General-electorate monotonicity of alienation-induced polarization

Prove that increasing the alienation parameter \(\alpha\) strictly increases equilibrium platform separation, and characterize its limiting behavior, for general voter-ideal-point distributions rather than only for a uniform electorate.

Background

For a uniform electorate and super-regular voting-cost distribution, the paper proves that the equilibrium gap is strictly increasing in alienation and converges to full separation as α→∞\alpha\to\infty. Numerical results suggest a similar effect for concentrated non-uniform electorates, but the proof does not extend to arbitrary voter distributions.

The unresolved issue is therefore whether the comparative-static result is a general property of the model or depends essentially on uniform voter density.

References

On alienation: a general-electorate version of \Cref{thm:alien-mono}, whose conclusion holds numerically for the concentrated electorate of \Cref{rem:deadzone} but is proved only for uniform voters.

— Costly Voting in the Hotelling-Downs Model  (2609.29869 - Wolf et al., 24 Sep 2026) in Section 6, paragraph “Open questions”; Section 4, subsection “Higher \(\alpha\) Polarizes”

On the margin game with alienation: the exact non-existence region in $(\alpha,\beta,v)$, and whether the quantile path of \Cref{cor:quantile} has an $\alpha>0$ analogue.

— Costly Voting in the Hotelling-Downs Model  (2609.29869 - Wolf et al., 24 Sep 2026) in Section 6, paragraph “Open questions”; Remark rem:no-quantile, Section 4.1