  We study mechanism design when agents may have hidden secondary goals which
will manifest as non-trivial preferences among outcomes for which their primary
utility is the same. We show that in such cases, a mechanism is robust against
strategic manipulation if and only if it is not only incentive-compatible, but
also nonbossy -- a well-studied property in the context of matching and
allocation mechanisms. We give complete characterizations of
incentive-compatible and nonbossy mechanisms in various settings, including
auctions with single-parameter agents and public decision settings where all
agents share a common outcome. In particular, we show that in the single-item
setting, a mechanism is incentive-compatible, individually rational, and
nonbossy if and only if it is a sequential posted-price mechanism. In contrast,
we show that in more general single-parameter environments, there exist
mechanisms satisfying our characterization that significantly outperform
sequential posted-price mechanisms in terms of revenue or efficiency (sometimes
by an exponential factor).
