  We study the inefficiency of mixed equilibria, expressed as the price of
anarchy, of all-pay auctions in three different environments: combinatorial,
multi-unit and single-item auctions. First, we consider item-bidding
combinatorial auctions where m all-pay auctions run in parallel, one for each
good. For fractionally subadditive valuations, we strengthen the upper bound
from 2 [Syrgkanis and Tardos STOC'13] to 1.82 by proving some structural
properties that characterize the mixed Nash equilibria of the game. Next, we
design an all-pay mechanism with a randomized allocation rule for the multi-
unit auction. We show that, for bidders with submodular valuations, the
mechanism admits a unique, 75% efficient, pure Nash equilibrium. The efficiency
of this mechanism outperforms all the known bounds on the price of anarchy of
mechanisms used for multi-unit auctions. Finally, we analyze single-item
all-pay auctions motivated by their connection to contests and show tight
bounds on the price of anarchy of social welfare, revenue and maximum bid.
