“…In contrast, with a heavy-tailed distribution, adding bidders makes it more likely that an extreme right outlier bid will occur, boosting the winner's surplus as well as the seller's revenue. Valuation distribution shapes play a role in some other results about auctions, including determining whether imposing Pigovian taxes [21] on bidders to account for their negative externalities [1,4,9] can increase surplus for bidders as a class [27]. Heavy tails are also a topic of interest in the study of complexity theory [5,11,31,32], in marketing [2], in characterizing distributions based on samples [14,15,28], and in the study of a variety of physical and social phenomena [20,22,25,26].…”