[EM] Majority winner set

[LAYTON Craig]

Mike wrote (in part):

>Take another look at my definition of sincere voting. It doesn't
>say that sincere voting must be nonstrategic. With a rank method
>the only sincere ballot is a sincere ranking of all the candidates.

Why do you have to rank all the candidates in order for it to be a sincere
ballot?  In fact, I think that the opposite could be true (see below).

>In Plurality, the only sincere ballot is one that votes for one's
>sincere favorite. But in Approval, I'm not sure what you'd want to call
>a sincere ballot.

I think that a sensible definition of sincere voting, encompassing all
voting methods, is as follows;

A voter votes sincerely where no candidate x is voted below another
candidate y, if the voter actually prefers x to y.  In situations where a
voter does not prefer one candidate to another, and is allowed by the method
to express the equality of those candidates on the ballot without
compromising the validity of that vote or preventing the voter for
expressing other sincere preferences, then the vote must show those
candidates as equal on the ballot.

This definition would mean that in a rank ballot with three candidates
A,B,C, and the voter preferring A to B and C, but not having a preference
between B and C, then voting A first, and not numbering the others, is the
only sincere vote possible.  If you add candidate D, whom the voter likes
less than all the other candidates, then there are five (or seven) sincere
ways of voting;
A1
A1,B2
A1,C2
A1,B2,C3 (with or without D4)
A1,C2,B3 (with or without D4)

In a cumulative voting situation, it would allow a voter to give two
candidates the same number of votes, even if that voter prefers one
candidate to another, but doesn't allow a voter to give a different number
of votes to two candidates liked the same.  I could extrapolate to the other
methods, but I won't bother at the moment.

The definition is broad enough to define a wide range of voting choices as
sincere, and also allows for a distinction between sincere strategic voting
and insincere strategic voting.  Whether this is a good thing or not, I
don't know, but I think the definition is intuitively correct, as well as
applicable to all methods & situations.