Two Indicators, One Threshold: The Cost of Dependence under Per-Coordinate Spectral Reads

Abstract

A state has two indicators. Compliance is decided by the posterior mean of the first, while a verifier reads each disclosed posterior through a per-coordinate spectral statistic of both marginals — a lower Expected Shortfall or a lower quantile of each indicator — and the sender minimises the expected read subject to a deterrence level. The paper asks when the second indicator is free: when the frontier equals the one-dimensional frontier of the first indicator plus the floor of the second. Under Expected-Shortfall reads (Part I) two posteriors suffice, the frontier is bounded below by that sum with equality under independence, equality is characterised by a Hall-type interval condition decided by one feasibility linear programme, and a two-dimensional price formula follows from duality for any finite joint prior. Under quantile reads (Part II) two posteriors do not suffice, a read-indexed linear programme is exact, and equality is equivalent to a tiling of the second indicator; at zero deterrence the tiling reduces to a corner condition with a closed-form dependence threshold for Fréchet mixtures, and the nesting “quantile equality implies Expected-Shortfall equality” holds in the floor region under a stated hypothesis that is automatic on the continuum when the first level is at most one half, and fails outside it. With positive deterrence, fixing the one-dimensional structure of Hanaeda (2026a) — thin shares and a ceiling — gives a closed form for equal levels at one half and, for a common level below one half under positive dependence and without truncation, a closed-form threshold that is necessary and, at the row level defined below, sufficient; the conditions first proposed for negative dependence and for unequal levels are necessary, and their sufficiency fails in general. The rest of Part II replaces them by a tiling programme that is exact for grid priors and by criteria that are exact for a row-level programme — a coupling to two kinds of bins that Strassen’s theorem reduces to a convex programme, a closed form under complete negative dependence, and a two-bank polymatroid condition whose binding sets are threshold sets; these are proved necessary for equality, while the step from the row level to a tiling of the second indicator is not proved and is supported by grid computations — finds that families of prefixes and suffixes, a relaxation of the exact family, give the same thresholds in the cases computed first but not in general (inside the window their edge exceeds the exact edge at two cells on every lattice computed and, by certificates, at three cells in the continuum), locates the failure — on the compliers’ side it starts exactly where the top thinned row becomes over-served (proved in one region, where the criterion does not depend on the threshold, and, to leading order, in a band near the limit of equal levels for every first level below one half), it does not occur under weak dependence, and beyond the onset in that region the gap grows at least with the cube of the over-service — and studies the structure of the thresholds: three-segment type profiles, a sliding argument, singular arcs solved by quadrature, a finite algebraic system, an order theorem, and transition rules at atoms. The conjecture that a three-segment profile is always optimal is refuted by computation — exact programmes on finite grids together with numerical solutions of the continuum problem — both at jumps of the low-region share of the rows and away from them. The last sections prove, on the family of prefixes and suffixes and under two stated hypotheses, that when the non-compliers take the low region of the thinned rows no corner is optimal among the three-segment profiles and a gradual switch through a single arc carried by the compliers is not optimal (for positive dependence, provided optimal multipliers exist as measures). Every statement carries its status: proved on the continuum, on finite grids or on a stated family; exact computation on finite priors; numerical; conjectured; or refuted. The computations are reported with their tolerances, including the checks that failed.