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).

翻译：我们考虑空间投票场景，其中候选者位于欧几里得$d$维空间中，每位选民根据其理想点与候选者的距离对候选者进行排序。我们探讨选民理想点位置信息不完整的情况：对于每个维度，我们已知一个可能的取值区间。我们研究了针对位置计分规则计算可能赢家与必然赢家的计算复杂度。结果表明，在选民偏好为偏序的经典模型中，我们保留了易于处理的情形。此外，我们发现存在某些位置计分规则，在这些规则下，对于偏序偏好，可能赢家问题难以处理，但在一维空间设定中是可处理的（而在更高固定维度中则难以处理）。