Multi-winner Voting: a question of Alignment

post by Jameson Quinn (jameson-quinn) · 2018-04-17T18:51:09.062Z · score: 59 (15 votes) · LW · GW · 19 comments

Contents

  Why isn't multi-winner voting theory just a generalization of the single-winner kind?
  Proportionality
  Values and beliefs
  Parties
  Voter strategy: free riding and vote management
  Pragmatics (1)
  Proportional Method Lego
  Pragmatics (2): I think PLACE is awesome
None
19 comments

This is my third (and for now, last) essay about voting theory for rationalists. In the first two, I focused primarily on single-winner voting theory; that is, methods for aggregating group preferences into a final verdict on some choice. Ideally, single-winner methods would be used in cases where decisions are inherently collective, while other mechanisms such as markets are better for cases where decisions are more individual. (As I touched on in the earlier articles, Sen's theorem puts limits on how precisely that distinction can be made; but that's not the point here. I'm going to take it as given that there are some cases where collective action is called for and others where action should be left up to individuals, and I don't want to spend time here arguing about the relative frequency or importance of those two kinds of situations.)

Why isn't multi-winner voting theory just a generalization of the single-winner kind?

If we've covered the best means for collective decisions, and individual decisions are out of the scope of voting theory, then what's left? Governance. That is, cases of collective action that aren't a single decision point but an ongoing series of decisions.

Such cases probably aren't best served by a series of separate single-winner elections, for a few of reasons. To begin with, it's not cognitively efficient; it would be silly for every citizen to need the expertise in order to make decisions about the minutia of every policy area. In fact, direct democracy tends to favor negative-sum rent-seeking: small groups extracting concentrated benefits by imposing diffuse costs, merely because they're the only ones motivated enough to sweat the details. And finally, it's not predictable: in many cases, governance should be coherent even at the sacrifice of some responsiveness.

In cases of governance, voting is not the final step, but merely one step in a larger process of decision-making. Thus, traditional multi-winner voting theory would look at ways to resolve this by electing a set of representatives to take those decisions.

To the rationalist community, such a multi-step process immediately raises the question of alignment. Just as designers of artificial intelligence should worry about whether their initial goals will be warped into a contrary outcome through the process of design and improvement, so should people like me designing multi-step mechanisms of governance worry about how values are preserved, lest small misalignments in each step add up to major disconnects in outcome.

Proportionality

Of course, if you're worried about preserving some property over a multi-step process, the first thing to do is to define that property. In this case, the key property is the proportions of decision-makers with each given set of utilities. Proportional multi-winner voting methods are those that are designed to (roughly) preserve these proportions. Thus, collective decisions can be made by smaller groups; the ugly dynamics of mass argument can replaced by the hopefully-healthier ones of a smaller group. (Though flawed, the concept of "Dunbar number" is relevant here.)

Note that voting theory itself has nothing to say about how to define the original group whose proportions should be preserved. That is, it doesn't answer questions of who should be able to vote or how many votes each voter should have, in defining the original proportion. I'd argue that the safest and ultimately best rule today is for each human above a certain age to be allowed to vote; but that's out of the main scope of this article.

In stating the goal of "proportionality", I've been deliberately a bit vague about defining it. If voters come pre-sorted into comprehensive and mutually-exclusive partisan sets, it's relatively easy to define "Droop proportionality", in which each party gets a minimum proportion of seats in the legislature. But what if divisions of opinion are more complicated than that — continuous and/or multidimensional? In that case, there are various desirable proportionality properties, and some degree of tradeoff between them.

As a statistician, I should mention that there is one democratic "voting" method which will satisfy every possible proportionality property, at least "asymptotically" as the legislature grows in size. I'm talking about random sampling, or, as it's called when used for governance, sortition. By the law of large numbers, a random sample will tend to resemble the underlying population in proportions as to any and all individual characteristics, at least if the sample is large enough. In practice, sortition is rarely used for governance, though advocates of "citizens' assemblies", "citizen juries", "deliberative polls", and the like are trying to change that.

If we require voting methods to be deterministic, there are still a number of methods that have been designed to ensure proportionality; all such methods are called "proportional representation". (Since that's a mouthful and PR has too many meanings already, the best abbreviation is prop-rep.) In general, though perfect proportionality is impossible, most prop-rep methods come close enough that their other, more-pragmatic differences are more important.

