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In social choice theory, May's theorem states that simple majority voting is the only anonymous, neutral, and monotone choice function when there are two candidates. Further, this procedure is resolute when there are an odd number of voters and ties (indecision) are not allowed. Kenneth May first published this theory in 1952.^[1] Various modifications have been suggested by others since the original publication; in particular, Mark Fey ^[2] extended the proof to an infinite number of voters.

Arrow's theorem in particular does not apply to the case of two candidates, so this possibility result can be seen as a mirror analogue of Arrow's impossibility theorem. (Note that anonymity is a stronger form of non-dictatorship.)

Formal statement

• Condition 1. The group decision function sends each set of preferences to a unique winner. (resolute, unrestricted domain)
• Condition 2. The group decision function treats each voter identically. (anonymity)
• Condition 3. The group decision function treats both outcomes the same, in that reversing each preferences reverses the group preference. (neutrality)
• Condition 4. If the group decision was 0 or 1 and a voter raises a vote from -1 to 0 or 1 or from 0 to 1, the group decision is 1.

Theorem: A group decision function with an odd number of voters meets conditions 1, 2, 3, and 4 if and only if it is the simple majority method.

Notes

1. ^ May, Kenneth O. 1952. "A set of independent necessary and sufficient conditions for simple majority decisions", Econometrica, Vol. 20, Issue 4, pp. 680–684.
2. ^ Mark Fey, "May’s Theorem with an Infinite Population", Social Choice and Welfare, 2004, Vol. 23, issue 2, pages 275–293.

References

• Alan D. Taylor (2005). Social Choice and the Mathematics of Manipulation, 1st edition, Cambridge University Press. ISBN 0-521-00883-2. Chapter 1.
• May’s theorem and Bureaucracy