Symmetric majority rules

In the standard arrovian framework and under the assumption that individual preferences and social outcomes are linear orders on the set of alternatives, we suppose that individuals and alternatives have been exogenously partitioned into subcommittees and subclasses, and we study the rules that satisfy suitable symmetries and obey the majority principle. In particular, we provide necessary and sufficient conditions for the existence of reversal symmetric majority rules that are anonymous and neutral with respect to the considered partitions. We also determine a general method for constructing and counting those rules and we explicitly apply it to some simple cases.