Most decision problems can be understood as a mapping from a preference space into a set of outcomes. When preferences are representable via utility functions, this generates a mapping from a space of utility functions into outcomes. We say a model is continuous in utilities (resp., preferences) if small perturbations of utility functions (resp., preferences) generate small changes in outcomes . While similar, these two concepts are equivalent only when the topology satisfies the following universal property: for each continuous mapping from preferences to outcomes there is a unique mapping from utilities to outcomes that is faithful to the preference map and is continuous. The topologies that satisfy such a universal property are called final topologies. In this paper , we analyze the properties of the final topology for preference sets. This is of practical importance since most of the analysis on continuity is done via utility functions and not the primitive preference space. Our results allow the researcher to extrapolate continuity in utility to continuity in the underlying preferences.
 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 concepts are equivalent only when the topology satisfies the following universal property: for each continuous mapping from preferences to model's outputs there is a unique mapping from utilities to model's outputs that is faithful to the preference map and is continuous. The topologies that satisfy such a universal property are called final topologies. In this paper we analyze the properties of the final topology for preference sets. This is of practical importance since most of the analysis on continuity is done via utility functions and not the primitive preference space. Our results allow the researcher to extrapolate continuity in utility to continuity in the underlying preferences.
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Most decision problems can be understood as a mapping from a preference space into a set of outcomes. When preferences are representable via utility functions, this generates a mapping from a space of utility functions into outcomes.
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outcomes
the model's outputs
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outcomes
model's outputs
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outcomes
model's outputs
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