[Paper Review] Bribery as a Measure of Candidate Success: Complexity Results for Approval-Based Multiwinner Rules

Piotr Faliszewski, Piotr Skowron+1 more|arXiv (Cornell University)|Apr 19, 2021

Game Theory and Voting Systems30 references5 citations

TL;DR

This paper introduces bribery as a novel measure of candidate success in approval-based multiwinner elections, where the cost to make a candidate win via minimal vote modifications reflects their electoral performance. It provides a comprehensive complexity analysis across six approval-based multiwinner rules, revealing a rich landscape from polynomial-time solvable to inapproximable problems, with FPT algorithms for parameterized settings.

ABSTRACT

We study the problem of bribery in multiwinner elections, for the case where the voters cast approval ballots (i.e., sets of candidates they approve) and the bribery actions are limited to: adding an approval to a vote, deleting an approval from a vote, or moving an approval within a vote from one candidate to the other. We consider a number of approval-based multiwinner rules (AV, SAV, GAV, RAV, approval-based Chamberlin--Courant, and PAV). We find the landscape of complexity results quite rich, going from polynomial-time algorithms through NP-hardness with constant-factor approximations, to outright inapproximability. Moreover, in general, our problems tend to be easier when we limit out bribery actions on increasing the number of approvals of the candidate that we want to be in a winning committee (i.e., adding approvals only for this preferred candidate, or moving approvals only to him or her). We also study parameterized complexity of our problems, with a focus on parameterizations by the numbers of voters or candidates.

Motivation & Objective

• To address the lack of meaningful performance metrics for losing candidates in multiwinner approval elections.
• To overcome limitations of traditional score-based evaluation methods, which can be arbitrary, non-informative, or inapplicable in multiwinner settings.
• To propose a new conceptual framework using bribery problems to quantify how close a candidate was to winning, based on minimal changes to votes.
• To analyze the computational complexity of bribery problems under various approval-based multiwinner rules, including AV, SAV, GAV, RAV, ABC, and PAV.
• To explore parameterized complexity with respect to the number of voters and candidates, identifying tractable cases and FPT algorithms.

Proposed method

• Formalize bribery problems in approval-based multiwinner elections, allowing three actions: adding an approval, deleting an approval, or moving an approval between candidates.
• Model each bribery instance as a min-cost max-flow network with source, candidate nodes, type nodes, and sink, where flow represents approval changes.
• Use node types to represent possible approval patterns per voter, and assign costs to transitions between types based on allowed bribery actions.
• Apply polynomial-time min-cost max-flow algorithms to compute the minimum bribery cost for a target candidate to win, ensuring feasibility via flow constraints.
• Prove FPT results by showing that the problem is solvable in time f(k) · poly(n) for parameter k (number of voters or candidates), using dynamic programming over candidate types.
• Leverage structural properties of approval ballots and flow networks to reduce the number of types to 2^n, enabling efficient computation for small voter counts.

Experimental results

Research questions

• RQ1 How can we meaningfully measure the success of a losing candidate in an approval-based multiwinner election, beyond traditional scores?
• RQ2 What is the computational complexity of determining the minimum number of approval changes needed to make a given candidate win, under various approval-based multiwinner rules?
• RQ3 Which bribery problems are tractable (e.g., polynomial-time or FPT) when parameterized by the number of voters or candidates?
• RQ4 How does the complexity vary depending on the allowed bribery actions—specifically, when restricting changes to only benefit a preferred candidate?
• RQ5 Can min-cost flow techniques be effectively applied to model and solve bribery problems in approval-based multiwinner elections?

Key findings

• The bribery cost provides a robust, intuitive measure of candidate success: lower cost indicates stronger performance.
• For all studied rules—AV, SAV, GAV, RAV, ABC, and PAV—the bribery problem is in XP when parameterized by the number of candidates or voters.
• The problem is FPT with respect to the number of voters, as the min-cost flow formulation runs in time polynomial in the number of candidates and exponential in the number of voters.
• The complexity landscape is rich: problems range from polynomial-time solvable to NP-hard with constant-factor approximation, to inapproximable.
• Problems become significantly easier when bribery actions are restricted to only help the target candidate (e.g., only adding approvals for the preferred candidate), due to reduced solution space.
• The min-cost flow model successfully captures all allowed bribery actions and enables efficient computation of minimum-cost transformations, especially under parameterized constraints.