Bi-Criteria Metric Distortion

Kiarash Banihashem; Diptarka Chakraborty; Shayan Chashm Jahan; Iman Gholami; MohammadTaghi Hajiaghayi; Mohammad Mahdavi; Max Springer

arXiv · 2412.10671

Bi-Criteria Metric Distortion

Kiarash Banihashem·Diptarka Chakraborty·Shayan Chashm Jahan·Iman Gholami·MohammadTaghi Hajiaghayi·Mohammad Mahdavi·Max Springer

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

cs.GT cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Kiarash Banihashem, Diptarka Chakraborty, Shayan Chashm Jahan, Iman Gholami, MohammadTaghi Hajiaghayi, Mohammad Mahdavi, Max Springer. 2024-12-14. Bi-Criteria Metric Distortion. https://arxiv.org/abs/2412.10671

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Coverage Games

We introduce and study coverage games - a novel framework for multi-agent planning in settings in which a system operates several agents but does not have full control on them, or interacts with an environment that consists of several agents. The game is played between a coverer, who has a set of objectives, and a disruptor. The coverer operates several agents that interact with the adversarial disruptor. The coverer wins if every objective is satisfied by at least one agent. Otherwise, the disruptor wins. Coverage games thus extend traditional two-player games with multiple objectives by allowing a (possibly dynamic) decomposition of the objectives among the different agents. They have many applications, both in settings where the system is the coverer (e.g., multi-robot surveillance, coverage in multi-threaded systems) and settings where it is the disruptor (e.g., prevention of resource exhaustion, ensuring non-congestion). We first study the theoretical properties of coverage games, including determinacy, and the ability to a priori decompose the objectives among the agents. We then study the problems of deciding whether the coverer or the disruptor wins. Besides a comprehensive analysis of the tight complexity of the problems, we consider interesting special cases, such as the one-player cases and settings with a fixed number of agents or objectives.

cs.GT↗

arXiv2026

Trading Proportionality for Strategic Robustness in Multi-Winner Approval Voting

Classical strategyproofness assumes a manipulator either knows how everyone else votes or is willing to gamble as if they did. Real voters rarely do. The recently introduced RAT-degree measures how many other participants' reports an agent must actually observe before a manipulation becomes strictly safe, interpolating between full truthfulness and immunity to blind manipulation. While previously explored in auctions and single-winner settings, we bring this measure to multi-winner elections. We apply it to Approval-Based Committee (ABC) rules under free-riding: a voter drops approved candidates from their truthful ballot to concentrate weight on marginal ones. We first analyze Proportional Approval Voting (PAV). Knowledge of $\lceil n/k \rceil$ ballots already enables a safe and strictly profitable drop, whereas knowledge of at most $\lfloor n/(k+1) \rfloor - 1$ ballots leaves the rule completely immune; an explicit instance shows the latter bound cannot be raised in general. Since a manipulator informed about roughly a $1/k$ fraction of the electorate therefore suffices, we ask how much proportionality must be surrendered to buy strategic robustness. We introduce $d$-RPAV, a parameterized family of Thiele rules with weights $d/(j+d-1)$ that recovers PAV at $d = 1$ and approaches Approval Voting (AV) as $d$ grows. We prove that $d$-RPAV satisfies $α$-Justified Representation ($α$-JR) for $α= d$, and is immune to safe free-riding given up to $\lfloor dn/(k+2d-1) \rfloor - 1$ known ballots, yielding a clean and tunable trade-off between proportional representation and strategic robustness.

cs.GT↗

arXiv2026

Online Fair Division: Pushing the Frontier of Approximate Proportionality

Online fair division captures allocation problems in which indivisible resources arrive over time and must be assigned before future resources are known. Understanding what fairness remains achievable when allocation decisions are immediate and irrevocable is a fundamental question in this setting. We study deterministic online allocation among $n$ agents with nonnegative additive valuations, where the number of goods is unknown and the adversary can adapt to previous allocation decisions. We focus on proportionality up to one good (PROP1) and examine how advance information affects the achievable guarantees. Without additional future information, we give a deterministic algorithm that guarantees $Ω(\frac{1}{\log(nm)})$-PROP1 after every round, where $m$ is the number of goods at termination. We complement this result by showing that, for every $n$ and sufficiently large $m$, every deterministic algorithm has an adaptive instance with $m$ goods on which its allocation has PROP1 approximation guarantee $O(\frac{\log \log m}{\log m})$. Thus, when the number of agents is fixed, our upper and lower bounds on the competitive ratio differ by at most an $O(\log\log m)$ factor. These results answer an open question proposed by Choo et al. on whether a nontrivial deterministic approximation for PROP1 can be obtained. We also study the setting where the algorithm knows the predictions of the maximum item value for every agent. When the predictions are accurate, we give a deterministic $\frac{1}{2}$-PROP1 algorithm, improving the $\frac{1}{n}$ guarantee of Choo et al. to a constant. We further establish an explicit upper bound below one on the competitive ratio, even for two agents with accurate predictions. Finally, we give a single deterministic algorithm that guarantees $\frac{1}{2}$-PROP1 when predictions are accurate and $Ω(\frac{1}{\log (nm)})$-PROP1 for arbitrary predictions.

cs.GT↗