Metric Distortion of Line-up Elections: The Right Person for the Right Job

TL;DR

This work defines the ($m$,$\ell$)-line-up election in a common metric space, where the goal is to choose $\ell$ candidates and assign them to $\ell$ positions to minimize the total voter cost, which combines distances to chosen candidates and candidates’ fitness to their positions. It develops deterministic mechanisms with constant distortion under three information regimes: ordinal voter preferences, ordinal position preferences, and exact candidate/position locations, each achieving $3$, $5$, or $3$ respectively in various settings, with further improvements (e.g., $7$ for general cases with ordinal voter preferences and $2$ or $5/3$ in selective scenarios) when additional structure is leveraged. The methods synthesize social-choice techniques (e.g., single-winner distortions, serial dictatorship) with matching theory, exploiting the shared metric space to bound the combined cost of selection and assignment. The results demonstrate that surprisingly little information is needed to achieve outcomes close to the full-information optimum, and they identify tight lower bounds in several regimes, while outlining directions for future work, including randomized mechanisms and alternative cost notions.

Abstract

We provide mechanisms and new metric distortion bounds for line-up elections. In such elections, a set of $n$ voters, $m$ candidates, and $\ell$ positions are all located in a metric space. The goal is to choose a set of candidates and assign them to different positions, so as to minimize the total cost of the voters. The cost of each voter consists of the distances from itself to the chosen candidates (measuring how much the voter likes the chosen candidates, or how similar it is to them), as well as the distances from the candidates to the positions they are assigned to (measuring the fitness of the candidates for their positions). Our mechanisms, however, do not know the exact distances, and instead produce good outcomes while only using a smaller amount of information, resulting in small distortion. We consider several different types of information: ordinal voter preferences, ordinal position preferences, and knowing the exact locations of candidates and positions, but not those of voters. In each of these cases, we provide constant distortion bounds, thus showing that only a small amount of information is enough to form outcomes close to optimum in line-up elections.

Figures (13)

• Figure 1: Consider a distribution network where we have farms $\mathcal{P}$, warehouses $\mathcal{C}$, and stores $\mathcal{V}$. We show a single store connected to $3$ warehouses, each of which can handle a single type of item. The cost for a single store is represented by the distances to the assigned warehouses and from the warehouses to the farms. Each distribution center can process a single good. We need to form a matching between producers and warehouses so that the total distance all goods need to travel between producers, warehouses, and stores are minimized.
• Figure 2: Metric space $d_1$.
• Figure 3: Metric space $d_2$.
• Figure 5: The initial matching with the lines representing the cost for each voter $v$ where $\operatorname{cost}(v,M) = d(A,1) + d(B,2) + d(C,3) + d(v, A) + d(v,B) + d(v,C)$
• Figure 6: In line (\ref{['kk3label1']}) we use the triangle inequality on $d(A,1) \leq d(A,v) + d(v,B) + d(B,1)$

Theorems & Definitions (47)

• Definition 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Lemma 3.4
• proof
• Corollary 3.5