[EM] Majority winner set

[Markus Schulze]

Dear Craig, dear Martin, dear Mike,

Craig wrote (1 Dec 2000):
> Markus wrote (30 Nov 2000):
> > I suggest the following definition of sincere voting in Approval Voting:
> >
> >   A voter votes "sincerely" when he approves all those candidates
> >   he prefers to the incumbent and disapproves all those candidates
> >   to which he prefers the incumbent.
>
> Ahh.  I was halfway into writing a message against this when I realised that
> it's actually quite a good definition.  The incumbent represents a perceived
> utility value of zero (no change) while the other candidates are above or
> below that value.  I'm assuming that whether or not he votes for the
> incumbent does not effect the sincerity of the vote (?).

I agree.

Craig wrote (1 Dec 2000):
> Does this mean, however, that if the incumbent is no longer running, and
> the incumbent is more liked by a voter than any of the candidates (00 pres
> election?), then that voter cannot cast a sincere vote?

Even when the incumbent doesn't run for re-election the vNM utility of the
incumbent is a good guess for what a given voter can expect from the new
winner. When the incumbent doesn't run for re-election and a given voter
strictly prefers the incumbent to every running candidate, then -to my
opinion- this voter disapproves all candidates when he votes sincerely.

Martin wrote (1 Dec 2000):
> Is there not a difference between voting insincerely and voting
> strategically? For example, if I like Alice and Bob equally in a
> plurality system, then a cross-mark for Alice, and a cross-mark for
> Bob are both sincere votes. Suppose I use a dice to initially decide,
> and pick Alice. then I get info from polls and discover that the race
> is between Bob and Charlie, and change my vote. This, by Mr. Schulze's
> requirement, is insincere.
>
> But if in the same case the dice happens to pick Bob, then presumably
> this would be a sincere vote? But to have the sincerity of my vote
> effectively decided by a dice seems, well... odd at least. Don't like it.

To my opinion, "sincerity" must be defined in such a manner that there is
a unique (but not necessarily deterministic) way of voting "sincerely" for
a voter with a given opinion.

Example:

   When FPP is used then the unique way of voting sincerely is to
   make a cross-mark for that candidate who is the favorite candidate
   due to this voter's sincere opinion.

   Of course it is possible that a given voter has no unique favorite
   candidate. In this case he will randomly or arbitrarily decide which
   of his favorite candidates gets his cross-mark. But who gets his
   cross-mark must not change when this given voter gets additional
   information about the voting behaviour of the other voters.

Mike wrote (24 Nov 2000):
> A voter votes sincerely if he doesn't vote a preference
> that isn't a sincere preference or leave unvoted a sincere
> preference that the balloting system would have allowed him
> to vote in addition to the preferences that he actually did
> vote.

To my opinion, this is only a necessary but not a sufficient condition
for "sincerity." To my opinion, it is also necessary that the voting
behaviour of a "sincere" voter does not depend on his information
about the voting behaviour of the other voters.

Mike wrote (1 Dec 2000):
> Markus wrote (30 Nov 2000):
> > I doubt that those who don't promote Approval Voting
> > will agree to your definition of "sincerity."
>
> I'm reasonably sure that Brams & Fishburn have said, in their book
> _Approval Voting_, that, for Approval, they define sincere voting
> as voting without reversing a preferences, without falsifying a
> preference.

Brams and Fishburn are well known supporters of Approval Voting.

Mike wrote (1 Dec 2000):
> Their definition means the same thing that my definition means when
> it's applied to Approval. My definition agrees with how Brams &
> Fishburn would define sincere voting for Approval, and it also
> agrees with how we'd all define sincere voting with rank balloting
> and Plurality.

When Approval Voting is used then the optimal strategy looks as
follows:

1. Approve all those candidate you prefer to the expected winner.
2. Disapprove all those candidate to which you prefer the expected
   winner.
3. Approve the expected winner when rather a less preferred than
   a more preferred candidate is elected. Otherwise disapprove him.

Due to Mike's definition of "sincerity," a voter who uses this
strategy votes "sincerely." To my opinion, this voter votes
"insincerely" at least in those cases where he would have voted
in a different manner when he had no information about the voting
behaviour of the oher voters.

Markus Schulze