The motivation for BTR-score (Bottom-Two-Runoff with score) is that it is a Condorcet method that is easy to explain and count, while avoiding obvious problems like e.g. Copeland has.

1. Score candidates on a scale from 0 to 5.
2. Order the candidates from highest to lowest score.
3. Compare the lowest two candidates by pairwise preference. Eliminate the looser. Repeat.
4. Elect the last remaining candidate.

This always elects a candidate from the Smith-set. For cycles with 3 candidates, it always elects the score winner from within the cycle. In a 4-way cycle it becomes more complicated, but elects the score winner from within the cycle in 18 of 24 possible permutations. In the remaining 6 it elects the candidate with the second highest score from within the cycle. This means, in most (practically almost all?) cases it gives the same results as Smith//score.

Note that in describing the method there is no need to explain what a Condorcet winner or Smith-set is and no need for a pairwise matrix. It also reduces the inferential steps needed when you already explained STAR voting. This means, in terms of voting advocacy, you can explain score, then STAR as "score with runoff" and BRT-score as "STAR but for all candidates".

BTR stands for Bottom-Two-Runoff, like in BTR-IRV. I did an edit on the first post, fixed the typo.

With no need for a matrix, I mean we only have to compare a few pairs, not all possible pairings (N instead of N²). In practice may still be useful to use a runoff matrix "behind the scenes".

@jack-waugh
It might still be too complex for voters without prior education. As someone immersed in these discussions, it is easy to loose sight of how little people know about voting and how many conceptual steps people have to take before understanding the method. I've argued here that it might be more useful to promote a spectrum of compatible methods (approval, score, STAR and BTR-score in this case).

This idea is quite similar to MARS. And in the examples you list, MARS would yield the same results each time (assuming a range 0-3). The main difference is that it seems you imply strict rankings. This creates a teaming effect. The A voters fail to win strategically, because of the number of candidates supported by the other group, not because of a resistance to polarization inherent in the method. When you think about it, it's quite odd if a majority has no way to vote that ensures them winning (majority criterion).

If you randomly remove one ballot, then A wins in 3 of 6 possible cases, B wins 2 of 6, and one is a B-C tie.

I have no opinion on who should win. For me this is just a 3-way tie.

One could see it as ironic that experts in voting can't manage to agree on the best voting method. Then laugh and put the topic down.
But, we know that no voting method can force a majority if there is non in the electorate. The same is true for a Condorcet winner or finding a consensus. And going one step further back, maybe there isn't a single answer to that question.

I yesterday started to write two articles about the question, what makes a good voting system? Two articles because I arrived two answers.

The first one very practical.
Let's look at some obvious desirable criteria: satisfaction, fairness and simplicity (Jameson Quinn already wrote about that somewhere). In theory (if we had enough information) we could plot all voting systems on a three dimensional graph. But then the question is, which is the "best"? One might be very fair (resistant to strategy), but give medium satisfaction and is medium complex (e.g. MJ). While another gives good satisfaction and is simple, but easy to manipulate (e.g. score).
Instead of looking for the best one we could exclude every system which is worse than another in all three categories. This leaves us with a pareto front of several voting systems that are reasonably good.
Without having enough data, I guess this could look something like this (from simple to more complex):
approval, score, STAR or 3-2-1, Smith//score, Woodall's or Benhams' method

The second argument is more philosophical.
Since we know that no deterministic voting method can be strategy free we can ask our selves: How do we deal with strategy? While trying to map which voter strategies are possible I came to the conclusion that most fall in one of three categories: Dishonest preferences (favorite betrayal), Chicken Dilemma and Exaggeration.
The last of these is pretty much unavoidable. The other two can be avoided and I think it's important we do. The both represent a failure to cooperate. In terms of game theory, the first as prisoners dilemma, the second as the game of chicken. Failing to cooperate isn't only some obscure quirk in the niche science of voting theory, but actively harmful to society. If we could fully fix this, then this would be a huge advancement to humanity.
So I think, the voting reform movement currently tries to fix the spoiler effect and somewhat improve the selection of the winner. But when we have achieved this and the whole world is using some Condorcet or score method, then we will find that FB and CD are real problems we have to deal with. That this isn't utopia yet.
Finding methods that avoid both problems is hard, because solving CD in a strong sense implies having FB. It is however possible to solve CD in a weak sense.
These method, as far as I can see, achieve this:
ICT, MAV, MCA (some variants), MMPO, SIV
(The next step would be to filter out those that are subject to clones.)

