[EM] SARC definition improvement

[MIKE OSSIPOFF]

>Could you please post an example showing Borda violates SARC?

I'll write the full example, but first let me outline it. You know
that it's possible for your favorite to not have any chance of winning,
and that there could be a possibility of a close tie between your
2nd choice and your last choice. So close that you could make your
favorite win instead of your last choice by insincerely voting your
2nd choice in 1st place. By so doing, you can make your 2nd choice win,
where your 1st choice can't win no matter what you do, with that
particular configuration of the other people's votes that we're
talking about. Any way of voting that doesn't have your 2nd choice
in 1st place and your last choice in last place will produce a worse
result, because you're only able to make your 2nd choice win by
giving a maximum vote difference between your 2nd & last choices.
I'm saying let's assume that we have a configuration of the other
people's votes for which that statement is true.

Of course it makes no difference how you re-arrange the candidates
between your 1st & last choice positions, since it's a close race only
between your 2nd & last choices.

Since all the ways of voting that have your 2nd choice in 1st place
and your last choice in last place give a better result than every
other way of voting, with this partiular configuration of the other
people's votes, none of those ways of voting can be dominated by
ways of voting that don't have your 2nd choice in 1st place and your
last choice in last place. Maybe I'd better give those ways of voting
a briefer name, and call them "the possible best".

But that doesn't mean that those possible best ways of voting can't
dominate eachother. If you could show that every one of them is
dominated by others of them, I couldn't call any one of them
undominated.

If you've found that they all dominate eachother, then make things easier by 
saying so, because the possibility hadn't occurred to me
before you mentioned it.

For any arangement for rank positions between 1st & last, if we
make a change in that order, I could find some configuration of the
other people's votes where that change would worsen the result for me.
Whichever one I lowered could barely have a win. Or whichever one I
raised could barely lose to someone whom I like better. For that reason,
I don't think you can say that every possible best way of voting
is dominated by another possible best way of voting.

So then, there are undominated ways of voting that rank your 2nd choice
in 1st place.

SARC says that voting an undominated strategy should never defeat
your favorite, who'd have won if you hadn't voted. There are undominated
strategies that rank your 2nd choice over your 1st choice. What if
you voted one of those strategies, and it turned out that your 1st
& 2nd choices were in a close tie? Then the fact that you showed up &
voted in that way would make your 1st choice lose.

Isn't every part of that correct? Then doesn't that mean that Borda
fails SARC?

Example:

Your sincere ranking is ABC

100 voters: ABC
99 voters: BAC

Without you, A wins.

Then you show lup, and, believing that B needs your vote against C,
you vote BAC. Now B wins. You've defeated your favorite by voting
an undefeated strategy.

Mike



A, B, & C are your 1st, 2nd, & last choices in a 3-candidate election.

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