Sequence-Wise Temporal Voting

• Sequence-wise temporal voting is a model for repeated collective choices where the outcome is a sequence of winners, reflecting cumulative voter satisfaction.
• The framework introduces proportionality axioms like JR, PJR, and EJR, which assess fairness over the entire decision sequence rather than in isolated rounds.
• Key challenges include coNP-completeness in verification and strategic dynamics, with tractable cases emerging under monotonic preferences and specific candidate constraints.

Searching arXiv for the cited temporal voting papers and closely related work. Sequence-wise temporal voting is a model of repeated collective choice in which a fixed electorate participates over a finite time horizon and the social outcome is a sequence of winners rather than a single one-shot decision. In the approval-based single-winner setting, one fixes a time horizon $\ell$, a voter set $N$, a candidate set $P$, voter-specific approval trajectories $\mathcal{s}_i=(s_{i,1},\dots,s_{i,\ell})$, and a winner sequence $W=(w_1,\dots,w_\ell)$ with one winner per round; voter $i$ then derives satisfaction

$$\mathrm{sat}_i(W)=|\{t\in[\ell]: w_t\in s_{i,t}\}|.$$

Within this framework, proportionality is assessed over the entire sequence, not round by round, and the strongest temporal fairness requirements are explicitly called “sequence-wise” axioms (Elkind et al., 9 Feb 2025). In the broader temporal-fairness literature, the same sequence-oriented viewpoint also appears in repeated committee selection, sequential decision making with changing approvals, and related models of temporal governance and delegation (Elkind et al., 2023).

1. Conceptual scope and modeling perspective

Sequence-wise temporal voting arises from the observation that many electoral and decision systems are repeated: one candidate, alternative, or committee is selected at each period, while voter preferences may change over time. A unified temporal framework therefore treats the outcome as a sequence $W=(W^1,\dots,W^T)$ in the multiwinner case, or $D=(d_1,\dots,d_T)$ in the single-winner sequential-decision case, with ballots that may be approval-based, ranking-based, or cardinal, and with preferences that may be static, dynamic non-adaptive, or dynamic adaptive (Elkind et al., 2023).

In the approval-based sequential-decision formulation, each round $j$ has its own set of alternatives $N$0, voter $N$1 submits an approval ballot $N$2, and the utility of voter $N$3 under a decision sequence $N$4 is

$N$5

This representation makes temporal proportionality possible: rather than selecting the most popular option in each round independently, one asks whether groups that repeatedly agree receive a commensurate share of favorable decisions over time (Chandak et al., 2023).

The sequence-wise perspective is therefore not a minor extension of static voting. It changes both the object being evaluated and the relevant fairness question. A plausible implication is that per-round legitimacy does not by itself guarantee across-time proportionality, because the cumulative sequence may systematically underserve cohesive groups even when each individual round appears acceptable.

2. Formal sequence-wise proportionality axioms

In the approval-based temporal model with fixed candidate set $N$6, the key coalitional statistics are a coalition’s agreement and demand. For a coalition $N$7,

$N$8

and

$N$9

Here $P$0 counts the rounds in which the coalition has a common approved candidate, while $P$1 is its proportional demand over the horizon (Elkind et al., 9 Feb 2025).

The strong temporal axioms are JR, PJR, and EJR. JR requires that for every coalition $P$2 with $P$3, some member of $P$4 obtains positive satisfaction. PJR requires that for every coalition $P$5,

$P$6

EJR requires that for every coalition $P$7, there exists a voter $P$8 such that $P$9. Their weak counterparts, w-JR, w-PJR, and w-EJR, restrict attention to coalitions with $\mathcal{s}_i=(s_{i,1},\dots,s_{i,\ell})$0, that is, coalitions that agree in every round (Elkind et al., 9 Feb 2025).

