Bi-Criteria Metric Distortion

TL;DR

This work investigates bicriteria metric distortion by allowing a fixed-size committee of candidates and measuring voter costs to the nearest committee member. On the line metric, it shows that a constant-size committee (as small as $2$) can achieve the optimal single-candidate cost under the sum-cost objective, and extends to general metrics with a bound of $1+\frac{2}{m-1}$ using $m-1$ candidates; tight lower bounds are provided for line and $2$-D Euclidean spaces. For the max-cost objective, the authors establish that $4$, $3$, and $2$ committees yield distortions of $1$, $1.5$, and $2$ respectively on the line, with corresponding lower bounds, and show a $3$-distortion barrier persists in $2$-D Euclidean spaces even for large committees. A key methodological contribution is the core-based ordering framework, built from SplitLine, SortCandidates, and SortVoters, which enables precise structure-awareness on the line and underpins the upper and lower bounds. Together, these results advance the understanding of how small, well-chosen committees can outperform single-winner rules in metric distortion settings and have implications for practical committee selection in spatial voting models.

Abstract

Selecting representatives based on voters' preferences is a fundamental problem in social choice theory. While cardinal utility functions offer a detailed representation of preferences, ordinal rankings are often the only available information due to their simplicity and practical constraints. The metric distortion framework addresses this issue by modeling voters and candidates as points in a metric space, with distortion quantifying the efficiency loss from relying solely on ordinal rankings. Existing works define the cost of a voter with respect to a candidate as their distance and set the overall cost as either the sum (utilitarian) or maximum (egalitarian) of these costs across all voters. They show that deterministic algorithms achieve a best-possible distortion of 3 for any metric when considering a single candidate. This paper explores whether one can obtain a better approximation compared to an optimal candidate by relying on a committee of $k$ candidates ($k \ge 1$), where the cost of a voter is defined as its distance to the closest candidate in the committee. We answer this affirmatively in the case of line metrics, demonstrating that with $O(1)$ candidates, it is possible to achieve optimal cost. Our results extend to both utilitarian and egalitarian objectives, providing new upper bounds for the problem. We complement our results with lower bounds for both the line and 2-D Euclidean metrics.

Figures (10)

• Figure 1: This figure is the illustration of the succession of nearest of any voter.
• Figure 2: For a voter $v_i$, if we have $c_1 \succ_{i} c_k$, $c_2 \succ_{i} c_k$, and $c_k \succ_{i} c_k$, then $c_k$ and $c_j$ cannot both be in the same side of $p$. The figure above illustrates this contradiction, while the one below shows that they can be on opposite sides.
• Figure 3: For a voter $v_i$, if we have $c_1 \succ_{i} c_2$ and $c_k \succ_{i} c_2$, then $c_k$ is in $L$. The figure above illustrates that if $c_k$ were in $R$, then $c_2$ would be in the consecutive subsequence of $c_1$ and $c_k$ which is a contradiction. On the other hand, the one below shows that $c_k$ must be in $L$.
• Figure 4: A figure illustrating three candidates $c_1$, $c_2$, and $c_3$ along with the possible locations of voters closest to each candidate, $V_1$, $V_2$, and $V_3$. Voters $v_1$, $v_2$ and $v_3$ show examples of voters in each set.
• Figure 5: Illustration of the candidates $c_l$ and $c_r$, and the voters $v_l$ and $v_r$, demonstrating why choosing these candidates does not achieve a distortion of $1$

Theorems & Definitions (55)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 5: Core
• Definition 6: Pivot Voter and Candidates
• Definition 7: Pivot Point
• Definition 8
• Definition 9
• Theorem 10