Threshold MES

[Marcus Ogren]

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).

[A Former User]

@marcus-ogren Hi Marcus,

In the first example (chicken dilemma) that is going to be an issue for any proportional method. In fact, I would maybe even go as far to say that if such strategy is not a consideration, then your rule isn't proportional.

Fundamentally, if you have one quota of voters, half of whom like A > B and the other half of whom like B > A, then only one of them can get elected, and both factions will try to exaggerate their preference between the two.

To that end, I believe this rule deals with this kind of situation better than many of the others we've discussed. I believe that because of its guarantee to give what I am calling Split Justified Representation (SJR), which is basically lower-quota even for parties who face intra-party vote-splitting (like this chicken dilemma example). It says if there is a T-cohesive group, each voter approving B out of some set of T candidates, then they will get at least B winners.

Even in the worst case where they chicken-dilemma each other into oblivion and one half rates A: 5, B: 1, others: 0, and the other half rates A: 1, B: 5, others: 0, then still one of either A or B is guaranteed to win. The same is not true for, e.g. Allocated Score.

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.

By way of example, I strongly strongly feel that in the following instance (for 0 < x < 20)

40 + x%: A5, B3, C0
60 - x%: A0, B3, C5

That the best, most fair, most intuitive, and most proportional outcome is AABCC. If you select BBBBB like some more linear-utility-respecting rules would, then all you're doing is telling voters to make their ballots as exaggerated as possible.

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."

I do not mean to say that such incentives do no exist; they do, and they have to because you cannot have proportionality without them. But I would expect such issues to be at most as bad, and probably strictly less bad, than they are in the linear-utility-respecting rules like MES, SSS.

[Marcus Ogren]

@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.)

[A Former User]

@marcus-ogren said in Threshold MES:

STV avoids it, but I don't know what else does.

Right, exactly and STV does not satisfy Justified Representation, which is (personally speaking) my minimal demand for proportionality. Anything which gives Justified Representation will exhibit a chicken dilemma.

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.

I'm still not sure I'm seeing why there's an incentive to do so; if they all vote A5B0 or A0B5, then very possibly neither candidate will be elected, which is far worse for the group than either one.

Let's look at more abstractly the game of chicken because I think there is something a little subtle going on here.

Even in the case that we have

• Two identical groups of rational agents
• A larger payoff for either group if they defect (so defection is best-response for each individually)

That does NOT necessarily mean that the incentives will result in both groups defecting, because a simultaneous best-response is NOT always an equilibrium. Chicken is a classic example of an anti-coordination game, and there are multiple strategic equilibria involving unequal strategies for the two groups (and also mixed equilibria involving defecting / compromising with some probabilities).

How that's relevant to voting, is that just because there is a coalition split half A>B and half B>A, we shouldn't assume that the steady-state strategic equilibrium will be both sides burying the other. More likely is one side capitulates, and which side that will be is essentially random (in this example).

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

The rule of thumb you describe is not a strategic equilibrium.

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.

only rank candidates you'd vote for under Approval Voting

disincentivizing candidates from further alienating opposing voters is a big deal for depolarization

Well, kind of yeah, but this intentionally the design. I included a very brief similar discussion in the (pretty janky) electowiki page I made just to have something to reference, but this method entirely rejects the assumptions that scores should be linearly additive over sets, or averaged over voters, etc, and instead treats the scores as thresholds for lower quotas of representation.

Remember that with the linearly-additive interpretation of scores, parties are punished if their voters are not maximally cohesive and if they try to express any intra-party preference, so it is "polarizing" in a different way. 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.

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.

Agreed. My favorite kinds of examples are those that don't depend too particularly on the exact numbers (this is what I mean by "parameterizable" ) as I think they give some of the best pictures of how a method actually behaves. Maybe to distinguish we can call these "scenarios" rather than examples, and I like to give them names; probably we should compile a battery of these and make some kind of chart for what different proportional rules do.

• Chicken dilemmas
• Center squeezes
• Highly polarized clusters
• Justified representation
• Laminar vote splitting: a coalition all agree with its top ratings on some popular candidates, but lower ratings fragment into independents and sub-parties.
• Compromise vote splitting: a coalition's top ratings are fragmented over independents and sub-parties, but their lower ratings all agree on some popular candidates.
• Cyclic (down-ballot) vote splitting: some mix of the above two

etc.

