Final Topology for Preference Spaces (2004.02357v3)

Abstract: We say a model is continuous in utilities (resp., preferences) if small perturbations of utility functions (resp., preferences) generate small changes in the model's outputs. While similar, these two questions are different. They are only equivalent when the following two sets are isomorphic: the set of continuous mappings from preferences to the model's outputs, and the set of continuous mappings from utilities to the model's outputs. In this paper, we study the topology for preference spaces defined by such an isomorphism. This study is practically significant, as continuity analysis is predominantly conducted through utility functions, rather than the underlying preference space. Our findings enable researchers to infer continuity in utility as indicative of continuity in underlying preferences.