction with n bidders. Assume that v(j) is the valuation of bidder j and b(k) is the bid price of bidder k. Assume that no two bidders bid the same price. For each type of auctions: English auction, Dutch auction, first price sealed-bid auction and second-price sealed-bid auction, what is the utility payoff of each bidder.

utility of bidder = private value - price to pay

utility of seller = selling price - private value