Proof of Calibration — Reputation from Strictly Proper Scoring Rules: the Exact Price of Faking Skill

Abstract

Decentralized reputation mechanisms certify trust by making agents spend a scarce resource: computation (Proof of Work), capital (Proof of Stake), or identity (Proof of Personhood). In all three the resource is orthogonal to the quality being certified — a well-capitalized agent is not a skilled one. We study the construction in which the spent resource is the certified competence: Proof of Calibration, in which pseudonymous agents commit probabilistic predictions, outcomes resolve against ground truth, and reputation accrues through a strictly proper scoring rule. Building on the merit-gating results of Calibration-Gated Reputation (Alassa and Alashqar, SSRN 6505678), we replace that paper's concentration estimates with an exact theory. The realized mean score of an identity is a linear function of a single sufficient statistic — its empirical outcome frequency p̂ — so passing a merit bar is exactly the event that p̂ reaches an "alibi frequency" q_c, and the price of that event is an information divergence: the Sybil-lottery success probability is controlled, two-sidedly for all K and all N beyond a mild explicit threshold, by e^(−N·KL(q_c‖p)). Strict propriety is precisely positivity of this rate at every dishonest report, unifying the companion paper's impossibility and merit-gating theorems as the zero- and positive-rate regimes of one scalar. The theory is exact enough to reproduce, with zero free parameters, every Monte Carlo experiment of the companion suite, including the finite-size deviations from the asymptotic ε⁻⁴ cold-start law. Two structural results follow. First, the gate has two margins: in expectation, the adversary's best response is honesty and the margin is the generalized-entropy gap; at the event level, the optimal attack is skill-mimicry — report the bar's skill, not your own — and the margin is min{KL(p_h‖p_a), KL(1−p_h‖p_a)}, the divergence between the skills themselves, identical across all strictly proper, outcome-symmetric rules. The gate's security exponent is a property of the skills, not of the rule. Second, Sybil amplification requires independent evidence: identities scored on a shared outcome stream gain almost nothing from their number. Finally, the gate is not merely sound but optimal: no local pseudonymous mechanism that admits honest bar-skill agents with non-vanishing probability achieves a soundness exponent exceeding KL(p_h‖p_a), and the strictly proper gate attains it — it is the Neyman–Pearson test of the skill hypothesis. The composite Notch Score reduces to the same single statistic, yielding an exact calibration-weight floor.