Exploring Equilibrium under Fallback Voting and Preference Approval Voting

This note examines the conditions under which Preference Approval Voting (PAV) and Fallback Voting (FV) yield equilibrium outcomes. First introduced by Brams and Sanver (2009) and further explored by Kamwa (2023, 2019), FV and PAV are voting rules that combine rankings with approvals. Despite their appeal, a notable drawback of these systems is their occasional failure to produce equilibrium outcomes. An outcome is considered to be in equilibrium if the approval strategies of each voter class form a Nash equilibrium, meaning no class of voters has an incentive to unilaterally change their approval strategy, as doing so would not result in a better outcome and could even lead to a worse one. Our findings show that while PAV and FV often produce similar outcomes, significant differences can arise, particularly concerning equilibrium status. Using limiting probabilities, we analyze the susceptibility of AV, FV, and PAV to strategic manipulation in three-candidate elections, demonstrating that FV and PAV are generally more resistant to such manipulation compared to AV.