Online Temporal Voting: Strategyproofness, Proportionality and Asymptotic Analysis

Abstract: We study online temporal voting, where a group of voters submit 0/1 approvals on sets of alternatives that arrive online over multiple rounds and a single alternative is chosen in each round. We introduce online variants of two well-known game theoretic properties, strategyproofness (SP) and independence of irrelevant alternatives. We show that online independence of irrelevant alternatives (OIIA) is a sufficient condition for online strategyproofness (OSP), and that several known online voting rules satisfy OIIA and thus OSP, but that they are not SP. In particular, we show that Perpetual Phragmen, the only known online voting rule to satisfy PJR, satisfies OSP. The Method of Equal Shares (MES), a semi-online voting rule knwon to satisfy wEJR, also satisfies OSP. We then introduce the price of manipulability, which quantifies the effect of strategic behaviour on proportional representation guarantees. Finally, we introduce asymptotic satisfaction of proportional representation and show that an online voting rule, Serial Dictator, is fully strategyproof and satisfies proportional justified representation (PJR) up to an additive constant.

• The paper introduces online strategyproofness, shows that online independence of irrelevant alternatives guarantees it for WAMs, GreedyJR, MES, and Perpetual Phragmén, and clarifies how truthful behavior can coexist with proportional representation.
• The paper defines the price of manipulability and proves that one strategic voter can cause an Ω(T) loss in PJR guarantees, while JR remains protected against a single deviation.
• The paper presents Temporal Serial Dictator, a fully strategyproof and efficient rule whose PJR satisfaction ratio approaches 1 as the number of voting rounds grows, despite weak performance in short horizons.

Overview

This paper studies online temporal voting, in which $n$ voters submit 0/1 approval ballots over alternatives that arrive sequentially over $T$ rounds, with a single alternative selected per round. The work makes three contributions: (1) it introduces an online relaxation of strategyproofness (OSP) and shows that a sufficient condition for OSP — online independence of irrelevant alternatives (OIIA) — is satisfied by several well-known voting rules that carry strong proportionality guarantees; (2) it introduces the price of manipulability, quantifying how strategic behavior degrades proportional representation guarantees; and (3) it introduces asymptotic satisfaction of proportionality and exhibits a fully strategyproof online rule, Temporal Serial Dictator (TSD), that satisfies PJR up to an additive constant vanishing as $T$ grows.

The model assumes complete approvals (non-empty approval sets in all rounds), deterministic rules, and an adversarial online arrival of both alternative sets and approvals. Satisfaction of a group is the number of rounds in which at least one member approves the winner, and proportionality axioms (JR, PJR, EJR, in weak and strong forms) are lower bounds on group satisfaction expressed via $\ell$-cohesiveness: a group is $\ell$-cohesive if there exist $\ell$ rounds in which its approval intersection is non-empty.

Online strategyproofness

The paper defines OSP myopically: fixing history and other voters' current-round approvals, no voter can gain satisfaction in the current round by misreporting. Two structural results anchor the analysis. First, monotonicity (adding the winner to one's ballot cannot change the outcome) is necessary for OSP, though whether it is sufficient remains open. Second, OIIA — removing a non-winning alternative from any voter's ballot leaves the outcome unchanged — implies OSP. The proof proceeds by showing that under OIIA, any sequence of ballot edits either preserves the winner or changes it to an added alternative, which cannot benefit a voter who does not approve the original winner. OIIA is shown not to be necessary for OSP via a constructed rule ("IrrelevantDictator") that is SP but violates OIIA.

The central positive finding is that OIIA holds broadly:

• Weighted Approval Methods (WAMs): the entire class of rules that score alternatives by sums of voter weights and update weights after each round satisfies OIIA, hence OSP.
• GreedyJR (a WAM that zeroes out satisfied voters' weights): satisfies JR and is OSP, yet is not SP — a two-round counterexample shows a voter gaining satisfaction 2 versus 1 by withholding approval in round 1.
• Method of Equal Shares (MES) without a completion procedure: satisfies OIIA and hence OSP, but not SP — a four-round example shows a free-rider strategy raising satisfaction from 3 to 4. The paper notes MES is semi-online (it must know $T$), non-exhaustive, and that whether a completion procedure preserving OSP exists is open.
• Perpetual Phragmén: satisfies OIIA and hence OSP, but not SP.

