First-Price Sealed-Bid Auctions

We have N = {1, . . . , n} bidders, where each bidder has a private valuation vi drawn from distribution Fi 1 with strictly positive den1 Here, we assume that valuations follow the IPV model. sity, fi : Ti → R>0, for a good up for sale by the auctioneer, agent 0. In a first-price, sealed-bid auction, each agent has a type vi ∈ Ti, and submits a bid bi to the auctioneer, without revealing what the contents of the bid are to any other bidder. Using the submitted bids b = (b1, . . . , bn), the auctioneer then decides on the winner. In this setting, the winner of the auction is the bidder with the highest bid, maxi∈N bi. 2 Let i∗ ∈ arg maxN bi denote the winner of the auction. 2 This is the nth order statistic, b(n). The kth order statistic, X(k), is the kth smallest value amongst a set of n random variables X1, . . . , Xn, so mini∈N = b(1). Should there be a tie, i∗ is randomly determined by the auctioneer. The auctioneer could, for example, select a bidder uniformly at random, or break ties using an alphanumeric schema. The winner is allocated with probability xi∗(bi∗ , b−i∗) = 1, and all other bidders i∗ 6= i ∈ N are allocated with probability xi(bi, b−i) = 0. The winner, bidder i∗, pays her bid, bi∗ , and all other bidders pay nothing.