  The literature on the Price of Anarchy (PoA) of simple auctions employs a
no-overbidding assumption but has completely overlooked the no-underbidding
phenomenon, which is evident in empirical studies on variants of the second
price auction. In this work, we provide a theoretical foundation for the
no-underbidding phenomenon. We study the PoA of simultaneous 2nd price auctions
(S2PA) under a new natural condition of {\em no underbidding}, meaning that
agents never bid on items less than their marginal values. We establish
improved (mostly tight) bounds on the PoA of S2PA under no underbidding for
different valuation classes (including unit-demand, submodular, XOS,
subadditive, and general monotone valuations), in both full-information and
incomplete information settings.
  To derive our results, we introduce a new parameterized property of auctions,
termed $(\gamma,\delta)$-revenue guaranteed, which implies a PoA of at least
$\gamma/(1+\delta)$. Via extension theorems, this guarantee extends to coarse
correlated equilibria (CCE) in full information settings, and to Bayesian PoA
(BPoA) in settings with incomplete information and arbitrary (correlated)
distributions. We then show that S2PA are $(1,1)$-revenue guaranteed with
respect to bids satisfying no underbidding. This implies a PoA of at least
$1/2$ for general monotone valuation, which extends to BPOA with arbitrary
correlated distributions. Moreover, we show that $(\lambda,\mu)$-smoothness
combined with $(\gamma,\delta)$-revenue guaranteed guarantees a PoA of at least
$(\gamma+\lambda)/(1+\delta+\mu)$. This implies a host of results, such as a
tight PoA of $2/3$ for S2PA with submodular (or XOS) valuations, under no
overbidding and no underbidding. Beyond establishing improved bounds for S2PA,
the no underbidding assumption sheds new light on the performance of S2PA
relative to simultaneous 1st price auctions.
