Single-Peakedness Does Not Prevent Leapfrogging under Abstention
Parties in spatial competition rarely choose platforms that reverse their ideological order. Mutual leapfrogging is the strongest form of reversal: each party locates beyond the other party's ideal point. In voting models without abstention single-peakedness rules out such reversals. We show that this conclusion does not survive endogenous abstention. There is a spatial voting model in which voter and party preferences are single-peaked, yet mutual leapfrogging occurs in pure-strategy equilibrium. The equilibrium survives because some deviations change which voters participate. We prove that such equilibria are impossible under a sufficient ordinal condition: parties agree on how to rank leftward and rightward deviations from their ideal points. The condition is general enough to cover symmetric single-peaked utilities and common translated utility shapes.