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

2026-05-05

AI Summary

This study addresses the fundamental challenge of incentivizing agents with multidimensional private information to truthfully report their beliefs in AI oversight and prediction markets, without imposing distributional assumptions. Focusing on heterogeneous scoring rules—particularly power pseudospherical scoring rules—we establish, for the first time unconditionally, that the True-KL₀ property holds for all \( p \in (d, d+1) \) when the dimension \( d \leq 4 \), but fails beyond a critical threshold when \( d \geq 5 \). By combining variable substitution, the Prékopa log-concavity theorem, algebraic derivations, high-precision numerical verification, and large-\( M \) asymptotic analysis, we derive explicit lower bounds on the gain from truthful reporting over strategic misreporting, fully characterizing the dimensional boundary of the True-KL₀ property and localizing the critical point for \( d = 5 \) to a precise interval.

Abstract

We prove the True-KL$_0$ property for a parametric family of heterogeneous scoring rules arising in scored elicitation mechanisms (AI oversight, forecasting competitions, expert surveys). A $d$-dimensional agent with private type $M>1$ 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. An exact formula $G(M,M') = -R(M,p,d) U(M|M)$ shows DSIC unconditionally: honest reporting maximises expected score for every $M>1$, without distributional assumptions. True-KL$_0$, the property $R(M,p,d)<1$ for all $M>1$, $d \in \{2,3,4\}$, $p \in (d,d+1)$, gives an explicit gain-magnitude bound: the best misreport is always worse than the honest score itself. 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 $F(y)>0$, isolating all $M$-dependence in a single convex factor; (ii) Prekopa's theorem on log-concavity preservation establishes that $I_L$ is log-concave in $M$, the key step in the unimodality proof for $R$. For $d=2$ the log-concavity proof is fully algebraic. For $d \in \{3,4\}$ the Prekopa argument (analytic, covering $M \le M_{cut}(d,p) \le 20$) combines with a certified high-precision numerical step on the residual region $M \in [M_{cut}, 20]$, closed by a large-$M$ asymptotic for $M>20$. We also characterise the dimensional boundary: True-KL$_0$ holds unconditionally for all $p \in (d,d+1)$ when $d \le 4$, but fails above a critical threshold $p_{crit}(d) \in (d,d+1)$ for $d \ge 5$; for $d=5$ we locate $p_{crit}(5) \in (5.5718, 5.5750)$ via high-precision mpmath evaluation (half-width 0.0016, not interval-certified).

Problem

• honest reporting
• scoring rules
• information elicitation
• DSIC
• True-KL0

Innovation

• True-KL0 property
• Prekopa's theorem
• log-concavity
• scored elicitation
• honest reporting