Explicit form of neutral social decision rules for basic rationality conditions

n The paper deals with aggregation operators F :(5 ) → 5 which satisfy the classical require1 2 ments of binariness (independence), neutrality to alternatives, non-imposition, and which transform any n-tuple of individual relations of the given class 5 into a collective relation of the 1 given class 5 . We consider as 5 and 5 the classes: + of linear orders, 0 of weak orders, 6 2 1 2 of semiorders, ( of interval orders, 3 of partial orders, 7 of transitive relations, ! of acyclic relations. For all 27 possible pairs 5 ,5 [ h+,0,6,(, 3,7,! j such that 5 # 5 , we bring the 1 2 1 2 n explicit form of operators (5 ) → 5 . The results are obtained on the basis of the following 1 2 approach. Using a logical form of operators, we associate to each F a so-called ordinal binary n n relation r on R (for any x,y [ R one is uniquely determined by signs of coordinate differences F n x 2 y , 1 # i # n). We prove that if 5 satisfies some mild conditions then F maps 0 into 5 if i i 2 2 n and only if r [ 5 . So the description of the operators 0 → 5 amounts to the description of F 2 2 the ordinal relations of 5 . The approach can be adapted to some classes 5 ± 0.  2000 2 1 Elsevier Science B.V. All rights reserved.