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On the Extension and Decomposition of a Preorder under Translation Invariance

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  • Mabrouk, Mohamed

Abstract

We prove the existence, for a translation-invariant preorder on a divisible commutative group, of a complete preorder extending the preorder in question and satisfying translation invariance (theorem 1). We also prove that the extension may inherit a property of continuity (theorem 2). This property of continuity may lead to scalar invariance. By seeking to clarify the relationship between continuity and scalar invariance under translation invariance, we are led to formulate a theorem that asserts the existence of a continuous linear weak representation under a certain condition (theorem 3). The application of these results in a space of infinite real sequences shows that this condition is weaker than the axiom super weak Pareto, and that the latter is itself weaker than the axiom monotonicity for non-constant preorders. Thus, theorem 3 is a strengthening of theorem 4 of Mabrouk 2011. It also makes it possible to show the existence of a sequence of continuous linear preorders whose lexicographic combination constitutes the finest combination coarser than the preorder in question (theorem 4). This decomposition makes it possible to handle continuous functions instead of preoders when one looks for optima, which may be more practical. Finally we apply this decomposition to the preorder catching-up. Several examples are provided.

Suggested Citation

  • Mabrouk, Mohamed, 2018. "On the Extension and Decomposition of a Preorder under Translation Invariance," MPRA Paper 90537, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:90537
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    File URL: https://mpra.ub.uni-muenchen.de/90537/1/MPRA_paper_90537.pdf
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    References listed on IDEAS

    as
    1. Mohamed Ben Ridha Mabrouk, 2011. "Translation invariance when utility streams are infinite and unbounded," International Journal of Economic Theory, The International Society for Economic Theory, vol. 7(4), pages 317-329, December.
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    4. Jaffray, Jean-Yves, 1975. "Semicontinuous extension of a partial order," Journal of Mathematical Economics, Elsevier, vol. 2(3), pages 395-406, December.
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    More about this item

    Keywords

    preorder extension; translation invariance; lexicographic decomposition; continuity; scalar invariance;
    All these keywords.

    JEL classification:

    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • D7 - Microeconomics - - Analysis of Collective Decision-Making
    • D9 - Microeconomics - - Micro-Based Behavioral Economics

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