Ballot Exhaustion in Multiwinner Single Transferable Vote Elections

We study ballot exhaustion in multiwinner single transferable vote (STV) elections using a dataset of 1,070 Scottish local government elections comprising over 5.4 million ballots. While ballot exhaustion has been studied extensively in single-winner elections, comparatively little work examines exhaustion in the multiwinner setting. We introduce formal definitions of several types of exhaustion in STV elections, distinguishing between exhausted ballots, non-first-choice exhausted ballots, unrepresented exhausted ballots, and weight exhaustion. These definitions clarify important conceptual differences between ballots that cease to transfer and ballots that fail to contribute meaningfully to representation. Our empirical analysis shows that 27.9\% of ballots are exhausted by the final round of counting, although the corresponding weight exhaustion rate is only 7.1\%, indicating that many exhausted ballots have already contributed to the election of a candidate. Moreover, most exhausted ballots correspond to voters who achieve some form of representation, either because their first-ranked candidate wins or because a candidate ranked among their top choices is elected. These results suggest that raw exhaustion rates alone substantially overstate the extent to which voters lose their influence or fail to obtain representation under STV. We also investigate whether exhaustion can affect electoral outcomes by extending partial ballots under several completion models. Under extreme assumptions, exhaustion can potentially alter a substantial number of outcomes, but under a proportional ballot-completion model only 3.5\% of seats change. Finally, we show that a substantial number of winners fail to reach quota, even after the elimination of all losing candidates. These results help clarify the practical and normative significance of ballot exhaustion in real-world STV elections.