Voting Systems (Part 1)
The Lifeblood of Democracy
The idea of democracy goes hand in hand with voting. The ability to vote and choosing a good method of voting is critically important in a democracy. The strength of a democracy is intrinsically linked to what percentage of the population have a meaningful say in the voting process and how that say is translated into collective action.
Ballots: Capturing Preferences
The most tangible part of the voting process is the process of capturing voters’ preferences — ie when you actually write your vote on a ballot. There are two broad categories of how this can be done: Ranked Voting and Rated Voting.
Ranked Voting
In a Ranked Voting system, aka Ordinal Voting, voters rank candidates in the their order of preference. Sometimes systems like this might just ask voters to rank their top three candidates, and a system like first-past-the-post voting that most of us are familiar with is a special case of ranked voting where voters only specify their first choice. There are a number of voting systems that used ranked preference capturing. The most common ones are Instant-runoff voting (aka Ranked Choice Voting, RCV), Single Transferable Vote (STV), and Borda count.
Ranked Voting allows voters to more accurately express their preferences than the common vote-for-one approach because voters give each candidate a rank, which gives a lot more information about their preferences that can be used to choose a winner that voters are happier with.
One problem with Ranked Voting is Arrow’s Impossibility Theorem. Kenneth Arrow proved that all Ranked Voting systems will fail at least one of three fairness criteria. This leads to the various ranked voting systems having weird quirks and edge cases that make people (including myself) feel uneasy.
Rated Voting
Rated Voting, aka Cardinal Voting, is a preference capturing method where voters give a score to each candidate. Voters can give each candidate an independent score, meaning that they can give multiple candidates the same score if they like. An example is scoring candidates from 0–3. The winner of a single-winner Score vote is usually chosen by adding up each candidates scores from all the ballots, and the winner is simply the one with the highest score.
Rated Voting allows for a more accurate preference capturing than Ranked Voting, making score voting the most accurate way to capture voter preferences. You can not only describe a ranking of candidates, but you can specify how *much* more you want one candidate vs another.
Score Voting may be the oldest recorded method of voting, having been used in Ancient Sparta in votes decided by the loudest collective shouting in a crowd of voters. Approval voting is a special case of score voting where voters can only give scores of 0 or 1. Score voting is the most well known version of this. But there is also STAR Voting, Cumulative Voting, and Quadratic Voting, among others.
One design issue with a system like Score Voting is that people don’t always have a linear range they vote within. Score Voting systems treat a score of 2 to be twice as strong as a score of 1, and a score of 4 is twice as strong as a score of 2, etc. But sometimes people’s mental processes aren’t linear like that. People often give a great review to a restaurant on yelp, but only give that restaurant 3/5 stars. For some people, that last star or two is reserved for only the best of the best, and that might not be quite linear. One solution to this is to choose a narrower range, like 0–3, where the scaling oddities of individual voters are minimized simply because the range requires them to fit all their candidates into it. STAR Voting attempts to fix this by normalizing votes to decide between the two top candidates, which might legitimately help a lot. Highest Median Voting attempts to fix it by finding the median vote for each candidate and choosing the candidate with the highest median, but this comes with significant problems.
The choice of range for Score Voting is also sometimes contentious. Should -2 through +2 be used, or 1–5, or 0–4? As far as vote counting, all of these are identical, but negative numbers might confuse some people, and having a range of 1–5 might mislead people into thinking that 1 is “first choice” as in ranked voting and so misvote. That’s why I suggest using 0–3, since its a small range and the 0 makes it clear that the lower numbers are a lower score rather than an enumerated rank.
In any case, these aren’t very serious problems with Rated Voting relative to Ranked Voting, but are simply issues to be considered when designing an election. I’ll talk more in depth about several Rated Voting systems in my next post.
Default Vote and “No-opinion” votes
One other consideration for both Ranked and Rated voting systems is what to do when the voter doesn’t give a candidate any rank or score. Sometimes, its decided that not voting for a candidate will be the same as giving them the lowest possible rank or score, but other times its treated differently.
Sometimes voters are given the option to vote “no-opinion” on a candidate, which allows the voter to ensure that their vote doesn’t affect that candidate’s final score. This can also be chosen as the default vote, tho the unknown fanatic effect can be caused by doing this (this is explained more below).
Single-winner, Multi-winner, and Direct Representation
We’re all familiar with single-winner voting systems. Presidents, mayors, governors, and the like are all elected via single-winner voting systems. In many countries, the legislature is also elected this way. But this isn’t the only way to do it.
Many countries have methods that elect multiple candidates from a given jurisdiction. Often this is done to make representation more proportional to the population. For example, if your district has 40% democrats, 35% republicans, and 25% libertarians, electing a democratic representative means that 60% of the district will feel unrepresented. Multi-winner systems allow, for example, 4 democrats and 3 republicans, and 2 libertarians to win, making representation match the electorate a lot more closely. This is called Proportional Representation.
A very different category of multi-winner systems is called Direct Representation, where each elected candidate has a weighted vote depending on how many voters from their jurisdiction voted for them. I’ll talk more about this next post.
