Extending Utility Functions on Arbitrary Sets
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We consider the problem of whether a function f^_P defined on a subset P of an arbitrary set X can be extended to X monotonically with respect to a preorder ≽ defined on X. We prove that whenever ≽ has a utility representation, such an extension exists if and only if f^_P is gap-safe increasing. An explicit construction for a monotone extension of this kind involving an arbitrary utility representation of ≽ is presented. The special case where P is a Pareto subset of X is considered. The problem under study does not include continuity constraints.
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