Margins Example final part

[David Marsay]

The following is an important example that still awaits a response.

> From:          Mike Ositoff <ntk at netcom.com>
> Subject:       Margins Example final part
> Date:          Sat, 19 Sep 1998 16:44:55 -0700 (PDT)

I have changed it slightly as Mike's alignments don't work for me. I
hope I've interpreted correctly. If not, it is still an important
example. 

> 100 voters.
> 
> Sincere preferences:
> 
41 A(BC)
39 B(AC)
20 BCA
> 
(where braces denote a tie)
> 
> Actual votes (A voters insincerely order B & C):
> 
41 ACB
39 B(AC)
20 BCA

> 
> With Margins, A voters' strategy succeeds, & A wins.

I have previously posted: 
Subject:          Re: Margins Example Continued
Date sent:        Mon, 21 Sep 1998 11:24:03

Here I suggested that we should focus on spatial voting, since 
tactical voting problems are unavoidable in the general case.

Mike kindly replied off-line:
>All of my examples for Votes-Against vs Margins have been
>spatial examples, with a 1-dimensional policy-space or
>political spectrum.

For a spatial interpretation of Mike's example:
Options can be ordered with B or C in the middle, but not A (else CBA
is non-spatial). Then we suppose that the supporters of A and B are
blind to the space.

I propose a more spatial example, with B in the middle of A, C.

12 ABC
10 BAC
8   CBA

B is the Condorcet winner (CW).
Now, suppose those who prefer A actually vote A(BC). Then one has a 
cycle of pluralities. If one takes plurality as 'for-against' then 
the Condorcet tie-breaker discounts AB to give A as the winner. Note 
that C cannot defend against this. Thus it seems a 
sensible way for A to vote. The danger would be if C gained support, 
in which case C might win.

More generally, if there were one large extreme party and many minor 
parties, one might expect the large party's supporters not to 
declare their preferences.

I haven't understood the 'votes against' proposal yet. I propose a 
'votes-for' tie-breaker. Maybe its the same in the end, though.
I suggest that where there is a cycle, we discount the link that has 
the least votes-for support.

Suppose we have A,B,C as before.
Let a, b,c denote the number of voters who prefer A,B,C respectively.
Let ba, bc denote those who prefer b first and a (resp. c) second.
Let |XY| denote the pairwise support For 'X > Y'.

Then |BA| = b + c, and |CB| = c, so for b > 0, |BA| > |CA|.

Note that A's supporters do not contribute to these, and could only 
do so by voting C ahead of B. Hence |BA| will not be discounted by a 
'votes-for' tie-break.

I don't know if this brings any other problems. For now, I wonder how 
we might rationalise it.

Using 'margins' is equivalent to using a Young/Kemeny distance that 
has a tie (AB) half way between AB and BA. This fits Dodgson too.
The idea is to minimize the conflict between a voter's ranking and 
our overall ranking. 'Votes-for' seems to only take account of 
explicit rankings. Thus a tie is treated as 'do not mind' rather than 
'half way between the two.'

Is 'votes-against' really the same, or does it have a similar 
rationalisation?

Cheers.


--------------------------------------------------
Sorry folks, but apparently I have to do this. :-(
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