[EM] James--Your example shows that my claim wasn't correct

[James Green-Armytage]

James replying to Mike...

>You asked why I'd say that the supporters of some candidate A can't steal 
>the election for A by offensive order-reversal unless A is the sincere 
>Plurality winner. Good question. I said it because I was only considering 
>examples in which the CW, B, is between A and C, in the following sense:
>Everyone who prefers A to C prevers B to C. Everyone who prefers C to A 
>prefers B to A.
>If a CW is between the other 2 c andidates in a 3-candidate example, I
>call 
>that CW a "middle CW".
>So tha;t's why I made the claim that I made: I was only considering
>examples 
>in which the CW is a middle CW.
>My guarantee about A needeing to be sincere Plurality winner, in order
>for 
>the offensive order-reversal to succeed holds then. I don't know if it
>holds 
>in every spatial example. If so, that would be a good thing to find out. 
>Does anyone know?

	My earlier example could probably be conceived of as a spatial model in
three dimensions, but perhaps you would like one in two dimensions as
well. So, here goes. I'm not too familiar with doing explicitly spatial
examples, so please bear with me if it's a bit clunky.
	Imagine that there are 101 evenly spaced points, marked sequentially from
0 to 100. Thus, there are 100 intervals between the points. There are 100
voters, with one on each interval. All voters rank candidates who are
closer to them above those who are further away. The candidates themselves
are located on the following points:
A: 10
B: 20
C: 40
D: 50
E: 80
F: 84
	First choice votes:
A: 15
B: 15
C: 15
D: 20
E: 17
F: 18
	Candidate D is the middle CW, and also the plurality winner, with 20
first choice votes. Here are the key pairwise comparisons, according to
the sincere votes:

D>C: 55-45
D>E: 65-35
C>E: 60-40

	The 45 voters who prefer C to D can elect C by burying D under E, i.e. by
voting C>E>D instead of C>D>E. The pairwise comparisons would then be as
follows:

D>C: 55-45
E>D: 80-20
C>E: 60-40

	It's possible that I've made one or two small calculation errors in this,
but I know that the general answer to your question is: No, it doesn't
hold; a burying strategy can succeed in favor of a non-plurality-winner,
even when candidates and voters are arrayed along a 1 dimensional spectrum.

	I think that your statement might hold under the following conditions:
1. There are only 3 candidates.
2. The CW is exactly on the median point, not just closer to the median
than all other candidates.

	I may be wrong, of course... I only just started thinking about this when
I got your note.

Sincerely,
James
>
>