New voting method: Linear medians

Lime

@paretoman I'm interested in hearing your take on this.

Lime

This gets me to thinking what we really want from an electoral system and why STAR seems appealing despite its catastrophic edge cases.

My hypothesis is what we really want from a voting system is for it to:

Satisfy IIA. This means that with perfect or near-perfect information, the optimal strategy is minmaxing around the Condorcet winner, like in approval voting, so we elect the Condorcet winner.
Zero-information optimal—if each voter has no information about the election at all, the method should elect the utilitarian winner, or hopefully at least the Condorcet winner.

Why 0-info honesty? In situations where we're choosing between two very similar candidates, and a third (very different, less popular) candidate—i.e. a Burr dilemma—we want to have voters be honest about which of the two they prefer.

STAR tries to split this in half. It has a first stage that satisfies IIA and is therefore weakly sincere, and a second stage that guarantees honesty. The combination satisfies neither in theory, but the hope is it will satisfy both in practice (and it will in clone elections).

Maybe a better alternative is to look for a method that satisfies both at once?

Lime

Also mentioning @ChocoPi and @Jameson-Quinn .

(I've seen you around on r/EndFPTP; hi ChocoPi!)

ChocoPi

@lime

So you have to be careful when hearing about "strategy-proofness" in broader contexts of game theory, set theory, and mechanism design. It varies in definition pretty substantially, both in terms of the literal definition being applied and the domain of the problem space it is being applied to.

Many of these statements have no direct relevance to the field of democratic voting.

For example, many cases define "strategy-proof" as a local maxima outcome for an individual voter. But we could hardly care less about that in a political context, in which virtually all political activity is coalitional.

I know a lot of Buttigeig supporters. Not one of them voted for Buttigeig in the 2020 presidential election. Somehow, every single one of them voted for Joe Biden. Every single one! Our top investigators are still trying to get to the bottom of this conspiracy. (Apparently they held a secret meeting in Milwaukee, that was broadcast on every major news channel? The plot thickens.)

This sort of semantical confusion surrounding strategy tends to come up a lot with medians. This is because medians have mathematical properties enforcing majority rule within single-peaked domains.

For instance, say we are voting on how much to allocate for a budget, and everyone has a single ideal they'd prefer to be as close to as possible. Everyone says a number, the median value will have majority support over any other--and no voter or group can change that result in their favor by altering or exaggerating their votes.

This is possible because of the domain restriction. All the candidates--in this case, numbers--are perfectly ordered. And all voters only care about a single ideal preference, within that order. The moment a single voter says "Repave the parking lot or bust, go big or go home, I prefer $20,000 > $0 > $10,000" then all these assumptions go poof.

Assuming that the electorate's natural preferences are single-peaked and one-dimensional is already notoriously dubious. But some of these domain restrictions go a step further and operate on the basis that the restriction applies to the expressable preference space, the ballot. That not only do there exist zero Trump supporters who prefer Sanders over Biden, but that it is impossible for someone to fill out a ballot that way at all, as if Biden is purely a number that exists between Trump and Sanders.

Various domain restrictions can make a lot of sense in other appropriate contexts: auctions, scheduling, network routing... But applying them and their implications of "strategy-proofness" to elections is a sort of circular logic. ("Trump voters have no incentive to exaggerate and bury Biden under Sanders if our assumptions declare they can't." Wow, you don't say?)

Median judgement rules are pretty vulnerable to strategy (the political definition we actually care about), roughly on the level of Plurality and Approval. Shuffle through random elections and Median judgement will be vulnerable to some strategy ~40% of the the time for 4 candidates. You can mouseover the "simple wasted votes" for Median to highlight the voters who are most punished for voting honestly.

This paper's proposal of uniformity with regards to Moulin's mechanism (applying a "grading curve") doesn't change the coalitional incentives. All groups ultimately have the same motivations and resources.

And unrelated: "Phantom ballots?" "Grading curve for votes?" "Vote transformation?" Man, Dinesh D'Souza and Mike Lindell can smell the money.

Lime

@chocopi said in New voting method: Linear medians:

Median judgement rules are pretty vulnerable to strategy (the political definition we actually care about), roughly on the level of Plurality and Approval. Shuffle through random elections and Median judgement will be vulnerable to some strategy ~40% of the the time for 4 candidates. You can mouseover the "simple wasted votes" for Median to highlight the voters who are most punished for voting honestly.
This paper's proposal of uniformity with regards to Moulin's mechanism (applying a "grading curve") doesn't change the coalitional incentives. All groups ultimately have the same motivations and resources.

