  The Coalitional Manipulation problem has been studied extensively in the
literature for many voting rules. However, most studies have focused on the
complete information setting, wherein the manipulators know the votes of the
non-manipulators. While this assumption is reasonable for purposes of showing
intractability, it is unrealistic for algorithmic considerations. In most
real-world scenarios, it is impractical for the manipulators to have accurate
knowledge of all the other votes. In this paper, we investigate manipulation
with incomplete information. In our framework, the manipulators know a partial
order for each voter that is consistent with the true preference of that voter.
In this setting, we formulate three natural computational notions of
manipulation, namely weak, opportunistic, and strong manipulation. We say that
an extension of a partial order is if there exists a manipulative vote for that
extension.
  1. Weak Manipulation (WM): the manipulators seek to vote in a way that makes
their preferred candidate win in at least one extension of the partial votes of
the non-manipulators.
  2. Opportunistic Manipulation (OM): the manipulators seek to vote in a way
that makes their preferred candidate win in every viable extension of the
partial votes of the non-manipulators.
  3. Strong Manipulation (SM): the manipulators seek to vote in a way that
makes their preferred candidate win in every extension of the partial votes of
the non-manipulators.
  We consider several scenarios for which the traditional manipulation problems
are easy (for instance, Borda with a single manipulator). For many of them, the
corresponding manipulative questions that we propose turn out to be
computationally intractable. Our hardness results often hold even when very
little information is missing, or in other words, even when the instances are
quite close to the complete information setting.
