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Approval-Based Committee Voting under Incomplete Information

Aviram Imber, Jonas Israel, Markus Brill, Benny Kimelfeld

TL;DR

This paper advances the theory of approval-based committee voting under incomplete information by introducing three formal models of incompleteness—poset approval, three-valued approval (3VA), and linear incomplete approval—and by analyzing the computational complexity of core problems for Thiele-style rules (notably CC and PAV). It provides a detailed complexity landscape for computing possible and necessary committees and for determining possible/necessary membership, establishing NP-hardness results in general while identifying polynomial-time solvability when the committee size $k$ is fixed. The work also investigates how incomplete information interacts with representation axioms, showing polynomial-time decidability for JR, and for stronger axioms PJR+ and EJR+ in several models, with some problems remaining open in others. The findings have practical implications for elicitation strategies and decision-making in multiwinner elections when voter information is incomplete, offering tractable paths to representative committees and candidate inclusion decisions. Overall, the paper bridges multiple models of uncertainty with a unified set of problems, yielding a rich map of tractability boundaries and guiding future exploration of additional rules and uncertainty models.

Abstract

We investigate approval-based committee voting with incomplete information about the approval preferences of voters. We consider several models of incompleteness where each voter partitions the set of candidates into approved, disapproved, and unknown candidates, possibly with ordinal preference constraints among candidates in the latter category. This captures scenarios where voters have not evaluated all candidates and/or it is unknown where voters draw the threshold between approved and disapproved candidates. We study the complexity of some fundamental computational problems for a number of classic approval-based committee voting rules including Proportional Approval Voting and Chamberlin-Courant. These problems include determining whether a given set of candidates is a possible or necessary winning committee and whether a given candidate is possibly or necessarily a member of the winning committee. We also consider proportional representation axioms and the problem of deciding whether a given committee is possibly or necessarily representative.

Approval-Based Committee Voting under Incomplete Information

TL;DR

This paper advances the theory of approval-based committee voting under incomplete information by introducing three formal models of incompleteness—poset approval, three-valued approval (3VA), and linear incomplete approval—and by analyzing the computational complexity of core problems for Thiele-style rules (notably CC and PAV). It provides a detailed complexity landscape for computing possible and necessary committees and for determining possible/necessary membership, establishing NP-hardness results in general while identifying polynomial-time solvability when the committee size is fixed. The work also investigates how incomplete information interacts with representation axioms, showing polynomial-time decidability for JR, and for stronger axioms PJR+ and EJR+ in several models, with some problems remaining open in others. The findings have practical implications for elicitation strategies and decision-making in multiwinner elections when voter information is incomplete, offering tractable paths to representative committees and candidate inclusion decisions. Overall, the paper bridges multiple models of uncertainty with a unified set of problems, yielding a rich map of tractability boundaries and guiding future exploration of additional rules and uncertainty models.

Abstract

We investigate approval-based committee voting with incomplete information about the approval preferences of voters. We consider several models of incompleteness where each voter partitions the set of candidates into approved, disapproved, and unknown candidates, possibly with ordinal preference constraints among candidates in the latter category. This captures scenarios where voters have not evaluated all candidates and/or it is unknown where voters draw the threshold between approved and disapproved candidates. We study the complexity of some fundamental computational problems for a number of classic approval-based committee voting rules including Proportional Approval Voting and Chamberlin-Courant. These problems include determining whether a given set of candidates is a possible or necessary winning committee and whether a given candidate is possibly or necessarily a member of the winning committee. We also consider proportional representation axioms and the problem of deciding whether a given committee is possibly or necessarily representative.

Paper Structure

This paper contains 24 sections, 25 theorems, 32 equations, 1 figure, 7 tables.

Table of Contents

  1. Introduction
  2. Overview.
  3. Related Work
  4. Possible and necessary winners.
  5. Approval voting with incomplete information.
  6. Trichotomous preferences.
  7. Preliminaries
  8. Models of Incompleteness
  9. Poset approval.
  10. Three-valued approval (3VA).
  11. Linear incomplete approval.
  12. Possible and necessary committees.
  13. Computing Possible Committees
  14. Three-Valued Approval
  15. Linear Incomplete Approval
  16. ...and 9 more sections

Key Result

Lemma 1

Let $w_1, w_2$ be two weight functions. Assume there are two integers $k, t \geq 0$ and a strictly increasing linear function $g$ such that $w_2(x+t) = g(w_1(x))$ for every $x \in \left\{ 0, 1, \dots, k \right\}$. Then, in each of the three models of uncertainty, there is a polynomial-time reduction

Figures (1)

  • Figure 1: Models of incomplete approval profiles: Poset approval and the special cases of 3VA and linear incomplete approval. Dashed frames depict candidates that are approved in a valid completion.

Theorems & Definitions (51)

  • Example 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 41 more