Anonymous, non-manipulable binary social choice

Let V be a society whose members express weak preferences about two alternatives. We show simple representation formulae that are valid for all, and only, the elements of various classes non-manipulable social choice functions on V. represent entire class functions, its subclasses corresponding to further properties. focus mainly anonymity. Efficiency neutrality up one voter also considered. As consequence formulae, cardinalities some these established. Notably, we number anonymous is 2n+1 if contains n members.