Values and beliefs

Of course, representation (proportional or otherwise) is a goal regarding values, but decisions are also based on beliefs; and when it comes to beliefs, the goal should be truth, not representation. The idea of futarchy is about creating a political system that separates values and beliefs, so that values are resolved using a voting method (presumably one where any sub-steps preserve proportionality), while beliefs are resolved using prediction markets. While I'm skeptical of the possibility of designing markets that are immune to bubbles, values-based manipulation, or other systematic distortions, I think that the idea of trying to design a system that respects the separate logics of both values and beliefs is a good one.

Note that the current US voting system actually does try to do this to some extent, it just does a really crappy job. If political parties were groups of people with perfectly homogeneous values, then party primaries would not be the worst way of selecting smart, knowledgeable people with those values and thus of getting a slightly extrapolated volition as compared to mere sortition. Of course, we know that in many real-world cases, primaries are more about ideological litmus tests than qualifications like expertise or intelligence.

Still, that suggests that proportional voting methods should probably include mechanisms for both intra-party and inter-party selection of candidates. In particular, closed-list proportional methods, which offload intra-party selection to some partisan mechanism probably dominated by insiders, are a bad idea.

(A related dichotomy is that between instrumental rationality, which involves both values and beliefs, and epistemic rationality, which involves only beliefs. So this issue can be seen as about finding ways to decrease the misalignment between the incentives for an instrumental and an epistemic rationalist.)

Parties

Another important question about voting methods is the party system they encourage.

First question: should there be parties at all? Though some people would disagree, I'd suggest that parties play an inevitable, and in some regards a positive, role in a political process. Yes, they do have bad effects, such as mind-killing tribal thinking; but they also have good ones, such as serving as useful cognitive heuristics for voters, and possibly allowing intraparty sorting to have more of a focus on qualifications and ability rather than ideology. Furthermore, even if you do believe they are bad on net, getting rid of them is really hard. Metaphorically speaking, if you try to design a voting system that bars the door against parties, you may find that they just make a hole in a load-bearing wall as they force their way in anyway.

Second question: how many parties should there be? Too few, and you get a stagnant "monopolistic" or "duopolistic" system in which zero-sum thinking leads to negative-sum outcomes. (For a real-world example, look at the USA.) Too many, and you encourage politicians who make narrow, single-issue appeals. (For a real-world example, look at Israel.)

Political scientists often view the distinction of few or many parties as by considering the representative voting method as just one step in a larger process of forming a majority coalition to take a societal decision. In other words, they speak of systems which encourage few parties as encouraging pre-election coalition-building, and those that encourage many parties as encouraging post-election coalition-building. In my view, it's good to have a little of both.

A useful way to measure number of parties is "effective number of parties" (ENP). The formula is

, where s_i is the size of party i as a fraction of all voters. Intuitively, this is the reciprocal of the fraction of voters in the party of the average voter (thus naturally weighting larger parties more). In other words, if the average voter's party size is 1/3 of the electorate, then ENP is 3. I'd aim for something between 3 and 4 as ideal.

I'd argue that choosing a voting method that tends towards such a moderate ENP will also tend to encourage better rationality within the legislature. As I said above, in a two-party system, with only one ideological dimension, winning or losing the eternal battle against the other side is all that matters, and so norms of debate (including rationality norms) go out the window. And a highly fragmented world of single-issue parties actually has exactly the same problem; since each party is focused on just one issue, they have no reason to subscribe to overarching norms. It's only when there are more than two parties which each care about more than one issue that norms become selfishly worthwhile to each; though the norms might work against them on any one issue, insofar as they're positive-sum norms they will tend to work for each party's interests more than they work against them.

Voter strategy: free riding and vote management

In essentially all proportional methods (except weighted/proxy systems), an individual voter has an incentive not to vote for a candidate whom they know will win anyway, in order to avoid having any of their voting power "used up" by that foregone conclusion. But even though this incentive exists to some extent across many methods, its strength varies. All else equal, it's better to look for methods where this incentive is relatively weak.

On a collective level, this incentive is somewhat self-limiting. That is, if nobody votes for a popular candidate just because they're a "sure thing", then that candidate won't win after all. So collectively this incentive isn't so much for "free riding" as for "vote management": giving each candidate exactly the minimum number of votes they need to win. For instance, a party might try to equalize the number of votes that favor each of their candidates by instructing voters to vote based on their birthday.

