Auros (auros) wrote,

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Occasionally-updated lecture on voting systems.

I recently posted this to the Silicon Valley Dean Democratic Club mailing list, because somebody asked me for a primer on election systems. I keep re-editing and expanding this text, to make it more comprehensive, and, I hope, clearer. I'm trying to create something that's accessible to election-system novices who have at least a basic grasp of arithmetic and logic. In particular, I hope to convert IRV supporters away from that quixotic pursuit, towards better ideas. After I posted it, another SVDDC member suggested it should go up on my blog, to have a permanent public home. It's a complicated topic, but worth understanding, so I hope you'll bear with me.

Everyone understands that there are problems with Plurality Voting, in which the guy with the most votes wins, even when there are many candidates, and the "winner" has less than 50% of the votes and is seen as a "lesser evil" by many of those voting for him. The most common current way to deal with this is a Top-Two Runoff system, and the most commonly suggested reform is the Instant Runoff Vote, in which voters list their order of preference among the candidates. Candidates are eliminated based on having the fewest first-choice votes, and voters whose current top choice is eliminated have their vote transferred to their next preferred candidate. Because there are some fairly well-organized existing organizations backing IRV -- including within DFA -- I'd like to dispense with that first.

I'm aware that plenty of well-intentioned people support IRV. They'd generally be better off backing either Approval Voting or some variety of Condorcet (on both of which, I'll say more lower down). The proportional representation variant of IRV is pretty good for sets of four or more seats (again, we'll come back to this topic lower down). But for single-winner elections, IRV is almost as bad as what we have now, and in one non-trivial regard, it's worse.

Here's the most famous theoretical example of IRV's failings:

Imagine a world in which the Green Party manages to pull over most liberal Democrats, and the Democrats pull in a bunch of moderate Republicans. So, most Greens will vote Green over Democrat over Republican; a fair number of Democrats (who are now conservatives from the old Dems, plus moderates of the old Repubs) will vote Dem over Rep over Green; and most Reps will vote Rep over Dem over Green. Not too implausible, right? (Note that the math below doesn't change in any meaningful way as long as at least 50% of the Dems say D over R over G; and even today, I'd say 50% of Dems would vote Republican over Green.)

On election day, in a town with 100 voters, the returns thus far, with 98 votes counted, look like this:

35 Republicans:R > D > G
32 Democrats:D > R > G
31 Greens:G > D > R

Before I count the ballots under IRV, please note that both the Reps and Dems prefer D over G; and both the Greens and Dems prefer D over R. Since each party has roughly a third of the voters, we can see that candidate D has a whopping two-thirds majority, in a one-on-one race against either of his opponents. Clearly, he is the majoritarian winner under any sane system.

Now, to count the votes: G has the least first choice ballots, so he gets eliminated; 31 votes transfer to D, and he wins. Great!

But wait: there were two more votes! Turns out, they're both Greens:

35 Republicans:R > D > G
32 Democrats:D > R > G
33 Greens:G > D > R

Now, when we run the IRV algorithm, D is eliminated first; R wins.

Note that the situation mentioned above, where about two-thirds of the voters prefer the Democrat over the Republican, has not changed. The Dem should, by rights, be seen as a solid majority choice.

But even worse, the voters who changed the results, themselves preferred the Democrat over the Republican! They would've been happier with the outcome if they'd just stayed home, rather than participating in the election. This "Participation Failure" problem, where a voter's own ballot can make things worse for that voter, even though the "correct" winner is obvious, is the place where IRV is actually worse than our current system. Sure, the scenario may seem a little contrived, but it isn't totally implausible, and if something like this ever happened, the two-thirds who got shafted would (or at least should!) be mad as hell. Unfortunately, most people would probably just say, "Let's go back to what we had," rather than exploring better options. This could set back the cause of real reform for a generation or two.

Here's a second example. Suppose we have five candidates, spanning a spectrum: FarLeft, CenterLeft, CenterModerate, CenterRight, and FarRight. Individual voters have a favorite, and then their preferences expand evenly "outward" from that position. I'll use the equals-sign to show places where the ballots are split roughly evenly on how individuals within a bloc order a pair of candidates. The votes look like this:

35 FL > CL > CM > CR > FR
65 CL > FL = CM > CR > FR
99 CM > CL = CR > FL = FR
65 CR > CM = FR > CL > FL
35 FR > CR > CM > CL > FL

FarLeft and FarRight, having the least first-choice votes, get eliminated first. Their votes flow inwards, to CenterLeft and CenterRight, who now each have 100 votes... meaning that CenterModerate, with 99 votes, gets eliminated.

