Finding Possible and Necessary Winners in Spatial Voting with Partial Information

We consider spatial voting where candidates are located in the Euclidean $d$-dimensional space and each voter ranks candidates based on their distance from the voter's ideal point. We explore the case where information about the location of voters' ideal points is incomplete: for each dimension, we are given an interval of possible values. We study the computational complexity of computing the possible and necessary winners for positional scoring rules. Our results show that we retain tractable cases of the classic model where voters have partial-order preferences. Moreover, we show that there are positional scoring rules under which the possible-winner problem is intractable for partial orders, but tractable in the one-dimension spatial setting (while intractable in higher fixed number of dimensions).