These results expose a sharp gap between the online and offline settings: rules known or suspected to be manipulable in general are unconditionally OSP. Since Perpetual Phragmén satisfies PJR [(2603.26504)'s cited prior work] and MES satisfies wEJR, OSP is compatible with strong proportionality even though full SP appears incompatible with any non-trivial proportionality axiom, mirroring Peters' computer-assisted impossibility result for multi-winner approval voting.

Price of manipulability

Because proportionality guarantees are functions of declared approvals, manipulation can dissolve $\ell$-cohesiveness and void guarantees. The paper bounds this degradation when a single voter deviates. For JR, a single manipulator causes no violation: if the manipulator belongs to group $G$, their gain of at least one unit of satisfaction restores the JR bound of 1. For PJR, however, the price of manipulability is $\Omega(T)$: a single strategic voter can reduce a group's guaranteed satisfaction by roughly $T$0 per cohesive round, which is significant when $T$1 or $T$2 is small. The analysis is a worst-case lower bound only; tight bounds require per-rule analysis, and extending to multiple manipulators would require equilibrium analysis, which the paper explicitly leaves open.

Asymptotic proportionality

The paper defines asymptotic satisfaction of an axiom PR as the ratio of achieved group satisfaction to the PR lower bound tending to 1 as $T$3 grows. Temporal Serial Dictator — a round-robin rule where voter $T$4 dictates the winner in round $T$5 — fails even weak JR for small $T$6, yet satisfies PJR asymptotically: every group obtains satisfaction at least $T$7 where $T$8, while the PJR bound is at most $T$9, giving ratio $T$0. TSD is also fully SP, since each round's outcome depends on exactly one voter's ballot, who trivially cannot gain by misreporting. This yields the paper's notable claim: a naive, efficiently computable, fully online rule is both completely strategyproof and arbitrarily close to PJR given enough rounds — a combination no known rule achieves exactly.

The asymptotic lens also softens the price of manipulability: for $T$1, JR becomes trivially satisfied under any manipulation, and for PJR-satisfying rules, any $T$2-cohesive group retains at least a $T$3 fraction (hence at least $T$4) of its PJR bound asymptotically. This factor-of-$T$5 benchmark is loose compared to TSD's additive-constant guarantee, and the paper concedes it is unclear whether actual rules exhibit such degradation.

Limitations and open questions

Several caveats qualify the results. The complete-approvals assumption excludes abstention and indifference, restricting applicability. MES results hold only without a completion procedure, and premature termination is a real possibility. Whether monotonicity suffices for OSP, whether a (semi-)online rule satisfies wEJR (and if so, whether it is OSP), and whether TSD characterizes SP rules achieving asymptotic justified representation — analogous to the Like mechanism in online fair division — all remain open. Convergence rates of asymptotic proportionality and notions of partial convergence ("most" groups converging within some rounds) are likewise unaddressed, as is multi-winner temporal voting.

Conclusion

The paper establishes that myopia substantially enlarges the class of truthful voting rules: OIIA is a clean sufficient condition for OSP satisfied by WAMs, GreedyJR, MES, and Perpetual Phragmén, so OSP coexists with JR, PJR, and wEJR even though exact SP with proportionality remains elusive. Complementing this, the price-of-manipulability framework quantifies fragility of PJR under single-voter deviation ($T$6), while asymptotic analysis shows that strategyproofness and near-PJR representation are simultaneously achievable by Temporal Serial Dictator when $T$7, suggesting that the temporal setting admits escape routes from impossibilities that bind in multi-winner approval voting.