Functional Voting Operators: The Non-Monotonic Case

We extend the non-binary framework of social choice introduced by Aizerman and Aleskerov (1986) , in which individual choice functions are aggregated into a social choice function, by considering non-monotonic operators. We characterize the class of "local" operators and provide the explicit forms of local operators satisfying various combinations of normative and rationality conditions in the absence of monotonicity. Surprisingly, the restriction of monotonicity is not binding for operators satisfying the usual rationality conditions. We identify two rationality restrictions which do admit non-monotonic operators. One restriction admits every sovereign and neutral operator, and the other admits only dictatorship and anti-dictatorship operators. This last result is a direct non-binary counterpart to Wilson's (1972) theorem.