As a bonus argument, one could say that we can avoid strategy altogether with non-deterministic methods. They also are more "fair" in the sense that they give every voter an equal chance of being heard. Therefor the best methods would be:
random ballot, random pair

Now with these arguments alone there are over a dozen methods that could qualify as "best", depending on the definition. So maybe there isn't really an answer to that question and which methods would be wise to use, depends very much on the situation.

@lime said in MARS: mixed absolute and relative score:

Hmm, I really like this idea in theory. Have you worked out what criteria it complies with?

I haven't looked closer into it, but because it mixes score and Condorcet, it's trivial to see that it also fails most criteria either of those fails. In the same way, it very likely satisfies criteria that are satisfied by both Condorcet and score.

I'm pretty sure it satisfies: monotonicity, independence of clones, reversal symmetry, equal vote criterion

@isocratia
Not exactly.
Article 38 of the GG (the constitution) says: "Members of the German Bundestag shall be elected in general, direct, free, equal and secret elections."
The important part is the "direct". Which is interpreted in the way that votes have to translate to outcomes in a transparent and unaltered way. e.g. no electoral college.
There was a ruling in a special case where non-monotonicity was the issue, but in this case the non-monotonicity came about through an unusual situation and was not expected. That means, the way the electoral law was intended and how it functioned where in conflict.
There was a recent report where they looked at the question if IRV would be constitutional and concluded that it would, making the argument that when non-monotonicity is expected and accepted, then it's okay (or something like that).

Most likely, yes. The ideas here are a theoretical exploration.

@cfrank
I was using "expected" as from the view of the law makers. If it was a known issues while the law was made, then it has been accepted as part of the law. If it turns out that a law exhibits the problem contrary to expectation, then it's a problem.

The report says in the summary (starting at page 135):

Die integrierte Stichwahl ist anfällig für einen paradoxen Sondereffekt, der Monotonieversagen genannt wird. Nicht ausgeschlossen ist nämlich, dass ein Wähler zwar für seinen bevorzugten Kandidaten stimmt, ihm aber noch mehr genützt hätte, wenn er ein ganz bestimmtes alternatives Stimmverhalten gezeigt hätte. Dieser Effekt muss sich anhand von Maßstäben beurteilen lassen, die das BVerfG für die Verfassungsmäßigkeit eines ähnlichen Paradoxes, nämlich des sog. negativen Stimmgewichts, aufgestellt hat. Dieses Paradox besteht darin, dass ein Zuwachs an Zweitstimmen für eine Partei dazu führen kann, dass dieselbe Partei gerade dadurch Mandate verliert. Das BVerfG hält dies für verfassungswidrig, es sei denn, der Effekt ist dem Wahlsystemtyp eigen und lässt sich deshalb nur durch dessen vollständigen Wechsel vermeiden. So verhält es sich hier: Der Gesetzgeber müsste auf die integrierte Stichwahl ganz verzichten, wenn er die Anfälligkeit für Monotonieversagen vermeiden wollte. Daher ist die integrierte Stichwahl mit den Maßstäben des BVerfG vereinbar.

Die integrierte Stichwahl ist auch mit dem Grundsatz der Unmittelbarkeit der Wahl vereinbar. Das BVerfG entnimmt ihm, für Wähler müsse vorhersehbar sein, wie sich die eigene Stimmabgabe auf den (Miss-)Erfolg der Kandidaten auswirken könne. Das ist bei integrierten Stichwahlen nicht in jeder Hinsicht der Fall, weil infolge der Anfälligkeit für Monotonieversagen für die Wähler nicht immer vorhersehbar ist, ob mit Blick auf das Endergebnis der Wahl paradoxerweise nicht ein ganz bestimmtes alternatives Stimmverhalten für den gewünschten Kandidaten günstiger wäre. Ein Verstoß gegen den Grundsatz der Unmittelbarkeit folgt daraus jedoch nicht, denn auch hier wirkt sich verfassungsrechtlich aus, dass die Anfälligkeit für Monotonieversagen innerhalb des Wahlsystemtyps nicht zu vermeiden ist.

In English (I highlighted the important part):

The instant runoff election is susceptible to a paradoxical special effect known as monotonicity failure. It is possible that a voter, even though they vote for their preferred candidate, could have actually benefited their candidate more by choosing a completely different voting behavior. This effect must be assessed using the criteria established by the Federal Constitutional Court (BVerfG) for the constitutionality of a similar paradox, namely the so-called negative voting weight ["negatives Stimmgewicht"]. This paradox occurs when an increase in second votes for a party can lead to that party losing seats as a result. The BVerfG considers this to be unconstitutional unless the effect is inherent to the type of electoral system and can therefore only be avoided by completely changing the system. This is the case here: the legislature would have to abandon the integrated runoff election entirely if it wanted to avoid susceptibility to monotonicity failure. Therefore, the integrated runoff election is compatible with the standards of the BVerfG.

