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Honest Reporting in Scored Oversight: True-KL0 Property via the Prekopa Principle
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Honest Reporting in Scored Oversight: True-KL0 Property via the Prekopa Principle
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Paper
Honest Reporting in Scored Oversight: True-KL0 Property via the Prekopa Principle
2026
Overview
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 ın (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 ın \\2,3,4\\\), \\(p ın (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 \\(ınt_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 ın \\3,4\\\) the Prekopa argument (analytic, covering \\(M M_cut(d,p) 20\\)) combines with a certified high-precision numerical step on the residual region \\(M ın [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 ın (d,d+1)\\) when \\(d 4\\), but fails above a critical threshold \\(p_crit(d) ın (d,d+1)\\) for \\(d 5\\); for \\(d=5\\) we locate \\(p_crit(5) ın (5.5718, 5.5750)\\) via high-precision mpmath evaluation (half-width 0.0016, not interval-certified).
Publisher
Cornell University Library, arXiv.org
Subject
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