The possible winner with uncertain weights problem

The original possible winner problem consists of an unweighted election with partial preferences and a distinguished candidate c and asks whether the preferences can be extended to total ones such that c wins the given election. We introduce a novel variant of this problem: possible winner with uncertain weights. In this variant, for a given weighted election, not some of the preferences but some of the preferences' weights are uncertain. We introduce a general framework to study this problem for nonnegative integer and rational weights as well as for four different variations of the problem itself: with and without given upper bounds on the total weight and with and without given ranges to choose weights from. We study the complexity of these problems for important voting systems such as scoring protocols, (simplified) Bucklin and fallback voting, plurality with runoff and veto with runoff, ${\text{Copeland}}^{\alpha }$, ranked pairs, and Borda.