May's theorem

In social choice theory, May's theorem, also called Arrow's possibility theorem,^[1] says that majority vote is the unique ranked social choice function that satisfies the following criteria:

• Anonymity – equality of voters, i.e. one man, one vote.
• Neutrality – equal treatment of candidates, i.e. a fair election.
• Positive responsiveness – votes have a positive (not negative) value.

The thoerem was first published by Kenneth May in 1952.^[1]

Various modifications have been suggested by others since the original publication. If rated voting is allowed, a wide variety of rules satisfy May's conditions, including score voting or highest median voting rules.

Arrow's theorem does not apply to the case of two candidates (when there are trivially no "independent alternatives"), so this possibility result can be seen as the mirror analogue of that theorem. Note that anonymity is a stronger requirement than non-dictatorship.

Another way of explaining the fact that simple majority voting can successfully deal with at most two alternatives is to cite Nakamura's theorem. The theorem states that the number of alternatives that a rule can deal with successfully is less than the Nakamura number of the rule. The Nakamura number of simple majority voting is 3, except in the case of four voters. Supermajority rules may have greater Nakamura numbers.^{[citation needed]}

Formal statement

• Condition 1. The group decision function treats each voter identically. (anonymity)
• Condition 2. The group decision function treats both outcomes the same, in that reversing each set of preferences reverses the group preference. (neutrality)
• Condition 3. 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. (positive responsiveness)

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. JSTOR 1907651
2. ^ Mark Fey, "May’s Theorem with an Infinite Population", Social Choice and Welfare, 2004, Vol. 23, issue 2, pages 275–293.
3. ^ Goodin, Robert and Christian List (2006). "A conditional defense of plurality rule: generalizing May's theorem in a restricted informational environment," American Journal of Political Science, Vol. 50, issue 4, pages 940-949. doi:10.1111/j.1540-5907.2006.00225.x

References

1. May, Kenneth O. (1952). "A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision". Econometrica. 20 (4): 680–684. doi:10.2307/1907651. ISSN 0012-9682.

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