Anchor-proofness in Voting

We study how order-dependent individual evaluations affect collective decision-making. We consider a voting environment in which voters have intrinsic preference rankings and acceptability thresholds, but form approval ballots sequentially as alternatives are presented by a social planner. Because alternatives encountered early on serve as anchors, approval ballots may depend on the presentation order even when intrinsic preferences remain unchanged. We ask when approval-based voting rules are robust to this order dependence, a property we call anchor-proofness. We first show that no non-constant approval-based voting rule is anchor-proof. We then study weaker forms of robustness and characterize anchor-proof profiles for the classical approval voting rule and the nomination rule. Finally, we examine the strategic role of the social planner under different information structures. We show that, while a planner with no information about voters' preferences cannot safely manipulate any rule satisfying weak and total unanimity, even limited information about voters' acceptability or plurality assessments can restore manipulation. Overall, we highlight presentation order as an important design feature of collective decision-procedures when voters are subject to anchoring.