[EM] May's Theorem

[Markus Schulze]

Dear Craig,

you wrote (3 Jan 2005):
> Here is a brief description of the 1952 May 'theorem' I got from the
> Internet:
>
> | May's theorem: When choosing among only two options, there is only one
> | social decision rule that satisfies the requirements of anonymity,
> | neutrality , decisiveness and positive responsiveness, and it is the
> | majority rule.

May also presumed that the result depends only on whether the individual
voter strictly prefers candidate A to candidate B, strictly prefers
candidate B to candidate A or is indifferent between candidate A and
candidate B, but it must not depend on the ratings of the individual
voters for the different candidates (Kenneth O. May, "A Set of
Independent Necessary and Sufficient Conditions for Simple Majority
Decision," Econometrica, vol. 20, pp. 680--684, 1952).

However, Hylland proved that when there are only two candidates and
the used single-winner election method is strategyproof then the
result depends only on whether the individual voter strictly prefers
candidate A to candidate B, strictly prefers candidate B to candidate A
or is indifferent between candidate A and candidate B (Aanund Hylland,
"Strategy Proofness of Voting Procedures with Lotteries as Outcomes
and Infinite Sets of Strategies," University of Oslo, 1980).

Therefore, I interpret May's theorem in connection with Hylland's
theorem as follows:

   When there are only two candidates then the unique anonymous,
   neutral, decisive, and strategyproof single-winner election
   method is FPP. Therefore, every single-winner election method
   should satisfy the following criterion:

   When there are only two candidates and the number of voters who
   strictly prefer candidate A to candidate B is strictly larger
   than the number of voters who strictly prefer candidate B to
   candidate A, then candidate A should be elected with certainty.

Markus Schulze