A new theoretical study finds that fair representation guarantees for round-by-round elections fall apart once voters rank candidates instead of simply approving them.
The paper studies temporal voting, where one candidate wins each round across a series of elections. Past work on this problem assumed approval ballots, where voters simply mark who they like. This paper switches to ranked ballots, which raises a harder question: how many of a voter's top-ranked candidates should count as 'approved,' and does that cutoff stay fixed, shift by round, or vary voter by voter? The authors test every plausible cutoff rule against a standard ladder of fairness axioms, justified representation (JR), proportional JR (PJR), and extended JR (EJR) as increasingly strict versions of the same guarantee, plus a separate benchmark, proportionality for solid coalitions (PSC), built for voters who agree in every round.
EJR, the top of that ladder, turns out to be unreachable under ranked preferences, no matter how much a rule knows about future rounds. JR, PJR, and PSC survive, but only under shrinking conditions: all three hold with a fixed, shared cutoff if the rule sees the entire election schedule up front, and PSC alone needs no advance knowledge at all when voters agree every round. Loosen the cutoff further and most guarantees vanish, though a rule as basic as serial dictatorship still matches the best possible outcome for PJR under fully individual cutoffs.
It is a useful corrective to fairness research that leans on approval ballots because they are mathematically convenient. Real voters rank things, and this paper's message is that those tidy guarantees look a lot shakier once preferences stop getting rounded down to yes or no.