[Marcus Ogren]

@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.

[A Former User]

@marcus-ogren said in Threshold MES:

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

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. Even if one is a staunch utilitarian, I think ultimately voters should be modeled to have utilities over decisions that the committee makes and not the individual members of the committee; that is, I think it is much too strong an assumption to presume that it is even possible in the first place for a voter to assign 'sincere' utilities to each candidate such that their utility for any set of those candidates is equal to the sum of candidate utilities.

I prefer to think of this method (which I am now calling Threshold Equal Approvals because @Keith-Edmonds suggested a rename and that spells TEA , haha) as instead interpreting each score as answering the question "would you like to be inside or outside this candidate's 5 (or 4/3/2/1)-star coalition"

Anyway let's consider the scenario you suggest (I guess we can name it "Single Peaked 1d" ?)

Far Left: A > B > C > D
Center Left: B > A > C > D
Center Right: C > D > B > A
Far Right: D > C > B > A

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

1. Left gets at least (w+x)% seats and Right gets at least (y+z)% seats
2. 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
3. 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)

With TEA, I actually agree with your assessment that it will usually be best for Left to only rate Left candidates, and the same for Right (unless they feel strongly about certain independents or certain candidates in the other party). But that's not a function of the method, that's just due to the way the profiles are set up where there are two very clear coalitions. 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.

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. If I am not mistaken, it seems you are mostly concerned that TEA does not incentivize voters to express much preference over candidates in opposing parties; I have the converse concern that Allocated Score does not incentivize voters to express much preference over candidates in their own party.

I have to admit, when I read statements like

I think it's only Allocated Score and Allocated Score with runoffs that favor moderates an appreciable amount

It feels like we are jumping straight to guarantee 3. from above, and skipping 1. and 2. which I personally view as more significant. Again this might come down to a philosophical disagreement, but I do not think that "favoring moderates" is a good thing when proportionality is the goal; I do not think any types of candidates should be favored except for those best representing the electorate in miniature, so to speak.

Although, I am worried we might be starting to go in circles , not that this is any fault of yours or mine, it is just the nature of these kinds of discussions when they're one-on-one. Maybe we should try to get more (and fresher) eyes on the problem.

[Marcus Ogren]

@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.

[A Former User]

@marcus-ogren said in Threshold MES:

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.

Not to start bikeshedding, but

Red (33%): A5 B0 C0 D2
Blue (33%): A0 B5 C0 D2
Green (33%): A0 B0 C5 D2

To elect 3 candidates. To me, clearly ABC is the correct committee, and this is what TEA elects. However, with linearly additive utilities, one might be fooled into thinking that DDD is the correct committee (in fact, it Pareto dominates ABC !), and this is what AS or SSS or MES elects (with or without runoffs).

Otherwise yes I think it seems we both understand each other's viewpoint and still disagree; this is ok, such is the burden of judgement, as Rawls would put it.

[Marcus Ogren]

@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.

[A Former User]

Added a Julia implementation to the electowiki page. If people want it in Python I can translate.

[Keith Edmonds]

@andy-dienes said in Threshold MES:

Added a Julia implementation to the electowiki page. If people want it in Python I can translate.

My preference is python with heavy use of the pandas library. This should help it be short an clean. Or at leas shorter than what you have there. We do not really want production code but a very precise definition.

[A Former User]

@keith-edmonds DataFrames.jl is essentially the same as pandas so unfortunately I do not think the logic can be made appreciably shorter (without golfing the code and making it harder to follow)

There are some pieces I could cut though if you just want it visually shorter

1. cut whitespace & comments
2. remove the isapprox checks---these in theory should not be necessary, but without them you can get bad results when some candidate gets precisely one quota (because of floating point imprecision)
3. remove the option to allow clones
4. remove the example with Tennessee capital cities

I'll put a Python version up and you let me know which components if any should be cut

[A Former User]

Ok, I added a Python version as well. As requested, I made as liberal use of Pandas functionality as I could. Personally, I find the more spelled-out implementation easier to follow, but pd magic is undeniably concise in some places. It's probably a good idea regardless to have two very different implementations---this way we can validate them against each other to find errors.

[Keith Edmonds]

@andy-dienes Awesome. Pandas definitely can make things concise if done right. There are lots of functions.