The self-dual core and the anti-self-dual remainder of an aggregation operator

In most decisional models based on pairwise comparison between alternatives, the reciprocity of the individual preference representations expresses a natural assumption of rationality. In those models self-dual aggregation operators play a central role, in so far as they preserve the reciprocity of the preference representations in the aggregation mechanism from individual to collective preferences. In this paper we propose a simple method by which one can associate a self-dual aggregation operator to any aggregation operator on the unit interval. The resulting aggregation operator is said to be the self-dual core of the original one, and inherits most of its properties. Our method constitutes thus a new characterization of self-duality, with some technical advantages relatively to the traditional symmetric sums method due to Silvert. In our framework, moreover, every aggregation operator can be written as a sum of a self-dual core and an anti-self-dual remainder which, in some cases, seems to give some indication on the dispersion of the variables. In order to illustrate the method proposed, we apply it to two important classes of continuous aggregation operators with the properties of idempotency, symmetry, and stability for translations: the OWA operators and the exponential quasiarithmetic means.