Verifying Proportionality in Temporal Voting
Edith Elkind, Svetlana Obraztsova, Jannik Peters, Nicholas Teh
TL;DR
The paper studies verification of proportional representation in temporal voting with a fixed horizon, revealing that coNP-hardness holds for verifying JR, PJR, and EJR (including weak variants) and that the temporal setting is strictly harder than the traditional multiwinner model. It delineates a landscape of tractability via natural parameters: XP algorithms in the number of rounds $\ell$, W[1]-hardness in $\ell$, and fixed-parameter tractability in the number of voters $n$, along with a polynomial-time tractable special case where candidates join over time but never leave with static preferences. The authors provide constructive methods to obtain EJR outcomes through a two-stage Greedy Cohesive Rule, an ILP formulation, and prove an impossibility result for achieving EJR in online/semi-online settings. They also show that monotonic preferences yield efficient verification for both weak and standard variants, highlighting a practical domain where strong representation guarantees can be efficiently checked. Overall, the work maps the complexity frontier for temporal representation verification and offers algorithms and formulations for finding EJR outcomes under favorable conditions.
Abstract
We study a model of temporal voting where there is a fixed time horizon, and at each round the voters report their preferences over the available candidates and a single candidate is selected. Prior work has adapted popular notions of justified representation as well as voting rules that provide strong representation guarantees from the multiwinner election setting to this model. In our work, we focus on the complexity of verifying whether a given outcome offers proportional representation. We show that in the temporal setting verification is strictly harder than in multiwinner voting, but identify natural special cases that enable efficient algorithms.