Voting Systems

In a ranked voting system , each person ranks a set of options and then sine procedure produces either either an "overall" ranking of candidates or at least a single winning candidate.

Probably the most famous example of an RVS is plurality voting, where the winner is whoever most people rank #1 on their ballots.

I'm going to invent the phrase "reasonable voting system" (RVS) to mean a ranked voting system that

  • Supports more than two candidates.
  • Is not a dictatorship.
  • Is deterministic.

Limiting Results

As you will see, the entire field of voting theory is littered with limiting results: proofs that it is impossible for a RVS to have two desirable properties.

I go over three limiting results here.

Limiting Result: Tactical Voting

Gibbard's theorem and the Gibbard–Satterthwaite Theorem both essentially state that any RVS will have situations where voters are motivated to lie about those rankings. This is called tactical voting. There are no exceptions.

As a brief aside, another exception is if our system uses randomness. For instance, a system that randomly selects a ballot and then chooses whichever candidate is ranked #1 on it is not susceptible to tactical voting.

The Condorcet Paradox

Alice is the Condorcet winner if she would beat all other candidates in a 1-on-1 election. It seems intuitive that if Alice could do this, then a RVS should choose Alice as the winner.

Unfortunately, there is no guarantee that a group of voters will yield such winner. In other words, it's totally possible for Alice to beat Bob in a 1-on-1 match-up, Bob to beat Carol, and then for Carol to beat Alice.

This means the best a RVS can do is guarantee that if there is a Condorcet winner, then the system will choose that winner. If there is no Condorcet winner, the RVS must do something else.

Later No Harm

When a voter moves one candidate's ranking up from #412 to #251, it seems desirable for that to not harm any of the candidates that voter ranked above #251. If a RSV guarantees this, we say it satisfies the later-no-harm criterion.

Unfortunately, no RVS can always pick the Condorcet winner (assuming one exists) and also guarantee the later no harm criterion.

We must pick one (or neither).

As an aside, the Condorcet criterion is also incompatible with the Participation criterion.

Independence of Irrelevant Alternatives

Many theorists would like a voting system to satisfy the Independence of Irrelevant Alternatives (IIA). To simplify a bit, this means that if our RVS says Alice is better than Bob and we then add Carol to the election, the RVS must still say Alice is better than Bob.

The slightly more complex explanation is that if you have a 100 candidates and the RVS says candidate #59 is better than candidate #19, then adding a new candidate to the mix can't change that.

Unfortunately, it is impossible for a RVS to guarantee the IIA and also guarantee that if a majority of people make Alice their first choice, then Alice will win.

Discussion of Limiting Results

In short, we have three limiting results:

  1. Every RVS must have tactical voting.
  2. A RVS can either always choose the Condorcet winner if one exists or you can guarantee the later no harm criterion.
  3. A RVS can either always choose the majority winner if one exists or you can guarantee the IIA.

Note: if a RVS satisfies the Condorcet criterion, it must also satisfy the majority criterion. Therefore, (2) and (3) imply that a RVS that guarantees the Condorcet criterion can not guarantee the IIA. This means, each RVS must choose one of:

  1. majority (e.g. Bucklin voting or Coombs rule)
  2. majority & Condorcet (e.g. the Schulze method, the Kemeny-Young method, or ranked pair voting)
  3. majority & later-no-harm (e.g. instant-runoff voting or plurality voting)
  4. later-no-harm (???)
  5. IIA (e.g. Majority judgment)
  6. IIA & later-no-harm (???)
  7. none (e.g. Anti-plurality voting)

Best Voting System By Criteria

The Pareto Candidates

Choosing the best voting system is something of a fools errand. Most voting systems are better at somet things and worse at other things. However, Wikipedia provides a convenient table showing how various systems do on various metrics.

No system matches all the desirable criteria, but some systems are clearly superior to others. For instance, among the listed systems that satisfy the Condorcet criterion, the ranked pair voting method falls short on only three criteria, which no other Condorcet system has either, suggesting it is (vaguely) the best Condorcet system we've found.

With similar reasoning, we can conclude that if you really want later-no-harm, there's no reason to to pick a system that isn't instant-runoff voting or plurality voting. Every other system is strictly worse than one of these two.

Using similar reasoning, we can narrow down the best systems to one of five from Wikipedia's table:

  • Ranked pair voting
  • Instant-runoff voting
  • Plurality voting
  • Borda count
  • Bucklin voting

To reiterate: choose any voting system in the table besides one of these five. One of these five systems is strictly better than the voting system you chose.

Finalist

Choosing the best of these five systems is more subjective.

Let's start by pitting ranked pair voting (RPV) against Bucklin voting. Bucklin beats out RPV in one way: it satisfies the later-no-help criterion. In contrast, RPV beats out Bucklin in 7 ways, 3 of which are essentially criterion that prevent similar candidates from "splitting the vote" and harming each other (ISDA, LIIA, clone independence), 3 of which are related to it being a Condorcet system (Condorcet, Condorcet loser, Smith), and 1 of which is reversal symmetry.

Now, this is still a subjective call, but, in my opinion, preventing unviable candidates from stealing elections and electing Condorcet winners are each more important than slightly reducing tactical voting. Together, it looks like a no-brainer to eliminate Buckling voting.

The argument for ranked pair voting over using the Borda count is similar, though Borda count count also brings two other criteria to the table (participation and consistency), neither of which I've found anyone really cares about.

Likewise, I don't think anyone would really argue that plurality voting is better than instant-runoff voting. I'm not going to go in-depth on why, just google any of the scores of people talking about instant-runoff voting. So, to my mind, there are really two candidates for best voting system:

  • Ranked pair voting
  • Instant-runoff voting

These two systems really guarantee very different properties, which makes them difficult to compare. However, in my opinion, it really comes down to whether you care more about (a) reducing tactical voting or (b) electing Condorcet winners and not punishing similar candidates.

Table 2

Unfortunately, Wikipedia has a second table that suggests a couple other competitive candidates. However, most of these involve randomness or ratings (as opposed to rankings). Ignoring those groups, the table really only adds majority judgement as a competitive system.

Compared to ranked pairs, majority judgement drops Condorcet-related guarantees for reducing the effectiveness of voters and candidates from engaging in tactics (IIA, later-no-help, no favorite betrayal).

In theory, then, I see three "reasonable voting systems" that have a good claim for the top spot: ranked pair voting, instant-runoff voting, and majority judgement.

Non-Ranked Systems

There are a couple of non-ranked voting systems that give these top three systems a run for their money. From Wikipedia's second table, we can see the stiffest competition comes from score voting and approval voting (STAR voting is just worse than Borda count). However, these two really only count as a single voting system since approval voting is just a special case of score voting.

Finals List

This leads me to my list of four plausibly best non-random voting systems:

  • ranked pair voting
  • instant-runoff voting
  • majority judgement
  • score voting