Anchored Likelihood-Ratio Geometry of Anonymous Shuffle Experiments: Exact Privacy Envelopes and Universal Low-Budget Design

Abstract

We develop a geometric framework for anonymous shuffle experiments based on an anchored affine likelihood-ratio law: a mean-zero measure on the regular simplex polytope. Every finite-output d-ary channel corresponds, up to refinements, to a unique anchored law, and conversely. On privacy: among all epsilon_0-LDP channels, binary randomized response universally extremizes all convex f-divergences and hockey-stick profiles after shuffling. A rigidity converse shows that saturation of both directed envelopes at finite n forces the binary endpoint law. On design: under the pairwise chi_* budget, we prove exact trace-cap and two-orbit frontier theorems. Every frontier point is realized by a mixture of at most two orbit laws. In the low-budget regime, augmented randomized response is minimax-optimal to the sharp constant over all channels and estimators. Under the raw LDP cap, the problem reduces to subset-selection with explicit optimal subset size. The arguments are self-contained and independent of the author's trilogy.