arXiv subjects

Jianhao Jia

Publications and source records attributed to Jianhao Jia.

4 recordsLinked to original sources

arXiv2026

Strategyproof Aggregation in Euclidean Spaces: Rigidity and Median Optimality

We study deterministic strategyproof aggregation in finite-dimensional Euclidean spaces. For every odd number $n\ge3$ of agents and every finite dimension, we prove that the coordinate-wise median minimizes the worst-case approximation ratio for total Euclidean distance among all continuous, anonymous, deterministic strategyproof mechanisms. The same optimality result holds for every even $n\ge4$ when each coordinate uses a fixed choice of the lower or upper middle rank. The proof combines a rigidity theorem with a normalization that does not increase the approximation ratio: any hypothetical mechanism outperforming the median has a normalized representative that is a fixed coordinate-wise order-statistic rule in a single orthonormal frame. A reflection argument then shows that no such rule improves on the median.

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arXiv2026

Robust Lottery Compression for Metric Voting: A Transfer Principle for Bounded Randomness

We study metric distortion in randomized social choice under bounded randomness: on every preference profile, the voting rule must deterministically identify a multiset of $K$ candidates and then select a uniformly random entry. Previous work showed that this restricted model can beat the optimal deterministic distortion of $3$. We show that it can in fact approach the current best unrestricted upper benchmark of $5/2$. For every integer $K\ge 802$, there exists a bounded-randomness rule with distortion at most $\frac{5}{2} +3\left(\fracπ{8K}\right)^{1/3} +2\sqrt{\fracπ{8K}}$. Consequently, $O(\varepsilon^{-3})$ entries suffice for distortion $5/2+\varepsilon$, independently of the numbers of voters and candidates. We also show that $164$ entries already achieve distortion strictly below $3$, giving $2\le N^\star\le 164$ for the minimum list size needed to break the deterministic barrier. Our main technical contribution is a dimension-free compression theorem: if a lottery has distortion at most $ρ$ and every candidate in its support has deterministic distortion at most $H$, then it admits a uniform $K$-entry approximation with distortion at most $ρ+(H+1)\sqrt{π/(8K)}$. Thus, lotteries whose possible outcomes are already well behaved incur only $O(K^{-1/2})$ compression loss. Mixed Integrated Veto does not satisfy this support condition, so we first remove early-eliminated outcomes, trading $O(τ^2)$ distortion loss for an $O(1/τ)$ bound on the deterministic distortion of every supported candidate. Balancing this repair cost against compression yields the $O(K^{-1/3})$ convergence rate.

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arXiv2026

Product Gap Mechanisms for Multi-Facility Location

We study randomized strategyproof mechanisms for locating multiple facilities on the real line. We introduce the \emph{Product-Gap mechanism}, which selects $k$ reported locations with probability proportional to the product of the consecutive gaps between them and opens facilities at the selected locations. We prove that, for every $k\geq 2$, the mechanism achieves a tight approximation ratio of $2k$ for social cost. We then study its incentive properties and show that it is strategyproof in expectation for $k=2$ and $k=3$, but is not strategyproof for $k\geq 4$. In particular, the mechanism gives strategyproof $4$- and $6$-approximations for two and three facilities, respectively. Finally, for two facilities, we combine Product-Gap with the Proportional mechanism of Lu et al. We show that an optimized report-independent mixture is strategyproof and has a tight approximation ratio of $(74+4\sqrt{3})/23\approx 3.519$ on the line, improving upon the previous factor of $4$.

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arXiv2025

Approximation guarantees of Median Mechanism in $\mathbb{R}^d$

The coordinate-wise median is a classic and most well-studied strategy-proof mechanism in social choice and facility location scenarios. Surprisingly, there is no systematic study of its approximation ratio in $d$-dimensional spaces. The best known approximation guarantee in $d$-dimensional Euclidean space $\mathbb{L}_2(\mathbb{R}^d)$ is $\sqrt{d}$ via embedding $\mathbb{L}_1(\mathbb{R}^d)$ into $\mathbb{L}_2(\mathbb{R}^d)$ metric space, that only appeared in appendix of [Meir 2019].This upper bound is known to be tight in dimension $d=2$, but there are no known super constant lower bounds. Still, it seems that the community's belief about coordinate-wise median is on the side of $Θ(\sqrt{d})$. E.g., a few recent papers on mechanism design with predictions [Agrawal, Balkanski, Gkatzelis, Ou, Tan 2022], [Christodoulou, Sgouritsa, Vlachos 2024], and [Barak, Gupta, Talgam-Cohen 2024] directly rely on the $\sqrt{d}$-approximation result. In this paper, we systematically study approximate efficiency of the coordinate-median in $\mathbb{L}_{q}(\mathbb{R}^d)$ spaces for any $\mathbb{L}_q$ norm with $q\in[1,\infty]$ and any dimension $d$. We derive a series of constant upper bounds $UB(q)$ independent of the dimension $d$. This series $UB(q)$ is growing with parameter $q$, but never exceeds the constant $UB(\infty)= 3$. Our bound $UB(2)=\sqrt{6\sqrt{3}-8}<1.55$ for $\mathbb{L}_2$ norm is only slightly worse than the tight approximation guarantee of $\sqrt{2}>1.41$ in dimension $d=2$. Furthermore, we show that our upper bounds are essentially tight by giving almost matching lower bounds $LB(q,d)=UB(q)\cdot(1-O(1/d))$ for any dimension $d$ with $LB(q,d)=UB(q)$ when $d\to\infty$. We also extend our analysis to the generalized median mechanism in [Agrawal, Balkanski, Gkatzelis, Ou, Tan 2022] for $\mathbb{L}_2(\mathbb{R}^2)$ space to arbitrary dimensions $d$ with similar results.

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