I tested this framework using limited simulations of strategic voter behavior, and found significantly improved outcomes against benchmarks. For example, successful burial rates dropped from ~15% to ~7%, with >50% of buried Condorcet winners being rescued. This needs further investigation and more rigorous analysis.

The idea is as follows: in the first round, voters submit rank-score ballots, which determine two sets of candidates: a Majoritarian set M, and an Alternative set A.

For example, M could be the Smith set or Bipartisan set. A could be the Approval winner set or the IRV winner set.

Next, voters are given information about M and A. For example, M may be made public. A may also be made public, or some relevant statistics about A may be made public. For example, maybe instead of revealing the score winner, only the score is shown.

The key is, voters can use the information about M and A to do two things at once: (1) vote for whether to proceed by electing a candidate from A, or a candidate from M. And (2) possibly submit a “Rescue” signal, designed to counter burial tactics that may have manipulated M.

For example, the Rescue signal may be implicit, as the M-outsider with the highest top-rank support among outsiders. The design goal for the Rescue signal is to re-introduce a buried candidate into M with high efficiency, without ruining a sincere election. The cost is strict Smith compliance (although, if the Smith set is already manipulated and insincere, that may not be a bad thing).

If the electorate chooses M, then the pre-specified completion rule is run on M with the Rescue set R. For example, Benham could be run on M, and the winner could go head-to-head against a winner in R.

Otherwise, if they choose A, then the pre-specified completion rule is run on A.

The goal of MARE is to enable information/incentive engineering by selecting the rules for M, A, and R. In some experiments, I had M be Smith, A be IRV, and R be top outsider. There are various other considerations to explore within this framework. I just wanted to share it to see if others wanted to experiment with simulations.

I think Smith//STAR (implicit) as mentioned above has some problems, but I think Smith//Approval (explicit) as denoted in the electorama vote is a very good method despite taking two rounds. I’m half surprised Smith//Approval (implicit) didn’t do better.

I understand the general aversion to two-stage methods, but in the USA, we often effectively do that with primaries anyway, and it seems commonly used elsewhere. I think introducing alternative methods into party primaries could be pretty significant change in itself.

In fact, it may be that most strategic benefits of SSVV are caused by the score/approval in the Smith set being informed by the Smith set itself, which would mean Smith//Smith-informed-Approval is a reasonable comparator, and it may be superior.

I also agree that in the singleton case with sincere voters, the validation step is redundant. If the Score set is ({A}) and the Smith set is ({B}), then a majority vote consistently with sincere preferences should choose the Smith set, since that same majority sincerely prefers (B) to (A). Probably, often the Score winner and Condorcet winner will coincide anyway.

Where I think SSVV differs most substantially from Smith//Approval is in strategic behavior rather than sincere outcomes, particularly when strategic voting changes the composition of the reported Smith set. Smith methods like Smith//Approval automatically commit the electorate to choosing from the reported Smith set, whereas SSVV does not—since voters first see the Score and Smith sets, the majority can decide whether they actually want to validate a Smith set that may have been distorted by tactical voting.

That may matter specifically for burial. Suppose burial changes the Smith set from something relatively compelling into a larger or stranger set. The burial strategy has succeeded in manipulating the pairwise structure, but has also changed the object voters are being asked to validate. Voters who prefer the Score winner to every member of the Smith set obviously choose Score, but even some voters who prefer one Smith candidate to the Score winner might choose Score rather than risk one of the other Smith candidates winning.

So I think your observation points toward an interesting strategic question about the method. Manufacturing a cycle can indeed get a candidate into the Smith set or remove another candidate from it, but unlike under Smith//Approval, that does not automatically get the preferred candidate into the decisive election, nor does it necessarily eliminate the buried candidate from consideration altogether, since the manipulated Smith set has to survive validation by majority vote.

I don’t know yet whether that additional hurdle is strong enough to make enough of a difference to warrant it, but I think that’s the main distinction from Smith//Approval.

My simulations so far have been exploratory and conducted interactively through ChatGPT 5.6-Sol, using spatial and nonspatial electorates with coordinated factions iteratively trying various score-compression, compromise, and burial strategies, so I would not treat them as an equilibrium analysis. They do suggest that the validation stage can sometimes neutralize successful manipulation of the Smith set (primarily accomplished through burial), but that result needs more systematic investigation.