Primaries
“I don’t care who does the electing, so long as I get to do the nominating.” — William Magear Tweed (circa mid 1800s)
Primaries aren’t usually a necessary part of voting itself, but are a way to filter fringe candidates so that voters can spend more time with each candidate that has a real chance of winning. Voters can learn more about each of 3 candidates than they could about each of 20 candidates, for example.
But primaries are also a mechanism by which political parties exert control over an election. As the above quote by William Tweed suggests, the power to control the nominated candidates in an election can be far more important than controlling the final winner. Tho many US states do have open primaries, US political parties usually silo off primaries by individual party — ie republicans will have a separate primary from the democrats. This gives a lot of power to the party in determining or influencing who is able to run for office.
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Primaries themselves are a useful tool to make elections easier to understand for voters, but primaries run by private organizations like political parties are an enormous hindrance to an effective election, as George Washington clearly felt when he said:
“the alternate domination of one faction over another, sharpened by the spirit of revenge, natural to party dissension…is itself a frightful despotism.” — George Washington
Judging Voting Systems
There a lot of mathematical and logical criteria that have been devised over the ages to compare and contrast voting systems. Some are simple, some are complex. Some are criteria that make sense, and other times they can be misleading. If you’d like to check out a chart comparing voting systems, see a chart here.
The following describes some of the most important criteria to consider.
Criteria of a Good Voting System
The winner of a Clone-Proof voting system won’t change even if you add a non-winning candidate similar to the winning candidate. Systems that have the spoiler effect, where a 3rd party candidate similar to say a Democrat in the US will make it harder for that Democratic candidate to win, are not clone-proof. Borda Count and First-past-the-post both fail this criteria.
A Monotonic voting system is one where ranking a candidate higher will never be bad for the candidate. Its frustrating to work in a system like Instant-Runoff Voting where giving your preferred candidate a higher rank than others could cause them to lose.
Similar to Monotonicity, a voting system satifies the Later-No-Help criterion if giving a less-preferred candidate a higher ranking or score might cause a more-preferred candidate to win. For example, if someone has a ballot ranking Alice 1st, Bob 3rd, and Carol 5th, as long as the voting system satisfied monotonicity, changing the ranking of carol to 4th could never result in Bob winning the election when they didn’t before (if all other ballots are held the same). Ranked Pair voting and the Schulze method fail this criterion.
Similar to Monotonicity and Later-No-Help, a voting system satisfies the Participation Criterion if adding a ballot that prefers A to B could cause B to win instead of A. This criterion is actually incompatible with the Condorcet criterion — no Condorcet voting system satisfies Participation. IRV, Ranked Pairs, and Schulze fail this criterion.
A Consistent voting system is one where if the election is divided into parts and each part elects the same choice, then putting them all together elects the same choice. It would be strange if adding together multiple groups that all agree could choose a different candidate than the individual groups agreed on. Perhaps surprisingly, however, many commonly used and proposed voting systems fail this criterion including IRV, Ranked Pairs, and Schulze.
A voting system is Independent of Irrelevant Alternatives (IIA) when changing a vote for a candidate that won’t win either way (an irrelevant alternative) doesn’t change the winner. Basically, there are situations in some voting systems where someone can change their vote for candidate C, and that change causes B to win over A, instead of A winning over B. In that case, C is the “irrelevant” alternative, because C will never win. Despite that it’s “irrelevant”, votes for C can still affect which “relevant” candidate (A or B) will win. The independence of irrelevant alternatives is a property of a system that doesn’t allow that to happen. Oddities around irrelevant alternatives plague the many systems that fail this criterion like Borda Count, IRV, Plurality, Schulze, and Ranked Pairs.
A voting system satisfies the Unanimity Criterion if a candidate A who is less preferred than candidate B by every voter will never win over that preferred candidate B. In other words, you would expect that if everyone prefers one candidate to another, the more preferred candidate will have better election results than the less-preferred candidate. Kenneth Arrow proved that this is incompatible with independence of irrelevant alternatives for ranked-choice ballots.
The Voter Satisfaction Index (also known as Voter Satisfaction Efficiency) and its inverse, Bayesian Regret, are statistical methods to measure how close a voting method gets to perfect utility (voter satisfaction) over a sample of simulated elections. Basically, given a group of voters who have certain quantified preferences, these methods measure how satisfied voters will be with the candidates elected via the voting system being tested, and compares it against the best-case scenario (or the worst case scenario in the case of Bayesian Regret). This can be a very powerful way to compare voting systems in a way that’s much more quantitative than using logic-based criteria like the above.
Criteria of a Bad Voting System
There are also some criteria that have been devised with the best of intentions, but are actually detrimental to a voting system. Note that many people still believe these criteria should be striven for, but I’ll make the case that those people should change their mind.
