An auction with n bidders. Assume that

v(j) is the valuation of bidder j

b(k) is the bid price of bidder k.

If there is no two bidders bid the same price.

What is the utility payoff of each bidder in each type of auctions: English auction, Dutch auction, first price sealed-bid auction and second-price sealed-bid auction.

utility of bidder = private value - price to pay

utility of seller = selling price - private value

Here is the example answer as below (I am not sure it is correct or not):

In english Auction,if the valuation is v(j), the bid price is b(k). The utility of english auction is v(j)- b(k)