It seems that if a PR method chose slate {X, Y} for a two winner election, and
only X or Y received
increased support in the rankings or ratings, then {X, Y} should still be
chosen by the method.
But consider the following approval profile (for a two winner election):
3 X
1 XY
2 Y
2 Z
It seems pretty clear that the slate {X, Y} should be elected, and that is the
PAV decision.
Now suppose that X gets additional approval on some ballots but the Y and Z
approvals stay the same:
2 X
3 XY
2 Z
Now PAV elects {X, Z}, and this seems like the right choice, because this slate
completely covers the
electorate, unlike any other pair. Candidate Y has more approvals than Z, but
everybody that approves
Y also approves X, so given that X is part of the slate, Y would only
contribute half a satisfaction point
per ballot, while Z adds a full point per ballot. Since 2>1.5, Z wins over Y
for the remaining position on
the slate.
This violates the strong monotonicity ideal of the first paragraph, but does
not violate a weaker version
that says if only one candidate X of the winning slate gets additional support
on some ballots (and all
other candidates have the same or less support as before on all ballots) then
that one candidate X
should be a part of the new slate.
Now let's look at this example from the point of view of the Ultimate Lottery:
In the before scenario, the Ultimate Lottery probabilities x, y, and z for the
respective candidates X, Y,
and Z are obtained by maximizing the product
x^3*(x+y)*y^2*z^2 subject to the constraint x+y+z=1
The solution is exactly (x,y,z)=(45%, 30%, 25%) .
After the increase in support for x the Ultimate Lottery probabilities are
obtained by maximizing the
product
x^3*(x+y)^3*z^3 subject to the same constraint x+y+z=1.
The solution is precisely (x, y, z) = (75%, 0, 25%) .
Note that (in keeping with the strong ideal expressed at the beginning of this
message) the only
candidate to increase in probability was X, the one that received increased
support. It did so at the
expense of Y whose probability decreased to zero. So Z passed up Y without any
change in its
probability. That's basically why Z took Y's place on the slate without any
increased support on the
ballots.
So this helps us understand (in the PR election) why the weaker member of the
two winner slate
changes from Y to Z, and why we cannot expect the strong monotone property for
a finite winner PR
election; the discretization in going from the ideal proportion of the Ultimate
Lottery to a finite slate
allows only a crude approximation to the ideal proportion.
In other words, it is just one of the classical apportionment problems in
disguise.
How do other PR methods stack up with regard to monotonicity?
Since IRV is non-montone, automatically STV fails even the weak montonicity
sartisfied by PAV.
How about the other common methods?
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