Some characterisations of self-dual aggregation functions when relative shortfalls are considered

Whenever shortfalls are defined as the absolute difference between the upper bound and the level of attainments the characterisation of aggregation functions that rank attainment and shortfall distributions mirroring one another, i.e. self-dual aggregation functions, is a widely discussed issue. In this paper we consider an alternative definition of shortfalls as the relative difference between the upper bound and the level of attainments and extend some characterisation results to this new framework. Moreover, we propose a particular dual decomposition for each aggregation function and apply it to two major classes of homogeneous aggregation functions: α-power means and OWA operators.