The effectiveness of dishonest strategies in different voting methods
Which voting methods best encourage voters to vote honestly? There are many reasons we care:
- Dishonest voting artificially distorts vote totals; consider, for instance, voters who prefer third-party candidates to major-party candidates but strategically vote for the more tolerable major-party candidate instead. This makes third parties appear less popular than they actually are, making it more difficult for them to gain a foothold.
- When dishonest strategies determine election outcomes it can appear especially “scumbaggy” and may weaken confidence in democracy.
- The kinds of strategies that can completely destroy a voting method tend to be outright dishonest.
- We may just intrinsically prefer it when honesty pays off and voters are honest.
In this post, we’ll give a workable definition of what it means for a strategy to be dishonest. Next, we’ll look at the results for how advantageous dishonest is under different voting methods and evaluate the reasons for the differences. Then we’ll get into the nitty-gritty of the methodology (for the readers who are curious), and the end of the post I’ll share my main takeaways.
An ordinal voting method is one in which you can show that Candidate A is preferred to Candidate B, but you can’t show how much better A is than B. Vote-for-one Plurality and all forms of Ranked Choice Voting are examples of ordinal voting methods. By contrast, a cardinal voting method is one in which you can rate candidates on some scale independently of one another. Approval and STAR Voting are examples of cardinal voting methods.
It’s trivial to define dishonesty under an ordinal voting method: A ballot is dishonest if it indicates that some candidate A is preferred to some other candidate B, when in fact the voter who cast this ballot thinks B is better than A. In fact, so long as the voter uses all the allowed rankings, doesn’t give candidates tied rankings (doing so is not allowed under Ranked Choice Voting), and considers no two candidates to be exactly equal, there is only one way to cast an honest ballot.
Cardinal methods allow for much stricter definitions of honesty. Consider an Approval Voting election in which only A and B have a chance of winning, but a voter thinks they’re both terrible and greatly prefers C to either of them. She views B as the lesser evil. Is it dishonest for her to vote for both B and C (which means her vote would matter) rather than only voting for C (which means it wouldn’t)? It feels at least defensible to answer “yes”. However, this would entail holding cardinal methods to a much higher standard than what we would ordinal methods to — it’s not suitable for comparisons between cardinal and ordinal voting methods.
So we’ll go with the same definition of honesty for cardinal methods that we use for ordinal methods: A ballot is dishonest if it indicates that some candidate A is preferred to some other candidate B, when in fact the voter who cast this ballot thinks B is better than A. However, this means that, for cardinal voting methods, voters can cast many different honest ballots.
One final note: we want to define (dis)honesty for strategies and not just ballots. (A strategy can be thought of as a function that, when given a voter’s preferences and (maybe) some polling data, yields a ballot for that voter to cast.) We will say that a strategy is dishonest if it ever results in the voter casting a dishonest ballot.
I’ll get into the results before detailing the methodology because the methodology (including the precise strategies that were used) takes a while to describe and I expect most readers are just interested in the results. We will use Expected Strategic Influence Factors (ESIF) to quantify the extent to which dishonest strategies are or are not effective. An ESIF of 1.2 for a dishonest strategy means that a voter’s vote accomplishes 120% as much in expectation (from the voter’s perspective) as if the voter used an honest strategy instead. So higher numbers mean greater rewards for dishonesty.
We will consider two kinds of dishonest strategies. Favorite betrayal strategies involve trying to gain an advantage by giving your favorite candidate (or your favorite amongst the three top-polling candidates) less than maximal support. Pushover strategies involve giving a hopefully-weak candidate a dishonestly high level of support in an attempt to get this candidate into either a runoff or a late round of tabulation, where one of your favorite candidates will (hopefully) defeat this pushover candidate.
Without further ado, here are the actual results (more precise numbers in this spreadsheet):
- Plurality: 1.25 for favorite betrayal
- Plurality Top 2: 1.10 for favorite betrayal, 1.16 for favorite betrayal + pushover
- Instant Runoff Voting (single-winner Ranked Choice Voting): 1.05 for favorite betrayal; 1.05 for the combination pushover + favorite betrayal. (Pure favorite betrayal was slightly better, though you’d need more digits to see it.)
- Approval: 0.91 for favorite betrayal
- Approval Top 2: 1.00 for both favorite betrayal, 1.01 for pushover
- STAR: 0.98 for favorite betrayal, 0.99 for pushover
Note that the exact numerical values are not especially meaningful since they are greatly influenced by the details of the strategies that are used and by the amount of polling noise. Whether a given ESIF is less than or greater than 1 is not especially meaningful either. What is meaningful is how the numbers compare to one another. For example, we cannot conclude from this that STAR never incentivizes favorite betrayal — but we can conclude that favorite betrayal strategies make less sense under STAR than they do under Approval Top 2 or IRV.
