I'm worried about the strategic implications of this. In the single-winner case this is Majority Approval Voting (assuming you're using the Droop quota), and even with more winners this has all of those strategic issues.

• Suppose you are part of a faction that comprises ~1.2 quotas and has fielded two candidates, one of whom you like more than the other (you hate every candidate outside of your faction). Your faction is very likely to win one seat, but it won't win two. Here you want to have your ballot count only as an approval for your favorite as far into the tabulation as possible, so you shouldn't give your second choice a score greater than 1. Even bullet voting is reasonable; if the vast majority of voters in your faction are giving their second choice a 1, bullet voting is the only way to make your ballot count in the decisive round. (This is, of course, the chicken dilemma.)
• Suppose there's a two-party system. The other party has both moderates you tolerate and extremists you hate, and you'd rather they elect more moderates than extremists. Still, the main thing you want is for your party to win more seats than them. Here, giving even a 1 to a moderate candidate in the opposing party is a big mistake. It won't make a difference at all until all scores of 1+ are counted as approvals, and at that point you still want to favor your party's candidate over the opposing moderates.

These are simplified examples, but the simplicity is not necessary for such problems to manifest. I'm not claiming that it's never strategically optimal to give a candidate a 4, but it's pretty atypical. It's strategically optimal to use lower scores than other voters do, such that they're supporting both their own favorite(s) and your favorite(s) while you're only supporting your favorite(s). The only equilibrium involves 5s for a voter's favorite(s) and low scores for everyone else.

Under Threshold MES, voters who ignore strategy lose a lot of influence, and when voters are strategic the expressive power of the 5-star ballot is mostly wasted. I am pessimistic about it reducing political polarization any more than any other PR method would; I actually expect it to be worse than STV in this regard since parties would be heavily incentivized to get their voters to not give candidates from any other party a score greater than 0 and this would encourage divisive attacks. Similarly, candidates would be incentivized to play exclusively to a party's base (and perhaps to some "sucker" voters in other parties who vote as would make sense in single-winner STAR).

@andy-dienes said in Threshold MES:

I do really want to emphasize that one of my main theses for this design is that the mere existence of 'sincere' utilities for each candidate, when in a proportional multiwinner context, feels somewhat nonsense to me.

In principle, I fully agree with you. In practice, I think assigning utilities to each individual candidate works pretty well. An individual voter will only have a marginal effect on the outcome; questions that cause the individual-candidate-utilities model to break down (such as comparing between electing 5 candidates you love, 3 candidates you love and 2 candidates you hate, and 2 candidates you love and 3 you hate to a five-person committee) become irrelevant (at least usually). If a single ballot can cause at most one of the winners to be different, I can't think of an example off the top of my head where the model of having individual utilities of each candidate and maximizing the sum over all the winners breaks down.

Let's say the percentages of the electorate are respectively w, x, y, z%. This is (kind of) an instance of laminar vote splitting---at least if there is some candidate popular among the entire Left wing, and likewise for the Right. To me, I would prioritize the guarantees in this order

• Left gets at least (w+x)% seats and Right gets at least (y+z)% seats

• Within Left seats, the Far and Center factions get seats in a ratio w:x, within Right seats, the Far and Center factions get seats in a ratio z:y

• Residual preferences respected (i.e. cross-party preferences and low scores come into play to flip small win margins or change election order within party)

I have two major points of disagreement. First, I place no intrinsic value whatsoever on having guarantees; all I care about are results and incentives. I am completely indifferent between having a result occur with it being guaranteed to occur and it occurring without a guarantee. Second, I consider the second point to be undesirable. In my example, voters have somewhat stronger preferences for who wins within the opposing party than within their party, and I don't think the weaker preferences should take precedence over the stronger preferences. Also, points 2 and 3 are in direct conflict, and I care about point 3 because it encourages depolarization.

