floor( vote / droop_quota ) is the lower quota, ceil( votes / droop_quota ) is the upper quota.

]]>So what is that algorithm? I mean, it could be Condorcet, and I have no problem with that, but I can't see how the term "proportional representation" applies. It just sounds like multi-winner.... there isn't anything proportional about it.

The only way the term "proportional representation" would apply (by my understanding of the term) is if we assume that there are some number of parties, and each candidate and each voter is in one and only one party. If a 3rd of the voters are in the Bull Moose party, then a third of those elected should be in the Bull Moose party. The further you get away from that, the less "proportional representation" seems to be a meaningful descriptor.

All of my complaints regarding PR (and with so many people's insistence that it is so much better than single winner methods such as Condorcet methods) are based on the assumption that voter X below is considered to have "better representation" if d, f and e are elected (because candidate d is very close to voter X), than if a, b and c are elected.

For the algorithm, you will be aware of Single Transferable Vote, which gives PR without any mention of parties. If voters happen to vote along party lines, it gives party PR, of course. That's just one example, without having to mention obscure methods invented by people on this forum.

And as for a, b, c versus d, e, f, I discussed that in the other thread here. I'm not sure it's worth quoting though because it's quite long.

]]>If the ballot set consists only of

C1>C2>…>CN

CN>C(N-1)>…>C1

which are perfect reversals of each other, then a system satisfies the compromise criterion if and only if it elects either C((N+1)/2) if N is odd, or one of C(N/2) or C(N/2+1) if N is even.

For example, IRV does not satisfy the compromise criterion: if the ballots are

x>y>z

z>y>x

then the compromise candidate y is immediately eliminated.

Bucklin voting does satisfy the compromise criterion. Borda count does not, but concave score systems do. Plurality fails. Generally Condorcet methods are agnostic on the compromise criterion, and whether they satisfy it or not depends on how situations without Condorcet winners are handled. For example, Black's system (Condorcet//Borda) fails. Approval voting “conditionally" passes or fails depending on “approval thresholds” (technically the criterion does not apply, since it is defined for rank order systems).

For three candidates we can generalize and strengthen the compromise criterion:

For every ballot set of that is a multiple of

xyz+Pyzx+(1-P)zyx

for any P in [0,1], the winner is y. In this case, a weak Condorcet method that prioritizes candidates with positive margins of victory over ties and that satisfies the weak compromise criterion will also satisfy this stronger compromise criterion.

]]>@rob I revisited this and I think Delegated Condorcet is a really interesting idea, what are your latest thoughts about that concept?

I haven't given it a lot of thought in the years since. It seemed like a good idea, and probably would work pretty well if adopted, but I doubt people would trust it enough to adopt it in the first place.

It might be best for local elections where there isn't a huge amount of publicity and the public tends to only know one or two candidates out of a larger field.

]]>Agreed that voters' ballots will likely change depending on the candidates in the race in Approval more with a higher probability than in most other voting rules.

@Toby-Pereira to answer your question I'm looking into it. Arrow's Theorem actually has quite a few (slightly different) formalizations, and it looks like what I said is technically not true for the version defined on Wikipedia since that one only allows (strict) linear orders, but I feel quite sure I saw a formalization where the domain was all (weak) linear orders. I will try to find it.

]]>In principle, this is an infinite recursion.

And hence, "hall of mirrors."

In practice, my iterative vote simulator stops when the ordering of candidates doesn't change between two rounds. Since the ordering is all the "vote caster" algorithm looks at, there is no point continuing because it will always get the same result.

But keep in mind, that doesn't mean it has reached the one and only equilibrium. There can be multiple equilibria.

But again, I'd rather have a method that doesn't encourage this sort of thing. If I did a vote simulator for Condorcet, there would be very little, if any, iterating, since there isn't an obvious way to adjust your ballot even if you know how others will vote.

]]>[edit]

Suppose we have the system I described above, with the rule of matching ballots and throwing them out in pairs.

Let's say the candidates are Gore, Bush, and Nader and I vote Nader, 511. Let's account for the votes in terms of their effects on the pairs of candidates. In fact, let's ignore Nader-Bush and concentrate on Nader-Gore. What is the effect of my vote on the Nader-Gore accumulator? It has to be canceled by a vote of Nader, -511.

]]>Suppose there is a way to interpret each ballot as assigning a number to each ordered pair of candidates, such that doing this preserves all the information the ballot provides that would be relevant to the electoral outcome.

Further suppose that without changing the outcome, we can rewrite the tally in such a way that its first step is to sum up for each ordered pair of candidates, the numbers given to that ordered pair by the ballots. The rest of the tally then depends only on those results, and reproduces the outcome that the tallying procedure originally described for the voting system would have produced, electing the same candidate, and reporting equally about how well or poorly the losing candidates did.

Then the voting system meets a constraint I am introducing here, of "pairwise additivity".

I suggest that any pairwise-additive system that also meets Frohnmayer balance suffices to defeat the absolute dictatorship of capital (in large elections).

I believe the following systems are among those that are pairwise additive and Frohnmayer balanced:

- Ranked Robin
- Score
- STAR
- Reverse STAR