[EM] Finding IRV UUCC example, contd.

[MIKE OSSIPOFF]

Because an inequality can be made into an equation by adding an
unspecified number, as when A>B is written as A=B+N, for some
unspecified positive N, and if these N are a lot bigger than one vote,
then it seems reasonable to expect that the values of the
variables can be changed enough to change B or Bc by one vote,
, while still satisfying the original inequalities,
if we wanted to. So if we get a solution of a system of equations
that gives B & Bc the same vote total, then we'd be able to
make enough change in the other variables to change B or Bc by
one vote, so that Bc has one more vote than B.

Here are my names for the relevant vote totals:

A: people at A position who rank A in 1st place.
Ac: people at A position who rank Bc in 1st place
B: people in B position who rank B in 1st place
Bc people in B position who rank Bc in 1st place.
C: people in C position

In this example outline, if there were just one candidate at the
B position, he'd be the middle Condorcet candidate, and the
smallest of the 3.

That means that everyone at the A position + everyone at the B
position > everyone at the C position. And everyone at the C
position + everyone at the B position > everyone at the A position.

So:

A+Ac+B+Bc > C
C+B+Bc > A

And, in this example outline, the B position is the least populated
of the 3 positions:

A+Ac>B+Bc
C>B+Bc

And, so that, when B gets eliminated and transfers to Bc, Bc
will have more votes than A:

B+Bc+Ac > A

Lastly, B & Bc are supposed to have equal vote totals:

Bc+Ac = B

These inequalities can be made into equations:

A+Ac = B+Bc+N1
C = B+Bc+N2
A+Ac+B+Bc = C+N3
C+B+Bc = A+Ac+N4
B+Bc+Ac = A+N5
Bc+Ac = B

Those are the equations that I'd solve to find the numbers for
the IRV UUCC badexample. 6 linear equations in 10 unknowns.

Since there are more equations than unknowns, and the equations
don't appear to contradict eachother, then doesn't that mean that
there are more than one solution to the system of equations?

I realize that there are systems of equations that have no solution,
like:

A+B = 2
A+B = 3

But these 6 don't seem to obviously contradict eachother.

Because there are lots of solutions, and because the N can be
much larger than 1 vote, it seems that if we had a solution where
Bc+Ac=B, we could change the other variables enough to make
Bc+Ac=B+1, and still satisfy the original inequalities.

When I can take the time, and when I have several big pieces of
paper, I'll try the solution of those 6 equations. But in the
meantime, it certainly seems that they have solutions, and
that there are IRV UUCC failure examples.

Again, I realize that there are mathematicians on this list who
would deal with this problem in a quicker way, but they don't always
have the inclination to do the same subjects that I'm inclined to do.
Anyway, Blake asked me for an example where IRV fails UUCC. I
intend to find one, but for now, haven't I shown that it looks as
if there are such examples?

Mike Ossipoff



There are


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