The proportionality guarantees mean that no matter how much strategic jankery happens, as long as Far Left voters give Far Left cands higher scores than Center Left cands and vice versa, and all Left voters give Left cands higher scores than Right cands (and again, vice versa), then both 1. and 2. have to hold.

The stringent conditions (e.g. "all Left voters") make these guarantees seem weak to the point of irrelevance; "strategic jankery" that is well-justified and outside your allowed parameters will void these guarantees entirely. And strictly speaking, I don't think these guarantees are strong enough to prove what you want. Like, if most Left voters give the Left candidates a score of 3 and Right candidate a score of 0, but slightly less than a full quota of Left voters give the Left candidates a score of 5 and the Right candidates a score of 4, these latter Left voters will function like Right voters who will fill the quotas of Right candidates. Contrived, I know, but still. you need to assume a lot, including some pretty unreasonable things, in order to get a mathematical guarantee.

Conversely, in Allocated Score (with or without runoffs), we can only get guarantees 1. and 2. if every Left voter min-maxes their candidates. And in fact if they start peppering 2s to the opposing party I think it's quite likely that they will lose seats.

True with respect to guarantees (though I don't care about guarantees). As for the latter point, more precisely they can lose seat. Most of the seats will be decided by the filling of quotas; so long as their voters don't fill the quotas of opposing candidates, giving 2s to the opposing party is only harmful for winning the final seat. Still a solid argument against giving out these 2s, but I don't think it's an overwhelming one.

I think our big disagreements are (1) Should a voting method favor moderates over extremists? and (2) Are formal guarantees valuable? Our disagreements seem to be more over what a voting method should do than over what certain voting methods will do. I am not particularly optimistic about coming to an agreement on these points, but I think the agreements we have reached are valuable.

@sass said in Test it yourself! A new Score PR method from Sass:

Divide each candidate's Power Score by their Unity Score to get their Critical Score.

I haven't fully wrapped my head around this method yet, but division by the unity score seems extremely problematic. For instance, suppose Rita Writein gives herself a 5 and every single other voter gives her a zero. This gives her a Critical Score of 5^2/5 = 5 and causes her to immediately win a seat.

@andy-dienes I agree with you about that example. That said, if there were 6 winners I'd go with ABCDDD, and if they all gave D a three instead of a two I'd prefer DDD to ABC. Yeah, I think we understand one another's positions pretty well now.

@andy-dienes said in Threshold MES:

I do really want to emphasize that one of my main theses for this design is that the mere existence of 'sincere' utilities for each candidate, when in a proportional multiwinner context, feels somewhat nonsense to me.

In principle, I fully agree with you. In practice, I think assigning utilities to each individual candidate works pretty well. An individual voter will only have a marginal effect on the outcome; questions that cause the individual-candidate-utilities model to break down (such as comparing between electing 5 candidates you love, 3 candidates you love and 2 candidates you hate, and 2 candidates you love and 3 you hate to a five-person committee) become irrelevant (at least usually). If a single ballot can cause at most one of the winners to be different, I can't think of an example off the top of my head where the model of having individual utilities of each candidate and maximizing the sum over all the winners breaks down.

Let's say the percentages of the electorate are respectively w, x, y, z%. This is (kind of) an instance of laminar vote splitting---at least if there is some candidate popular among the entire Left wing, and likewise for the Right. To me, I would prioritize the guarantees in this order

• Left gets at least (w+x)% seats and Right gets at least (y+z)% seats

• Within Left seats, the Far and Center factions get seats in a ratio w:x, within Right seats, the Far and Center factions get seats in a ratio z:y

• Residual preferences respected (i.e. cross-party preferences and low scores come into play to flip small win margins or change election order within party)

I have two major points of disagreement. First, I place no intrinsic value whatsoever on having guarantees; all I care about are results and incentives. I am completely indifferent between having a result occur with it being guaranteed to occur and it occurring without a guarantee. Second, I consider the second point to be undesirable. In my example, voters have somewhat stronger preferences for who wins within the opposing party than within their party, and I don't think the weaker preferences should take precedence over the stronger preferences. Also, points 2 and 3 are in direct conflict, and I care about point 3 because it encourages depolarization.

