Metric Distortion with Preference Intensities

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.