Sharp Verification Boundary for Pairwise Social Choice under Strategic Evidence Distortion

This paper isolates the exact condition under which pairwise social choice survives strategic evidence distortion and solves that condition completely in three layers. The setting is a contested pair of alternatives with a finite set of latent frames, each frame supporting a different ranking of the pair. Each frame is associated with a compact set of post-distortion evidence laws reflecting the strategically feasible manipulation the actor can apply to the baseline evidence channel. The pairwise verification problem reduces to robust binary classification between two compact uncertainty sets: the union of feasible evidence laws across all frames favoring the first alternative, and the union of feasible laws across frames favoring the second.

An operational dichotomy theorem is proved for arbitrary compact uncertainty sets. If the two sets intersect, then for every sample size and every decision rule the worst-case typewise misranking error is at least one-half, because an adversary can route both a pro-first and a pro-second frame to the same evidence law and no statistical rule can distinguish them. If the two sets are disjoint, the total-variation gap between them is strictly positive and a nearest-set empirical-distribution classifier achieves uniformly exponentially decaying error with an explicit exponent controlled by the gap squared divided by eight.

The paper then specializes to reverse-KL distortion budgets, where each frame's feasible set is the KL ball of radius C_S around the baseline law. For this model the existence boundary is proved to be exact: the two uncertainty sets intersect if and only if the obfuscation budget meets or exceeds the pairwise witness capacity, defined as the minimum Chernoff information between the baseline laws across all cross-frame pairs. The Chernoff radius identity, proved via the tilted equalizer lemma, establishes the bidirectional equivalence.

Below the Chernoff threshold the hidden-adversary problem has an exact asymptotic minimax exponent, proved via a sandwich of upper and lower bounds. The upper bound is obtained using the generalized likelihood-ratio classifier against the feasible uncertainty sets and the method-of-types, achieving the minimum feasible Chernoff information over cross pairs as the exponential rate. The lower bound uses the type-class construction at the Chernoff tilted point to show that no rule can achieve a better exponent. Together these establish the exact minimax exponent as the minimum feasible Chernoff information over cross pairs at the given budget. A finite-sample polynomial sandwich around this exponent is also derived. A worked binary Bernoulli example tabulates the sharp exponent and the universal Pinsker exponent across the feasible range, showing the latter is numerically close but uniformly smaller.

For dynamic adversaries with ergodic feasibility constraints, the paper proves that time variation cannot improve on the static geometry: the average law of any ergodically feasible strategy must lie in the coordinatewise convex hull of the feasible sets, and the Bhattacharyya bound for that average law controls the achievable exponential rate. A switching-cost corollary shows that actor-side switching costs are irrelevant for the existence boundary because any aliasing strategy can be chosen constant, while auditor-side constraints can reduce probe diversity.

Under a common-preference misalignment model where all types share the same utility function over decisions but differ in evidence laws, the same Chernoff threshold governs mechanism design: overlap makes strict robust implementation impossible because the two types have identical incentives and evidence laws under any common post-distortion law, while separation admits an explicit constructive mechanism that achieves strict incentive compatibility using the universal classifier and a sufficiently large transfer penalty.