Exact Privacy--Distortion Design of Support-Constrained Randomized Response

Abstract

Full-support local randomizers can incur unnecessary reporting distortion on ordered alphabets. We study randomized response whose output is restricted to a radius-$t$ neighborhood of the private input. Input-dependent supports create privacy leakage at their boundaries, so ordinary likelihood-ratio calibration is no longer sufficient. Under approximate local privacy restricted to input pairs at distance at most $H$, we derive the exact hockey-stick defect of the mechanism. We then solve the joint design of the truthful probability, off-truth probability, and support radius. The fixed-radius optimum is available in closed form, while the global optimum requires only a one-dimensional integer search. We characterize both the privacy floor and its inverse, namely the smallest defect attainable under a reporting-distortion budget. We further prove that stringent approximate privacy forces the support size to scale as $Ω(H/δ)$. Deterministic evaluations illustrate the resulting tradeoff.