A voting system satisfies the Later-No-Harm criterion if giving a higher ranking or rating to a less-preferred candidate can’t cause a more-preferred candidate to lose. Later-no-harm has been criticized by the Center for Election Science as having a misleading name and dubious value as a voting system criterion. Mathematics PhD and voting system expert Warren D. Smith also harshly criticizes this criterion as “silly”, self-inconsistent, and opines that “any hope the LNH criterion will prevent strategic voting, is just garbage in the real world.” I have to agree. The idea that rating a less preferred candidate higher should not be allowed to cause that higher-ranked candidate to win certainly seems absurd to me. What use is rating a candidate higher if there is no chance that higher-rated candidate could win if enough people do so?
“To rely on majority rule is to resign oneself to indecision, manipulation, or arbitrariness.” — Jac Heckelman, Professor of Economics
A voting system satisfies the Majority criterion if the system always elects the candidate preferred by more than 50% of the voters if such a candidate exists. The problem with this system is that it can cause a tyranny of the majority that disenfranchises minority viewpoints. For example, if just over half of the electorate loves both candidate X and candidate Y (but X slightly more) and just under half hate X but love Y, its clear that while X would win in a system that satisfies the Majority criterion, Y would actually make more of the electorate happy. Here’s a simple example that demonstrates this. But there are other problems with this criterion.
A voting system satisfies the Traditional Condorcet criterion if it will always elect a Traditional Condorcet winner if one exists, where a Traditional Condorcet winner is defined as the candidate that would win in a head-to-head election with each other candidate using plurality rules. Any Traditional Condorcet method will also satisfy the Majority criterion, and thus has similar issues. Another place this criterion goes wrong is that it tends to elect the candidate that voters hate least, rather than the candidate they like most. Yet another problem with this criterion is its insistence on using plurality rules in those head-to-head elections. Condorcet is traditionally defined assuming ranked order voting and the head-to-head elections are always assumed to be simple plurality votes. With score and approval voting, this definition makes no sense and yet is still used. A better definition for a Condorcet winner would be “A candidate that would win head-to-head against every other candidate using the same rules as the full election.” I’d like to call this the Modern Condorcet winner, and thus the associated criterion the Modern Condorcet criterion. This modern definition is a much better criterion that is applicable to voting systems for both ranked and scored preference capture. For example Score and Approval voting both satisfy the Modern Condorcet criterion as defined here, tho they don’t satisfy the Traditional version.
Mo’ Voting Systems, Mo’ Problems
The Spoiler Effect is when a candidate “steals” votes from another candidate who would have won if that “vote stealing” candidate had not run. A famous example is the 2000 US Presidential election, where Ralph Nader gained a significant fraction of votes that would have otherwise gone to Al Gore. The prevalence of this effect in a voting system gives voters a huge reason to vote tactically.
Tactical Voting (aka strategic voting) is one major problem that voting systems try to minimize. This is where someone votes in a way that misrepresents their true preferences of candidates in order to give their preferred candidate (or perhaps several preferred candidates) an additional advantage over their true honest vote. Gibbard’s Theorem proves that any election with more than 2 candidates is susceptible to tactical voting. But not all voting systems are created equal here. Some voting systems give voters a lot more incentives to engage in tactical voting than other systems. Minimizing these incentives makes up a big portion of the debate around which voting system might be best.
Favorite Betrayal is where voters are incentivized to not vote for their favorite candidate, or give them a lesser vote. For example, systems where the spoiler effect exists cause voters to engage in favorite betrayal in order to avoid their vote going to a spoiler candidate.
Bullet Voting is when voters rate or rank only one or two candidates despite their ability to rate or rank more than that. This might be done if a voter doesn’t take the time to evaluate more than one or two candidates, or if they are tactically voting in order to increase the likelihood that their preferred candidate will win. This is often brought up as problem with Score Voting and Approval Voting, but the question is really one of incentives to tactically vote. Since Allan Gibbard proved that all voting systems are susceptible to tactical voting, bullet voting isn’t as much a separate problem as much as it would be a symptom of incentives to tactically vote, which can be analyzed without considering the idea of bullet voting itself.
The choice of voting method can also lead to a two-party system. Duverger’s Law states that any jurisdiction using plurality tends to naturally fall into a two-party system. Because such a system essentially by definition marginalizes other parties and groups, it’s certainly not a desirable thing to have happen. I’ll talk more about this next post.
The Unknown Fanatic effect (aka the cult-film effect) is seen in situations where the default vote is “no opinion”. In such a situation, most voters might not rate candidates they don’t know, while voters that do know that candidate are more likely to vote that candidate very highly. If the default score is a “no opinion” rather than a 0, this can lead to a small number of voters causing a fringe candidate to win the highest score. While this effect is certainly possible, its rarely seen in practice and is solvable either by defaulting unscored candidates to 0, giving all candidates an initial number of 0-scores they need to overcome, by or by removing the ability to vote “no opinion” entirely.
Summary
Voting has been thought about for millennia, but has only recently been given any kind of rigorous methodological thought. There are many issues with popular methods of voting, and next post I’ll be using the categorizations defined above to evaluate modern voting methods and discuss the future of voting.
