Multi-Audience Verification: What not to Build The Gaming-proof Cone and the Prohibitions it Implies

Abstract

A single disclosure is read by several audiences that summarise it differently: a market by its posterior mean, a court by a coherent downside floor, a prudential regulator by an expected shortfall of losses, an ESG rater by a dispersion statistic. Existing results characterise gamingproofness one reader at a time-a reader is safe exactly when its functional is convex in the posterior-and say nothing when readers disagree in curvature, which is the normal case. We show that the object governing the many-audience problem is not a direction but a cone: the set of liability calibrations under which no disclosure can shelter is the dual cone of the gaming-gain vectors, and it is polyhedral whenever the reads are piecewise linear on a common complex. Membership is a linear feasibility problem in the calibration and a supporting affine minorant, not an optimisation over disclosures. The count of audiences never enters: two convex readers aggregated by a "satisfy any one of them" rule are gameable, while five convex readers at arbitrary weights are not. Six prohibitions follow, each a corollary rather than a caution. Never aggregate verifiers by an infimum-and note that this defect survives the removal of every ingredient the ratings-shopping literature blames. Never let verifiers read separate documents; the sign here is opposite to the familiar recommendation that verifiers be sealed from the market. Never attempt to repair a gameable read with a harsher liability schedule, since a tent in the read defeats every increasing schedule. Never give a dispersion audience positive weight, since its gaming gain is exactly the mean-informativeness the deterrence audience is meant to receive. Comparative statics are available but not where one expects them: the tipping ratio is not a function of the prior's dispersion, is a monotone threshold in the prior mass on the worst state, and is not monotone in the court's own tail level-widening it helps until it overshoots that mass, and then tips the lattice.