Truthfulness of Decision-Theoretic Calibration Measures

Calibration measures quantify how much a forecaster's predictions violates calibration, which requires that forecasts are unbiased conditioning on the forecasted probabilities. Two important desiderata for a calibration measure are its decision-theoretic implications (i.e., downstream decision-makers that best-respond to the forecasts are always no-regret) and its truthfulness (i.e., a forecaster approximately minimizes error by always reporting the true probabilities). Existing measures satisfy at most one of the properties, but not both. We introduce a new calibration measure termed subsampled step calibration, mathsfStepCE^textsfsub, that is both decision-theoretic and truthful. In particular, on any product distribution, mathsfStepCE^textsfsub is truthful up to an O(1) factor whereas prior decision-theoretic calibration measures suffer from an e-Omega(T)-Omega(√T) truthfulness gap. Moreover, in any smoothed setting where the conditional probability of each event is perturbed by a noise of magnitude c > 0, mathsfStepCE^textsfsub is truthful up to an O(√log(1/c)) factor, while prior decision-theoretic measures have an e-Omega(T)-Omega(T1/3) truthfulness gap. We also prove a general impossibility result for truthful decision-theoretic forecasting: any complete and decision-theoretic calibration measure must be discontinuous and non-truthful in the non-smoothed setting.