Journal article: In this paper, we extend the generalized Yaari’s dual theory for multidimensional distributions, in the vein of Galichon and Henry’s paper (Galichon and Henry in J Econ Theory 147:1501–1516, 2012). We show how a class of generalized quantiles—which encompasses Galichon and Henry’s one or multivariate quantile transform [see Arjas and Lehtonen (Math Oper Res 3(3):205–223, 1978), O’Brien (Ann Prob 3(1):80–88, 1975) or Ruschendorf (Ann Probab 9(2):276–283, 1981)]—allows to derive a general representation theorem.