When strategic voting destroys a voting method

Marcus Ogren
7 min readApr 6, 2022

For most widely-discussed voting methods, strategic voting results in better outcomes for the electorate. However, it is absolutely devasting for a few lesser-known methods. Suppose there is a strategy under a given voting method that satisfies three properties:

  1. If a small number of voters use this strategy they will wield significantly more influence as a result.
  2. If a large fraction of voters use this strategy it will lead to an outcome that almost everyone considers terrible. (Example: electing Lord Buckethead.)
  3. Scenarios in which this strategy is viable are reasonably common.

If all of these conditions hold, and the voting method doesn’t somehow encourage other voters to counteract the problem, the voting method will be liable to yield universally hated results and be unacceptable for use in governmental elections.

First of all, it’s a given that some voters will use this strategy (unless you’re in an organization in which telling people to vote honestly will reliably lead to them voting honestly). If nobody used this strategy it would be a golden opportunity for voters looking to increase their influence, which many voters will want to do even if it seems dishonest or otherwise unpalatable.

Second, there will be escalation. Stories about the nefarious tactics of an opposing candidate’s supporters will spread, and voters who might be unwilling to use the strategy without prompting will feel the need to use it so that the other side doesn’t have an unfair advantage.

Third, there will be a risk of the terrible outcome. If there wasn’t, the escalation would continue unchecked, and candidates would tell their supporters to use the strategy. This does not mean that the terrible outcome will always, or even usually, occur; just that it will occur occasionally and that the risk of it occurring will be a major consideration in how people think about the election and cast their ballots.

Examples

Consider the DH3 pathology under Borda Count:

How Borda Count works: Rank all of the candidates. In a 10-candidate election, your first choice gets 9 points, your second choice 8 points, and so on, with your second-to-last choice getting 1 point and your last choice getting 0 points. The candidate who gets the most points wins. (Many implementations of Borda Count do not require voters to rank all the candidates, but I’ll focus on this case where ranking every candidate is required.)

Basic scenario for the DH3 (dark horse + 3) pathology: There are three reasonably popular candidates (A, B, and C) and several candidates that nobody wants to win. For concreteness, let’s say that these candidates are members of the Official Monster Raving Loony Party (OMRLP).

Everyone’s sincere preference is something like A>C>B>OMRLP or B>A>C>OMRLP. However, a voter would be foolish to vote in such a manner since, under Borda Count, it sacrifices the majority of their voting power. By voting A>B>C>OMRLP you give one more point to a than to B and two more points to A than to C; by dishonestly voting A>B>OMRLP>C you give the two more preferred candidates many more points than C. If you prefer A to B by more than you prefer B to C it probably makes sense to vote A>OMRLP>B>C so that A receives far more points than B as well. But, if enough voters vote like this, someone from the Official Monster Raving Loony Party will win the election. This doesn’t mean the OMRLP will actually win; it just means that there will be a real enough possibility of it that it affects voter behavior.

For a second example, consider a variant of Bucklin Voting suggested by Michael Wayne Sawyer.

How Bucklin Voting works: Rank as many candidates as you like. Tabulation proceeds in rounds and terminates when a candidate has majority support (i.e. has a vote from a majority of ballots cast) or as many rounds have been conducted as there are available rankings. In the first round, a candidate receives a vote for every voter who ranked them first. In the second round, a candidate receives a vote for every voter who ranked them either first or second, and so on. This means that, from the second round onward, voters are supporting their second choices exactly as much as they’re supporting their first choices. (If multiple candidates reach a majority on the same round, the winner is the candidate with the most votes.)

The variant: If a voter has left part of their ballot blank, such that there are unranked candidates and tabulation has proceeded past the rankings they have filled in, their next vote goes to whichever of the candidate they didn’t rank got the most votes in previous rounds. For example, if a vote only fills in their first choice, in the second round they will vote for both their first choice and the candidate who got the most votes in the first round (or the second-most votes if the voter voted for the candidate who got the most votes). This variant ensures that the winning candidate will receive support from a majority of ballots. (A simpler variant, which has the same strategic issues, is to require voters to rank all the candidates.)

