Proportionality from Sampled Approvals

Abstract

How much voter input is necessary in order to ensure representation in multiwinner elections? If voters are randomly selected from an underlying population, how many draws are necessary to find a proportional committee of $k$ candidates, with high probability? Sample-based adaptations of standard multiwinner voting rules that satisfy the justified representation (JR) proportionality axiom use $\tilde{O}(k^5\log \frac{m}{\hat{I}}´)$ sampled approval ballots over $m$ candidates, where $\hat{I}´$ is a probability of failure and $\tilde{O}$ suppresses $\mathrm{polylog}(k)$ factors. We present a rule for which the sample complexity of JR-family proportional committee selection is $\tilde{O}(k^4\log \frac{m}{\hat{I}}´)$. This separates the sample complexity of JR from that of the natural corresponding additive approximation to the voter coverage (Chamberlin-Courant) objective, which we show requires $\hat{I}˜(k^5\log \frac{m}{\hat{I}}´)$ samples. For lower bounds, we present a family of instances with $m,\frac{1}{\hat{I}}´\in \mathrm{poly}(k)$ for which $\hat{I}\text{c◯}(k^3)$ sampled ballots are necessary in order to identify a JR committee. We also show a dependence on $\log m$ is necessary. This lower bound is versatile, and also applies to Hare proportionality for solid coalitions (PSC) for ranked ballots. Unfortunately, no number of sampled ballots suffices to satisfy the slightly stronger Droop JR and Droop PSC axioms with high probability. But mild relaxations of JR require fewer samples, as do the beyond-worst-case domains and actual approval preferences we evaluate.

44 pages, 9 figures