I also just experimented with public B and hidden A, or revealing only the mean score of A rather than candidate identity. This mostly improved behavior even more, because manipulating the Smith set becomes more risky in validation if the Score winner set is kept hidden. But it opens a can of worms in terms of “information engineering” to improve incentives. Even if the Score “escape hatch” isn’t used, its mere existence changes strategic incentives.

If the property is independently and coherently defined, it may still be mathematically useful to establish that demanding it is equivalent to demanding IRV. But no normative conclusion about IRV follows from that equivalence alone. Treating the characterization itself as a normative defense of IRV would be a non sequitur.

But quickly: you're first person to emphasize this. "Choose any" only makes sense in contrast to "choose one". I suppose indeed totally out-of-context it could be misinterpreted, and your point is valid, and it does make me less totally confident about the label. Thanks

explicitly says they are equivalent if everyone submits full rankings, but not equivalent if partial rankings are allowed:

In a Borda count, bc, M. de Borda suggested the last preference cast should receive 1 point, the voter’s penultimate ranking should get 2 points, and so on. Today, however, points are often awarded to (first, second,..., last) preferences cast as per (n, n−1, …, 1) or more frequently, (n−1, n−2,…, 0). If partial voting1 is allowed, and if a first preference is to be given n or n −1 points regardless of how many preferences the voter casts, he/she will be incentivised to rank only one option/candidate. If everyone acts in this way, the bc metamorphoses into a plurality vote… which de Borda criticized at length. If all the voters submit full ballots, the outcome—social choice or ranking—will be the same under any of the above three counting procedures. In the event of one or more persons submitting a partial vote, however, outcomes may vary considerably. This preliminary paper suggests research should consider partial voting. The author examines the consequences of the various rules so far advocated and then purports that M. de Borda, in using his formula, was perhaps more astute than the science has hitherto recognised.

The classic example is three voters using rank ballots over three candidates:

V1: A>B>C
V2: B>C>A
V3: C>A>B

You can see that A>B 2:1, B>C 2:1, but C>A 2:1. So A>B>C>A is a Condorcet cycle, which is a generalized “rock-paper-scissors” situation. Whichever candidate you choose as the winner, there is some majority of the voters who would have preferred a different candidate. That’s the unfortunate thing that happens when a Condorcet winner doesn’t exist…

Regardless, a Condorcet method is any method that guarantees electing the Condorcet winner when one exists. Condorcet methods differ in how they reconcile choosing a winner when the Condorcet winner does not exist, I.e. in effect how they determine which majority group(s) to jilt.

So for example, if Ranked Robin doesn’t specify how it resolves when there is no Condorcet winner, then it’s really a blanket term for Condorcet methods in general. Or maybe it’s a label for a particular curated subset of Condorcet methods.

There are many Condorcet methods, including Ranked Pairs, Schulze’s method, Copeland’s method, Minimax, and Bottom-Two-Runoff (by Tideman).

I like Bottom-Two-Runoff because it’s efficient and also equivalent to a seemingly (but not actually) more robust system: https://www.votingtheory.org/forum/topic/564/bottom-n-and-bottom-2-runoffs-are-equivalent

That’s unfortunate, but also makes sense.

With ties in other methods, the resolution of ties is typically standard because the set over which the ties occur is discrete—uniformly sample one from among the tied candidates. But for maximal lotteries, when non-uniqueness holds there is a continuum of admissible lotteries as you indicated. The analog of a uniform distribution in this case would be using Jeffrey’s prior, which is why I think that’s the “right” way to go.

But yes it ultimately doesn’t really matter how the maximal lottery used is chosen since they are all maximal, but that’s also kind of the issue, because a choice has to be made.

I made a mistake in an earlier post on this thread:

I stated that in a BTR-Score election with a 3-way cycle, the highest seed in the cycle would have a 50% chance of winning, and the two lower seeds would each have a 25% chance of winning.

This is incorrect. The highest seed in the cycle would always win.

Consider this BTR-Score tournament with a 3-way cycle:

Candidate A is the highest seed in the cycle.
Candidate B is the 2nd highest seed in the cycle.
Candidate C is the 3rd highest seed in the cycle.

In every such election, B and C will face off in the tournament; the winner will then face A in the deciding contest.

In a 3-way cycle, each of the candidates wins one match and loses one match to the other two candidates. The winner of the B vs C contest uses its only win to prevail; therefore, that winner has no chance of defeating A.

