Functional local operators in the voting theory. I

The paper is concerned with voting operators which transform sets of individuals choice functions into a “collective” choice function. The concept of dosed regions in the space of choice functions with respect to such operators is introduced. The local nature of the voting operator is defined. In the space of local operators the operator ranges are identified which satisfy the characteristic properties natural for voting systems. Principles of alternative neutralities are formulated which lead to incompatibility of characteristic conditions which are imposed on the operators when the closeness conditions hold for “natural regions” in the space of choice functions. The causes of this incompatibility are studied. Functional analogs are established of the “classical”. Arrow paradox which arises in studies of operators for transformation of orderings of the options which guide the voters into a collective ordering of options.