The proportionality guarantees mean that no matter how much strategic jankery happens, as long as Far Left voters give Far Left cands higher scores than Center Left cands and vice versa, and all Left voters give Left cands higher scores than Right cands (and again, vice versa), then both 1. and 2. have to hold.

The stringent conditions (e.g. "all Left voters") make these guarantees seem weak to the point of irrelevance; "strategic jankery" that is well-justified and outside your allowed parameters will void these guarantees entirely. And strictly speaking, I don't think these guarantees are strong enough to prove what you want. Like, if most Left voters give the Left candidates a score of 3 and Right candidate a score of 0, but slightly less than a full quota of Left voters give the Left candidates a score of 5 and the Right candidates a score of 4, these latter Left voters will function like Right voters who will fill the quotas of Right candidates. Contrived, I know, but still. you need to assume a lot, including some pretty unreasonable things, in order to get a mathematical guarantee.

Conversely, in Allocated Score (with or without runoffs), we can only get guarantees 1. and 2. if every Left voter min-maxes their candidates. And in fact if they start peppering 2s to the opposing party I think it's quite likely that they will lose seats.

True with respect to guarantees (though I don't care about guarantees). As for the latter point, more precisely they can lose seat. Most of the seats will be decided by the filling of quotas; so long as their voters don't fill the quotas of opposing candidates, giving 2s to the opposing party is only harmful for winning the final seat. Still a solid argument against giving out these 2s, but I don't think it's an overwhelming one.

I think our big disagreements are (1) Should a voting method favor moderates over extremists? and (2) Are formal guarantees valuable? Our disagreements seem to be more over what a voting method should do than over what certain voting methods will do. I am not particularly optimistic about coming to an agreement on these points, but I think the agreements we have reached are valuable.

@andy-dienes I agree with you that the chicken dilemma is nowhere near as bad as it's often made out to be; I've written about it for single-winner voting methods and the mitigating factors for Approval Voting there should apply to a lot of chicken dilemmas under PR as well.

Well, maybe, but one could also speculate that if you score your backups a 2, but other voters use higher ratings for their candidates, then all seats might be filled before the threshold reaches 2, so you don't have that "security." I'm not saying that this is an unreasonable concern per se, but I don't think the incentives are nearly so cut-and-dry as you are making them out to be.

There are obviously some cases where you should give a candidate who isn't one of your favorites a high score, such as when your favorites are dark horses with virtually no chance of winning a seat. But in single-winner Approval Voting, when most voters are voting for a lot of candidates it makes sense for you to vote for fewer candidates and vice versa. I'm pretty sure the dynamic is still present here, and, conditional on all the seats being filled before the threshold reaches 2, doing something akin to bullet voting seems like it should typically be a good strategy.

I think it would help me if you give specific-yet-parameterizable examples of types of situations where this rule will do poorly, but a more utility-based one will do well.

There are two parties, Left and Right, and two sub-factions within each party, Moderate and Extremist. So you have Far Left, Center Left, Center Right, and Far Right. There are 2 candidates within each sub-faction. The sincere preferences for a Far Left voter, on a 0-5 scale, average 4.5 for Far Left candidates, 3.8 for Center Left, 1.5 for Center Right, and 0.1 for Far Right (different voters within each sub-faction have slightly different preferences). For Center Left voters it's an average of 3.8 for Far Left candidates, 4.5 for Center Left, 1.5 for Center Right, and 0.1 for Far Right, and for voters on the Right it's symmetrical. The is substantial uncertainty over how many voters are in each sub-faction, so we don't know how many seats the Left will win altogether. (We could also suppose that ~10% of voters are completely non-partisan and will ignore the factions altogether, for the purpose of incentivizing free-riding, though we don't really need that for this analysis.) You can treat the numbers of voters and candidates in each sub-faction as tunable parameters, and the uncertainty is another parameter.

