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.

本文探究了在不完整信息情况下的联合操纵问题及其计算性质，并提出了三种自然的操纵计算概念。我们提出的操纵问题在很多情况下都是计算上难以处理的，即使在很少信息缺失的情况下也是如此，这也使得本文的研究有着重要的实际应用意义。