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The lexicographic complexity of asymmetric binary relations

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  • Knoblauch, Vicki

Abstract

The lexicographic complexity of an asymmetric binary relation on a finite set is defined to be the minimal number of ternary criteria sufficient to construct a 1–1 lexicographic representation of that relation. Lexicographic complexity provides a measure of the degree of non-representability by utility function of an asymmetric binary relation. Tight upper and lower bounds are established for the lexicographic complexity of an asymmetric binary relation on a set of cardinality n. Four examples launch an exploration of the relationships between lexicographic complexity and intransitivity and between lexicographic complexity and incompleteness. Two examples of 1–1 lexicographic representations exhibit a strong lexicographic flavor in that, for each of the two examples, any two distinct reorderings of its coordinate functions result in representations of distinct binary relations.

Suggested Citation

  • Knoblauch, Vicki, 2022. "The lexicographic complexity of asymmetric binary relations," Mathematical Social Sciences, Elsevier, vol. 117(C), pages 6-12.
    • Handle: RePEc:eee:matsoc:v:117:y:2022:i:c:p:6-12
    DOI: 10.1016/j.mathsocsci.2022.02.002
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    Cited by:

    1. Knoblauch, Vicki, 2023. "Lexicographic preference representation: Intrinsic length of linear orders on infinite sets," Journal of Mathematical Economics, Elsevier, vol. 105(C).

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