  We study the efficiency of sequential first-price item auctions at (subgame
perfect) equilibrium. This auction format has recently attracted much
attention, with previous work establishing positive results for unit-demand
valuations and negative results for submodular valuations. This leaves a large
gap in our understanding between these valuation classes. In this work we
resolve this gap on the negative side. In particular, we show that even in the
very restricted case in which each bidder has either an additive valuation or a
unit-demand valuation, there exist instances in which the inefficiency at
equilibrium grows linearly with the minimum of the number of items and the
number of bidders. Moreover, these inefficient equilibria persist even under
iterated elimination of weakly dominated strategies. Our main result implies
linear inefficiency for many natural settings, including auctions with gross
substitute valuations, capacitated valuations, budget-additive valuations, and
additive valuations with hard budget constraints on the payments. Another
implication is that the inefficiency in sequential auctions is driven by the
maximum number of items contained in any player's optimal set, and this is
tight. For capacitated valuations, our results imply a lower bound that equals
the maximum capacity of any bidder, which is tight following the upper-bound
technique established by Paes Leme et al. \cite{PaesLeme2012}.
