Explicit form of neutral social decision rules for basic rationality conditions

Abstract The paper deals with aggregation operators F:( R 1 ) n → R 2 which satisfy the classical requirements of binariness (independence), neutrality to alternatives, non-imposition, and which transform any n-tuple of individual relations of the given class R 1 into a collective relation of the given class R 2 . We consider as R 1 and R 2 the classes: L of linear orders, W of weak orders, S of semiorders, I of interval orders, P of partial orders, T of transitive relations, A of acyclic relations. For all 27 possible pairs R 1 , R 2 ∈{ L , W , S , I , P , T , A } such that R 1 ⊆ R 2 , we bring the explicit form of operators ( R 1 ) n → R 2 . The results are obtained on the basis of the following approach. Using a logical form of operators, we associate to each F a so-called ordinal binary relation ρF on R n (for any x , y ∈ R n one is uniquely determined by signs of coordinate differences xi−yi, 1≤i≤n). We prove that if R 2 satisfies some mild conditions then F maps W n into R 2 if and only if ρ F ∈ R 2 . So the description of the operators W n → R 2 amounts to the description of the ordinal relations of R 2 . The approach can be adapted to some classes R 1 ≠ W .