Pragmatics (1)

So from the above, we're looking for a voting method that's reasonably proportional; that allows voter input on both within-party and between-party choices; that encourages a moderate number of parties; and that has a relatively weak free-riding incentive. That is an underdetermined set of constraints; there are a number of methods which do all of those to (what I'd consider) a pretty good degree. To choose between those proposals, we can add in pragmatic questions. Which methods are easiest for voters? Which are easiest to count? Which are likely to be most politically viable (which includes being non-disruptive to incumbents, at least, when disruption doesn't serve a useful purpose for any of the values above)? Which have the best track record?

Proportional Method Lego

Most proportional methods can be thought of as combinations of a few basic building blocks:

Combining the above building blocks, we can build various voting methods:

Pragmatics (2): I think PLACE is awesome

I'm going to switch from just explaining multi-winner voting theory to advocating for a specific method, so I should start out by explaining where I'm coming from. I'm a US activist for voting reform; on the board of the Center for Election Science (electology.org). My object-level politics, and my social milieu, tend to be pretty much on the left of the spectrum, but I also have real meta-level politics in favor of democracy. Ask me about any given issue and I'll happily explain why my own views are smarter than those of the median voter; but across all of those issues I know that the crowd is probably wiser than I am as often as not.

I've been thinking seriously about voting theory for over 20 years, and it's the main reason I am now getting a doctorate in statistics. In that time, I've designed many voting methods. The ones I consider best (3-2-1, PLACE, EPH, and SODA) are designed to optimize on the characteristics I think are important. When I argue for these methods, of course I'm biased. But I'd suggest that when I argue "My method is best normatively because it optimizes characteristic X", you should question my bias more by disputing whether X as I've defined it is important than by wondering whether the method actually optimizes X.

So what do I think is important in a practical proposal for a multi-winner method? It should:

PLACE voting was designed with these characteristics in mind; it does reasonably well on all of them. All other methods I know of fail significantly on several characteristics. (In fact, it took me decades of learning about voting theory, followed by almost a year of concentrated design work for hours a week, to settle on PLACE.)

Down in comments, before I finished this article, there was already a comment criticizing PLACE (from somebody who knows me from elsewhere). I understand that the criticism, that voters may find delegated methods distasteful, is real. I don't think it's as serious as it would be to fail on the other characteristics above.

If you're interested in activism on this, contact me. PLACE is compatible with the US constitution and current law, so it could be done by either state or federal legislation. I'm looking to get this passed somewhere (Somerville, MA?) at a municipal level first (there's a nonpartisan version that's appropriate). My email is firstname dot lastname at google's public email service. I'd also encourage you to support the Center for Election Science. Even if you're in the UK or Canada (especially BC), I can help hook you up with local movements for reform.

19 comments

Comments sorted by top scores.

comment by zulupineapple · 2018-04-19T08:11:11.785Z · score: 13 (3 votes) · LW · GW

Surely proportionality is not the correct goal for elections. At least, when you remember that people aren't identical copies of one of a few "kinds", it becomes less obvious what proportionality means.

I propose this. First, suppose a perfectly educated population would vote on every bill (using some single winner method X) and remember their choices. Second, let the same population elect a government (using some multi winner method Y) and then let that government vote on every bill (using some single winner method Z). The multi winner method Y is "good" if the outcomes of the first simulation (running X) are matched by the outcomes of the second one (running Y followed by Z) - I believe this is the correct justification for electing a government. Or you could just measure utilities more directly, to be fair.

Of course, proportionality correlates with this definition of "goodness" (and if there were perfectly homogenous voting blocks, and if X=Z, I think the correlation would be 1). But, by Goodhart's law, correlation isn't good enough.

One important observation here is that the "goodness" of the multi winner voting system is strongly related to the "goodness" of the voting system used by the elected officials. I.e. if the senate's voting system has some pathologies, then the we might want to correct for those pathologies in the general election.

