Possible Winners in Noisy Elections

Predicting election winners (or, election possible winners) is an important topic in computational social choice. Very generally put, we consider the following setting: There is some set of candidates C and some set of voters V (with preferences over C). We either do not know which candidates will take part in the election or we do not know which voters will cast their votes. However, for each set C ′ ⊆ C (each set V ′ ⊆ V ) we know probability PC(C ′) that exactly candidates in C ′ participate in the election (probability PV (V ′) that exactly voters in V ′ cast their votes). Our goal is to compute the probability that a given candidate c ∈ C wins the election. In its full generality— with unrestricted probability distributions PC and PV —these problems can very easily become computationally hard. We provide natural restrictions on PC and PV that allow us to obtain positive results for several election systems, including plurality, approval, and Condorcet’s rule. On the technical side, our problems reduce to counting solutions to the problems of election control.