The instant runoff election is also compatible with the principle of directness in elections. The Federal Constitutional Court (BVerfG) interprets this principle to mean that voters must be able to foresee how their vote will affect the (success or failure) of the candidates. This is not always the case with integrated runoff elections because, due to the susceptibility to monotonicity failure, it is not always predictable for voters whether, paradoxically, a completely different voting behavior might be more favorable for the desired candidate with regard to the final election result. However, this does not constitute a violation of the principle of directness, as the susceptibility to monotonicity failure cannot be avoided within this type of electoral system.

I'm neither defending this view, nor IRV. But apparently, according to this report, IRV is constitutional. The GG intentionally puts almost no constraints on the voting system and therefore does not go into the weeds of voting theory.

@isocratia
Good question. According to Electowiki:

Methods that pass dominant mutual third burial resistance [DMTBR] provide no incentive to bury under a dark horse.

Following the argument here it is clear that BTR-score fails DMTBR

In election (A) and (B) A wins. In (D) the winner is C, but in (E) the winner is A again. (I assumed score from 0 to 2.)

where A is the CW, and so (E)->(D) is a DMTBR failure.

But this does not exclude the possibilty that the score part reduces the strategic incentive to bury.

@ex-dente-leonem
I wanted to create some figures for an article on MARS voting that I plan to write, but fail to run vse-sim. Is it okay for you when use your results?

@cfrank
I was using "expected" as from the view of the law makers. If it was a known issues while the law was made, then it has been accepted as part of the law. If it turns out that a law exhibits the problem contrary to expectation, then it's a problem.

The report says in the summary (starting at page 135):

Die integrierte Stichwahl ist anfällig für einen paradoxen Sondereffekt, der Monotonieversagen genannt wird. Nicht ausgeschlossen ist nämlich, dass ein Wähler zwar für seinen bevorzugten Kandidaten stimmt, ihm aber noch mehr genützt hätte, wenn er ein ganz bestimmtes alternatives Stimmverhalten gezeigt hätte. Dieser Effekt muss sich anhand von Maßstäben beurteilen lassen, die das BVerfG für die Verfassungsmäßigkeit eines ähnlichen Paradoxes, nämlich des sog. negativen Stimmgewichts, aufgestellt hat. Dieses Paradox besteht darin, dass ein Zuwachs an Zweitstimmen für eine Partei dazu führen kann, dass dieselbe Partei gerade dadurch Mandate verliert. Das BVerfG hält dies für verfassungswidrig, es sei denn, der Effekt ist dem Wahlsystemtyp eigen und lässt sich deshalb nur durch dessen vollständigen Wechsel vermeiden. So verhält es sich hier: Der Gesetzgeber müsste auf die integrierte Stichwahl ganz verzichten, wenn er die Anfälligkeit für Monotonieversagen vermeiden wollte. Daher ist die integrierte Stichwahl mit den Maßstäben des BVerfG vereinbar.

Die integrierte Stichwahl ist auch mit dem Grundsatz der Unmittelbarkeit der Wahl vereinbar. Das BVerfG entnimmt ihm, für Wähler müsse vorhersehbar sein, wie sich die eigene Stimmabgabe auf den (Miss-)Erfolg der Kandidaten auswirken könne. Das ist bei integrierten Stichwahlen nicht in jeder Hinsicht der Fall, weil infolge der Anfälligkeit für Monotonieversagen für die Wähler nicht immer vorhersehbar ist, ob mit Blick auf das Endergebnis der Wahl paradoxerweise nicht ein ganz bestimmtes alternatives Stimmverhalten für den gewünschten Kandidaten günstiger wäre. Ein Verstoß gegen den Grundsatz der Unmittelbarkeit folgt daraus jedoch nicht, denn auch hier wirkt sich verfassungsrechtlich aus, dass die Anfälligkeit für Monotonieversagen innerhalb des Wahlsystemtyps nicht zu vermeiden ist.

In English (I highlighted the important part):

The instant runoff election is susceptible to a paradoxical special effect known as monotonicity failure. It is possible that a voter, even though they vote for their preferred candidate, could have actually benefited their candidate more by choosing a completely different voting behavior. This effect must be assessed using the criteria established by the Federal Constitutional Court (BVerfG) for the constitutionality of a similar paradox, namely the so-called negative voting weight ["negatives Stimmgewicht"]. This paradox occurs when an increase in second votes for a party can lead to that party losing seats as a result. The BVerfG considers this to be unconstitutional unless the effect is inherent to the type of electoral system and can therefore only be avoided by completely changing the system. This is the case here: the legislature would have to abandon the integrated runoff election entirely if it wanted to avoid susceptibility to monotonicity failure. Therefore, the integrated runoff election is compatible with the standards of the BVerfG.

