Sharp sufficiency thresholds and existence beyond uniform costs

Determine sharp density-ratio thresholds for the sufficiency umbrellas governing pure Nash equilibria in the costly-voting Hotelling–Downs model, and develop a sufficiency theory for non-uniform voting-cost distributions, particularly when candidate competitiveness is positive.

Background

The paper establishes sufficient density-ratio conditions under which first-order balance conditions characterize equilibria, including the baseline bound ρv≤4\rho_v\le4, its alienation extensions, and a general-cost result for non-increasing cost densities when β=0\beta=0. However, the authors do not establish whether these thresholds are sharp in general.

For positive competitiveness and non-uniform costs, differentiating the opponent-payoff term introduces a bulk term involving k′k' whose sign and magnitude cannot be controlled using only density ratios. The paper therefore leaves open both sharper threshold results and a broader sufficiency theory beyond the cases treated analytically.

References

Several questions remain open. On sufficiency: sharp thresholds for the density-ratio umbrellas of \Cref{tab:summary}, and a sufficiency theory beyond uniform costs (\Cref{n:prop:A4} covers non-increasing $k$ at $\beta=0$; the obstruction at $\beta>0$ is isolated in \Cref{m:rem:D1k}).

— Costly Voting in the Hotelling-Downs Model  (2609.29869 - Wolf et al., 24 Sep 2026) in Section 6, paragraph “Open questions”; Remark m:rem:D1k, Section m:sec:D

We conjecture that a pure equilibrium always exists in the baseline ($\alpha=\beta=0$) under \Cref{ass:reg}. The case the list leaves conspicuously open is uniform costs with $\rho_v>4$, where we have neither a proof nor a counterexample: the four-cluster electorate of \Cref{app:counterexample} has $\rho_v=120$ and does possess equilibria, even though not every profile solving the conditions is one.

— Costly Voting in the Hotelling-Downs Model  (2609.29869 - Wolf et al., 24 Sep 2026) in Remark 2.14, Section 2.3.3; Section 6, paragraph “Open questions”; Appendix “Existence and Uniqueness of Equilibria”

On existence: under \Cref{ass:reg} we conjecture pure existence for every $(v,k)$ in the baseline and, at $\alpha=0$, for every $\beta<1$ (\Cref{rem:exist,m:conj:alpha0}), the first of these open already for uniform costs once $\rho_v>4;

— Costly Voting in the Hotelling-Downs Model  (2609.29869 - Wolf et al., 24 Sep 2026) in Section 6, paragraph “Open questions”; Remark m:rem:E1scope

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”; Section 5, subsection “The Margin Game”

Whether this pattern persists under other distributions is left to future work.

— Costly Voting in the Hotelling-Downs Model  (2609.29869 - Wolf et al., 24 Sep 2026) in Appendix, Section “Turnout Comparative Statics: A Worked Example”