Instant runoff voting (IRV) is an increasingly-popular alternative to traditional plurality voting in which voters submit rankings over the candidates rather than single votes. In practice, elections using IRV often restrict the ballot length, the number of candidates a voter is allowed to rank on their ballot. We theoretically and empirically analyze how ballot length can influence the outcome of an election, given fixed voter preferences. We show that there exist preference profiles over $k$ candidates such that up to $k-1$ different candidates win at different ballot lengths. We derive exact lower bounds on the number of voters required for such profiles and provide a construction matching the lower bound for unrestricted voter preferences. Additionally, we characterize which sequences of winners are possible over ballot lengths and provide explicit profile constructions achieving any feasible winner sequence. We also examine how classic preference restrictions influence our results--for instance, single-peakedness makes $k-1$ different winners impossible but still allows at least $\Omega(\sqrt k)$. Finally, we analyze a collection of 168 real-world elections, where we truncate rankings to simulate shorter ballots. We find that shorter ballots could have changed the outcome in one quarter of these elections. Our results highlight ballot length as a consequential degree of freedom in the design of IRV elections.
翻译:即时决决决投票(IRV)是一个日益流行的替代传统多元化投票方式,选民在这种投票中对候选人而不是单一选票提交排名。实际上,使用IRV的选举往往限制选票长度,允许选民在选票上排数。我们从理论上和从经验上分析投票长度如何影响选举结果,因为有固定选民的偏好。我们显示,优等比例高于k美元的候选人比例,以至于不同的候选人在不同投票长度上赢得了高达k-1美元。我们从中得出关于这种选民概况所需选民人数的精确下限,并提供一个与不受限制选民偏好的下限相匹配的建筑。此外,我们确定胜者顺序可能超过选票长度,并且提供清晰的轮廓结构,以达到任何可行的获胜顺序。我们还研究了传统的优惠限制如何影响我们的选举结果,例如,单价1美元使得不同的获奖者不可能获得,但是仍然允许至少达到$\Omega(sqrtkkk)美元。我们分析的是168个真实世界选举的集集,其中的排名比分数要低得多,我们把选举的排名排在每四分之一的票数上显示我们的选票。