The instant runoff election is also compatible with the principle of directness in elections. The Federal Constitutional Court (BVerfG) interprets this principle to mean that voters must be able to foresee how their vote will affect the (success or failure) of the candidates. This is not always the case with integrated runoff elections because, due to the susceptibility to monotonicity failure, it is not always predictable for voters whether, paradoxically, a completely different voting behavior might be more favorable for the desired candidate with regard to the final election result. However, this does not constitute a violation of the principle of directness, as the susceptibility to monotonicity failure cannot be avoided within this type of electoral system.

I'm neither defending this view, nor IRV. But apparently, according to this report, IRV is constitutional. The GG intentionally puts almost no constraints on the voting system and therefore does not go into the weeds of voting theory.

@isocratia
Not exactly.
Article 38 of the GG (the constitution) says: "Members of the German Bundestag shall be elected in general, direct, free, equal and secret elections."
The important part is the "direct". Which is interpreted in the way that votes have to translate to outcomes in a transparent and unaltered way. e.g. no electoral college.
There was a ruling in a special case where non-monotonicity was the issue, but in this case the non-monotonicity came about through an unusual situation and was not expected. That means, the way the electoral law was intended and how it functioned where in conflict.
There was a recent report where they looked at the question if IRV would be constitutional and concluded that it would, making the argument that when non-monotonicity is expected and accepted, then it's okay (or something like that).

@jack-waugh
For the purpose of the article it was more than needed. The requirement of allowing equal ratings/ranks already covers many cases. I'm unsure if it is really "too restrictive", but I also can't tell at the moment if it is necessary as a minimum requirement.

@GregW
That's good. You may even remove the "Hiveism substack" in the text and just keep the foot note if this makes it more readable.

@ex-dente-leonem said in ABC voting and BTR-Score are the single best methods by VSE I've ever seen.:

Have I got great news for you...

Thank you very much. That's even better than I expected.

@cfrank
That's the issue that prompted me to think about related methods like BTR-score. Later however, I came across a good argument that convinced me this isn't so much of a problem. STAR becomes like score with strategic nomination, but that is the expected behavior. The main election is the score part, the runoff is there as a safe-guard and to make it compliant with laws that require a majority. Basically, the difference between score and STAR isn't that big, but practically it's easier to get STAR implemented. In the VSE simulation by John Huang, they even perform the same under "honest" and "two-sided strategic" assumptions.

I have to admit that I'm confused by the results. I did expect BTR-score to perform well, but I also thought that the difference to Smith//score would be to small to measure. I wonder why there is such a big difference.

Also, when looking at viability-aware, STAR performs a lot better, which is also surprising. Is it that it better utilizes the score part and therefor gets the "best of both worlds"? I don't know.

How long does it take for you to run those simulations?

If it's not too much work, could you try out MARS? It's basically like BTR-score, but each round does not only compare votes, but votes+scores (both measured in % of maximum possible). Previously I thought that that would be even better than BTR-score (on the cost of being complicated), but now I'm no longer sure.

One thing to be cautious of is that these results present very small differences in a high range. It might be that the choice of model has too much of an influence to make decisive claims.

ABC is a good idea and I think could be really a viable method when fleshed out. Most importantly I think there may be more intuitive ways to preset it. Maybe something in between scores and approval, like "ABC-123" where letter ratings count as 0 and numbers as scores.

@jack-waugh
Balance is one feature that the four methods share, but as a general minimal requirement for me to support a voting system it's to restrictive. It might exclude methods that are reasonable. On the other hand, it does even include some that are not so good in my opinion. A for-or-against variant of plurality, where you can either vote for only one, or against only one candidate, would still pass balance, but would be hardly better than plain plurality.
There has been a quite elaborate discussion about this on /r/endfptp which convinced me to remove that paragraph.

@cfrank
Thanks for the feedback. I will add that to the article.
What is your reason against STAR?

@jack-waugh
It might still be too complex for voters without prior education. As someone immersed in these discussions, it is easy to loose sight of how little people know about voting and how many conceptual steps people have to take before understanding the method. I've argued here that it might be more useful to promote a spectrum of compatible methods (approval, score, STAR and BTR-score in this case).