Here's how I see it going under some different voting methods:

• STV: Voters can't really benefit from strategy. The preference for voters on the Left for Center Right over Far Right is irrelevant, except maybe for the final seat.
• PAV, SPAV, MES with Approval ballots, etc.: Voters are best off either voting for all the candidates in their party (since they care more about winning more seats for their party than about optimizing who wins within their party) or only voting for their sub-faction (since these voters are the ones who affect which candidate within a party will win, and doing so doesn't reduce your party's expected number of seats won by all that much). I don't have a great feel for what the equilibrium looks like quantitatively, but voting for a moderate in the opposing party is a terrible move. The preference for voters on the Left for Center Right over Far Right is even less important than under STV.
• Threshold MES: Analyzing strategic voting under Bucklin-based methods is a pain, in general, since it depends so heavily on the details of what other voters are doing. Here though, I think we can get a good approximation to optimal strategy just by noting that every round is basically an MES election, and since voting for an opposing moderate is a bad idea under MES it's a bad idea in every round here. For a Far Left voter, I think it's best to give 4s and 5s to Far Left candidates, some combination of 0s, 1s, and 2, to Center Left candidates, and 0s to everyone else. A rigorous analysis is extremely difficult, but I'm confident that opposing moderates should be given 0s. This favors moderates no more than the voting methods that use Approval ballots.
• Allocated Score: First, let's suppose that voters are giving scores of 3-5 to all the candidates in their party. In this case, giving 2s to the opposing moderates is sound; you're unlikely to end up in their quota, but it will help them a lot against the opposing extremists. That said, giving the opposing moderates 2s does hurt your party in the final round. And you could help your sub-faction the most by giving everyone in it a 5 and everyone else a 0 (and if enough voters do this it means that giving opposing moderates 2s could easily land you in their quotas). There are several competing interests to consider, and ultimately I think voting about honestly makes the most sense so long as other voters are doing the same.
• Allocated Score with runoffs in each round: Similar to Allocated Score, but the benefits of giving candidates (including those in the opposing faction) different scores are heightened. The runoffs also reduce the cost of giving 4s to moderates in your faction if you're an extremist since there could be a runoff between one of your moderates and one of your extremists.

Of these methods, I think it's only Allocated Score and Allocated Score with runoffs that favor moderates an appreciable amount when voters are competent at strategic voting.

I like your idea of creating a bunch of named scenarios for proportional methods akin to the chicken dilemma and center squeeze for single-winner voting methods. I think computer simulations are ultimately best for making quantitative comparisons (at least usually), but named scenarios are great for understanding what's going on.

@andy-dienes Okay, you're right about the chicken dilemma and other voting methods - STV avoids it, but I don't know what else does. But note that, if voters "chicken-dilemma each other into oblivion" it doesn't mean everyone votes A: 5, B: 1, (or the reverse) it means that everyone votes A: 5, B: 0 (or the reverse). The incentive for the more extreme version is pretty significant, though this isn't all that different from Allocated Score in the same scenario.

Perhaps this is a philosophical disagreement more than a mathematical one, but I believe throughout this entire design process of evaluating cardinal PR rules there has been too much bias towards 'centrist' or 'moderate' candidates, and this is another example of that. If the voting method selects too many moderates, then voters will start exaggerating their positions more and more; if the opposing party is half moderates and half crazies, then that's unfortunate, but it means I would expect the elected committee to be 1/4 crazies, since that's what proportionality means.

This depends on the exact preferences of the crazies. If they all insist on being represented by another crazy and the moderates are viewed as being virtually the same as the other side, then yes, proportionality means electing the crazies. There's no getting around this. However, if the crazies would feel very well-represented by a crazy and also reasonably well-represented by a moderate, electing moderates instead of crazies seems consistent with PR. In the latter case, I think electing moderates is strongly preferable to electing crazies; not because of the labels, but because I think the preferences of voters in other factions should have some nonzero influence on which of them is elected. And I think this is very important; disincentivizing candidates from further alienating opposing voters is a big deal for depolarization, and I believe the vast majority of the benefits of electoral reform stem from depolarization.