2 Possibilities:
A > B > C > A - B beats C then B loses to A
C > B > A > C - C beats B then C loses to A

A BTR-Score election with a Condorcet winner rewards majority.

A BTR-Score without a Condorcet winner rewards utility.

It would be possible, but extremely weird, for a BTR-Score election to have a 3-way cycle that does not include the top seed, the candidate with the highest total score.

I’m not saying the “Condorcet adversary” should be the score winner, just that they should be a strong alternative. Your approval-seeded Llull ballot could accommodate an approval winner as the Condorcet/Smith-method’s adversary, for example.

I think a “rank with approval cutoff” ballot makes sense. Then there could be the approval winner, and, say, the B2R (Smith compliant) winner, followed by an independent head-to-head.

To be precise, if Z+ is the set of natural numbers, let s:Z+ to R ("score") and hth:Z+xZ+ to R (“head-to-head”) be fixed real-valued functions, where s is injective, and where hth(y,x)=-hth(x,y) and hth(x,y)=0 if and only if x=y (“no ties”). In our case we consider a set C of “candidates” to be a finite subset of Z+.

Let B2R(C) denote the “Bottom 2 Runoff” winner in C, determined by repeatedly pitting the bottom two scoring candidates against each other in a head to head, and eliminating the loser until one candidate (the winner) remains.

Also, let RCLE(C) denote the “Recursive Condorcet Loser Elimination” winner in C, determined by repeatedly looking at the set of N lowest-scoring candidates, starting with the full set C, and eliminating any Condorcet loser among them (if none, update N↦N−1 and retry, resetting to the full remaining set with each elimination), until one candidate (the winner) remains.

Both B2R and RCLE are Condorcet compliant and Condorcet loser compliant methods. It also turns out that for fixed score and head-to-head functions with no ties, they are equivalent.

PROOF:

We can proceed by induction on the size of C. With 1 or 2 candidates, the equivalence is trivial. So we consider |C|>=3.

Without loss of generality, we can assume that there is no Condorcet winner in C—with a Condorcet winner, B2R automatically coincides with RCLE, since both methods are Condorcet compliant.

We can also assume that there is no Condorcet loser in C, because that Condorcet loser will immediately be eliminated in RCLE, and will have no material effect on B2R. More formally, if L is the Condorcet loser in C, then B2R(C)= B2R(C omitting L), and RCLE(C) = RCLE(C omitting L).

Now, consider the first candidate eliminated by each method, and call them B2R1 and RCLE1, respectively. If B2R1=RCLE1, then we will be done by the inductive hypothesis. Thus, we have to examine the case where B2R1 differs from RCLE1. In that case, the only option is that RCLE1 is the Condorcet loser in the set of bottom-N scorers for some N >= 3. That is, RCLE1 loses head-to-head against every other candidate in that bottom-N scoring set.

In particular, RCLE1 loses head-to-head against every candidate whose score is lower than its own score. This means that in B2R, no matter which of the candidates whose scores are lower than RCLE1's emerges to face RCLE1, that candidate is guaranteed to beat out RCLE1 head-to-head. Therefore, the presence of RCLE1 in the candidate pool is immaterial to the outcome of B2R.

Formally, it follows that B2R(C)=B2R(C omitting RCLE1). But we also know that by definition, RCLE(C)=RCLE(C omitting RCLE1) as well. By the inductive hypothesis, B2R(C omitting RCLE1) = RCLE(C omitting RCLE1), and this implies that B2R(C)=RCLE(C).

QED

This is true no matter which scoring or head-to-head functions are used, as long as they are static functions and no ties are involved.

This means that, while RCLE superficially appears to be more conservative than B2R, this is actually not the case. B2R is always just as conservative as RCLE, since they are equivalent. B2R is actually just a more efficient implementation of RCLE.

Alternatively you might want to come up with a cloneproof measure to find the top 3, perhaps similar to the score excess method that I posted here, based on Chris Benham's approval opposition.

@lime said in On one-sided strategy:

@jack-waugh said in On one-sided strategy:

Give your favorites the top score and your most hated the bottom score. If you have a compromise candidate, and if you are convinced that your favorites are very unpopular or unknown, exaggerate the score of the compromise candidate almost up to the next higher candidates, but not quite up to them.

In practice, this is the same as thresholding, assuming you rate your compromise close enough to perfect.

What you mean, "rate"? In my heart, or on my ballot?

On your ballot.

