[EM] A "top 3" to replace the "top 2"

[Kristofer Munsterhjelm]

On 09/22/2014 11:56 PM, Dick Burkhart wrote:
> It is well known, of course, that in certain situations Borda will
> not give good results if one side votes strategically while the other
> side does not. This would not happen in the normal course of partisan
> politics.
>
> Take your example:
>
> 55%: A>B>C
> 35%: B>C>A
> 10%: C>B>A
>
> If B and C represent moderate and right wing Republicans, while A
> represents the Democrats, then the Republican parties are voting
> strategically by listing the Democratic candidate last. However the
> Democrats are ignoring the strength of their principle foe, the
> moderate Republicans, by listing their candidate second instead of
> last.

I didn't intend the example to be strategic, and there doesn't have to 
be any strategy involved in it. If we say that the Democrat is at 
position 0.1 on the axis, the moderate Republican is at position 0.6 on 
the axis, and the right-wing Republican is at position 0.8 on the axis, 
then all we need is:

55% of the voters near 0 on the axis,
35% of the voters near 0.5 on the axis
10% of the voters near 0.9 on the axis.

This gives:
	55% of the voters rank 0.1 first, then 0.5, then 0.9, i.e.
	A>B>C;
	35% of the voters rank 0.6 first (distance 0.1), then 0.8 (distance 
0.3), then 0.1 (distance 0.4), i.e. B>C>A;
	10% of the voters rank 0.8 first (distance 0.1), then 0.6 (distance 
0.3), then 0.1 (distance 0.8), i.e. C>B>A.

Now, you might object to that these voters are pinpoint distributed in 
the manner above, but you could probably modify the model so that the 
voters are represented by Gaussians of the right size centered on the 
right points. Then you could set up a story somewhat like, say, there's 
a polarized electorate around 0 and 0.6, their centers are near the 
positions of the Democratic and Republican candidates, but a Republican 
that lost the primary decided to run on his own. The numbers would be 
closer to even, but I'm pretty sure you could still get a majority 
failure that way. I just don't feel like doing all that work.

The thing is, Borda's decision is dragged too far to the right by the 
presence of C, because it pads B's score. In general, Borda is biased 
towards where more candidates are concentrated. As I understand your fix 
for greater than three candidates, you basically acknowledge that 
problem when there are lots of candidates. But in order to fix it 
completely, you'd have to do something about the n=3 case too.

> Actually, if the Dems do a little math, they would know that they
> could win easily by instructing their followers to randomly rank the
> B and C candidates second (2 points) or third (1 point), so that on
> the average they get 1.5 points.
> Then the points for the Democrats are proportional to 0.55 * 3.0 +
> 0.35 * 1 + 0.10 * 1 = 1.650 + 0.450 = 2.100 And for the moderate
> Republicans we get                        0.55 * 1.5 + 0.35 * 3 +
> 0.10 * 2 = 0.825 + 1.250 = 2.075 And for the right wing Republicans
> we get                        0.55 * 1.5 + 0.35 * 2 + 0.10 * 3 =
> 0.825 + 1.000 = 1.825
>
> So A wins.  More generally, we can devise a formula for splitting the
> Dems vote between B an C to guarantee that the Dems will always win
> in this kind of situation:
>
> frac/pts  A   B    C x               3  p   3-p y               1   3
> 2 z                1  2    3

Of course, with Borda, a majority can *force* their winner given enough 
coordination and polling data, just as they can in, say, Range. However, 
with Nanson or Baldwin you can have that *without* coordination.

Why should the voters need to use defensive strategy any more than they 
have to? Why should a majority have to coordinate when there are 
perfectly good methods where the majority wins outright? Just use a 
system with Majority support and the majority no longer has to 
coordinate to win. Or use a system with Mutual Majority (implied by 
Smith, which both Nanson and Baldwin pass) to have an even stronger 
guarantee: if a majority of the voters rank a set of candidates first, 
in no particular order,  the winner will come from that set. And all you 
have to do is take your elimination two more steps and remove your 
fractional redistribution fix.