Unsurprisingly, dishonesty is the most strongly rewarded for Plurality, where you need to vote for a frontrunner or your vote is completely wasted.
Plurality Top 2 provides lower rewards for dishonesty than Plurality without a runoff since wasting your vote in the first round is less of a big deal (your ballot will still matter in the runoff). However, the runoff opens the door to a pushover strategy. Imagine your favorite is a controversial incumbent who is almost certain to make it to the runoff, but could easily lose it to a strong challenge. Voting for a different divisive candidate who could make it to the runoff (but would be relatively weak in the runoff) in the first round can do more to improve your favorite’s odds of reelection than casting an honest vote for your favorite.
Instant Runoff Voting (IRV) provides weaker incentives for both favorite betrayal and pushover strategies than Plurality Top 2. The favorite betrayal part is obvious: Voting for a candidate who has no chance of making it to the runoff completely wastes your first-round vote under Plurality Top 2, but under IRV your vote will simply be transferred. The reason pushover makes less sense under IRV is a bit trickier. Under Plurality Top 2, you can vote for a pushover in the first round, and then, if all goes according to plan and your favorite and the pushover candidate advance to the runoff, you can vote for your favorite in the runoff election. Under IRV, your vote is guaranteed to count for the pushover (and therefore against your favorite) if the strategy succeeds to the point that the pushover makes it into the final round. Pushover strategies are much riskier under IRV than under Plurality Top 2.
Approval Voting offers no incentives for dishonesty, as is clear from thinking about the tabulation for more than 10 seconds. If you’re voting for A and you like B more than A, should you vote for B as well? Yes; whatever strategic arguments you have for voting for A also work for voting for B, except they’re even stronger because B winning is better than A winning. Approval Voting satisfies the favorite betrayal criterion, which says that it is always in a voter’s interest to provide maximal support to their favorite. Out of all the dishonest strategies considered here, the one for Approval Voting (which has the voter vote for one of the top two candidates Plurality-style) is unique in that there is no hand-wavey argument for why it might be effective. It’s just obviously stupid.
The addition of a runoff means that there are plausible ways in which dishonesty could be effective under Approval Top 2. If your favorite is poised to finish second in the first round and then lose the runoff, and you prefer the third-place finisher in the polls to the winner in the polls, maybe you should vote for this third-place finisher and not your favorite in order to send a more viable candidate who you find acceptable to the runoff. Pushover strategies akin to what you can do in Plurality Top 2 are also possible, though under Approval Top 2 you’ll want to vote for your favorite as well as for the pushover. But these dishonest strategies are much less effective than the dishonest strategies for Plurality Top 2. Compared to IRV, favorite betrayal makes much less sense under Approval Top 2, but pushover may be a bit less of a bad idea.
STAR is fairly similar to Approval Top 2 strategically, but, since you can’t change your ballot between the first round and the runoff, the dishonest strategies are much riskier. Only Approval Voting without a runoff rewards honesty more than STAR.
We can classify these six voting methods, and their incentives for dishonesty, in this chart:
In the left column are ordinal voting methods that only let you support one candidate at any given point in the tabulation. On the right are cardinal methods that let you support any number of candidates in the first round. This difference implies a much stronger incentive for favorite betrayal for the voting methods on the left; if your favorite has little chance of winning, you need to betray your favorite in order to support someone more viable. For the voting methods on the right, offering full support to a viable candidate is fully consistent with offering full support to your favorite; you just need to choose a good honest strategy.
Another difference between the columns: The voting methods on the left use Plurality in the first round — a poor proxy for how viable they’ll be in the final runoff. Those on the right use Approval and Score Voting — much better proxies for viability in the final runoff. When candidates who won’t be viable in the final round can easily advance to the final round, dishonest strategies have a lot more potential. This is much less common with the cardinal methods than with the ordinal methods.
In the top row are single-round voting methods whose tabulation involves nothing more than electing whoever has the most votes. The absence of a runoff means that strategy is more important in these methods than the other four since if you vote for none (or all) of the frontrunners your vote is wasted. The difference between Plurality and Approval is that Approval lets you achieve this while voting honestly, but Plurality doesn’t.
The voting methods in the middle row are the same as those in the top row, except they have a top-two runoff added in. This runoff reduces the overall importance of strategy (hence why Plurality Top 2 encourages favorite betrayal less strongly than plain Plurality), but opens the door to strategies that involve manipulating who gets into the runoff, including pushover strategies.