As part of an experiment to see if it's possible for the Democrats to hate someone more than Trump, or just to set a Guinness World Record for biggest political career implosion?

I didn't say it would be a good idea. As I mentioned, he'd have no hope, since voters are using the primary to settle on an equilibrium. The question is whether Democrats have a gun to their head that would keep them from voting for Buttigieg's third-party, even if Biden looked hopeless.

@chocopi said in New voting method: Linear medians:

I agree more with you that these edge cases can plausibly be disregarded. The very idea of anyone executing a pushover strategy is absurd--you would need exact polling, exact coordination, no counter-strategy, and face a worst-case backfire if you get any of that wrong.

I don't think voters supporting a pushover is where this really falls apart. The problem with Tideman's framework is the strategies he finds are often:

Every voting system has strategy. The real concern is whether voters playing their optimal strategy creates a bad result, e.g. a turkey winning. After all, in Borda, the Strong Nash equilibrium is still the Condorcet winner, but that doesn't happen in real elections. The reason Borda is bad is because the only proper equilibrium ends up selecting a winner at random.

In Benham's method, optimal strategy looks like a center-squeeze, because whenever you have a center-squeeze setup, the largest faction can bury the Condorcet winner and elect a candidate on the wings. (That's especially true if the wings tend to be overconfident.) By contrast, in cardinal methods, the optimal strategy looks like, well, the Condorcet winner being elected.

I'd like to clarify that I think party strategy does play a huge role in FPP, IRV, or Condorcet elections, because strategy is either too complex for typical voters or there are many equilibria (and voters have to coordinate on just one). In these specific situations, voters have to follow instructions on voting cards issued by their party. In approval or score, any idiot with a pulse can work out that your best strategy is to give as many points as possible to the best frontrunner.

@chocopi said in New voting method: Linear medians:

Baldwin's is a ying-yang similarity, a method that is practically only impacted by said esoteric NP-hard strategies. (Simple compromise-burial does almost nothing.) These non-trivial Baldwin strategies are the hardest to calculate of any method, even with perfect [everything]. I think it's a fair and interesting academic question to quantify these, but I'd also raise an eyebrow (or two) at anyone listing them as a point against Baldwin's.

This is a good example of why "NP-hard"ness results are not very useful in practice. What matters is what happens if voters execute their ideal strategy. For Baldwin's method, the strategy is an absolute disaster, just like for Borda: it ends in a turkey winning with high probability.

In practice, elections have 2-5 viable candidates, so even "NP-hard" manipulation is trivial in practice. If you have just 2 candidates and a turkey, Baldwin is Borda-with-runoff, with a simple strategy: bury the leader to make sure they can't make it out of the first round. Does this have a shot of backfiring? Yes. (It has a good shot of picking a turkey, in fact.) But at its core, it's still Borda, and has the same result.

eliminate any candidate who would lose in a runoff

If that's what it amounts to, then maybe a restatement of the definition in terms of predicting such a runoff would make the definition easier to read for a broad and skeptical audience.

I promise you that nobody in the election-methods mailing list is particularly positive on IRV.

Yes, I have noticed that.

Most of the support for IRV is from the Alaskan model (Top Four & Final Five) proponents and their ally Fair Vote.

Fair Vote is promoting Proportional Racked Choice Voting in the Fair Representation Act (Rep. Donald Beyer, D-VA-8).

The Fair Representation Act (FRA) calls for Ranked Proportional Voting (SVT), FairVote claims:

"It’s straightforward for voters: Rank candidates in order of choice. Voters can rank as many candidates as they want, without fear that doing so will hurt their favorite candidate’s chances. Ranking a backup choice will never hurt a voter’s favorite candidate, so voters have no reason to vote for only one candidate."

This year's version of the FRA includes provisions for states with blanket primaries.

As with previous versions, FRA protects Voting Right Act of 1965 set aside districts. Frankly I think fair voting systems, especially proportional representation, will help minorities far more than set aside districts. Set aside districts are perceived by Republicans as a perfectly legitimate excuse to gerrymander like all hell.

The FairVote FRA pages give the impression the the chief purpose of proportional representation is to get more people of color, women, LGBTQ candidates elected.

To get Proportional Representation enacted we will need support from a good number of conservatives and Republicans. We should sell voting system reforms as color blind (they are), and fair. They will help minority representation by virtue of being color blind.

The FRA is now in committee, the speaker will decide when to let it out of committee, smart money is on never.