A closely related formulation appears in sequential decision making under the names PJR, Strong PJR, EJR, and Strong EJR. There, a group $\mathcal{s}_i=(s_{i,1},\dots,s_{i,\ell})$1 of size $\mathcal{s}_i=(s_{i,1},\dots,s_{i,\ell})$2 that agrees in a $\mathcal{s}_i=(s_{i,1},\dots,s_{i,\ell})$3 fraction of rounds must be satisfied in at least $\mathcal{s}_i=(s_{i,1},\dots,s_{i,\ell})$4 rounds, and the stronger variants require that one member of the group personally obtains the corresponding number of approved outcomes (Chandak et al., 2023). This suggests a direct conceptual bridge between the “sequence-wise” axioms of temporal voting and strong proportionality notions for sequential public decisions.

The temporal-fairness literature also includes alternative axiom families beyond justified representation. Examples include average-utility guarantees, prefix-style requirements such as Temporal Justified Representation, and frequency-based no-starve conditions forbidding long stretches of low utility for a voter (Elkind et al., 2023). These variants broaden the scope of sequence-wise evaluation while retaining the central premise that fairness is a property of the full trajectory.

3. Verification problems, hardness, and tractable fragments

A central computational question is verification: given a temporal election

$\mathcal{s}_i=(s_{i,1},\dots,s_{i,\ell})$5

and a winner sequence $\mathcal{s}_i=(s_{i,1},\dots,s_{i,\ell})$6, does $\mathcal{s}_i=(s_{i,1},\dots,s_{i,\ell})$7 satisfy a chosen proportionality axiom? In the sequence-wise setting this verification problem is substantially harder than in static multiwinner voting. For each $\mathcal{s}_i=(s_{i,1},\dots,s_{i,\ell})$8, deciding whether $\mathcal{s}_i=(s_{i,1},\dots,s_{i,\ell})$9 provides $W=(w_1,\dots,w_\ell)$0 is coNP-complete; the hardness already holds when $W=(w_1,\dots,w_\ell)$1 for w-JR and w-PJR, and when $W=(w_1,\dots,w_\ell)$2 for w-EJR. For each $W=(w_1,\dots,w_\ell)$3, verification is also coNP-complete, already for $W=(w_1,\dots,w_\ell)$4 (Elkind et al., 9 Feb 2025).

The reductions isolate different sources of hardness. For w-JR, the proof reduces Clique to coalition checking by encoding vertices as voters and edges so that a large clique induces a coalition witnessing failure. For w-EJR, the reduction is from Independent-Set with $W=(w_1,\dots,w_\ell)$5. For the strong axioms, the reductions are from Maximum–Edge–Biclique to JR, PJR, and EJR (Elkind et al., 9 Feb 2025). These results show that temporal proportionality is computationally difficult even when the candidate space is very small.

At the same time, several natural fragments are tractable. When $W=(w_1,\dots,w_\ell)$6, checking w-JR is polynomial-time: one identifies all “grumpy” voters who never get any of the two winners and checks their demand. When $W=(w_1,\dots,w_\ell)$7 and all $W=(w_1,\dots,w_\ell)$8, even JR is polynomial-time. Verification is fixed-parameter tractable in the number of voters $W=(w_1,\dots,w_\ell)$9 by brute-forcing all $i$0 coalitions, and it is also FPT in $i$1 and XP in $i$2 by enumerating round-types and coalition-size patterns. By contrast, for the weak axioms, parameterization by $i$3 alone is W[1]-hard (Elkind et al., 9 Feb 2025).

A particularly important positive result concerns monotonic preferences, where approvals only grow over time, $i$4. In that case all six axioms become polynomial-time verifiable. The structural reason is that any coalition’s rounds of full agreement form one contiguous suffix, so it suffices to check $i$5 extremal coalitions (Elkind et al., 9 Feb 2025).

Verification/search setting	Status	Notable tractable case
w-JR, w-PJR, w-EJR	coNP-complete	w-JR is polytime for $i$6
JR, PJR, EJR	coNP-complete	JR is polytime for $i$7 and all $i$8
EJR-sequence search	Constructive algorithms exist	FPT in $i$9 via ILP

Beyond verification, the same work gives constructive methods for finding EJR sequences. A Greedy Cohesive Rule runs in $$\mathrm{sat}_i(W)=|\{t\in[\ell]: w_t\in s_{i,t}\}|.$$0 and always returns an EJR sequence; on monotonic inputs, candidate-coalition pruning yields a truly polynomial-time implementation. An ILP formulation with $$\mathrm{sat}_i(W)=|\{t\in[\ell]: w_t\in s_{i,t}\}|.$$1 variables and constraints characterizes EJR outcomes, and Lenstra’s theorem then yields an FPT algorithm in $$\mathrm{sat}_i(W)=|\{t\in[\ell]: w_t\in s_{i,t}\}|.$$2 that can additionally impose minimum-satisfaction constraints or maximize total welfare (Elkind et al., 9 Feb 2025).