There are two kinds of strategy here, and I think both really matter! Lots of voters are strategic with the intent of manipulating the outcome. However, lots of other voters are strategic with the intent of manipulating the score of each candidate.

Scores are what determine which candidates and positions are considered viable or popular, and they can make-or-break your whole career. Entire careers have been built around strong losing performances. Bernie Sanders' losing campaign kicked off the modern socialist movement in the US; Perot's losing campaign forced Clinton to focus on tackling the national debt; and Jon Ossoff turned a narrow 2017 loss in the House into a Senate seat.

Hell, it might even be that most voting is motivated by this kind of desire to "send a message" to politicians. France's 2002 runoff had the highest turnout in history, even though Le Pen was hopeless, because people wanted to defeat Le Pen as decisively as possible. I'd argue the impact of elections on public policy, especially in more competitive settings, is roughly 1/3 about who wins and 2/3 about whose ideas win (by proving their popularity).

Lime

@chocopi said in New voting method: Linear medians:

And unrelated: "Phantom ballots?" "Grading curve for votes?" "Vote transformation?" Man, Dinesh D'Souza and Mike Lindell can smell the money.

By the way, I should probably mention this rule has a simplish explanation—each candidate's score is equal to the largest value X, such that X% of voters give a score of at least X%.

ChocoPi

@lime said in New voting method: Linear medians:

Scores are what determine which candidates and positions are considered viable or popular, and they can make-or-break your whole career. Entire careers have been built around strong losing performances. Bernie Sanders' losing campaign kicked off the modern socialist movement in the US; Perot's losing campaign forced Clinton to focus on tackling the national debt; and Jon Ossoff turned a narrow 2017 loss in the House into a Senate seat.

Hell, it might even be that most voting is motivated by this kind of desire to "send a message" to politicians. France's 2002 runoff had the highest turnout in history, even though Le Pen was hopeless, because people wanted to defeat Le Pen as decisively as possible. I'd argue the impact of elections on public policy, especially in more competitive settings, is roughly 1/3 about who wins and 2/3 about whose ideas win (by proving their popularity).

Losing this political participation is the entire danger of coalitional manipulation.

All fully-informed+fully-rational two-sided strategies elect the Condorcet winner, and sometimes you will have someone arrive at this realization and declare that strategy isn't much of a concern if all roads lead to Rome. But this line of thinking fails to question how many political casualties there were along the way--how many subfactions were told to shut the hell up, get in line, and be a good little [our party] before the [other party] wins. (And how many voters were disenfranchised or disillusioned by this inevitable political pugilism?)

Coalitional manipulation is the difference between Nikki Haley, Bernie Sanders, and Pete Buttigeig being on the debate stage (and a full voice in the political discourse) or not. As you say, even when they don't win, that is a very important thing for the system to get right.

Lime

@chocopi said in New voting method: Linear medians:

Losing this political participation is the entire danger of coalitional manipulation.

Right, but this means voters (when casting their ballots) will tend to engage in grading strategy as well. There's basically two possibilities here:

Final grades are important, so people want to manipulate them. In that case we need to consider grade-strategy manipulation and the effects it can have on grades.
Final grades are unimportant, so nobody tries manipulating them and "all roads lead to Rome". In that case we should probably just stick to approval voting (which is simple and sincere).

@chocopi said in New voting method: Linear medians:

All fully-informed+fully-rational two-sided strategies elect the Condorcet winner, and sometimes you will have someone arrive at this realization and declare that strategy isn't much of a concern if all roads lead to Rome.

I do think this needs to be emphasized: if you think voters engage in something close to fully-informed coalitional manipulation strategy, that implies the Condorcet winner always wins (no matter the system). That seems wildly unrealistic to me, since I'm pretty sure I've seen lots of elections where the Condorcet winner lost.

That's why I think we need to be very careful to distinguish situations involving cooperation to coordinate on a single equilibrium (e.g. Democrats settling on Biden) from the idea of true "group strategy". The latter often requires voters to simply execute commands from their party, even when it violates their own self-interest. It also doesn't strike me as a very plausible model of strategy, since we have plenty of cases like Alaska 2022 where Republicans all should have supported Begich. In practice, I think it's very clear that voting systems don't always produce a maximal lottery.

