A Note on Numerical Representations of Nested System of Strict Partial Orders

This note provides two numerical representations of a nested system of strict partial orders. The first representation is based on utility and threshold functions. We generalize the threshold representation of menu-dependent preferences by allowing the threshold to depend not only on the menu but also on the pair of alternatives under comparison. The threshold function can be interpreted as the distance between alternatives. The second representation is based on the aggregation of multi-dimensional preference. This representation describes a decision-making procedure where multiple criteria are gradually aggregated into an overall assessment.

A Note on Reference-Dependent Choice with Threshold Representation

This note considers reference-dependent choice procedures shaped by a utility function and a threshold function. Alternatives are considered only when the utility gain of moving from the reference is bigger than the threshold. We impose no requirement on the threshold functions whereas previous works characterize the special case of strictly positive threshold functions. We also examine the case when the threshold function depends on both compared alternatives.

Does the threshold representation associated with the autoconversion process matter?

Different ad hoc threshold functions associated with the autoconversion process have been arbitrarily used in atmospheric models. However, it is unclear how these ad hoc functions impact model results. Here systematic investigations of the sensitivities of climatically-important properties: CF (cloud fraction), LWP (liquid water path), and AIE (aerosol indirect effect) to threshold functions have been performed using a 3-D cloud-resolving model. It is found that the effect of threshold representations is larger on instantaneous values than on daily averages; and the effect depends on the percentage of clouds in their transitional stages of converting cloud water to rain water. For both the instantaneous values and daily averages, the sensitivity to the specification of critical radius is more significant than the sensitivity to the "smoothness" of the threshold representation (as embodied in the relative dispersion of droplet size distribution) for drizzling clouds. Moreover, the impact of threshold representations on the AIE is stronger than that on CF and LWP.

Does the threshold representation associated with the autoconversion process matter?

Different ad hoc threshold functions associated with the autoconversion process have been arbitrarily used in atmospheric models. However, it is unclear how these ad hoc functions impact model results. Here systematic investigations of the sensitivities of climatically-important properties: CF (cloud fraction), LWP (liquid water path), and AIE (aerosol indirect effect) to threshold functions have been performed using a 3-D cloud-resolving model. It is found that the effect of threshold representations is larger on instantaneous values than on daily averages; and the effect depends on the percentage of clouds in their transitional stages of converting cloud water to rain water. For both the instantaneous values and daily averages, the sensitivity to the specification of critical radius is more significant than the sensitivity to the "smoothness" of the threshold representation (as embodied in the relative dispersion of droplet size distribution) for drizzling clouds. Moreover, the impact of threshold representations on the AIE is stronger than that on CF and LWP.