Math Forum Discussions

JohnF

Posts: 219
Registered: 5/27/08

Voting systems
Posted: Sep 7, 2013 1:43 AM

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Some references for discussion/question below are
http://en.wikipedia.org/wiki/Voting_system
and
http://en.wikipedia.org/wiki/Voting_system#Mathematical_criteria
In particular, the "Jane Doe -3" under
http://en.wikipedia.org/wiki/Voting_system#Rated_voting_methods
More generally, just google voting theory or see
http://en.wikipedia.org/wiki/Category:Voting_theory
(where I thought the "Negative vote weight" might be relevant,
but it's about something else).

A lot's been written about voting theory, and I know none of it
beyond a quick google search, as suggested above. That didn't
uncover what occurred to me, although the #Rated_voting_methods
may in fact subsume it. But I couldn't quite see how to demonstrate
that one way or the other.

It's the slimey (what else?) candidate field in the current
nyc mayoral race that suggested this to me. Maybe you don't
care so much who (among the slime) wins, but there's one
particularly slimey candidate whom you really want to see lose.

So give each voter a choice: you can either mark your ballot
"pro" for the candidate you want to win, or "anti" for the one
you want to lose. If you vote pro then that guy gets +1 votes,
or if you vote anti then -1 votes. The guy with the highest
(or least negative if all voters hate every candidate) score wins.
An alternative voting scheme might allow each voter to mark
their ballot with both one pro and one anti vote.

For n=2 candidates there's obviously no difference; an anti vote
for one is equivalent to a pro vote for the other. But for n>2
this scheme is clearly different than the usual one: an anti vote
accomplishes something different vis-a-vis the election outcome
than a pro vote. But I can't quite figure out how to characterize
that difference mathematically. And that #Mathematical_criteria
discussion didn't seem quite rigorous enough to do it, either.

So (a) how would you formulate this scheme (or any scheme, for that
matter) and the usual scheme is such a way that they can be compared?
And (b) is the above scheme equivalent to, or subsumed by, the
#Rated_voting_methods scheme (and please prove your answer, if possible)?
--
John Forkosh ( mailto: j@f.com where j=john and f=forkosh )