May's theorem

May's theorem

In social choice theory, May's theorem states that simple majority voting is the only anonymous, neutral, and monotone choice function between two alternatives. 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 that theorem. (Note that anonymity is a stronger form of 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.

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


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