Integral representations of a coherent upper conditional prevision by the symmetric choquet integral and the asymmetric choquet integral with respect to hausdorff outer measures

Complex decisions in human decision-making may arise when the Emotional Intelligence and Rational Reasoning produce different preference ordering between alternatives. From a mathematical point of view, complex decisions can be defined as decisions where a preference ordering between random variables cannot be represented by a linear functional. The Asymmetric and the Symmetric Choquet integrals with respect to non additive-measures have been defined as aggregation operators of data sets and as a tool to assess an ordering between random variables. They could be considered to represent preference orderings of the conscious and unconscious mind when a human being make decision. Sufficient conditions are given such that the two integral representations of a coherent upper conditional prevision by the Asymmetric Choquet integral and the Symmetric Choquet integral with respect to Hausdorff outer measures coincide and linearity holds.