  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.
