Honest Reporting in Scored Oversight: True-KL0 Property via the Prekopa Principle

Abstract

We prove the True-KL$_0$ property for a parametric family of heterogeneous scoring rules arising in scored elicitation mechanisms (AI oversight, forecasting, expert surveys). An agent with private type $M>1$, scored through a $d$-dimensional outcome interface, reports to a principal who evaluates via a power-$p$ pseudospherical scoring rule, $p \in (d,d+1)$; $M$ captures the agent's information quality relative to a reference. Honest reporting is dominant-strategy optimal for every $d$ and every $p>1$, without a prior over the agent's type: a consequence of strict properness and identifiability, with a quadratic misreport-loss rate. True-KL$_0$, the property $R(M,p,d)<1$ for all $M>1$, $d \in \{2,3,4\}$, $p \in (d,d+1)$, is the quantitative core: $R$ is the Rayleigh quotient of the radial misreport channel of an annular oversight model, and True-KL$_0$ certifies a uniform curvature-domination margin for that channel: $1-R \ge 0.26$ ($R \le 0.7324$, semi-rigorous numerical certificate). Two structural tools drive the proof: (i) a substitution $y=(x+1)/(x-1)$ rewrites the loss integral $I_L$ as $\int_1^M F(y)(M^2-y^2)^{d/2} dy$ with $M$-independent weight $F(y)>0$; (ii) log-concavity of $I_L$ in $M$: algebraic for $d=2$ up to a small certified compact verification, via Prekopa's theorem plus semi-rigorous certificates for $d \in \{3,4\}$. True-KL$_0$ then follows from elementary tail bounds plus a certified bound on $M \in [1.001, 20]$. We also characterise the dimensional boundary: True-KL$_0$ holds for all $p \in (d,d+1)$ when $d \le 4$; $d=5$ is the unique transition, with $p_{crit}(5) \in [5.5718, 5.5750]$ (mpmath, not interval-certified); for $d=6,7$ (and conjecturally all $d \ge 6$) no threshold exists: the bound fails at every sampled $p \in (d,d+1)$.