Strengthening Proportionality in Temporal Voting

TL;DR

This work extends justified representation concepts to temporal voting with approval ballots, building a comprehensive hierarchy of temporal proportionality notions such as EJR+, sEJR+, FJR, sFJR, FPJR, sFPJR, wFPJR, and Core variants. It provides algorithmic and existential results, notably showing that EJR+ is polynomial-time verifiable and always exists via $\varepsilon$-lsPAV (with $\varepsilon<1/\ell^2$), and that temporal FJR outcomes exist (via a Greedy Cohesive Rule variant). When $n$ divides $\ell$, SDR computes strong FPJR, and strong FPJR implies FPJR, PJR, and wFPJR, creating a tractable regime for robust temporal proportionality; by contrast, some stronger notions (e.g., naive EJR+ variants and certain core forms) may be unsatisfiable, and the authors provide a detailed separation analysis. The paper thus delivers a rigorous map of local and global proportional guarantees over time, clarifying implications, limits, and computational aspects, with practical implications for sequential decision processes and time-evolving representation systems.

Abstract

We study proportional representation in the framework of temporal voting with approval ballots. Prior work adapted basic proportional representation concepts -- justified representation (JR), proportional JR (PJR), and extended JR (EJR) -- from the multiwinner setting to the temporal setting. Our work introduces and examines ways of going beyond EJR. Specifically, we consider stronger variants of JR, PJR, and EJR, and introduce temporal adaptations of more demanding multiwinner axioms, such as EJR+, full JR (FJR), full proportional JR (FPJR), and the Core. For each of these concepts, we investigate its existence and study its relationship to existing notions, thereby establishing a rich hierarchy of proportionality concepts. Notably, we show that two of our proposed axioms -- EJR+ and FJR -- strengthen EJR while remaining satisfiable in every temporal election.

Figures (1)

• Figure 1: Axioms considered in our paper. A solid arrow from axiom $A$ to axiom $B$ means that $A$ implies $B$; an absence of a path from $A$ to $B$ means that the implication does not hold (even if each voter approves exactly one candidate per round, denoted by $\mathcal{E}_{=1}$). A dashed arrow means implication for $\mathcal{E}_{=1}$, but not in general. Thick arrows denote implications established in this paper. The axioms on a green plain background can be satisfied in each election. The axioms on a yellow dotted background can be satisfied if each voter approves at least one candidate per round and the number of rounds is divisible by the number of voters, but not in general; for the ones on a red striped background even in this special case there are instances where they are not satisfiable. For axioms on white background, satisfiability remains open. The axioms introduced in this paper are written in bold.

Theorems & Definitions (59)

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