Approval voting is a common method of preference aggregation where voters vote by ``approving'' of a subset of candidates and the winner(s) are those who are approved of by the largest number of voters. In approval voting, the degree to which a vote impacts a candidate's score depends only on if that voter approved of the candidate or not, i.e., it is independent of which, or how many, other candidates they approved of. Recently, there has been interest in satisfaction approval voting and quadratic voting both of which include a trade-off between approving of more candidates and how much support each selected candidate gets. Approval voting, satisfaction approval voting, and quadratic voting, can all be viewed as voting where a vote is viewed as analogous to a vector with a different unit norm ($\mathcal{L}^{\infty}$, $\mathcal{L}^{1}$, and $\mathcal{L}^2$ respectively). This suggests a generalization where one can view a vote as analogous to a normalized unit vector under an arbitrary $\mathcal{L}^p$-norm. In this paper, we look at various general methods for justifying voting methods and investigate the degree to which these serve as justifications for these generalizations of approval voting.

翻译：批准投票是一种常见的偏好聚合方法，选民通过"批准"某个候选人子集进行投票，得票最多的候选人获胜。在批准投票中，一张选票对候选人得分的影响仅取决于该选民是否批准该候选人，即独立于其批准的其他候选人的身份或数量。近年来，满意度批准投票和二次投票引起了学界关注，这两种方法都涉及"批准更多候选人"与"每位被选候选人获得的支持力度"之间的权衡。批准投票、满意度批准投票和二次投票均可被视作一种投票方式，其中选票被类比为具有不同单位范数（分别对应$\mathcal{L}^{\infty}$、$\mathcal{L}^{1}$和$\mathcal{L}^{2}$范数）的向量。这表明存在一种泛化形式：可将选票视为任意$\mathcal{L}^p$范数下的归一化单位向量。本文考察了多种用于验证投票方法的通用框架，并探究这些框架在何种程度上能为批准投票的泛化形式提供合理性依据。