Speaking of which, what voting system does the senate use? I think they vote on two options (yes/no), one bill at a time, but in reality there is more than one option at a time, which might lead to some unique pathologies. E.g. "I like the bill A, but I like bill B even more, and if we pass A, we won't even vote on B, so I'll vote against A".

comment by Kaj_Sotala · 2018-04-18T08:48:39.832Z · score: 12 (2 votes) · LW · GW
Defining "expected number of parties" (ENP) as the reciprocal of the fraction of voters in the party of the average voter

I keep running out of working memory trying to parse this sentence. :)

comment by Jameson Quinn (jameson-quinn) · 2018-04-18T10:38:14.063Z · score: 10 (2 votes) · LW · GW

Take N voters at random. Ask each of them their favorite party, and check to see how big that party is as a fraction of all voters. Take the average of those fractions. Take the reciprocal. Voila: an estimate of ENP. (If you want a better estimate, use a bigger N. But even N as low as 10 will get you a reasonable guess.)

Example: you choose 4 voters. 2 of them are from a party with 40%, one has 25%, one has 15%. Average is 30%, reciprocal is your estimate of ENP: 3.33.

Size of party counts twice: once explicitly, and once in the probability that a party will be sampled. So the formula is just one over the sum of party-size-squared. But I think my version gives a better feel for why it matters.

You can usually google up ENP estimates for most countries, both in terms of voters and in terms of parties in the legislature. The latter number is usually lower. It's around 2 in the US and over 6 in Israel.

comment by ChristianKl · 2018-04-18T10:13:22.186Z · score: 11 (2 votes) · LW · GW

When it comes to using sortion, it would be interesting to have a system where the congress is normally elected and then the upper house which can either approve or reject a bill is made up via sortion.

Sortion seems to a more satisfactory way to create an upper house that's independent of the political parties than UK's system of lords and bishops.

Given that one state two votes for the senate seems very undemocratic given population differences it might also be a good way to replace the US senate.

comment by Jameson Quinn (jameson-quinn) · 2018-04-18T10:40:49.922Z · score: 4 (1 votes) · LW · GW

I agree that that would be a good idea. But it would take a constitutional amendment, and politically that is really really hard. I'm trying to focus on proposals where I think that the chances of real progress within 10 years are over 1%.

comment by parker friedland (parker-friedland) · 2018-04-18T01:12:55.991Z · score: 9 (3 votes) · LW · GW
... to be continued

Let me guess: Your next article is going to mainly be about place voting, right?

Which are likely to be most politically viable

Before you just assume that PLACE voting is actually the most politically viable multi-winner voting method, perhaps you should conduct a poll to see what voters actually think about the method as well as what voters actually think about other multi-winner voting methods. You might also want to poll voters on how much they like the idea of delegated voting in general. And if voters are skeptical of delegated voting methods, are their skeptical opinions about such voting methods justified? Do delegated voting methods maintain the same level of political accountability as other proportional voting methods? Can slimy politician A delegate their votes to slimy politician B whom A (and A's donors) know will vote differently on many issues then they will, and then pretend to be surprised when slimy politician B does vote differently then they will and deceive his base about the issues he cares about with this acting? Because it takes far more work to learn about all of A's policy positions then all of the political "veiws" of every person A has delegated his votes to, will there be as much as a backlash among A's supporters when the people he has delegated his vote to vote against his "veiws" then when A himself votes against them. Can delegated voting lead to bribery where politicians make secret deals to delegate their left over votes to somebody else in exchange for campaign contributions? It is questions like these that make me (and will probably make many others) very skeptical of delegated voting methods. They are still far and away better then the status quo, but I don't believe that delegated voting methods such as PLACE are the best way to advocate for PR.

comment by Jameson Quinn (jameson-quinn) · 2018-04-18T02:08:16.630Z · score: 10 (2 votes) · LW · GW

Actually, this article is still a work in progress. Yes, about a third of the part that's missing will be about PLACE.

I have cogent responses to the points you raise, but let's wait until the article is finished, so that we're not just following a private conversation carried over from previous interactions.

comment by Sniffnoy · 2018-04-28T01:59:40.382Z · score: 8 (3 votes) · LW · GW

The formula for ENP seems to have gone missing?

comment by daozaich · 2018-04-22T01:03:57.327Z · score: 8 (2 votes) · LW · GW

Re PLACE: Interesting proposal. Have you considered the following problem (I'd guess you have; a link would be appreciated):

Candidates are not exchangeable. Candidate A has done a very good job in the legislature. An opposing faction may decide to coordinate to support his local opposing candidate B, in order to keep person A out of parliament.

