Metric Distortion for Tournament Voting and Beyond

TL;DR

We investigate metric distortion in voting under a proximity model where voters and candidates occupy a shared metric space. By extending tournament rules to $k$-tournaments and employing a biased metric framework, we derive a universal lower bound of $\$3.1128$ for deterministic rules and an upper bound of $\$3.9312$, while showing that increasing $k$ enables deterministic rules to approach distortion $3$ and that randomized $k$-tournament rules can beat $3$ even for $k=3$. The paper introduces stable lotteries and two concrete rules—Deterministic Simultaneous Lottery Veto and Randomized Pruned Double Lotteries—establishing distortion guarantees that progressively close the gap between the best deterministic and randomized rules. These results connect metric distortion to committee-selection techniques and RLHF-style information regimes, suggesting practical pathways for leveraging higher-order preferences. Finally, the work outlines a broad research program to develop a cohesive theory of $k$-tournament rules with potential applications in algorithmic social choice and AI alignment contexts.

Abstract

In the well-studied metric distortion problem in social choice, we have voters and candidates located in a shared metric space, and the objective is to design a voting rule that selects a candidate with minimal total distance to the voters. However, the voting rule has limited information about the distances in the metric, such as each voter's ordinal rankings of the candidates in order of distances. The central question is whether we can design rules that, for any election and underlying metric space, select a candidate whose total cost deviates from the optimal by only a small factor, referred to as the distortion. A long line of work resolved the optimal distortion of deterministic rules, and recent work resolved the optimal distortion of randomized (weighted) tournament rules, which only use the aggregate preferences between pairs of candidates. In both cases, simple rules achieve the optimal distortion of $3$. Can we achieve the best of both worlds: a deterministic tournament rule matching the lower bound of $3$? Prior to our work, the best rules have distortion $2 + \sqrt{5} \approx 4.2361$. In this work, we establish a lower bound of $3.1128$ on the distortion of any deterministic tournament rule, even when there are only 5 candidates, and improve the upper bound with a novel rule guaranteeing distortion $3.9312$. We then generalize tournament rules to the class of $k$-tournament rules which obtain the aggregate preferences between $k$-tuples of candidates. We show that there is a family of deterministic $k$-tournament rules that achieves distortion approaching $3$ as $k$ grows. Finally, we show that even with $k = 3$, a randomized $k$-tournament rule can achieve distortion less than $3$, which had been a longstanding barrier even for the larger class of ranked voting rules.

Figures (5)

• Figure 1: The biased metric integrals as stacked blocks. Each block is annotated with its corresponding ranking on the left, and voters with the same ranking are aggregated into a single block. The figures are exact representations of the integrals when there are only three candidates $\{i^*, j^*, k\}$, but when there are more candidates, the figure on the left can be viewed as a lower bound, and the figure on the right can be viewed as an upper bound. For example, if $i^*\succ_v j^*\succ_v k$, then $2d(i^*, v)\geq x_{j^*} - x_k$ and $d(j^*,v) - d(i^*, v) \leq x_k$.
• Figure 2: After sliding the blocks corresponding to voters that prefer $i^*$ over $k$, the height of mass between $[0, x_k]$ is at least $s_{k\succ i^*}$, and the height of mass between $[x_k, x_{j^*}]$ is at least $s_{j^*\succ k}$ (highlighted by the dashed boxes).
• Figure 3: Examples of local structures where \ref{['cor:local']} shows that $\mathop{\mathrm{SC}}\nolimits(j^*) \leq (1 + 2\lambda)\mathop{\mathrm{SC}}\nolimits(i^*)$.
• Figure 4: Left: the cycle $\mathcal{C}$. Right: the cycle $\hat{\mathcal{C}}$. We derive a contradiction by showing that some $i^*$ cannot blanket some $j^*$, using $k$ and $\ell$ as labeled.
• Figure 5: A tournament graph where each candidate has worst-case distortion at least 3.112.

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: $k$-tournament rules
• Definition 2.2: metric distortion
• Definition 3.1
• Theorem 3.2: necessary and sufficient integral condition DBLP:journals/jacm/CharikarRWW24
• Corollary 3.3: DBLP:conf/soda/CharikarR22
• proof : Proof sketch
• proof : 1
• Lemma 4.1