We develop new voting mechanisms for the case where voters and candidates are located in an arbitrary unknown metric space, and the goal is to choose a candidate minimizing social cost: the total distance of the voters to this candidate. Previous work has often assumed that only the ordinal preferences of the voters are known (instead of their true costs), and focused on minimizing distortion: the quality of the chosen candidate as compared to the best possible candidate. In this paper, we instead assume that a (very small) amount of information is known about the voter preference strengths, not just about their ordinal preferences. We provide mechanisms with much better distortion when this extra information is known as compared to mechanisms which use only ordinal information. We quantify tradeoffs between the amount of information known about preference strengths and the achievable distortion. We further provide advice about which type of information about preference strengths seems to be the most useful. Finally, we conclude by quantifying the ideal candidate distortion, which compares the quality of the chosen outcome with the best possible candidate that could ever exist, instead of only the best candidate that is actually in the running.
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