ChocoPi

@lime said in New voting method: Linear medians:

In that case we should probably just stick to approval voting (which is simple and sincere).

Wait what? Voting methods don't get less sincere than pure cardinal.

@lime said in New voting method: Linear medians:

I do think this needs to be emphasized: if you think voters engage in something close to fully-informed coalitional manipulation strategy, that implies the Condorcet winner always wins (no matter the system). That seems wildly unrealistic to me, since I'm pretty sure I've seen lots of elections where the Condorcet winner lost.

I think there must be some misunderstanding--this is the point I was making.

It doesn't matter much if optimal outcomes of strategic equilibirums are desirable. Imperfect strategy commonly leads to results that fall short of that, and even when it doesn't still leaves a trail of bloodstains and broken noses.

@lime said in New voting method: Linear medians:

That's why I think we need to be very careful to distinguish situations involving cooperation to coordinate on a single equilibrium (e.g. Democrats settling on Biden) from the idea of true "group strategy".

No, this is absolutely verbatim what a coalitional strategy is.

There is no strategy more pure than the selection of a party nominee, and no enforcement mechanism more straightforward than an agreement to only put that candidate on the ballot.

@lime said in New voting method: Linear medians:

since we have plenty of cases like Alaska 2022 where Republicans all should have supported Begich.

They should have! There are many, many, many, many partisan primaries in which a partisan echo chamber got greedy, overplayed their hand, and nominated an extremist who would lose rather than a moderate who would win. This has happened with both public primary electorates and the old days of men in backrooms with cigars.

Partisans and their donors are motivated to push the least-moderate candidate they think will still win, and it's easy to get that calculation wrong. Polls are weak, voters are fickle, and partisans are infamously prone to delusion.

Palin is a particularly extreme case of this. Her special election campaign was not only founded on the misconception that she was a good candidate, but spent the entire time attacking... Begich! Then the results are in, the false-dream was exposed, and the general election was looking far more difficult (pro-choice winds and higher turnout favored Peltola), Yet She Persisted. (Ultimately Begich would have lost in the general all the same, but it would have been a much better swing at the ball.)

Sarah Palin's House campaign was one long exercise in irrationality, and a great example of specific groups/leaders being pathologically incapable of compromise.

Lime

@chocopi said in New voting method: Linear medians:

Wait what? Voting methods don't get less sincere than pure cardinal.

Pure cardinal voting strategies are almost-always sincere, i.e. they never involve rating a worse candidate higher than a better one. (This is Brams and Fishburn's terminology.)

@chocopi said in New voting method: Linear medians:

No, this is absolutely verbatim what a coalitional strategy is.
There is no strategy more pure than the selection of a party nominee, and no enforcement mechanism more straightforward than an agreement to only put that candidate on the ballot.

OK, so if your point here is about strategic nomination, that's somewhat different.

I thought we were asking "why didn't Buttigieg run in the general?" to which the answer is "he wouldn't get any votes, because the party signaled to voters they should coordinate on Biden instead."

Party nominations work by announcing a strategy to voters, rather than enforcing it. In FPP, if one voter thinks everyone else will follow the lead of the DNC, it becomes rational for an individual voter to follow it as well: if I think other people will vote for Biden if he's the nominee, and I want to vote for one of the frontrunners so I vote for Biden.

However, this is because the Democratic nomination provides information, not because they can compel voters to follow a particular strategy. If I saw the next day that Buttigieg had formed a new third-party which was polling 30% instead of Biden's 10%, I'd ditch the Democrats and join the new third-party, since the Democrats can't convince me to stay against my own interests.

@chocopi said in New voting method: Linear medians:

They should have! There are many, many, many, many partisan primaries in which a partisan echo chamber got greedy, overplayed their hand, and nominated an extremist who would lose rather than a moderate who would win. This has happened with both public primary electorates and the old days of men in backrooms with cigars.

Partisans and their donors are motivated to push the least-moderate candidate they think will still win, and it's easy to get that calculation wrong. Polls are weak, voters are fickle, and partisans are infamously prone to delusion.

Yes, which is why I'd argue Tideman's perfect-information model is obsolete. All of the assumptions in Tideman's model are wrong: voters work from inside a decent (but imperfect) informational environment where polls are off by a few percentage points. They're also willing to engage in strategies that are detrimental to their own party if they increase the chances of a good outcome for themselves (e.g. ranking Palin first).

