Spatial Voting with Incomplete Voter Information

TL;DR

This paper studies spatial voting with incomplete voter information, where each voter's ideal point lies in a $d$-dimensional space and only intervals per dimension are known. It defines possible- and necessary-winner problems for positional scoring rules and approval voting, and develops a framework to compare partial spatial profiles with partial-order profiles. The main results show that necessary winners are computable in polynomial time for all fixed $d$ and scoring rules, while possible winners are tractable in many 1D cases (notably for plurality, veto, and several two-valued rules) but become hard in higher dimensions for rules like $k$-approval; in approval voting, necessary winners are tractable for $d\le2$ but possible winners are NP-hard for any $d\ge1$. The findings illuminate how incomplete spatial information interacts with computational complexity, revealing scheduling-like reductions and geometric enumeration techniques as key tools, and they extend the analysis to approval-based committee voting, with implications for spatially informed electoral design.

Abstract

We consider spatial voting where candidates are located in the Euclidean $d$-dimensional space, and each voter ranks candidates based on their distance from the voter's ideal point. We explore the case where information about the location of voters' ideal points is incomplete: for each dimension, we are given an interval of possible values. We study the computational complexity of finding the possible and necessary winners for positional scoring rules. Our results show that we retain tractable cases of the classic model where voters have partial-order preferences. Moreover, we show that there are positional scoring rules under which the possible-winner problem is intractable for partial orders, but tractable in the one-dimensional spatial setting. We also consider approval voting in this setting. We show that for up to two dimensions, the necessary-winner problem is tractable, while the possible-winner problem is hard for any number of dimensions.

Figures (4)

• Figure 1: Example of a spatial voting profile with $d=2$: a set $C = \left\{ c_1, c_2, c_3 \right\}$ of candidates and a single voter $v$.
• Figure 2: An illustration of the proof of \ref{['res:boundedRankingsHighDim']} for $d=2$ and $C = \left\{ c_1, c_2, c_3 \right\}$. A voter can be positioned at any point in the rectangle $[\ell_1, u_1] \times [\ell_2, u_2]$. Each line $H_{i,j}$ partitions $\mathbb{R}^2$ into 2 regions: the points closer to $c_i$, and the points closer to $c_j$. The top-left region --- above $H_{1,2}$ and to the left of $H_{2,3}$ and $H_{1,3}$ --- corresponds to the possible positions of the voter where the preference ranking equals $c_1 \succ c_2 \succ c_3$.
• Figure 3: An example of two voters in a completion of the partial profile $\mathbf{P}$ from the proof of \ref{['thm:pwdApproval']}. The voter $v$ represents a job of length $k$, and approves the $k$ candidates closest to it among $c_1, \dots, c_{\Tilde{d}}$. The voter $v'$ represents a job of length $k-1$ and approves $c^*$ and the $k-1$ candidates closest to it among $c_1, \dots, c_{\Tilde{d}}$.
• Figure 4: Example of an arrangement for $d=2$, candidate set $\left\{ c_1,c_2,c_3 \right\}$, and a voter $v$. The possible approval sets of $v$ are $\emptyset, \{c_1\}, \{c_2\}, \{c_3\},$ and $\{c_1, c_2\}$, depending on the actual position of $v$ inside the rectangle. The red points are all event points from the sweep line algorithm in \ref{['res:iterateApprCompletions']}.

Theorems & Definitions (22)

• Theorem 1: DBLP:journals/jcss/BetzlerD10DBLP:journals/jair/XiaC11DBLP:journals/ipl/BaumeisterR12
• Theorem 2
• Lemma 1: jamieson2011active
• Lemma 2
• proof
• Theorem 3
• proof
• Definition 1: Non-preemptive multi-machine scheduling with arrival times and deadlines
• Theorem 4
• proof