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Multi-winner approval elections are seen in a variety of settings rangingfrom academic societies and associations to public elections. In suchelections, it is often the case that ballot-length restrictions are enforced;that is, where voters have a limit on the number of candidates which they canvote for. Despite this common feature, there does not seem to be anytheoretical justification for ballot-length restrictions (Laslier and Van derStraeten, 2016).  This work endogenously derives the set of voter best-response ballot lengthsunder complete information and with general assumptions on voter utilities andvoting rules. These results provide justification for some ballot-lengthrestrictions observed in practice, however when considering equilibriumoutcomes our analysis shows that this justification is no longer valid.Equilibrium analysis is considered for voters with lazy and truth-biassecond-order tendencies and the equilibrium solution concept is pure-Nashequilibria.  The key insights show that ballot-length restrictions or institutionalfeatures which make voting costly may lead to instability in election outcomeswhen voters have diverse preferences, via the non-existence of equilibria. Onthe other hand, when equilibria do exist they satisfy desirable propertieswhich are not guaranteed by equilibria attained under costless voting and inthe absence of ballot-length restrictions. In summary our results highlight astark trade-off between stable and desirable election outcomes.

How long is a piece of string? An exploration of multi-winner approval voting and ballot-length restrictions