Persuasive Calibration

TL;DR

The paper studies persuasive calibration, where a principal designs predictions that remain credibly calibrated under an allowed calibration error budget to steer a downstream agent’s decisions. It develops a foundational two-step view: start from a perfectly calibrated predictor and apply an admissible miscalibration plan, yielding a tractable linear-program formulation that characterizes optimal predictors under an $\\ell_t$-ECE budget. In the event-independent case, it provides a precise structural characterization: the optimal predictor uses three regions (under-confident, perfectly calibrated, over-confident) with a symmetric linear-tailed convex payoff envelope $\\Gamma$ governing the tradeoff between calibration error and principal payoff; algorithmically, it achieves a FPTAS in general settings and polynomial-time exact solutions for $K_1$ and $K_\\infty$. The paper then connects persuasive calibration to a variant of Bayesian persuasion with signal-dependent bias, proving a revelation principle and enabling LP-based polynomial-time algorithms for the standard ECE metrics, thereby unifying calibration and information-design perspectives. Overall, the work advances both the theory and algorithms for designing trustworthy yet strategically misaligned predictions, with implications for calibration, persuasion, and decision-making under uncertainty.

Abstract

We introduce and study the persuasive calibration problem, where a principal aims to provide trustworthy predictions about underlying events to a downstream agent to make desired decisions. We adopt the standard calibration framework that regulates predictions to be unbiased conditional on their own value, and thus, they can reliably be interpreted at the face value by the agent. Allowing a small calibration error budget, we aim to answer the following question: what is and how to compute the optimal predictor under this calibration error budget, especially when there exists incentive misalignment between the principal and the agent? We focus on standard Lt-norm Expected Calibration Error (ECE) metric. We develop a general framework by viewing predictors as post-processed versions of perfectly calibrated predictors. Using this framework, we first characterize the structure of the optimal predictor. Specifically, when the principal's utility is event-independent and for L1-norm ECE, we show: (1) the optimal predictor is over-(resp. under-) confident for high (resp. low) true expected outcomes, while remaining perfectly calibrated in the middle; (2) the miscalibrated predictions exhibit a collinearity structure with the principal's utility function. On the algorithmic side, we provide a FPTAS for computing approximately optimal predictor for general principal utility and general Lt-norm ECE. Moreover, for the L1- and L-Infinity-norm ECE, we provide polynomial-time algorithms that compute the exact optimal predictor.

Figures (4)

• Figure 1: In both figures, blue dots are the predictions $p$ generated from the optimal predictor $\tilde{f}^*$, brown squares are the corresponding true expected outcomes $\kappa(p)$ conditional on the prediction $p$, \ref{['fig:general']} is an illustration of \ref{['thm:event-inde']} -- black solid curve is indirect utility $U$ and blue curve is the convex function $\Gamma$ in \ref{['prop:convex env']}. \ref{['fig:reliable-diagram-general']} is the reliable diagram (a standard visualization of the miscalibration structure) -- black solid line is diagonal line.
• Figure 2: Illustration of the principal with a S-shaped indirect utility (\ref{['example:s-shape']}). In both figures, blue dots are the predictions $p$ generated from the optimal predictor $\tilde{f}^*$, brown squares are the corresponding true expected outcomes $\kappa(p)$ conditional on the prediction $p$. \ref{['fig:general']} is an illustration of \ref{['thm:event-inde']} -- black solid curve is indirect utility $U$ and blue curve is the convex function $\Gamma$ in \ref{['prop:convex env']}. \ref{['fig:reliable-diagram-general']} is the reliable diagram -- black solid line is diagonal line.
• Figure 3: An illustration of hierarchical relation between $\mathcal{F}_{\varepsilon}$, $\mathcal{F}^{\rm EI}_{\varepsilon}$ and $\mathcal{F}_{0}$. The dashed arrow line from predictor $\tilde{f}$ to perfectly calibrated predictor $\tilde{g}$ is established by \ref{['prop:two-step equivalence']}, and the dashed arrow line from $\tilde{g}$ to the predictor $\tilde{f}^{\dagger}$ is established in \ref{['defn:miscal procedure']} and \ref{['prop:miscal procedure error guarantee']}.
• Figure 4: The solid (resp. dashed) line is the expected utility of the principal (resp. agent) under the optimal $\varepsilon$-calibrated predictor with ECE budget $\varepsilon\in[0, 0.05]$ in \ref{['example:win win example']}.

Theorems & Definitions (77)

• Example 2.1: Perfect calibration
• Definition 2.2: Expected calibration error
• Example 2.3: $\varepsilon$-calibrated predictor
• Definition 2.4: Indirect utility
• Definition 3.1: Under-/over-confidence
• Theorem 3.1: Structure of optimal $\varepsilon$-calibrated predictor
• Example 3.2: Binary-action setting
• Example 3.3: S-shaped indirect utility
• Definition 3.4: Symmetric linear-tailed convex function
• Proposition 3.2