In voting with ranked ballots, each agent submits a strict ranking of the form $a \succ b \succ c \succ d$ over the alternatives, and the voting rule decides on the winner based on these rankings. Although this ballot format has desirable characteristics, there is a question of whether it is expressive enough for the agents. Kahng, Latifian, and Shah address this issue by adding intensities to the rankings. They introduce the ranking with intensities ballot format, where agents can use both $\succ\!\!\succ$ and $\succ$ in their rankings to express intensive and normal preferences between consecutive alternatives in their rankings. While they focus on analyzing this ballot format in the utilitarian distortion framework, in this work, we look at the potential of using this ballot format from the metric distortion viewpoint. We design a class of voting rules coined Positional Scoring Matching rules, which can be used for different problems in the metric setting, and show that by solving a zero-sum game, we can find the optimal member of this class for our problem. This rule takes intensities into account and achieves a distortion lower than $3$. In addition, by proving a bound on the price of ignoring intensities, we show that we might lose a great deal in terms of distortion by not taking the intensities into account.

翻译：在排序投票中，每位参与者提交形如 $a \succ b \succ c \succ d$ 的严格排序，投票规则根据这些排序决定获胜者。尽管这种投票格式具有理想特性，但存在一个问题：它是否足以充分表达参与者的偏好。Kahng、Latifian 和 Shah 通过为排序添加强度来解决这一问题。他们引入了带强度排序的投票格式，参与者可以在排序中同时使用 $\succ\!\!\succ$ 和 $\succ$ 符号，以表达排序中相邻备选方案之间的强度偏好与普通偏好。虽然他们的研究集中在功利主义失真框架下分析这种投票格式，但在本文中，我们从度量失真的视角探讨使用这种投票格式的潜力。我们设计了一类称为位置评分匹配规则的投票规则，该规则可用于度量设置中的不同问题，并证明通过求解零和博弈，可以为我们的问题找到该类别中的最优规则。该规则考虑了偏好强度，并实现了低于 $3$ 的失真度。此外，通过证明忽略强度的代价上界，我们表明若不考虑强度信息，可能会在失真度方面造成显著损失。