Three types of strategic voting

For any reasonable voting method, there are multiple strategies voters could use in hopes of gaining greater influence. However, we can put every strategy into one of three categories:

• Uniformly advantageous: Voters benefit from using this strategy regardless of how many other voters are also using it.
• Saturating: Voters benefit from using this strategy if only a few other voters are also using it, but are harmed by using it if a lot of other voters use it.
• Coordinated: Voters benefit from using this strategy if a lot of other voters are also using it, but are harmed by using it if only a few voters use it.

Implicit in these definitions is the idea of a background strategy that voters will use if they don’t use the strategy in question. It’s natural to focus on honest background strategies, though with voting methods that admit multiple honest strategies (e.g. Approval) more specification is necessary. Also, we don’t have to imagine the possibility of the entire electorate using a particular strategy; it’s often more natural to imagine some subset of the electorate (for example, all the members of a given political party) using the strategy instead. Finally, voters are operating under uncertainty; they don’t know how everyone else will vote, and while a candidate may be heavily favored to win, no candidate is certain to win (or to not win, though their odds of victory may be very slight).

Uniformly Advantageous Strategies

Uniformly advantageous strategies are always worth using. Some examples:

• Voting for a frontrunner under Plurality if your favorite is a dark horse
• Giving every candidate either the minimum or maximum score in Score Voting.
• Using the full range of scores for viable candidates in STAR Voting.

Voting methods that don’t perform reasonably well when all the voters are using a uniformly advantageous strategy should be rejected out of hand. However, it’s generally hard for the consequences of a uniformly advantageous strategy to be extremely detrimental to society since, if they were, a lot of voters would be better off not using that strategy. But Plurality offers an exception. Here’s a contrived example:

There are 90 good candidates, each of whom is the first choice of 1% of voters, and two bad candidates, each of whom is the first choice of 5% of voters. Every voter who prefers a good candidate also prefers all of the other good candidates to the bad candidates, and even the voters who prefer a bad candidate prefer all of the good candidates to the other bad candidate. Either bad candidate would be defeated 95% to 5% head-to-head by any good candidate, but voting for the more tolerable of the bad candidates is still a uniformly advantageous strategy as compared to the baseline of everyone voting honestly. If every voter uses the baseline strategy, one of the bad candidates will win, but it’s unclear which of them will win, so voting for one of the instead of your favorite good candidate will, on average, yield an outcome that is better in your eyes. And the more voters rally to vote strategically for one of the bad candidates, the stronger the incentive for everyone else.

(I expect that plenty of people believe the previous paragraph describes the problem with American democracy, but I’m not one of them. I’m just using it as a contrived example to show that the universal use of a uniformly advantageous strategy can still lead to an atrocious outcome for society (albeit mainly because the baseline is so bad)).

A word of caution: just because a strategy is uniformly advantageous doesn’t mean the entire electorate will actually use it instead of the baseline strategy. Not all voters behave strategically, and a voting method needs to perform well with non-strategic as well as strategic voters.

Saturating Strategies

Using a saturating strategy is a good idea — but only when not too many other voters are using it. Saturating strategies are effective when only a few voters employ them, but backfire when a large enough chunk of the electorate does. Examples:

• Free riding under proportional voting methods like STV (proportional Ranked Choice Voting): If you’re virtually certain your favorite will be elected anyway, you’re best off ranking other candidates you like ahead of him to preserve the power of your ballot. But if enough other voters who also like your favorite do this, your favorite will be eliminated.
• The pushover strategy in Plurality + top-two runoff. Suppose your favorite candidate is a controversial incumbent but the vote is split between a lot of other candidates; you expect the incumbent to easily exceed 35% of the vote in the first round and none of the other candidates to exceed 25%, but she could easily lose to a broadly popular challenger in the runoff. But one of the challengers is widely seen as unacceptable despite having substantial core support, and you expect your favorite to handily defeat this polarizing challenger in such a runoff. Therefore, you can help the incumbent’s chances the most by voting for the controversial challenger in the first round instead of her. Your support for the incumbent in the first round would almost certainly be superfluous, but getting an easy-to-defeat challenger matters a lot. Of course, the incumbent wouldn’t make it to the runoff if all of her supporters voted like this. (I’ve never heard of this strategy actually being used even though it seems quite powerful in theory.)

In analyzing the effects of a saturating strategy on a voting method, it doesn’t matter how badly it performs if everyone uses that strategy since doing so goes against the interests of the strategic voters. However, it is important for a voting method to do reasonably well when the number of voters using this strategy is in equilibrium, i.e. when the expected harm from an additional voter using this strategy and it backfiring equals their expected gain when they use it and it works. When voters are strategic strategically savvy and use a saturating strategy, usually things go fairly well and either the strategy works or it has no effect. Negative outcomes also occur, but if they’re exceptionally bad then they are necessarily rare. (If they weren’t rare then it wouldn’t be worth risking them.)