So, CenterModerate started out with the most first-choice votes, and, if you compare him against any other contender, he has about two-thirds of the votes in any head-to-head race. Yet he still loses! This effect is called the "Center Squeeze" (and when generalized to all systems that have this property of favoring a permanent Duopoly, where only two parties can remain in contention, it's called "Duverger's Law"). Basically, it shows that IRV does not help with the problem of polarization, and in fact will encourage vicious campaigning against ideological "neighbors", because it's critical that your candidate get first-choice votes, not second-choices which may never be counted if you're eliminated early. (In the example just above, CM has a strong incentive to run negative ads against CL and CR. He doesn't even care whether first-choices shift from them to him, or to FL and FR; if he can batter them enough to get them eliminated early, at least a few second and third choices will get transferred to him, allowing him to beat FL and FR.)

So: All the ill effects we associate with Plurality -- polarization, negative campaigning, people focusing more on "us vs them" division than on good policy, etc -- will be unaffected by IRV. And IRV will, in the long run, fail to encourage cooperation amongst parties, because those parties will still have the potential to become spoilers (see: GOP funding a Green Senate candidate in PA). As a friend of mine puts it, in IRV, it's safe to vote a third party as your favorite, as long as they have absolutely no chance of winning.

In the real world, we see Australia, with IRV, having two real parties. They technically have more than that, but in fact, the Labor party generally holds roughly half the electorate, and the next two parties (Liberal and National), plus one small party (the Country Liberals), cooperate as a "super-party" called the Coalition, which also holds roughly half the electorate.

The Coalition parties usually try to avoid running candidates against each other in vulnerable seats (which would be similar to a hostile primary challenge, in one of the US parties). In order to make sure they avoid enacting the Participation Failure situation, the Coalition parties (as well as minor parties like the Greens) distribute How-To-Vote Cards. The cards very occasionally encourage people to vote their own party as preference 2, instead of 1. (Or at least, that is what I've been told by a woman who grew up in Australia, and had been a registered Green.)

The biggest example of "runoff failure" one can point to -- though granted, this was a top-two runoff, not IRV -- is the French Presidential Election of 2002. I'm partially guessing at who should be called right and who left -- in particular, the Union for French Democracy has a mixed history, sometimes supporting the right; but they're strongly pro-EU, and dislike the right's xenophobic tendencies. So, the exact breakdown of the UDF voters' preferences between Jospin and Chirac aren't easy to guess. In any case, here's the breakdown from the first round. You can get this data from a wide variety of sources.

Parties that look like they're on the right:
19.88% Rally for the Republic (Jacques Chirac)
16.86% National Front (Jean-Marie Le Pen)
04.23% Hunt, Fish, Nature, Traditions
02.34% National Republican Movement (extreme splinter from National Front)
01.19% Forum of Social Republicans (ran on an anti-gay platform)

Total on the right: 44.5%

On the left (the names largely speak for themselves):
16.18% Socialist Party (Lionel Jospin)
06.84% Union for French Democracy
05.72% Workers' Struggle
05.33% Citizens' Movement
05.25% The Greens
04.25% Revolutionary Communist League
03.91% Liberal Democracy
03.37% French Communist Party
02.32% Left Radical Party
01.88% Citizenship, Action, Participation for the XXIst Century
00.47% Party of the Workers

Total on the left: 55.5%

So, a center-left candidate should win, handily, against a center-right candidate, if you held the race between them; and any centrist candidate should win against any more-extreme candidate.

And yet, if you suppose that even in an Instant runoff, any of the left parties might've broken 16.18% before votes got transferred to Jospin, he would've been eliminated.

In actual fact, because it was top-two, the second round was held between Chirac and LePen, instead of Chirac and Jospin; it's entirely possible that Jospin could've beaten Chirac, but we'll never know for sure.

IRV also has the property that a candidate can win every precinct, yet lose the election. Say there are just two precincts:

6 X > B > A
9 B > X > A
5 A > X > B

6 X > A > B
5 B > X > A
9 A > X > B

In each of these precincts, X wins. Yet, if you merge the two races together, A and B have 14 first-choice votes each, while X has only 12, so he gets eliminated, and we have a coin toss between A and B. Welcome to electoral bizarro-land.

There are a lot of systems out there that are better than IRV, and many of them aren't even particularly complicated.

