= 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, $P(C_{\rm correct}>C_{\rm error})+\tfrac12P(C_{\rm correct}=C_{\rm error})$ 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.
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