Distortion of Multi-Winner Elections on the Line Metric: The Polar Comparison Rule

We consider the problem of selecting a committee of $k$ alternatives among $m$ alternatives, based on the ordinal rank list of voters. Our focus is on the case where both voters and alternatives lie on a metric space-specifically, on the line-and the objective is to minimize the additive social cost. The additive social cost is the sum of the costs for all voters, where the cost for each voter is defined as the sum of their distances to each member of the selected committee.