Preference-Class Gap and Pathwise Distortion in Dynamic Decision Models

Abstract We provide an axiomatic foundation for two indices intended to quantify dynamic discrepancy between alternative decision criteria under the canonical invariance principle of decision theory and social choice: welfare representations are meaningful only up to positive affine transformations. The first index, the Preference-Class Gap (PPG), measures representation-level mismatch between two instantaneous welfareprimitives on a common compact admissible domain, by computing the uniform-norm distance from one primitive to the positive-affine equivalence class generated by the other, with a scale-free normalization ensuring well-posedness under near degenerate benchmark scales. The second index, the Pathwise Distortion Index (PDI), measures outcome-level discrepancy along realized optimal trajectories: it aggregates the benchmark-evaluated welfare wedge induced by the alternative criterion and normalizes it in benchmark units, with a strictly positive floor regularization required for stability when baseline magnitudes become small. For each index we propose transparent axioms and prove representation theorems: up to a monotone transform, PPG is uniquely characterized by the uniform-norm distance to a positive-affine class, and PDI is uniquely characterized by a floor-regularized normalized cumulative wedge along realized paths. We also establish a sharp logical separation: representation-level proximity and pathwise distortion are not equivalent objects. Small preference-class gaps can be dynamically amplified by constraints and feedback, while large primitive mismatches may be irrelevant on realized paths. These results provide a measurement-theoretic basis for a general Dynamic Discrepancy Theory of regime comparison in dynamic decision problems.