Let’s suppose there are two popular candidates, A and B, who each have ~40% first-choice support. In addition, there is a fringe candidate F with 20% first-choice support and little second-choice support and a real estate agent R who isn’t running a serious campaign and only got onto the ballot to boost her name recognition. Let’s suppose most people vote honestly. In this case, voting A>B>F>R (or only for A, in which case the ballot would behave the same under the variant) will have no effect on the outcome. In the first round, nobody gets a majority. In the second round, this ballot gives one vote to each of the viable candidates, A and B — each of whom receives a majority in this round. Only voters who cast a ballot such as A>F>R>B will have their votes matter. And if enough people vote this way, either F or R will win. If you have the preferences A>B>F>R you have two choices: vote dishonestly and contribute to the risk that a blatantly terrible candidate will win, or throw your vote away altogether (except insofar as it prevents F or R from winning).

Strategic voting means that, under both Borda Count with mandatory ranking and this variant of Bucklin Voting, there can be no such thing as a non-viable candidate. (Unless a candidate has a majority of first-choice support, at least.) It doesn’t matter if the candidate is running purely as a joke; voters are strongly incentivized to give them enough support to become competitive.

Comparison to the chicken dilemma

The chicken dilemma in Approval Voting bears many similarities to the examples above.

(The chicken dilemma, briefly summarized: There are two equally popular candidates aligned with a party that has ~60% support (let’s say it’s the Democrats) and one candidate aligned with a party with ~40% support (let’s say the Republicans, though of course the party labels are arbitrary). One of the Democrats will win — unless nearly all of the electorate bullet votes, i.e. only votes for one candidate.)

The first of the three properties listed at the top of this post are easily satisfied; so long as most of the Democratic voters vote for both of their candidates, bullet voting will give a Democrat far more influence. The third property is also easily satisfied; the chicken dilemma may not occur in the majority of elections but is hardly contrived. The second property isn’t satisfied to anywhere near the extent that it’s satisfied in the examples above. “Whoops, elected a Republican!” in a district that’s mostly Democratic is an unpopular outcome, but it’s not an outcome that obliterates confidence in elections like “Whoops, elected Lord Buckethead!”. It’s true that the Republican ends up being at least somewhat competitive here, but, since he’d win by a 10-point margin under Plurality, this does not seem especially ridiculous.

Another big difference is that, in the chicken dilemma, there’s a big chunk of the electorate that is incentivized to vote in a way that prevents the undesireable outcome. The Democrats are heavily favored, so Republican voters tend to be best off voting for the more tolerable of the two Democrats in addition to the Republican.

Why this isn’t an issue for Condorcet

Warren Smith claims that the DH3 pathology also applies to Condorcet methods. He is correct that, if voting blocs achieve perfect coordination, they could gain an advantage by ranking a widely-disliked candidate second and creating a Condorcet cycle. But this strategy doesn’t satisfy the first of the properties I listed (“If a small number of voters use this strategy they will wield significantly more influence as a result.”). Instead, it requires outlandish levels of coordination. Smith’s example, for instance, involves every single voter in two opposing blocs working together. If just a few voters go for the dishonest strategy there is no chance of success, and, if it involves a “dark horse” who is popular enough that creating Condorcet cycle doesn’t require a coalition of enemies, there is a real chance of the strategy backfiring even if only a few voters use it. (See, for example, my discussion of Condorcet methods in the chicken dilemma.)

Since there is a high threshold for dishonest strategies to (maybe) become effective and a real cost to using them prior to reaching this threshold, such strategies would have an extreme hard time getting “off the ground”. If a few people use it and tell their friends it won’t yield a response of “That’s a good idea, I’ll try it myself” or “What?! That’s completely unfair! I’ll use it myself so to deny these cheaters their advantages!”; because it’s ineffective without extremely widespread buy-in, a more appropriate response is bemusement.

The DH3 pathology is only an issue for Condorcet methods in an idealized world in which coordination is easy, voting blocs are monolithic, and voters move to embrace unintuitive strategies in lockstep. In the real world, convincing massive swaths of the electorate to vote in a weird way requires an enormous effort. Coordination problems are hard. Expecting voters to embrace this tactic en masse after hearing one argument in favor of it is like expecting everyone to migrate from Facebook to some other social media platform at the drop of a hat. Real changes in social behavior, be it in voting patterns or social media usage, start small. A few people do things a new way, find that it’s better, and so their friends switch over as well. If you get rid of the part where early adopters see benefits and make the change difficult-to-understand and crazy-sounding, it has no chance of taking off.

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