A characterization theorem for collective choice in binary decisions: among all deterministic voting rules that map individual binary votes to a single collective binary outcome, simple majority rule is the unique rule satisfying anonymity (treating voters symmetrically), neutrality (treating the two alternatives symmetrically), and positive responsiveness (if a change in one vote turns a tie into a majority, the outcome changes accordingly).

Principle

For binary choices under the stated axioms, any rule that is anonymous, neutral and positively responsive must coincide with majority rule; the normative fairness constraints determine the rule uniquely in that domain.

Demonstration

Illustrative scenario → Binary proposal with two outcomes. A rule that treats all voters the same (anonymity), treats both options symmetrically (neutrality), and rewards additional support when it changes the balance (positive responsiveness) must select the alternative preferred by more voters; therefore the rule behaves as simple majority. Recognition: check axioms; Action: apply rule to a profile; Consequence: majority winner selected when axioms hold.

Misapplication

Extending the theorem’s uniqueness claim beyond binary choices or to voting rules that permit randomized outcomes or side payments. The error is to assume the theorem justifies majority rule in contexts (multi‑option ballots, cardinal voting, deliberative processes) to which its axioms do not apply.

Consequence

Within its scope, May’s theorem supplies a compact normative justification for simple majority as the only rule satisfying basic fairness and responsiveness conditions for binary decisions; it also clarifies the limits of that justification beyond binary contexts.

Reversal

If one of the axioms is relaxed (for example by allowing non‑anonymity, preferential weighting, or weakening positive responsiveness) or if choices are multi‑option rather than binary, other aggregation rules become permissible and uniqueness is lost.

Boundary

Clearly within: deterministic, binary collective choice functions over profiles of ordinal binary votes with anonymity, neutrality and positive responsiveness. Boundary case: rules that are probabilistic, quota systems with weights, or multi‑alternative settings where the axioms do not uniquely determine a rule. Clearly outside: aggregation of cardinal utilities, ranked multi‑option mechanisms, or procedures using additional information beyond the binary votes.

Semantic Tension

Minimal fairness axioms (anonymity, neutrality, responsiveness) ↔ Practical desiderata like stability, deliberation, or resistance to strategic manipulation; satisfying the fairness axioms does not address other normative or strategic concerns.

Synthesis

May’s theorem shows that the common normative appeal of majority rule in binary decisions is not accidental but follows from three simple fairness conditions; however, its force is strictly limited to deterministic binary contexts, so institutional designers must assess whether those conditions capture the relevant normative priorities in more complex settings.