Probabilistic Inference of Winners in Elections by Independent Random Voters

We investigate the problem of computing the probability of winning in an election where voter attendance is uncertain. More precisely, we study the setting where, in addition to a total ordering of the candidates, each voter is associated with a probability of attending the poll, and the attendances of different voters are probabilistically independent. We show that the probability of winning can be computed in polynomial time for the plurality and veto rules. However, it is computationally hard (#P-hard) for various other rules, including $k$-approval and $k$-veto for $k>1$, Borda, Condorcet, and Maximin. For some of these rules, it is even hard to find a multiplicative approximation since it is already hard to determine whether this probability is nonzero. In contrast, we devise a fully polynomial-time randomized approximation scheme (FPRAS) for the complement probability, namely the probability of losing, for every positional scoring rule (with polynomial scores), as well as for the Condorcet rule.