I also have to admit I might not be seeing the same strategy you are, but this assertion

in this regard since parties would be heavily incentivized to get their voters to not give candidates from any other party a score greater than 0

does not make a lot of sense to me. By the nature of the proportionality guarantees (t-SJR, for example), as long as the party can coordinate some threshold score above which to put their candidates, then they can be guaranteed their lower quota, but that threshold definitely need not be 0. For example, a rule of thumb could be "use scores 3,4,5 for intra-party preference, and scores 0,1,2 for other-party preferences."

Suppose every party uses this rule of thumb. Then, if I score the candidates in my party other than my favorite a 2, I can both get the security of supporting those candidates as a backup plan while having a much bigger influence on which candidates within my party win. I can also help my party more by refusing to give anyone outside my party more than a 1; that way, in the round of 2+, my party benefits from approvals from other parties while other parties don't benefit from my approvals, and I get the bit of added security of supporting more acceptable parties with a 1. The rule of thumb you describe is not a strategic equilibrium.

I see strategic voting in Bucklin voting as being approximately, "only rank candidates you'd vote for under Approval Voting". Threshold MES is similar, but with "only give candidate you'd vote for under a proportional form of Approval Voting a nonzero score." I expect any voting method for which this is true to be very slightly worse than STV at depolarization.

I think your example and my first example have a common problem: they're too clean. Realistic strategic voting involves a mix of getting your faction to win as many seats as possible, get the best people within your faction to win those seats, and getting the most tolerable people in opposing factions to win their seats. Strategic voting centers around making good tradeoffs, in toy examples with only three types of candidates can't capture the tradeoffs you'll find in more realistic elections. (This is really annoying, because anything that can properly capture these tradeoffs will be too complicated to be a good toy example.)

I'm worried about the strategic implications of this. In the single-winner case this is Majority Approval Voting (assuming you're using the Droop quota), and even with more winners this has all of those strategic issues.

• Suppose you are part of a faction that comprises ~1.2 quotas and has fielded two candidates, one of whom you like more than the other (you hate every candidate outside of your faction). Your faction is very likely to win one seat, but it won't win two. Here you want to have your ballot count only as an approval for your favorite as far into the tabulation as possible, so you shouldn't give your second choice a score greater than 1. Even bullet voting is reasonable; if the vast majority of voters in your faction are giving their second choice a 1, bullet voting is the only way to make your ballot count in the decisive round. (This is, of course, the chicken dilemma.)
• Suppose there's a two-party system. The other party has both moderates you tolerate and extremists you hate, and you'd rather they elect more moderates than extremists. Still, the main thing you want is for your party to win more seats than them. Here, giving even a 1 to a moderate candidate in the opposing party is a big mistake. It won't make a difference at all until all scores of 1+ are counted as approvals, and at that point you still want to favor your party's candidate over the opposing moderates.

These are simplified examples, but the simplicity is not necessary for such problems to manifest. I'm not claiming that it's never strategically optimal to give a candidate a 4, but it's pretty atypical. It's strategically optimal to use lower scores than other voters do, such that they're supporting both their own favorite(s) and your favorite(s) while you're only supporting your favorite(s). The only equilibrium involves 5s for a voter's favorite(s) and low scores for everyone else.

Under Threshold MES, voters who ignore strategy lose a lot of influence, and when voters are strategic the expressive power of the 5-star ballot is mostly wasted. I am pessimistic about it reducing political polarization any more than any other PR method would; I actually expect it to be worse than STV in this regard since parties would be heavily incentivized to get their voters to not give candidates from any other party a score greater than 0 and this would encourage divisive attacks. Similarly, candidates would be incentivized to play exclusively to a party's base (and perhaps to some "sucker" voters in other parties who vote as would make sense in single-winner STAR).