Using a ranked ballot, there's a vast family of "Condorcet" systems, named after the Marquis de Condorcet, who made the first serious attempt at mathematically analyzing electoral systems -- it's a great shame his work was published twenty years after our Constitution was written, because he's exactly the sort of philosopher the Founding Fathers would've read and appreciated. His basic system simply looks at each head-to-head race, and picks the guy who wins against everybody else. There's one minor catch -- such a candidate doesn't always exist! You need a way to deal with situations where A beats B beats C beats A:

5 voters say: A > B > C
6 voters say: B > C > A
7 voters say: C > A > B

A > B by 12-6 (first and third blocs)
B > C by 11-7 (first and second blocs)
C > A by 13-5 (second and third blocs)

But, there are many perfectly acceptable ways to deal with this. The most common is to drop the weakest defeat -- in this case, B's victory over C, which has the least votes in favor of B. So, after breaking that link, we just have C > A > B. So, C wins.

In any case, such situations are extremely unlikely to occur in practice -- mathematicians and political scientists who study these things estimate that less than 1 in 1000 real-world elections with three contenders produce cyclic defeats. There are also quite a few good metaphors for explaining Condorcet to the electorate, such as looking at it like a sports league -- each team plays each other team, and one of them emerges as champion.

I personally think that the best, easiest, and cheapest reform is Approval Voting, in which you simply vote Yes/No on each candidate (like a ballot measure), and the guy with most Yes marks wins. This lets us reform the system without spending a nickel on new equipment, and with only the most minimal effort required to explain the system to the general public.

You might complain that this doesn't let you express a preference between two people you're voting for, but I don't find that terribly persuasive; if somebody approves just the Democrat, that's their favorite; if they approve the Dem and the Green, it's almost certainly the case that the Green is their favorite, but they just don't think he's likely to win, so they're approving the Dem as well to help him beat the Republican.

There's also a technical issue involved here. If you allow rated ballots -- say, giving each candidate a score from 0 to 9 -- it turns out that there's a strategic advantage in only using the top and bottom scores. People who follow that strategy often have a greater chance of deciding the election than those who are "honest" with their scores. So, while I certainly would endorse Range Voting as a great improvement over Plurality, it's clearly simpler, and probably fairer, to provide only two scores (0 and 1) -- thus, we're back to Approval. Sure, this leads to some tough choices, from time to time, but it's not like we don't face those choices today (whether to cast a ballot for a true favorite, or for a compromise), and that will happen, at least, much less under Approval than under Plurality. Range offers a chance to give honest ratings, but then sometimes punishes honesty in favor of strategy. Similarly, IRV's promise of allowing honest rankings turns out to be false, as seen in the examples above -- as in Plurality, people are often better off betraying their true favorite, and giving their first-ranked vote to a lesser-evil or compromise.

In general, as long as there's any sort of decent polling available, you can vote effectively (and interpret votes) on an Approval ballot. The general rule is to figure out how you're voting amongst the frontrunners (same as you would under Plurality if you're willing to cast a "lesser-evil" vote), and then also approve anyone you like better. Sometimes, if there's more than two frontrunners, you may want to vote for more than one -- e.g. if your attitude is "anybody but the guy I hate!" you'd vote for all frontrunners other than that guy. In the Participation Failure example, you might see something like this:

The 35 Republicans break into:
25 Yes on R only
10 Yes on R and D

The 32 Dems break into:
10 Yes on D and R
22 Yes on D only

The 33 Greens break into:
15 Yes on G and D
18 Yes on G only

Final scores: R=45, D=57, G=33.

I often hear the complaint that Approval violates "one man, one vote". It's more accurate that Approval Voting says "one man, one vote per candidate." Approval doesn't give any more power to somebody who votes Yes on more than one alternative. If you think about it, voting Yes on every candidate has the exact same effect as voting No on every candidate. If your ballot doesn't change the relative positions of the candidates, it might as well not get cast.

There is also a lot of merit to the idea of Proportional Representation. Under PR, there are multiple seats at stake all at once, and we try to allocate them "fairly". In order to win at least one seat, you need to achieve a minimum "Quota" of votes, usually the Droop Quota of 1+( #votes/ (N+1) ), where N is the number of seats. So, if there are four seats, you need one more than 20% (one fifth) of the votes. The point is that no more than N candidates can hold a Quota's worth of votes -- if there are four candidates who each have more than a fifth of the votes, there can be no fifth candidate with that much; less than a fifth of the votes remain. (Note: I'm using the character ε -- Greek Small Letter Epsilon -- in the next bit; it's possible you might not see it, if you have a very out-of-date font.

