Risk Reduction Through Reverse Engineering of a Comparison Functional

A new concept referred to as a “comparison functional” is introduced for the first time. It provides a unified framework for treating both super-additivity and sub-additivity of a risk-related response. Reverse engineering of the comparison functional serves as a generator of risk-reducing partitions of the input resource. This includes identifying a summable controlling variable, a summable risk-related response, selecting physically feasible aggregation or segmentation operations, and translating these into risk-reducing interventions. In both cases—positive and negative comparison functionals—risk reduction can be achieved through either aggregation or segmentation of the controlling input resource. Only when the comparison functional is zero can no redistribution of the input resource result in risk reduction. Exact lower and upper bounds of the comparison functional have been derived when the risk-related output can be presented as a convex or concave twice-differentiable function continuous in a closed interval. The proposed methodology has been illustrated by a number of application examples covering the case of explicitly specified convex risk-related output and a convex risk-related output whose functional form is unknown.