4. Voting rules and constructive guarantees

Several rule families have been proposed to realize sequence-wise proportionality under different information regimes. In approval-based sequential decision making, Sequential Phragmén is online, maintains nondecreasing voter loads, chooses in each round the alternative minimizing a water-level objective, and satisfies Strong PJR but fails EJR. The Method of Equal Shares (MES) is semi-online, assigns each voter budget $$\mathrm{sat}_i(W)=|\{t\in[\ell]: w_t\in s_{i,t}\}|.$$3, sets price $$\mathrm{sat}_i(W)=|\{t\in[\ell]: w_t\in s_{i,t}\}|.$$4, chooses affordable alternatives via minimal uniform cost $$\mathrm{sat}_i(W)=|\{t\in[\ell]: w_t\in s_{i,t}\}|.$$5, satisfies EJR, fails Strong PJR, and may terminate early if no candidate is affordable. Proportional Approval Voting (PAV) is fully offline, maximizes a harmonic objective over the whole sequence, and satisfies Strong EJR; exact optimization is NP-hard, but a local-search variant is polynomial-time and preserves the same axiomatic guarantees (Chandak et al., 2023).

Rule	Information regime	Guarantee or limitation
Sequential Phragmén	Online	Strong PJR; fails EJR
MES	Semi-online	EJR; fails Strong PJR; may terminate early
PAV / LS-PAV	Offline	Strong EJR; exact PAV NP-hard

The temporal-fairness survey records related constructive results for repeated committee selection. Under static approvals and no availability constraints, there always exists a sequence of length $$\mathrm{sat}_i(W)=|\{t\in[\ell]: w_t\in s_{i,t}\}|.$$6 satisfying TJR. A Greedy Perpetual JR rule picks in each round the candidate that maximally increases the number of as-yet-unserved approved voters; each round takes $$\mathrm{sat}_i(W)=|\{t\in[\ell]: w_t\in s_{i,t}\}|.$$7 time, for a total of $$\mathrm{sat}_i(W)=|\{t\in[\ell]: w_t\in s_{i,t}\}|.$$8 (Elkind et al., 2023).

Constructive existence and verification are therefore closely linked but not identical. A sequence may be hard to certify in the worst case and yet still be computable by a dedicated rule on structured instances. This suggests that algorithm design in sequence-wise temporal voting is shaped less by a single universal procedure than by an explicit trade-off among online implementability, proportionality strength, and worst-case computational cost.

5. Online strategyproofness and strategic dynamics

Recent work on online temporal voting adds game-theoretic properties to the sequence-wise fairness agenda. In that model, each round $$\mathrm{sat}_i(W)=|\{t\in[\ell]: w_t\in s_{i,t}\}|.$$9 has its own alternative set $W=(W^1,\dots,W^T)$0, each voter submits a $W=(W^1,\dots,W^T)$1 approval vector, and group satisfaction is

$W=(W^1,\dots,W^T)$2

The paper defines classical strategyproofness (SP), myopic online strategyproofness (OSP), and online independence of irrelevant alternatives (OIIA), and proves the implication $W=(W^1,\dots,W^T)$3 (Borodin et al., 27 Mar 2026).

This framework yields a refined picture of strategic robustness. All Weighted Approval Methods satisfy OIIA and hence OSP, but none satisfy full SP in general. MES satisfies OIIA, hence OSP, and wEJR in the offline sense, but fails SP because free-riding is possible and may terminate early. Perpetual Phragmén satisfies OIIA, hence OSP, and PJR, but fails SP. Temporal Serial Dictator is fully SP because only the designated dictator’s ballot matters each round, yet it fails JR and PJR in the worst-case offline sense. At the same time, Temporal Serial Dictator satisfies PJR up to an additive $W=(W^1,\dots,W^T)$4 error and asymptotically satisfies proportional representation, with

$W=(W^1,\dots,W^T)$5

when $W=(W^1,\dots,W^T)$6 (Borodin et al., 27 Mar 2026).

