## May's Theorem

A most important theorem concerning majority rule was proved a half century ago by May (1952)

May begins by deﬁning a group decision function:

```    D = f (D1,D2,...,Dn)
```
where n is the number of individuals in the community. Each Di takes on the value 1, 0, −1 as voter i’s preferences for a pair of issues are xPy, xIy, and yPx, where P represents the strict preference relationship and I indifference. Thus, the Di serve as ballots, and f() is an aggregation rule for determining the winning issue. Depending on the nature of the voting rule, f() takes on different functional forms. Under the simple majority rule, f() sums the Di and assigns D a value according to the following rule:

May deﬁnes the following four conditions:

• Decisiveness: The group decision function is deﬁned and single valued for any given set of preference orderings
• Anonymity: D is determined only by the values of Di , and is independent of how they are assigned. Any permutation of these ballots leaves D unchanged
• Neutrality: If x defeats (ties) y for one set of individual preferences, and all individuals have the same ordinal rankings for z and w as for x and y (i.e., xRy → zRw, and so on), then z defeats (ties) w
• Positive responsiveness: If D equals 0 or 1, and one individual changes his vote from −1 to 0 or 1, or from 0 to 1, and all other votes remain unchanged, then D=1

The Theorem states that a group decision function is the simple majority rule if and only if it satisﬁes these four conditions

It is a most remarkable result. If we start from the set of all possible voting rules one can conceive of, and then begin imposing conditions we wish our voting rule to satisfy, we shall obviously reduce the number of viable candidates for our chosen voting rule as we add more and more conditions. May’s theorem tells us that once we add these four conditions, we have reduced the possible set of voting rules to but one, the simple majority rule. All other voting rules violate one or more of these four axioms

Decisiveness seems at ﬁrst uncontroversial. If we have a decision function, we want it to be able to decide at least when confronted with only two issues. But this axiom does eliminate all probabilistic procedures in which the probability of an issue’s winning depends on voter preferences.

Positive responsiveness is also a reasonable property. If the decision process is to reﬂect each voter’s preference, then a switch by one voter from opposition to support ought to break a tie

The other two axioms are less innocent than they look or their names connote.

The neutrality axiom introduces an issue-independence property. In deciding a pair of issues, only the ordinal preferences of each voter over this issue pair are considered. Information concerning voter preferences on other issue pairs is ruled out, and thereby one means for weighing intensities is eliminated.

Where the neutrality axiom guarantees that the voting procedure treats each issue alike, anonymity assures that each voter is treated alike. On many issues this is probably a desirable property.