Extreme Copulas and the Comparison of Ordered Lists

We introduce two extreme methods to pairwisely compare ordered lists of the same length, viz. the comonotonic and the countermonotonic comparison method, and show that these methods are, respectively, related to the copula T <Subscript> <Emphasis Type="Bold">M </Subscript> (the minimum operator) and the Å\x81 ukasiewicz copula T <Subscript> <Emphasis Type="Bold">L </Subscript> used to join marginal cumulative distribution functions into bivariate cumulative distribution functions. Given a collection of ordered lists of the same length, we generate by means of T <Subscript> <Emphasis Type="Bold">M </Subscript> and T <Subscript> <Emphasis Type="Bold">L </Subscript> two probabilistic relations Q <Superscript> <Emphasis Type="Bold">M </Superscript> and Q <Superscript> <Emphasis Type="Bold">L </Superscript> and identify their type of transitivity. Finally, it is shown that any probabilistic relation with rational elements on a 3-dimensional space of alternatives which possesses one of these types of transitivity, can be generated by three ordered lists and at least one of the two extreme comparison methods. Copyright Springer Science+Business Media, LLC 2007