Or, in other words: Two candidates running in the same district cannot both become part of parliament. This opens a huge amount of gaming, in order to squash small parties / factions that do not have a deep bench of good candidates. Party A and its voters have large influence on the composition of Party B's parliamentary faction, and can strategically plan this.

The standard solution, e.g. in German federal elections, is to have pools: Candidates can be elected locally (in-district) or statewide. Candidates who won statewide are expected to still try to represent their district (where they lost), if they ran locally at all.

comment by Jameson Quinn (jameson-quinn) · 2018-04-22T14:47:54.255Z · score: 4 (1 votes) · LW · GW

Yes, this is the downside of biproportionality.

This is a problem, but at least there is a defense. Assuming candidate A has more local votes than candidate B (which is pretty much a necessary condition for this to be a "problem" in the first place), then if candidate A gets enough direct out-of-district votes to reach a full quota (average-district-worth of votes) without transfers, they win, regardless of how well the anti-A voters coordinate. That's because if two candidates both reach a full quota simultaneously (in this case, in the first tally), the tiebreaker is local votes. So in practice, candidate A's party would need to give them about half a district-worth of out-of-district votes; perhaps they would decide not to run candidates in a few of their weakest districts, and campaign for their voters there to give their votes to A.

Still, biproportionality can give suboptimal results insofar as there are two candidates from district n who are both better/more-popular than any candidate from district m. This will happen by chance occasionally, but hopefully not too severely.

Having "leftover" statewide seats so that two candidates could win in a given number of districts would fix this problem. But some people would probably see that as unfair: "how come their district gets 2 representatives and mine only gets 1?" My feeling is that, at least in the context of proposing a system for electing the US House, it's probably better to keep PLACE "simple" and strictly biproportional — especially because that also makes it less disruptive to incumbents.

(I've given you a response rather than a link because this question, while it's one I've thought about, isn't one I've been asked before.)

comment by daozaich · 2018-04-22T16:18:38.074Z · score: 4 (1 votes) · LW · GW

I guess that this is due to the historical fact that candidates in the US are supposed to be district-local, not state-local, and districts are supposed to be as small as possible. I'm not an American, so I cannot say how strong this is as a constraint for modified electoral systems.

If you had a small party/faction, with say 10% of popular vote, reaching up to maybe 30% in their strongest districts, then I would definitely see a problem: Such a party simply does not fit purely district-local representation (one-to-one mapping between districts and representatives). Think e.g. an environmentalist party.

If your representative chamber is more about opposing ideologies and national governance than about opposing local interests, then why not ditch this one-to-one mapping?

I mean, this works even in the EU parliament, and you can't tell me that opposing local interests across US districts are harder than across states within EU countries? And you have a second chamber (the senate) that is explicitly about conflicting local interests.

I presume that you know how German federal elections, or European parliament elections are run; easy to google, far from perfect, but gets at least this right. I would definitely be opposed to a change to PLACE at home, for these reasons, but agree that PLACE beats FPTP by lengths.

And something that PLACE gets extremely right is to involve the general populace in the within-party selection. In Germany, only party members are allowed to vote in primaries (and members pay contributions and are expected to be activist, and parties can expel or reject prospective members). This slightly sucks; "too radical" candidates / factions are routinely squashed by the high party functionaries.

Of course this gives you an amusing game-theoretical problem in e.g. EU parliament elections: If countries were entirely free to select their voting system, they would have incentive to move to a winner-takes-all system, which strengthens the representation of their national interest. Same way you cannot get rid of your accursed winner-takes-all within states in US presidential elections, as long as states are free to design their voting systems. And, on the same lines, if you must have a winner-takes-all system, then each state should have votes roughly based on square root of population (I am yet again ashamed to be German for the way we treated the Polish, this time after they proposed the square-root for the EU council).

comment by ChristianKl · 2018-04-18T10:06:14.086Z · score: 5 (1 votes) · LW · GW
First question: should there be parties at all? Though some people would disagree, I'd suggest that parties play an inevitable, and in some regards a positive, role in a political process. Yes, they do have bad effects, such as mind-killing tribal thinking;

What evidence do you have that this is an effect of political parties?

