[EM] Fair majority voting, scored-output methods

[Kristofer Munsterhjelm]

I was thinking about fair majority voting and how it might be 
generalized to other election methods.

FMV (as I understand it) works like this: To elect the members of a 
council proportionately, you first do a statewide Webster (or similar) 
method to determine how many members should come from each party. Then 
you give each party a weight, so that the number of voters who voted for 
a candidate from some party p in a district is multiplied by this 
weight, call it w_p. Adjust the weight so that the statewide composition 
of the council is proportional.

(This is biproportional representation with single-member districts)

Then I got to thinking: this should work with any voting method that 
returns a candidate quality score, as long as it passes a certain scale 
invariance criterion. Plurality is just a particularly simple such 
method, where the candidate quality is the number of first preferences 
that candidate got.

The scale invariance property is: if you double the number of voters in 
the election without altering the proportions of voters voting for each 
ballot, then the ratios between scores should remain unchanged. More 
generally, multiplying the number of voters by some alpha > 0 should 
keep the ratios unchanged.

Then if the score of candidate c in district d is S(c,d), the adjusted 
score is just w_p * S(c, d).

The point of the scale invariance property is that a popular candidate 
shouldn't be helped (or harmed) by running in a district with lots of 
voters; he shouldn't move closer to (or further apart from) the other 
candidates just by virtue of a population increase.

In addition, we also need a nonnegativity constraint because multiplying 
a negative score with w_p < 1 would push it closer to zero, i.e. improve 
the candidate's standing, and we want a lower weight to make it harder 
to win, not easier.

So what methods pass this? Obviously Range (with a nonnegative scale) 
and Approval both do. MMPO also does because doubling the votes can't 
change which contest Y>X is binding for X; all it does is double its 
magnitude. Minmax(margins) should pass scale invariance, because if

S(c) = min over other candidates X: C>X - X>C
	= C>X_min - X_min>C

then doubling it just doubles both C>X_min and X_min>C, and this 
constant factor cancels out when considering S(c)/S(d). But it doesn't 
pass nonnegativity - everybody but the CW has a negative score. So let's 
try rescaling to [0 ... number of voters]:

S(c) = min over other candidates: X: (C>X - X>C + |V|)/2.

If C loses against everybody by an unanimity, then X>C = |V|, C>X = 0 
and the scaled value is 0. If C beats everybody by a unanimity, then C>X 
= |V|, X>C = 0, and the scaled value is |V|. Does this pass the ratio test?

Let S(c) = (m_1 + |V|)/2 and S(d) = (m_2 + |V|)/2 where m_1 and m_2 are 
the minmax margins values. Hence S(c)/S(d) = (m_1 + |V|)/(m_2 + |V|). 
Multiplying by alpha will multiply the margins variables m_1 and m_2 by 
alpha, and |V| also by alpha. So yes.

Copeland passes (since the number of candidates a given candidate beats 
doesn't depend on |V| at all), but it has weird consequences: it would 
be easier to lose in a district with lots of contenders than one with a few.

fpA-fpC? Well, sum fpA-fpC has the same candidate-dependence problem, so 
how about
	S(c) = fpC - max over candidates X beating C pairwise: fpX
? It fails non-negativity. Does the same trick as for minmax-margins 
work? The max is when C has unanimity, i.e. S(c) = |V|. The minimum is 
when someone else has unanimity: S(c) = -|V|. So strictly speaking, yes:
	S(c) = (|V| + fpC - max...)/2
by same pattern as minmax (margins).

First preference Copeland?
	S(c) = - sum over candidates X beating C pairwise: fpX
The maximum is 0, the minimum is -|V|. So
	S(c) = |V| - sum over candidates X beating C pairwise: fpX
should work.

There is a problem with minmax(margins): Condorcet losers can never win, 
as their score is always zero. So a FMV mechanism based on this may have 
a problem in a two-party setting with 51% in favor of one party, 49% in 
favor of the other, and every district a bare majority. The fpA-fpC 
situation is a bit more complex, but perhaps there's some kind of "bad" 
nonlinearity there too?

In any case, I thought it'd be interesting. I was inspired by Filip 
Ejlak's comment that he likes first preference Copeland because it 
provides a measure not just of who wins, but by how much.

-km