The same paper introduces the price of manipulability, measuring how strategic play degrades proportional-representation guarantees. Under JR, a single manipulator cannot violate the JR bound at all, so the price is $W=(W^1,\dots,W^T)$7. Under PJR, a single manipulator can force an $W=(W^1,\dots,W^T)$8 drop (Borodin et al., 27 Mar 2026). This separates robustness of minimal representation from robustness of stronger proportionality demands.

Strategic timing itself can also become endogenous when interim tallies are public. In a two-period online-election model with informed voters, a swing voter, costly voting, and temporal bandwagon effects, equilibria may display always-early voting, last-minute voting, or mixed timing. The model formalizes “last minute” behavior as an equilibrium region in which all informed voters wait until the second period and vote only if needed to affect the outcome (Yaish et al., 2024). Although this is a distinct model from proportional sequence-wise voting, it shows that temporal structure affects not only fairness guarantees but also participation incentives.

6. Related temporal formulations and terminological scope

Sequence-wise temporal voting is part of a broader family of temporal social-choice models. One neighboring line studies two-stage committee elections with aversion to change. There, a first-stage committee $W=(W^1,\dots,W^T)$9 is fixed, a second-stage committee $D=(d_1,\dots,d_T)$0 must again be chosen under a Thiele rule, and the objective is to maximize stage-two score while also maximizing overlap $D=(d_1,\dots,d_T)$1. The resulting Resilient Committee Elections problem is polynomial-time for Approval Voting and NP-hard for every non-AV Thiele rule, even when the two profiles differ by a single approval; experimentally, small perturbations can already cause several committee replacements, and tie-breaking materially affects resilience (Zech et al., 2024).

Another neighboring formulation is temporal liquid democracy. A temporal multigraph records time-stamped delegation edges, trust horizons $D=(d_1,\dots,d_T)$2, and $D=(d_1,\dots,d_T)$3-time-conscious paths. The final object is a sequence of per-round delegation states, and the core computational problem is to resolve delegations over time while respecting time-consciousness and, optionally, confluence. The time-conscious plus confluent problem is NP-hard, the time-conscious unconfluent problem is NP-hard even when there is a single delegator and $D=(d_1,\dots,d_T)$4, but dropping one of the two constraints yields polynomial-time algorithms, and insisting on both admits an $D=(d_1,\dots,d_T)$5 exact algorithm via Directed Steiner Tree (Markakis et al., 2023). This does not select winners directly, but it extends the sequence-wise paradigm to evolving representation paths.

Temporal governance on blockchains provides a further variant. Always on Voting divides the interval between main elections into randomized epochs, allows participants to update their private ballots at any point within an epoch, tallies only the latest ballot at epoch end, and uses a Bitcoin puzzle oracle plus verifiable delay function to make epoch endings unpredictable. The realized outcome is again a sequence of temporally separated tallies rather than a single static election (Venugopalan et al., 2021).

The phrase “temporal voting” also appears outside computational social choice. In large-scale video search, it denotes a queue-based voting scheme that accumulates evidence for temporally consistent matches without allocating a full voting table (Esen et al., 2016). In action localization, a Temporal Voting Network accumulates frame-level voting evidence for action boundaries and improves high-IoU localization (Wang et al., 2022). These uses share the idea of aggregating evidence across time, but they are terminologically rather than institutionally connected to sequence-wise temporal voting in social choice.

Across these strands, the unifying feature is that outcomes are evaluated as temporal objects. In the social-choice literature, the sequence-wise viewpoint has made proportionality, tractability, and strategic behavior genuinely temporal notions: coalitions are defined by cross-round agreement, guarantees are formulated over whole trajectories, and algorithmic or incentive analyses must account for the fact that fairness is accumulated over time rather than decided in a single snapshot.