  Since the 1990s spectrum auctions have been implemented world-wide. This has
provided for a practical examination of an assortment of auction mechanisms
and, amongst these, two simultaneous ascending price auctions have proved to be
extremely successful. These are the simultaneous multiround ascending auction
(SMRA) and the combinatorial clock auction (CCA). It has long been known that,
for certain classes of valuation functions, the SMRA provides good theoretical
guarantees on social welfare. However, no such guarantees were known for the
CCA.
  In this paper, we show that CCA does provide strong guarantees on social
welfare provided the price increment and stopping rule are well-chosen. This is
very surprising in that the choice of price increment has been used primarily
to adjust auction duration and the stopping rule has attracted little
attention. The main result is a polylogarithmic approximation guarantee for
social welfare when the maximum number of items demanded $\mathcal{C}$ by a
bidder is fixed. Specifically, we show that either the revenue of the CCA is at
least an $\Omega\Big(\frac{1}{\mathcal{C}^{2}\log n\log^2m}\Big)$-fraction of
the optimal welfare or the welfare of the CCA is at least an
$\Omega\Big(\frac{1}{\log n}\Big)$-fraction of the optimal welfare, where $n$
is the number of bidders and $m$ is the number of items. As a corollary, the
welfare ratio -- the worst case ratio between the social welfare of the optimum
allocation and the social welfare of the CCA allocation -- is at most
$O(\mathcal{C}^2 \cdot \log n \cdot \log^2 m)$. We emphasize that this latter
result requires no assumption on bidders valuation functions. Finally, we prove
that such a dependence on $\mathcal{C}$ is necessary. In particular, we show
that the welfare ratio of the CCA is at least $\Omega \Big(\mathcal{C} \cdot
\frac{\log m}{\log \log m}\Big)$.