The US has weak political parties where the head of the party is a minor political figure. Germany has a system where the head of a party has a lot more influence and there's also less tribalism.

comment by Jameson Quinn (jameson-quinn) · 2018-04-18T10:45:37.249Z · score: 4 (1 votes) · LW · GW

I don't have systematic evidence for that, so I guess I should qualify it. Personally, I'm not particularly anti-partisan, so I was just trying to quickly and fairly summarize the best anti-party arguments I've heard.

comment by Sniffnoy · 2018-04-28T02:36:40.026Z · score: 4 (1 votes) · LW · GW

Question: What about the intermediate case?

That is to say, your previous articles talked about single-winner elections. This one talks about elections where the number of winners is large enough that prop-rep makes sense as a goal. But there's a number of two-winner, three-winner, etc. elections out there. What's been devised for those?

comment by Jameson Quinn (jameson-quinn) · 2018-04-28T21:22:25.010Z · score: 8 (2 votes) · LW · GW

Generally, you can use prop-rep methods, but some of the considerations in choosing which method shift a bit. Complex ballots are not as much of a problem as in bigger prop-rep situations, while premature elimination isn't as big a problem as in single winner. Thus, for off-the-shelf methods, STV or EPH are pretty good in those cases. If I were to design a method for a specific case, I could probably do better, but it might not be worth it.

comment by Jameson Quinn (jameson-quinn) · 2018-04-24T15:13:01.855Z · score: 4 (1 votes) · LW · GW

I realize that lesserwrong is a place for theory, not activism, but as appropriate I'll add off-site links my more activist-mode writing on multi-winner voting here.

1. "The room where it happens" — an article about unrepresented voters, FPTP, STV, and PLACE.

comment by Benito · 2018-04-24T15:34:50.545Z · score: 6 (1 votes) · LW · GW

I feel an impulse to let you know we actually launched and this place is called 'LessWrong' now - you can find us at www.lesswrong.com.

(And yup, big thanks for making your posts not include activism stuff, else they wouldn't have been able to go in frontpage or curated, and I'm glad they're there. It's totally fine to exist elsewhere, just not what LW is about.)

comment by dennisV · 2018-06-27T17:02:37.658Z · score: -4 (2 votes) · LW · GW

Another view of voting is where an objective opinion or belief is being formed about ANYTHING that can be named. It might be a belief about utility where the utility can only be seen subjectively but an objective value would be useful in planning, for instance. Another example is determination of what a given set of users might buy given prior page views, or what the value of a given piece of intelligence is to a given audience at a given time and related to a given topic. Such a voting structure is needed in crowd sourcing and in 'training'/'supervision' in machine learning. It is also necessary for determination of what is or is not fake news, where certain 'voters' might be given a greater weighting to their votes on given topics. As with reducing ranked voting to utility maximization, the problems of using these are complex, even though the intention of statistics in general would seem to be directed to the same facilities. Like machine learning, the issue is the great 'dimensionality' of the problem, and statistical solutions don't work well. In fact, where dichotomies exist (failed praxis), statistics simply fail without the use of probabilistic techniques such as expectation analysis. A system for handling all of this type of consensus mechanisms is on the way. See Patent US20140075004 - System And Method For Fuzzy Concept Mapping, Voting Ontology Crowd Sourcing ... - Google Patents; Patent US20140075004 - System And Method For Fuzzy Concept Mapping, Voting Ontology Crowd Sourcing ... (2nd pointer is to the list of projects that were required to acknowledge the patent as prior art) .

One aspect of interest regarding the edge of statistics mentioned is that in a causality based world (see 'The Book of Why'), the causality must still be addressed even if the statistical approaches don't seem to work. Answers from the two may disagree (where 'common sense' seems to be).

comment by nonwasp · 2018-04-19T21:32:28.949Z · score: -12 (4 votes) · LW · GW

You write a bit about voting, but say nothing about how it represents some interests better than others.

The original intent of the senate system seemed to make political change slow, and to represent each state's legislature. That was changed with direct voting, although the six year terms remain.

You wrote three essays on essentially rehashing scholarly essays. You say some voting systems create moderate numbers of political parties, but I don't know what that is.

The whole point of voting is to represent interests to implement policies. We might as well have transparent elections with votes bought from the federal treasury, at least the money wouldn't be going towards buying all the ad slots in battleground states.