The voting methods in the bottom row use automatic runoffs, which means that voters cannot shift from dishonesty in the first round (where dishonesty can be helpful) to honesty in the final round (where dishonesty is never helpful). This makes dishonest strategies riskier, and therefore less incentivized.
In designing a study to ascertain the effectiveness of dishonest strategies in different voting methods I encountered two challenges. First, ascertaining the effectiveness of a dishonest strategy requires comparing it to an honest strategy, and cardinal voting methods admit multiple honest strategies. How to pick an appropriate one for comparison? Second, in order to have apples-to-apples comparisons between voting methods you need to use strategies that are comparably powerful. For example, using a hyperoptimized dishonest strategy for IRV that made full use of polling information but a very crude dishonest strategy for STAR would bias the results towards making IRV appear more susceptible to dishonest strategies (relative to STAR) than it actually is.
My solution to the first challenge was to compare the dishonest strategies to a baseline of using whatever honest strategy is the most similar to it. The second challenge was thornier. There are a lot of possible strategies out there. For example, one can have thresholds for how large various margins in the polls must be before doing something risky like voting for a (hopeful) pushover. There are also many options for what polls to use. (For example, strategists under IRV benefit significantly from having a pairwise comparison matrix that you get from doing polls with a Condorcet method.) I opted to give each strategy only the polling information from the voting method in question, such that under Plurality or Approval you just get the raw vote totals and under IRV you get the full IRV tabulation (with an equal amount of randomly generated polling noise included in all cases).
One potential way to get equally optimized dishonest strategies for all voting methods is to have just have voters vote optimally, i.e. in whichever way gets them the highest expected utility, taking into account the polling uncertainty. This approach has two problems. First, it’s intractable. Finding the exact optimal strategy isn’t doable outside of toy models. Second, it’s unrealistic. Actual voters aren’t perfectly honed decision-makers, and if there are extremely rare opportunities for (say) a pushover strategy to be effective under STAR, but it's difficult to identify these opportunities, fallible real-world voters would be better off not trying it at all.
The approach I settled on was to use strategies that only utilized polling information (and the voter’s preferences) in a very limited — and uniform — way. For this study, I used what I call “top-3 positional” strategies that only attempt to strategize among the three candidates who do best in the poll and pay no attention to the margins of victory or defeat in the polls; they are only sensitive to who finishes first, second, and third in a given round of tabulation. Similarly, the top-3 positional strategies ignore how much one of the top three candidates is preferred over another of the top three; only the preference order factors into the strategic calculation. Taken together, these constraints mean that only a handful of different strategies are possible.
Top-3 positional strategies
Here are the strategies used for the study. I will denote the voter’s favorite among the top three candidates in the poll as A, the second-best of these as B, and the worst as C.
(If you feel your eyes glazing over, I suggest scrolling down to the “Closing thoughts” section.)
- Honest strategy: Vote for your favorite.
- Favorite betrayal: Vote for your favorite of the top two candidates.
Plurality Top 2:
- Honest strategy: Vote for your favorite.
- Favorite betrayal: Always vote for one of the top three candidates. If the results in the first round are C>A>B, and A loses the runoff, vote for B. Otherwise, vote for A.
- Favorite betrayal + pushover: If A wins or finishes third, vote for A. If the results in the first round are C>A>B, and A loses the runoff, vote for B. If A polls the highest in the first round but loses the runoff, instead vote for the third-place candidate (regardless of whether that’s B or C). (I also experimented with only being willing to use B as a pushover, but a willingness to use C as well yielded a higher ESIF.)
Instant Runoff Voting:
- Honest strategy: Rank the candidates honestly.
- If the results in the penultimate round (the one with three candidates) are C>A>B, and A loses in the final round, vote honestly except with the rankings of A and B swapped (so B is ranked higher than A). Otherwise, vote honestly.
- Favorite betrayal + pushover: If A wins or finishes third, vote for honestly. If the results in the penultimate round are C>A>B, and A loses in the final round, vote honestly except with the rankings of A and B swapped. If the results in the penultimate round are A>C>B, vote honestly except with B ranked immediately ahead of A. (B is used as a pushover/compromise candidate; the two kind of blur into one another here.) If the results in the penultimate round are A>B>C, vote honestly.
- Favorite betrayal + hardcore pushover: If A wins or finishes third, or if the results in the penultimate round are B>A>C, vote honestly. If the results in the penultimate round are C>A>B, and A loses in the final round, vote honestly except with the rankings of A and B swapped. If the results in the penultimate round are A>C>B or A>B>C, vote honestly except with the third-place finisher ranked immediately ahead of A. This strategy performed substantially worse than the safer pushover.
