Full Justified Representation in Temporal Voting: Efficient Computation and Verification

2026-09-22

AI Summary

This study addresses the long-standing open problem of polynomial-time computation and verification of the strong proportionality axiom, Fully Justified Representation (FJR), in temporal voting. We propose a strengthened axiom, FJR+, which integrates multi-winner election theory with satisfaction threshold counting, and achieve efficient computation via a modified Proportional Approval Voting (PAV) local search algorithm. Our primary contribution is the first polynomial-time computable strengthening of FJR, which strictly dominates existing combinations of FJR and EJR+. Furthermore, we prove that under static preferences, FJR+ is equivalent to quota rules. Collectively, this work achieves a dual breakthrough in both social choice theory and algorithmic design for proportional representation in sequential decision-making.

Abstract

In temporal voting, a fixed set of voters makes a collective decision in each of several rounds, and proportionality requires that groups with shared interests be represented across these decisions. Full justified representation (FJR) is among the strongest proportionality axioms known to be satisfiable in this setting, since it allows different members of a group to be satisfied by different selected candidates. However, whether an FJR outcome can be computed in polynomial time has remained open. We resolve this question by introducing temporal FJR+, a strengthening of both FJR and extended justified representation+ (EJR+) that can be computed and verified in polynomial time. The axiom measures a group's agreement in each round by the largest number of its members who approve a common candidate, and requires some member to attain the integer part of the group's proportional share of its best attainable average satisfaction, with a shortfall of at most one unit when this share is an integer. We show that verification reduces to counting approvals among the voters below each satisfaction threshold. For computation, we observe that local search for Proportional Approval Voting (PAV) can stop at an outcome violating FJR. Somewhat counterintuitively, the remedy is to subtract a small multiple of the minimum voter satisfaction from the PAV score: every FJR+ violation then admits a single-round change whose improvement guarantees polynomial running time. We further show that temporal FJR+ is strictly stronger than FJR and EJR+ combined, and coincides with lower quota on party-list preferences. Finally, temporal FJR+ and multiwinner FJR+ are incomparable in general, but coincide when approvals are static and each candidate-round pair is treated as a separate candidate.

Problem

• temporal voting
• full justified representation
• proportionality
• polynomial time computation

Innovation

• Temporal Voting
• Full Justified Representation
• Proportional Approval Voting
• Polynomial-time Computation
• Local Search