@chocopi said in New voting method: Linear medians:

Sarah Palin's House campaign was one long exercise in irrationality, and a great example of specific groups/leaders being pathologically incapable of compromise.

On the other hand, I don't think this is all that irrational. It's all at least somewhat-reasonable behavior from Republicans, if they thought Begich and Palin had about the same chances of winning. That makes sense if you have:

A polarized electorate with very few centrists,
Elections decided more by national environment than individual candidate quality,
If you think the special election was a weird fluke caused by low turnout and being held right after Dobbs
Polls shifting from slightly favoring Democrats towards moderately favoring Republicans over the next few weeks
A substantial education campaign pushing to reduce the number of exhausted ballots

Remember: Palin only lost by 3 points in the special, and a majority of voters didn't even rank her!

ChocoPi

@lime said in New voting method: Linear medians:

Pure cardinal voting strategies are almost-always sincere, i.e. they never involve rating a worse candidate higher than a better one. (This is Brams and Fishburn's terminology.)

Right, but this is a non-functional defintion to use for anything. It's asserting that the ballots:

Sanders - 10
Biden - 10
Trump - 0

and

Sanders - 10
Biden - 0
Trump - 0

...are equally "sincere" applications of identical preferences, simply because there is no inversion.

I do think this is a broader semantics issue, in which referring to different cardinal allocations as "strategic" or "honest" is too rigid; both lead to circular definitions. In my work, I introduce a middle layer and use the word "disposition" to refer to an individual voter's natural mapping of their (honest) preferences to a cardinal ballot. (I.e. we might have exactly identical preferences, but my 6 is your 7. We just say we have different dispositions, rather than asserting that only one of us is honest (and the other is a liar), or insisting our unequal ballots are identical.)

@lime said in New voting method: Linear medians:

I thought we were asking "why didn't Buttigieg run in the general?" to which the answer is "he wouldn't get any votes, because the party signaled to voters they should coordinate on Biden instead."
However, this is because the Democratic nomination provides information,

No, it is literally illegal for Buttigeig to run (as a non-write-in) in 47 states. The primary is a legally binding mechanism under sore loser laws. It is far from being a merely helpful suggestion.

(You do have edge cases who did revolt as a write-in (Lisa Murkowski, Byron Brown) in extenuating circumstances. Murkowski had the largest geographic bloc of independent voters in the US supporting her, and Byron's race had no other opponent to spoil + the tacit backing of the state Democrats. Both were incumbents, and decently liked. Both, when successful, became deeply despised by the non-centrists of their party to this day; while McConnell eventually came around as a Murkowski ally, at the time he and the rest of GOP leadership denounced her and stripped her position as vice char of the RSC.)

@lime said in New voting method: Linear medians:

All of the assumptions in Tideman's model are wrong:

I'm not sure that Tideman ever asserted a model in which it is assumed that anyone is acting with perfect information. (That seems pretty off-brand for him tbqh) Most his publications I'm familiar with (pertaining to strategy) merely address whether there exists possible successful one-sided strategies for a given context.

Tideman is pretty humble in interviews, and is quick to dismiss questions of human behavior on grounds that he is not a political scientist.

@lime said in New voting method: Linear medians:

On the other hand, I don't think this is all that irrational. It's all at least somewhat-reasonable behavior from Republicans, if they thought Begich and Palin had about the same chances of winning. That makes sense if you have:

Elections decided more by national environment than individual candidate quality,

I think this is the lynchpin. This was 100% a very real gap that the lower 48 thought Palin would win, and no one in Alaska thought she had a chance.

I'm pretty familiar with Alaska and its politics for a few reasons (not a resident though), and man was the mood on the ground sour. Palin hovered at around a roughly -45% favorability rating. She was seen as a quitter, someone who used Alaska as a stepping stone to TV shows and fame in the lower 48. And this was a pretty widespread rural sentiment, specifically. As Palin got endorsements exclusively from big names in the lower 48 (while Begich got the support of virtually every conservative group and legislator in Alaska), it only cemented this attitude. Even the Trump and Haley endorsements did surprisingly little to the polls.

But it did further advance the attitude outside of Alaska that wow gee, Palin must be a strong candidate.

Palin did outperform her polls in the special election, and her brutal attacks on Begich certainly made an impact on his numbers. But I can't think of a comparable US election where a candidate could come so numerically close yet have such a miniscule chance of ever winning.