- Honest strategy: Vote for your favorite of the top two candidates and everyone else you like at least as much
- Favorite betrayal: Vote for your favorite of the top two candidates and nobody else. This strategy is unique among those tested because it is obviously suboptimal; testing it doesn’t really tell us much of anything.
Approval Top 2:
- Honest strategy: If the results in the poll are C>A>B, vote for everyone who is at least as good as B. Otherwise, vote for everyone who is at least as good as A.
- Favorite betrayal: Same as the honest strategy, but if C wins and the results of the first round are C>A>B, don’t vote for A (while still voting for B).
- Pushover: Same as the honest strategy, but if B wins and the results of the first round are A>B>C, vote for C (in addition to A and all candidates preferred to A).
I will use the notation “vote x-y-z” to mean “give x stars to A, y stars to B, and z stars to C.
- All strategies, unless specified otherwise: If A wins in the poll, vote in the manner of the expressive voters that comprise the rest of the electorate (described in the next section). If A finishes third, vote 5–1–0, and candidates who don’t finish in the top three are scored expressively such that a candidate who is about halfway between A and B gets a 3, a candidate who is as good as (or better than) A gets a 5, etc.
- Honest strategy: If A finishes second in the runoff and the results of the first round are C>A>B or A>C>B, vote 5–4–0. If A finishes second in the runoff and the results of the first round are B>A>C or A>B>C, vote 5–1–0.
- Favorite betrayal: Like the honest strategy, except that if the results of the first round are C>A>B, vote 1–5–0. All candidates other than A who are at least as good as B also receive a 5.
- Pushover: If A wins the first round and loses the runoff, give the third-place finisher 4 stars and the winner of the runoff 0 stars, regardless of who those candidates are. Aside from the top 3, all candidates who are at least as good as A get 5 stars, and those who are worse than the winner get 0 stars.
Other methodological notes
All of these simulations had 75 voters and 5 candidates, where the electorates were generated using Jameson Quinn’s hierarchical Dirichlet voter model described in this paper. The added polling noise was normally distributed with a standard deviation equal to 5% of the maximum support a candidate could possibly have. For voting methods with multiple rounds, the polling noise is designed to mimic a realistic tabulation in which some candidates have more or less support than they do in the noiseless tabulation.
The simulations consist of voters voting in two rounds. In the first round, all voters use the honest strategies described in my paper on Candidate Incentive Distributions. This yields the polling data. (For the cardinal methods, these are different from the honest strategies described above. The strategies used in the first round are more expressive and make no use of any polling data.) This also functions as the background strategy for ESIF. ESIFs comparing dishonest cardinal strategies relative to the honest positional strategies are calculated by dividing the ESIF of the dishonest strategy relative to the background strategy by the ESIF of the honest positional strategy relative to the background strategy.
You can find the source code here.
For the most part, the results were well in line with my expectations. Here are my main takeaways:
- Cardinal methods encouraged more honesty. Note, however, that the failures of the ordinal methods I tested don’t imply a broader failure of ordinal methods in general. I didn’t even try to include Condorcet methods in this study, and I predict that they would have rewarded honesty at least as strongly as STAR.
- Dishonest strategies were surprisingly effective. The fact that all of the strategies do foolish things like treating being down in the polls by 0.01% as being radically different from being up in the polls by 0.01% means that all of the strategies I tested are decidedly suboptimal. Nonetheless, they were about to outperform honesty for every method except Approval and STAR. This leads me to think that cases where it’s strategically optimal to vote dishonestly are at least somewhat common, at least for well-informed voters who understand the voting methods well.
- The difference between IRV and Plurality Top 2 is pretty big. For most purposes, Plurality Top 2 isn’t that different from IRV in 5-candidate elections. It’s pretty uncommon for the IRV winner not to be one of the top two vote-getters in the first round, so these methods tend to elect the same winners. But the difference is dramatic when it comes to strategic voting.
- Plurality Top 2 is vulnerable to pushover strategies. I have never heard of anyone actually using a pushover strategy in a Plurality Top 2 election, and I believe that it is pretty rare for voters to do this. Nonetheless, supporting a pushover can be highly effective. I view this as a strategic vulnerability that is theoretically catastrophic, yet in practice is probably close to irrelevant.
- Pushover isn’t as dumb in IRV as I thought. I had expected this strategy to do abysmally. As in, comparably to favorite betrayal under Approval Voting. But, while it doesn't seem to be an especially great strategy, I have to acknowledge that it’s viable.