  We study two standard multi-unit auction formats for allocating multiple
units of a single good to multi-demand bidders. The first one is the
Discriminatory Auction, which charges every winner his winning bids. The second
is the Uniform Price Auction, which determines a uniform price to be paid per
unit. Variants of both formats find applications ranging from the allocation of
state bonds to investors, to online sales over the internet, facilitated by
popular online brokers. For these formats, we consider two bidding interfaces:
(i) standard bidding, which is most prevalent in the scientific literature, and
(ii) uniform bidding, which is more popular in practice. In this work, we
evaluate the economic inefficiency of both multi-unit auction formats for both
bidding interfaces, by means of upper and lower bounds on the Price of Anarchy
for pure Nash equilibria and mixed Bayes-Nash equilibria. Our developments
improve significantly upon bounds that have been obtained recently in
[Markakis, Telelis, ToCS 2014] and [Syrgkanis, Tardos, STOC 2013] for
submodular valuation functions. Moreover, we consider for the first time
bidders with subadditive valuation functions for these auction formats. Our
results signify that these auctions are nearly efficient, which provides
further justification for their use in practice.
