Persuasive Calibration

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.