When Is Calibrated Confidence Sufficient for Optimal Stopping in Sequential Prediction?

Sequential predictors use confidence to decide whether to answer, abstain, or spend more computation. Calibration at each fixed time is insufficient for this task: it pools histories with the same score even when they differ in current error risk or future informativeness. We separate these two roles. For the terminal answer/abstain choice, the relevant quantity is the probability that the current answer is correct conditional on the observed history. We study scores equal to this conditional probability, the binary form of an ideal forecast relative to the available information. This condition holds at every fixed time exactly when it holds at every bounded stopping time, and it characterizes calibrated scores that support Bayes-optimal terminal decisions for every reject cost. For continuation, however, even an ideal confidence forecast need not be a sufficient statistic for control: equal scores can require opposite actions and leave a constant-order risk gap. A Bellman recursion based only on confidence is valid uniformly over bounded horizons, costs, and terminal losses exactly when the next-score law depends on history only through the current score. Calibration at each fixed time can also leave a constant-order gap. If the score is uniformly within $\epsilon$ of the history-conditional probability, optimizing the surrogate stopping problem on the same filtration incurs at most $2c_{\mathrm{err}}\epsilon$ regret; a tie-free construction makes the constant tight. Numerical examples recover the theoretical gaps. A real-data diagnostic reveals residual history dependence but no decision-cost gap under the tested costs.