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Finding Possible Winners in Spatial Voting with Incomplete Information

Hadas Shachnai, Rotem Shavitt, Andreas Wiese

TL;DR

This work studies the possible winner problem in spatial voting with incomplete voter information in $d$-dimensional Euclidean space under positional scoring and approval rules. It introduces a shapes scheduling reduction to solve $\mathsf{PW}\langle1\rangle$ for constant $k$-truncated rules and develops a dynamic-programming algorithm that runs in polynomial time when $k$ is fixed. It also provides a parameterized algorithm for $\mathsf{PW}\langle d\rangle$ with parameter $m$, extends the framework to approval voting, and analyzes weighted variants, establishing NP-hardness in several weighted settings (notably Borda with $m=4$ and certain $d$). Collectively, the results delineate the tractability frontier for possible winners under incomplete spatial information and lay groundwork for future exploration of higher-dimensional and weighted scenarios.

Abstract

We consider a spatial voting model where both candidates and voters are positioned in the $d$-dimensional Euclidean space, and each voter ranks candidates based on their proximity to the voter's ideal point. We focus on the scenario where the given information about the locations of the voters' ideal points is incomplete; for each dimension, only an interval of possible values is known. In this context, we investigate the computational complexity of determining the possible winners under positional scoring rules. Our results show that the possible winner problem in one dimension is solvable in polynomial time for all $k$-truncated voting rules with constant $k$. Moreover, for some scoring rules for which the possible winner problem is NP-complete, such as approval voting for any dimension or $k$-approval for $d \geq 2$ dimensions, we give an FPT algorithm parameterized by the number of candidates. Finally, we classify tractable and intractable settings of the weighted possible winner problem in one dimension, and resolve the computational complexity of the weighted case for all two-valued positional scoring rules when $d=1$.

Finding Possible Winners in Spatial Voting with Incomplete Information

TL;DR

This work studies the possible winner problem in spatial voting with incomplete voter information in -dimensional Euclidean space under positional scoring and approval rules. It introduces a shapes scheduling reduction to solve for constant -truncated rules and develops a dynamic-programming algorithm that runs in polynomial time when is fixed. It also provides a parameterized algorithm for with parameter , extends the framework to approval voting, and analyzes weighted variants, establishing NP-hardness in several weighted settings (notably Borda with and certain ). Collectively, the results delineate the tractability frontier for possible winners under incomplete spatial information and lay groundwork for future exploration of higher-dimensional and weighted scenarios.

Abstract

We consider a spatial voting model where both candidates and voters are positioned in the -dimensional Euclidean space, and each voter ranks candidates based on their proximity to the voter's ideal point. We focus on the scenario where the given information about the locations of the voters' ideal points is incomplete; for each dimension, only an interval of possible values is known. In this context, we investigate the computational complexity of determining the possible winners under positional scoring rules. Our results show that the possible winner problem in one dimension is solvable in polynomial time for all -truncated voting rules with constant . Moreover, for some scoring rules for which the possible winner problem is NP-complete, such as approval voting for any dimension or -approval for dimensions, we give an FPT algorithm parameterized by the number of candidates. Finally, we classify tractable and intractable settings of the weighted possible winner problem in one dimension, and resolve the computational complexity of the weighted case for all two-valued positional scoring rules when .
Paper Structure (16 sections, 16 theorems, 16 equations, 8 figures, 1 table)

This paper contains 16 sections, 16 theorems, 16 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Spatial Voting
  5. Voting Rules
  6. Partial Spatial Voting
  7. Spatial Approval Voting
  8. Parameterized Complexity
  9. $\mathsf{PW}\mathord{\langle{1\rangle}}$ with $k$-Truncated Voting Rules
  10. The Shapes Scheduling Problem
  11. Reduction from $\mathsf{PW}\mathord{\langle{1\rangle}}$ to Shapes Scheduling
  12. An Algorithm for Shapes Scheduling
  13. Hardness Results for Scheduling with Shapes
  14. Parameterized Algorithm for $\mathsf{PW}\mathord{\langle{d\rangle}}$
  15. Spatial Voting with Weighted Voters
  16. ...and 1 more sections

Key Result

Lemma 1

For each possible position $T_{j}\in[\ell_{j},u_{j}]$ for voter $v_{j}$, only candidates in $\{c_{i_L},...,c_{i_R}\}$ receive a score from $v_{j}$.

Figures (8)

  • Figure 1: Example of spatial voting in a two-dimensional space.
  • Figure 2: Illustration of a partial spatial profile and two different spatial completions.
  • Figure 3: Two schedule options for a job at time $t$ by shape $f=(2,1)$ with processing time $p=2$.
  • Figure 4: Example of $i_L$ and $i_R$ for a voter $v_j$ described by $P_j$.
  • Figure 5: Two positions of a voter $v_j$ and the corresponding shapes.
  • ...and 3 more figures

Theorems & Definitions (34)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 24 more