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.

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).

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.

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.

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).

The problem is that, even with cardinal methods, you can't escape the Condorcet paradox with all the problems it implies.

Say you have three candidates and nine voters with the following preferences:

3: A > B > C
3: B > C > A
3: C > A > B

Then when you consider for each group of voters the best strategy under approval voting and repeat that for some iterations always based on the previous poll, the winner will turn in cycles.

Yes, but by including everyone approved by a majority, this is already the case. If in the second round people narrow their support, that's the point of the second round.
The wording "the top two most approved candidates and anyone approved by a majority" ensures that there is always a majority. Either in the first round or the second. If there is no majority in the first round then the top two advance and it is like a regular runoff.

(It would be possible that a enough people submit an empty ballot in the second round and thereby cause both candidates to fail the majority, but one could count them as invalid - just like in every other runoff.)

When holding an approval vote, which for some (legal) reason requires a runoff. Should it include all candidates approved by a majority of voters?

I started thinking about this as I tried to simplify score interval voting (SIV). The idea behind SIV is to identify possible cases of the chicken dilemma and have those candidates only compete in a second round while excluding the other. By the nature of the chicken dilemma, in most cases there is a majority which prefers some set of candidates to everyone else but is divided about who of those candidates should win. Therefor, it should be possible to avoid the CD (in most cases) if we just take all candidates approved by a majority and advance them to the second round. This isn't a complete fix but should be good enough as a practical approximation.

With this template one could think of several voting methods.

majority approval, with approval runoff:
Approval voting in the first round. The top two candidates and everyone approved by a majority of voters advances to the second round of approval voting.

majority approval // score:
Using a score ballot. Every score above 0 is considered an approval. The top two candidates and everyone approved by a majority of voters advances to the second round. The second round is using the same ballot data for score voting.

majority score // BTR-score:
Like BTR-score, order the candidates by score and have a bottom-two-runoff until one candidate remains. But limit the set of candidates to all which have more than 50% of the possible score.

The idea is related to graded Bucklin methods.

I think the first one is especially useful because of its simplicity. Some places already require an obligatory second round (primary and general in the US). In such cases it's an easy way to address the most common criticisms of approval voting - that you have to think to much about strategy in order to vote. It also addresses the question why you have a runoff after an approval vote with two potentially very similar candidates.

@jack-waugh said in MARS: mixed absolute and relative score:

Your first example supposes that voters leave power on the table. We're not going to see that in a political election that matters.

Then again in your first example for MARS, you have 35 voters giving scores 4 4 0 when the top of the range is 5.

There are many voters who don't vote at all. Some intentionally because they don't like any of the alternatives. Likewise I think it's very likely that some voters will only use the lower part of the range, because they don't like any of the alternatives.
However, this method and the examples don't depend on that aspect.

The same example using the full range:

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.

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

Is there any reason you think blending Condorcet and "utility based" will help reduce strategic incentives?

It was my initial intuition, that just like in STAR, the incentives to min-maxing and expressing dishonest ranking (e.g. burying) would conflict and partly cancel out.

Strategy under Condorcet involves creating or breaking a cycle. In MARS you would have to create a cycle in combined votes and score. But because score is linear it's harder to do this. And because the two metrics are correlated it's hard to burry A under B, without giving B a higher score which might allow them to beat your preferred candidate C.

So there are two cases:

1. You try to predict the outcome and pick the best strategy. This requires a good prediction and using only one strategy might not be enough to shift the result.

2. You employ both strategies and accept the higher risk that either of them backfires (or both).

Overall I think the cases where a strategy works are rarer, harder to predict and more likely to backfire. While you have more options compared to either score or Condorcet, I think in sum it's more robust against strategies.

However, I have no evidence beside reasoning. It would be great to see it tested in some VSE simulation.

Edit:
I just realized there is a simpler argument to be made. If you min-max like in score, you miss out in using preferences. If you express a dishonest order, you miss out in using scores. Either way, using one metric to vote strategically prevents you from using the other to advance your goal. A strategy is only helpful if you know that you won't need the other metric, which again is rare, hard to predict, hard to coordinate and risky.

