 A strictly increasing confidence transformation preserves metacognitive resolution

If confidence $C$ is replaced by a strictly increasing function $g(C)$, the attainable threshold partitions have the same ordering and ties. Consequently the type 2 receiver operating characteristic and its area under the ROC curve are unchanged. Equivalently, $P(C_{\mathrm{correct}}>C_{\mathrm{error}})+\frac{1}{2}P(C_{\mathrm{correct}}=C_{\mathrm{error}})$ is invariant. Calibration need not be preserved: adding a positive constant to probabilities within their allowed range changes their absolute values while preserving ranks. A merely nondecreasing transformation can merge distinct reports and requires additional care.