Voting on How to Vote

Abstract

All societies need to make collective decisions, and for this they need a voting rule. We ask how stable these rules are. Stable rules are more likely to persist over time. We consider a family of voting systems in which individuals declare intensities of preference through numbers in the unit interval. With these voting systems, an alternative defeats another whenever the amount of opinion obtained by the first alternative exceeds the amount of opinion obtained by the second alternative by a fixed threshold. The relevant question is what should this threshold be? We assume that each individual’s assessment of what the threshold should be is represented by a trapezoidal fuzzy number. From these trapezoidal fuzzy numbers we associate reciprocal preference relations on [0,m]. With these preferences over thresholds in place, we formalize the notion of a threshold being “self-selective”. We establish some mathematical properties of self-selective thresholds, and then describe a three stage procedure for selecting an appropriate threshold. Such a procedure will always select a threshold which can then be used for future decision making.