As you all know Germany uses MMP for elections to the Bundestag (national parliament).
There currently is a commission for electoral reform reviewing changes to the system. In a recent paper they propose a fundamental shift in how the MMP system works and consider approval voting and IRV as possible methods for district mandates.

As background, the current system is a mixed member proportional system. There is nothing in the constitution demanding any particular system, but this was invented in 1949 and since only changed slightly.
Voters have two votes. One for a candidate in their district and one for a party. The party vote determines the proportion of parties in the parliament. There is a 5% threshold, so it's not entirely proportional and many votes are just ignored. Proportionality is calculated both at the level of the Länder (=states) and nation wide, but I spare you the details. (The electoral law has been called "the law that makes people crazy" by an expert in a previous commission.)

A main problem with this is that a party can win more districts than they have seats to fill by the proportional vote. The solution, until now, was to increase the total size of the parliament until this isn't an issue any more. However, with this there is no limit to parliament size. In the last election there where 736 seats instead of the regular 598. Which makes the Bundestag the second largest national parliament after China. This for one costs a lot of money, it reaches the limits of what the parliament building can physically support and it slows down the work.

The commissions task is to find a solution to this problem. It also discusses lowering the voting age and ways to better represent women.

Last week they voted on a paper outlining the direction for the new voting system (pdf, German) ("Eckpunktepapier" lit.: corner point paper).

There will be exactly 598 seats and 299 districts. Voters still have two votes. The party vote determines the number of seats for each party. When a party has won more districts than seats, they only send the candidates with the best results, the leftovers don't get a seat.
E.g. Party A has won 35 districts, but is only entitled to 30 seats, therefor the 5 worst winners aren't elected.

Then the question is what happens to those left over districts. They shouldn't be vacant, because then the voters aren't directly represented and also have less of a vote than everyone else. So they list four alternatives:
a) A "replacement" vote. Voters get to mark a secondary preference, but it is only counted when the candidate with most votes isn't covered by the party vote.
b) Keep plurality voting and elect the second best candidate.
c) Implement Approval voting and elect the second best candidate.
d) Implement IRV and elect the second best candidate.

My comment on this proposal:
I would have liked to see multi member districts, but was aware that the chance for is was very small. The new rule solves the main problem but does nothing to improve the voting system or address the many other problems it has.

a) This is confusing to voters and mostly useless. In the last election, it would only have been relevant in 38 of 299 districts. So why should voters care to add another mark on the ballot. They already don't care about who wins the district.
b) Plurality voting is bad, but in this context it's even worse. You not only don't elect the candidate with the most votes, but one with no intersecting set of votes, i.e. the opponent. A right leaning district would be represented by a left candidate (or vice versa).
c) This is what I advocated for. Approval voting natively provides an ordering of candidates by popularity. It's no problem to elect the next best. I also hope that once approval has a use in Germany it will over time replace all other plurality elections and all runoff elections with approval+runoff.
d) There is no movement for IRV in Germany, but it still is the most widely known of the alternative voting methods. It doesn't provide a real ordering of candidates. The last candidate to be eliminated isn't necessarily the "second best", it very much depends on who is running and in what order they are eliminated. One object of the commission is to make the voting law simpler. With the above changes it takes a huge step towards this goal, IRV however would be detrimental to it.

In a previous petition and a recent letter (pdf, German) to the commission I advocated for the use of approval voting in the candidate vote, but also as cumulative voting in the party vote. If you would vote for four parties, each would get ¼ of your vote. I'm generally against a party vote threshold, but proposed that this could be used to reduce the number of ignored votes. In a first step count the votes and eliminate all parties that fail to reach the threshold, then count the votes again, but distribute among the remaining parties. So if you voted for 4 parties, but one got eliminated, then your votes goes with ⅓ to your remaining three parties.

That's the same idea the German NGO Mehr Demokratie proposes. I proposed a similar modification to MMP that uses approval voting for the direct mandate and multi party choice. The party vote is distributed equally among all approved parties, then all parties that don't reach the threshold are excluded and the vote is distributed equally among the remaining parties.
It has the same effect as a second choice, but with the added benefit that voters don't have to limit their vote to one party.