May's theorem
Encyclopedia
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. Various modifications have been suggested by others since the original publication; in particular, Mark Fey 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.
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.
Social choice theory
Social choice theory is a theoretical framework for measuring individual interests, values, or welfares as an aggregate towards collective decision. A non-theoretical example of a collective decision is passing a set of laws under a constitution. Social choice theory dates from Condorcet's...
, May's theorem states that simple majority voting is the only anonymous, neutral, and monotone
Monotonicity criterion
The monotonicity criterion is a voting system criterion used to analyze both single and multiple winner voting systems. A voting system is monotonic if it satisfies one of the definitions of the monotonicity criterion, given below.Douglas R...
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
Kenneth May
Kenneth O. May was an American mathematician and historian of mathematics, who developed May's theorem. The Kenneth O. May Prize is awarded for outstanding contributions to the history of mathematics....
first published this theory in 1952. Various modifications have been suggested by others since the original publication; in particular, Mark Fey 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
Nakamura number
In cooperative game theory and social choice theory, the Nakamura number measures the degree of rationalityof preference aggregation rules , such as voting rules....
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
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it is the simple majority method.