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Representing preorders with injective monotones.

Pedro Hack1, Daniel A Braun1, Sebastian Gottwald1

  • 1Institute of Neural Information Processing, Ulm University, 89081 Ulm, Germany.

Theory and Decision
|October 17, 2022
PubMed
Summary
This summary is machine-generated.

We introduce injective monotones, a new class of real-valued functions in preordered spaces. These functions enhance the classification of preordered spaces and connect to statistical inference and machine learning principles.

Keywords:
MajorizationMaximum entropyMulti-utility representationRichter–Peleg functionUncertainty preorder

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Area of Science:

  • Decision Theory
  • Mathematical Economics
  • Information Theory

Background:

  • Real-valued monotones are crucial for understanding preferences in preordered spaces.
  • Existing classifications rely on strict monotones and countable multi-utilities.
  • Further refinement of these classifications is needed for advanced applications.

Purpose of the Study:

  • Introduce and define injective monotones as a new class of real-valued functions.
  • Establish the position of preorders admitting injective monotones within existing classifications.
  • Extend known results from strict monotones to this new class.

Main Methods:

  • Develop a novel class of real-valued functions termed injective monotones.
  • Analyze the properties and existence conditions for injective monotones in preordered spaces.
  • Construct injective monotones from countable multi-utilities.

Main Results:

  • Demonstrate that injective monotones exist for a class of preorders situated between those with strict monotones and countable multi-utilities.
  • Generalize key results of strict monotones (Richter-Peleg functions) to injective monotones.
  • Establish connections between injective monotones, Debreu denseness, order separability, Shannon entropy, and the uncertainty preorder.

Conclusions:

  • Injective monotones offer a refined classification of preordered spaces.
  • These monotones provide new insights into the relationship between information theory and decision theory.
  • Generalizing Jaynes' maximum entropy principle using injective monotones has implications for statistical inference and machine learning regularization.