[EM] Borda and Centrist Dominance

[Alex Small]

It has been observed that if candidates and voters adhere to a 1D
political spectrum then Condorcet grants the center a near-monopoly
(unless the left or right commands an absolute majority of the voters). 
It has also been opined that monopoly of any sort is undesirable in the
market, whether it's a marketplace of ideas or any other sort of
marketplace.  Finally, it has been observed that in IRV the left and right
can position themselves just close enough to center that the center is
permanently excluded (having the fewest first-place votes).

Well, this has sparked some controversy on the list.  Let's try to heal
wounds by ganging up on a whipping boy we can all hate:  Borda!

In Borda, it is easy to show that if the center is the CW (which will
always be the case unless left or right has an absolute majority) then the
center always wins.  OK, so far so good.

But, even if left or right has an absolute majority, there are still
circumstances where center wins.  I think we can all agree that this is
bad.

Say we have a left-right spectrum, so the voters are of the types:

A>B>C, C>B>A, B>A>C, B>C>A where B is the center.

Votes for A:  2*ABC+BAC
Votes for C:  2*CBA+BCA
Votes for B:  2*(BAC+BCA)+ABC+CBA

Compare A with B.  B wins if:

2*(BAC+BCA)+ABC+CBA > 2*ABC+BAC

Subtract like terms to get

BCA + (BCA+BAC+CBA) > ABC

Restate this as

BCA + #of people for whom B is favorite > #of people for whom
                                                A is favorite

Even if A has an absolute majority of first place votes, B can still beat A

e.g.

52 A>B>C                   A gets 110 votes
 6 B>A>C                   B gets 126 votes
20 B>C>A                   C gets  64 votes
22 C>B>A

So Borda would make the centrist monopoly even more iron-clad, even in
cases where the centrist most definitely does NOT deserve to win.

Just food for thought.



Alex