[EM] What kind of monotonicity whould we exspect from a PR method?

[fsimmons at pcc.edu]

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?