Lime

@chocopi said in New voting method: Linear medians:

No, it is literally illegal for Buttigeig to run (as a non-write-in) in 47 states. The primary is a legally binding mechanism under sore loser laws. It is far from being a merely helpful suggestion.

Well, there you go: Buttigieg could've either run as a write-in, or he could've launched a third-party candidacy instead of joining the Democrats. Also, I don't think sore loser laws apply to presidential candidates in most states, since RFK Jr. is on the ballot in lots of them right now.

@chocopi said in New voting method: Linear medians:

I'm not sure that Tideman ever asserted a model in which it is assumed that anyone is acting with perfect information. (That seems pretty off-brand for him tbqh) Most his publications I'm familiar with (pertaining to strategy) merely address whether there exists possible successful one-sided strategies for a given context.

That's the perfect information assumption: the post-hoc manipulability rate checks whether a one-sided strategy could possibly succeed, if a group of voters had perfect information and perfect coordination (which requires knowing all other voters' preferences).

This is a kinda useless metric, because it doesn't consider any questions like:

Could I actually realize I'm in a situation where this strategy is viable, given limited polling information?
Could I pull it off with imperfect coordination—some defections, some voters refusing to be strategic, some voters making mistakes?
What are the actual effects of the strategy on the winner—is the new winner Hitler, or are we just swapping one clone for another?
How difficult would it be for another group of voters to stop it? Can I stop it by truncating or tweaking my approval threshold? Or do I need to execute favorite-betrayal, complex coordinated manipulation, etc.? (In which case we've effectively been forced back into a two-party system.) Would the polls even converge to the equilibrium—how would I even know someone is the Condorcet winner if they've been buried hard enough for all the polls to show them being eliminated in the first round?

ChocoPi

@lime said in New voting method: Linear medians:

Well, there you go: Buttigieg could've either run as a write-in, or he could've launched a third-party candidacy instead of joining the Democrats.

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?

@lime said in New voting method: Linear medians:

Also, I don't think sore loser laws apply to presidential candidates in most states, since RFK Jr. is on the ballot in lots of them right now.

Worm-brain man dropped out of the Democratic primaries last October, about 4 months before the first state primary.

He announced his independent candidacy the same day, and sore loser laws were his motivation for the preemptive move.

@lime said in New voting method: Linear medians:

That's the perfect information assumption: the post-hoc manipulability rate checks whether a one-sided strategy could possibly succeed, if a group of voters had perfect information and perfect coordination (which requires knowing all other voters' preferences).

Well, yes and no.

Pure compromise+burial is the optimal strategy in almost all methods, and essentially the only strategy in many. It requires minimal polling accuracy, voters find it intuitive, and is easily executed via ballot access given sufficient political infrastructure. Some authors have refered to it as "the trivial (but sufficient) strategy" or "the baseline strategy."

It is difficult to think of an election where either party failed to execute a compromise strategy at all. At best there are many examples where party leadership or public factors led to a greedy losing strategy, but even in the age of Trump endorsing Dr. Oz those cases are rare. And even then, these strategies didn't backfire, they were just probably worse than a better one--if the Pennsylvania Republican party nominated no one and told all their voters to just vote their honest gut, Fetterman wins even harder.

Tideman does include, in the papers I am thinking of, edge case strategies for applicable methods: pushover for IRV, extreme Borda optimization, and the same for Baldwin's. (I don't think he quantifies entry vulnerabilities, other than acknowledging them.)

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.

Borda's most extreme solutions are similar and sort of moot contextually--it's one of the most vulnerable methods in existence to just basic compromise+burial, so who cares if some esoteric NP-hard ballot calculation can technically squeeze out a couple more percent? Whether or not we count these vulnerabilities, Borda is still a poor method no one should use.

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.

I don't know that Tideman including these edge cases would upset anyone other than perhaps a FairVote IRV-defender-of-the-faith. If anything, the biggest objection would be that he does not include entry/teaming.

Lime

@chocopi said in New voting method: Linear medians:

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:

Individually unstable, and therefore couldn't occur with strategic voters. You need voters to do things like betray a favorite, even though that favorite has a good shot at winning. Sometimes they're impossible to pull off with imperfect coordination (improper equilibria, i.e. trembling-hands rule them out).
Prosocial or neutral—FPP has lots of opportunities for strategy, which is a good thing, because without strategy it turns into a random lottery.
Easily countered by basic defensive strategy.

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.