I think it makes sense to do Condorcet//Approval, in two separate rounds, the approval round restricted to the Smith set. But two round voting outright is a difficult sell in the USA (even though two-round voting is pretty common all over the rest of the world…).

People try to put the two together in a single round vote, but the strategic incentives of casting rank and approval/score indications on the same ballot cause issues.

My personal belief is that this system of two-round voting for single winner elections, I.e. approval conditional on already knowing the Smith set, would be most ideal. I think actually implementing the approval aspect first however is an easier sell than implementing the Condorcet aspect first. I can envision a natural progression as: (1) implement straight approval, (2) eventually indicate the shortcomings of approval in guaranteeing election of Smith set candidates, (3) reform to include a ranked primaries to restrict to the Smith set before the final approval vote.

I feel even having rank/Smith-based primaries makes way more sense than what we have if the subsequent system is approval. There’s no issue with vote splitting in that instance, and it fits at least partially into the political system we already have (although this would also require substantial changes).

I'm not a lawyer, but have been studying these things for years and we did win the case in the Oregon Supreme Court against the Oregon Legislative Council which found that RCV's ballot title in Oregon was misleading and inaccurate as it pertains to RCV's majority winner claims, though there is apparently no mechanism anymore to enforce such a ruling.

Two confounding data points:

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.

... So then shouldn't we be encouraging voters to give preferences as close to honesty as possible, to make sure we have as little error as possible?

No, in my opinion, we shouldn't. That's asking them to play the sucker, in the presence of a voting system that can get, I think, the right answer in case no party plays sucker. The proper use of Score Voting is to apply a tactic to maximize the expected value of the outcome. And I doubt whether STAR behaves significantly differently. It's just extra complexity for no gain.

@jack-waugh said in STLR - Score Than Leveled Runoff might not be too complex for voters:

"Honest voting" is a theoretical concept that can be useful in thought experiments and reasoning and the design of algorithms, etc. However, it does not describe a phenomenon that can happen in real elections in which something important rides on the outcome of the tally.

$20 says it does.

How are we going to test that? With the voters experiencing how many elections where the outcome matters to them? And how are we going to measure the importance of an election?

@jack-waugh said in STLR - Score Than Leveled Runoff might not be too complex for voters:

What grounds do you have for coming to such an opinion? I don't think it is correct.

I think that's just the definition of the "best result" under a given metric. Regardless of what social welfare function you pick, that social welfare function will be maximized if voters are honest.

What do you care about? Electing majority winners? A Condorcet method will always elect a Condorcet winner if voters are honest (but not necessarily if they're dishonest). Maximizing social utility? Score voting does that with honest voters (but not always for dishonest voters). Maximizing the number of voters who see their favorite candidate elected? FPP does that with honest voters (but once again, can't with dishonest ones).

No matter which social welfare function you come up with, that social welfare function will do better at its job if it has accurate information than if it has inaccurate information.

The input doesn't have to be accurate information about what the voters want. It suffices, in the case of Score Voting, if it is accurate information about the voter's tactical choice. If just one side votes "honestly" and the other is trying to maximize value, the result will be wrong and will skew to the side that is using the tactic. However, I contend that when all sides are using their respective best tactic, the "pull" balances out and the result will be the same as though all were "honest". The reason to think this is that the system is additive and balanced.

To paraphrase WDS: consider a voting system in which a vote consists of 32 bits. The tally takes the XOR of the ballots and then takes the result modulo the count of candidates to get the index of the winning candidate. How do I cast an "honest" vote in this system?

@lime said in Top-k primaries might be good?:
(regarding SNTV changes)

reserve a spot for the incumbent. Second, assign seats to parties using a rounded-down Hare quota based on party registration—e.g. if 45% of voters are registered Republicans, and there's 5 candidates on the ballot, 2 of the places go to the first- and second-place finishers on Republican ballots.

I am afraid that this would turn voter registration drives into a major industry. Abuses and mistakes could occur. Misunderstandings caused by language barriers could cause noncitizens to be registered by mistake, they would not know what happened but they could be prosecuted.

Perhaps I exaggerate, but people will pay dearly for any electoral advantage.

Also, a lot of people are not aware of which party the are registered with, if any.

BTW The common term for a payment to an person who gathers voter registrations is called a bounty. So if party R is paying $10 per registration from citizens residing in district x, the bounty for those registrations is $10.

There might be a small advantage to running more candidates (one of them might be a slightly stronger candidate than the others), but I'm guessing this is probably a small effect—generally, all the members of a party will do about equally well. There might even be a slight push in the opposite direction, because running more candidates splits funds and volunteer efforts between them. (Besides running the risk of party infighting.)

My guess is parties won't decide to do unethical things for such a small benefit.

1. Eliminate all candidates outside the pairwise-absolute majority Smith set. (i.e. equal-rating candidates creates cycles between the two candidates), unless this would eliminate all candidates.
2. (Optional.) Allow voters to specify second-round preferences for each candidate within the same party.
3. Elect the score winner.

Behaves much like STAR, in that voters have less incentive to equal-rank several candidates (they want to make sure their favorite can beat the others). However, it maintains Favorite Betrayal.

Reason: Originally, I expected STAR's gimmick runoff not to have much of an effect on score's good behavior. But ever since @SaraWolk mentioned the possibility of candidates not having clones in the race, I keep finding more and more situations where STAR reacts catastrophically badly to party-coordinated strategy. This is really bad, because parties exist for the explicit purpose of coordinating strategy. I'll give more examples elsewhere.

I was trying to think about burial but I don’t think my method addresses it quite as I conceived. My reasoning was that, by replacing absolute score differences with their more robust percentiles, burial (and bullet voting) strategies will suffer from severely diminishing returns compared with less risky and more honest ballots. For example, burying a second-favorite below a turkey to support a first favorite probably won’t significantly improve the score percentile provided by that voter to the first favorite’s runoff with the second favorite, but will significantly improve the chances of the turkey winning. This makes dishonest burial more severely punished and risky, meaning that fewer rational voters will choose to do it. Also, the effects of the fraction that do will be significantly reduced, since they will not only be fewer in number, but the magnitude of their indicated score differences will be majorly reeled back upon being replaced by their percentiles relative to the more honest bulk.

At the same time, the method is not restricted to Condorcet compliance, since, for example, it is possible for a [1-sqrt(1/2)]~0.2928… fraction minority of voters to overrule a sqrt(1/2)~0.707… fraction majority as long as the whole minority has the top quantile of absolute score differences and all of them have the same sign. That is the smallest possible minority that can overrule a majority in this method. It’s in one sense a generalized, more flexible extension of some of the reasonable measures we already have in the legislative houses, where for example a supermajority (2/3) is required for certain decisions.

Alternatively, each absolute score difference percentile could be measured relative to the distribution of all absolute score differences across all differences. The data set would consist of N*K(K-1)/2 values where N is the number of voters and K is the number of candidates.