Outcome-conditioned privacy without outcome invariance

Determine whether the outcome-conditioned lower-tail contextual-privacy requirement alone implies the general impossibility result for normalized, efficient, dominant-strategy incentive-compatible, ex post individually rational payment rules, without imposing lower-tail outcome invariance.

Background

For general incentive-compatible payment rules, the paper imposes lower-tail outcome invariance so that changing a low loser's value does not alter the allocation or any payment. Under this assumption, the authors establish an impossibility theorem for private-query implementations that are both lower-tail contextually private and prior-free credible.

The paper then notes that dominant-strategy incentive compatibility permits different winning payments across opponents' profiles, including payments that vary with lower losing reports. If such payment changes make the outcome-conditioned privacy premise inapplicable, the privacy condition may no longer force the early identification that drives the impossibility proof. Whether the impossibility survives without lower-tail outcome invariance is therefore left unresolved.

References

Whether the outcome-conditioned requirement eq:sa-privacy-general alone implies the general impossibility without outcome invariance is not resolved here.

eq:sa-privacy-general:

F(vi′,v−i)=F(v)  and  max⁡{vi,vi′}<ℓ(v)⟹zP(vi′,v−i)=zP(v).F(v'_i,v_{-i})=F(v)\ \text{ and }\ \max\{v_i,v'_i\}<\ell(v)\quad\Longrightarrow\quad z^P(v'_i,v_{-i})=z^P(v).

— Protecting Losers Undermines Auction Credibility  (2610.12204 - Nakamura et al., 8 Oct 2026) in Supplemental Appendix, Section S2, final paragraph of Section “General Incentive-Compatible Payments”