INTRODUCTION

In this chapter, we restrict our attention to the decisive voting rule with only two contesting candidates (or alternatives). A decisive voting rule maps for every possible vote (or preference) of the set of agents over the two contesting candidates (say, x and y) to either a winner or two non-losers. May’s theorem, which is the main subject matter of this chapter, specifies the importance of majority voting rule by arguing that it is the unique rule that satisfies four important democratic principles. These four key democratic principles are decisiveness of the voting rule, anonymity, neutrality, and positive responsiveness. Decisiveness of the voting rule requires that the voting rule must specify a unique decision even if the decision is indifference for any set of individual preferences over the two contesting candidates. Anonymity (or symmetry across agents) requires that a voting rule must treat all voters alike, in the sense that if any two voters traded ballots, the outcome of the election would remain the same. Neutrality (or anonymity across alternatives) requires that a voting rule must treat all candidates alike, rather than favor one over the other. Finally, positive responsiveness (a type of monotonicity property) requires that if the group decision is indifference or favorable to some alternative x, and if individual preferences remain the same except that a single individual changes his or her vote in favor of x, then the group decision should be x. Formally,May’s theoremstates that among the class of all decisive voting rules, the majority voting rule is the only one that satisfies anonymity, neutrality, and positive responsiveness.

The chapter is organized as follows: Section 3.2 provides the framework. In Section 3.3 we state and prove May’s theorem. In Section 3.4 we check the robustness of the axioms used in May’s theorem.

THE FRAMEWORK

We consider preferences of a finite set of agents N = ﹛1, … , n﹜ of a society voting over two alternatives x and y.