Establish matching interactive local-privacy lower bounds

Prove matching lower bounds in the interactive local-differential-privacy model for the polynomial dependence on the number of candidates in the decisive-margin guarantees for Single Transferable Vote and Condorcet voting.

Background

The paper’s interactive local-differential-privacy upper bounds for Single Transferable Vote and Condorcet still contain polynomial dependence on the number of candidates mm. The available lower-bound technique gives only a square-root dependence on the number of voters and does not match these candidate-dependent terms. The authors state that proving matching lower bounds is unresolved and may require techniques beyond existing approaches, which typically rely on mechanisms having exponentially many possible outputs rather than a single candidate output.

References

In terms of open questions, several gaps between upper and lower bounds remain in the local model. Specifically, in the interactive local model, our upper bounds for STV and Condorcet still depend polynomially on $m$. On the other hand, proving a matching lower bound seems beyond currently known techniques, since our output is simply a candidate among $m$ choices.

— Decisive Margins in Differentially Private Voting  (2608.18772 - Hillebrand et al., 19 Aug 2026) in Section 4, Section 4 conclusion