Robustness of Approval-Based Multiwinner Voting Rules

TL;DR

This work examines how robust approval-based multiwinner voting rules are to small perturbations in votes, introducing Add, Remove, and Swap operations, along with robustness levels and robustness-radius and counting variants. An analysis of seven rules (AV, SAV, CC, PAV, GreedyCC, GreedyPAV, Phragmén) reveals a dichotomy: robustness levels are either $1$ or the committee size $k$, with AV admitting polynomial-time radius decisions and FP counting for Add/Remove; SAV shows polynomial-time decision with ${\#P}$-hard counting for Add/Remove, while CC, PAV, GreedyCC, GreedyPAV, and Phragmén exhibit ${\mathrm{NP}}$-hardness or ${\mathrm{NP}}$-completeness for robustness problems. The unit-decreasing Thiele rules (including CC and PAV) inherit NP-hard robustness radii, and the greedier/sequential rules (GreedyCC, GreedyPAV, Phragmén) also reach NP-complete status for robustness-radius computations. The results map a comprehensive complexity landscape linking rule structure to stability under vote perturbations and have implications for auditing and designing robust multiwinner systems.

Abstract

We investigate how robust approval-based multiwinner voting rules are to small perturbations in the votes. In particular, we consider the extent to which a committee can change after we add/remove/swap one approval, and we consider the computational complexity of deciding how many such operations are necessary to change the set of winning committees. We also consider the counting variants of our problems, which can be interpreted as computing the probability that the result of an election changes after a given number of random perturbations of the given election.

Figures (5)

• Figure 1: The matrix notation for specifying elections and operations on them.
• Figure 2: Approval election matrices illustrating the proof of \ref{['pro:sav-add']}.
• Figure 3: Approval election matrices illustrating proofs of \ref{['prop-RL-UnitDecreasing']} and \ref{['prop-RL-GreedyUnitDecreasing']}, for $k=3$.
• Figure 4: Example of an election used in the proof of Theorem \ref{['propNPSwapUD']}, for X3C instance with universe set $U = \{u_1, \ldots, u_6\}$ and sets $S_1 = \{u_1,u_2,u_3\}$, $S_2 = \{u_3,u_4,u_5\}$, $S_3 = \{u_4,u_5,u_6\}$, and $S_4 = \{u_2,u_3,u_4\}$. Consequently, we have $m=4$ and $k = 2$. Symbol $\circ$ represents an approval for a candidate from a given voter, and $\bigcirc$ represents $\ell$ approvals coming from a group of voters.
• Figure 5: Timeline for the Phragmén rule acting on the election from Theorem \ref{['thm:phr']}.

Theorems & Definitions (33)

• Definition 2.1
• Definition 2.2
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Proposition 3.4
• proof