Let ε be an arbitrarily small positive number.
Q = 20% + ε
100% - 4Q = (20% - 4ε) < Q

The simplest PR system is Party Slate PR, in which people vote for a party, each party has a slate listing the order in which they will provide candidates, and the seats are allotted among the parties based on the percentage of the vote received. If a party gets one Quota's worth, they get one seat; with 2*Quota votes, they get two seats; etc. If there are four seats, the Quota is 20% plus a vote. Say the Dems get 49% (over two Quotas), the Repubs get 30% (above one Quota, but not two), and the Greens get 21% (ditto), the seats would be allocated to two Democrats, one Republican, and one Green.

PSPR, with slates determined by party bosses, is what was used to elect the Iraqi Parliament. How exactly you decide the order of candidates on the slate is irrelevant to PSPR; you could make the system a little more (small-d) democratic by having some sort of primary, letting the party's rank-and-file members choose the slate.

Single Transferrable Vote (STV or STV-PR) is the PR variant of IRV. The ballot is ranked, as with IRV, and the system works on a similar principle of dropping candidates who have the least first-choice votes. When a candidate reaches a Quota of first-choice votes, that candidate is elected, and any "surplus" of ballots (over the Quota) is transferred to next choices, proportional to what the next-choices actually are on the ballots backing the current win. You can see an actual example of this in Wikipedia. Single-winner IRV is equivalent to STV with one winner (and this a Quota of 50% plus one vote -- a majority). STV suffers from the standard IRV runoff problems, but only with respect to the final seat being filled. Even with just four seats, this starts looking like a relatively small problem, and by nineteen, it practically vanishes, taking with it almost all of the impact of gerrymandering.

Another way of thinking about STV is that voters who have already been partially satisfied, by receiving a victory in an earlier round, should have the "weight" of their ballots reduced for succeeding rounds. Standard STV-PR reduces the weight of all ballots contributing to a victory by the Quota percentage. There is an argument to be made that it is actually fairer to reduce their weight by a full 1/N instead (we're actually electing N candidates, so each one should pull one Nth of the total ballot weight out of the race). There's also an argument to be made that the weight being "removed" with a victor should be drawn more from those who actually listed the winning candidate as their first choice, than from those who've had two or three higher choices eliminated. But that's a deeply math-intensive discussion.

There are also party- and candidate-based PR systems based on Approval, and even more sophisticated systems using either the kind of pairwise-match data seen in Condorcet (one actually used by some large organizations, particularly in the Open-Source Software community, is Schwartz Sequential Dropping, also known as the Schulze Method) and more complex versions of the "reweighting" idea from STV-PR that try to deal with the problems I laid out in the previous paragraph.

The family of methods that operates in reweighted rounds (including STV-PR) is called something like Descending Layers of Redundant Representation (I've forgotten whether that's exactly right). It allocates support away from candidates who clearly have too little support to win, towards candidates still in the race, and in each round after the first, gives less value to the choices of people who've already had some of their choices elected. In variants more complex than STV-PR, rather than only counting first-choices towards a candidate's score, and always taking weight equally from all supporters of a winner, we may award some points based on lower-ranked choices, or take more weight from ballots that ranked a winner higher than from those who ranked him lower. There's a concept of a "similarity metric" based on the totality of ballots cast, which can be involved in how we score candidates at the beginning, how much weight we take away from each ballot after a victory, and how we transfer the remaining weight of the ballots between candidates' scores at the end of each round. Again, this is extremely math-intensive, so I won't get into deep detail. Good DLRR methods can draw on the best aspects of STV-PR, Condorcet, Approval, and Range. Ballots in sophisticated DLRR methods will generally collect either rankings with an explicit approval cutoff (numbered rankings, with "everyone below this rank is disapproved" as one of the things you're ranking), or ratings with an implicit approval cutoff (a score from 0 to 99, or a grade from A+ to F-, with the top half of the scale considered approved).

Even the simplest form of PR is much more complex than Approval Voting, and thus probably more difficult to pitch to voters; also, it can be complex to administer. This costs money, and can create confused, unhappy citizens.

I apologize for being so long-winded, but this is, of course, a complex topic. I hope, even if I haven't persuaded you that IRV is actually a bad and undesirable "reform", that I've at least convinced you that there are credible arguments against it, and some much better alternatives.

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