Coordinated Strategies

Coordinated strategies require coordination to be effective. If you’re the only voter using a coordinated strategy, you’d be better off just using the baseline strategy instead like everyone else. Only if enough voters collectively use a coordinated strategy can it be expected to benefit the strategists on average. Examples:

• Burial in Condorcet (specifically Minimax, but such scenarios can be designed for any Condorcet method): Suppose roughly 40% of the electorate prefers candidate A, 35% prefers B, 25% prefers C, and voters who prefer any one candidate are equally split for their second choices. A is the Condorcet winner and would beat B 52.5%–47.5% head-to-head and beat C 57.5%-42.5%. But, suppose all of the voters with the preferences B>A>C instead vote B>C>A. Then C would defeat A 60%-40%, resulting in a Condorcet cycle in which Minimax would elect B. However, in this scenario, at least 10% of the electorate — 60% of voters with the preferences B>A>C — would have to use this strategy for it to be effective. Without such widespread buy-in, the strategy can’t work — but it still backfires if C turns out to be more popular than anyone anticipated, resulting in a close contest between A and C.
• Uniting behind a good candidate in the contrived Plurality example above (with 90% of voters preferring a good candidate a 10% preferring a bad candidate): If the baseline strategy is voting for the more tolerable bad candidate, at least 1/3 of all voters would have to select a single good candidate and all vote for that candidate in order to elect her. If not enough voters sign on, everyone who does this is just throwing their vote away; they’re passing up the opportunity to prevent the election of the more intolerable bad candidate. If the baseline strategy is voting honestly, then rallying behind a single good candidate is still a coordinated strategy; if only one voter signs on then it won’t decrease the probability of a bad candidate winning by anywhere near as much as it decreases the probability of the best good candidate (in the eyes of the one voter) winning. Of course, the threshold at which this strategy becomes viable is far lower than with the “lesser of two evils” background strategy.

Uniformly advantageous and saturating strategies are going to be used. Coordinated strategies? Not necessarily. Coordination is hard. In the Condorcet example, just imagine the effort necessary to get 60% of voters who feel a particular way to use a dishonest strategy that can backfire on them. Are volunteers going to go door to door telling people to use it? Would there be TV ads encouraging dishonesty? Any such attempt would require immense resources and make the candidate benefiting from it seem Machiavellian (even if it’s all coordinated by a super-PAC or something). Conventional campaigning would be a far, far better approach. While I don’t think we should dismiss the possibility of coordinated strategies altogether, when a voting method is vulnerable to one, that shouldn’t be viewed as a fatal flaw.

Center Squeeze: IRV vs. Plurality

Let’s apply this categorization in more depth to strategies in a couple of interesting scenarios. First, a simplified version of the center squeeze, in which voters have the following preferences:

• 45%: L>C>R
• 35%: R>C>L
• 10%: C>L>R
• 10%: C>R>L

The centrist, C, has the least first-choice support but is the Condorcet winner.

Under Plurality, with an honest baseline strategy, there are two strategies that come to mind:

1. If you prefer C, vote for your favorite of L and R instead.
2. If you prefer R, vote for C instead.

(1) is uniformly advantageous. (2) is coordinated. We should not expect many voters to use (2).

Now consider Instant Runoff Voting (single-winner Ranked Choice Voting) with the honest baseline strategy. The only reasonable way anyone can get an advantage through strategic voting is for the R>C>L voters to vote C>R>L instead. This has the same effect as strategy (2) under Plurality — but under IRV, it’s a uniformly advantageous strategy instead of a coordinated one! This doesn’t mean it will be used; strategy under IRV is a lot less intuitive than under Plurality. But it has a much better chance.

The Chicken Dilemma

Next, let’s look at a simplified version of Approval Voting’s chicken dilemma, where voters have the following preferences:

• ~25%: D1>D2>R
• ~25%: D2>D1>R
• ~10%: D1=D2>R
• ~20%: R>D1=D2
• ~10%: R>D1>D2
• ~10%: R>D2>D1

Suppose the background strategy is that Democratic voters (60% of the electorate) vote for both Ds and Republicans (40% of the electorate) bullet vote (only vote for one candidate). Then, Democrats who have a preference between D1 and D2 have bullet voting as a saturating strategy: It works great, unless the vast majority of them do it, in which case R would win. So you’d naturally get an equilibrium where a Democrat wins the vast majority of the time, but enough Democrats bullet vote that R has a non-negligible chance.

Take this equilibrium as the new background strategy. Then Republicans who have a significant preference between Democrats have a uniformly advantageous strategy: Vote for the preferred Democrat in addition to voting for R. Having Republicans vote like this will shift the equilibrium among Democrats in favor of bullet voting.

Let’s say the background strategy is that everyone with a preference between the Democrats voters for their favorite of the two; Republicans still vote for R and the Democrats without a preference vote for both D1 and D2. So D1 and D2 get support on ~45% of ballots and R gets support on 40%. Now the Republicans have bullet voting as a coordinated strategy; if all of them do it, R will win, but if not enough do it then bullet voting means forfeiting influence. Since coordination is hard, we should expect this background strategy to be a relatively stable equilibrium.

(Honestly, I don’t think this analysis adds all that much to my previous analysis of the chicken dilemma; these categories of strategic voting are useful for understanding such scenarios but hardly necessary.)

Conclusion

Not all strategic voting is equivalent. Different strategies, and the voting methods that give rise to these strategies, give rise to different questions.

• Uniformly advantageous: What happens if almost everyone uses this strategy?
• Saturating: How good is the equilibrium, where some voters use it and some don’t?
• Coordinated: How high is the threshold to make this strategy effective? How likely is it to backfire when only a few voters use it? If we’re debating what voting method to use for a particular purpose, do we expect voters to be much better at coordinating than the general population?

Most notably, a voting method being vulnerable to a coordinated strategy is nowhere near as bad as being vulnerable to a uniformly advantageous or saturating strategy. Coordinated strategies matter less since they will get used less often.

This is an especially important point when designing metrics for how susceptible a voting method is to strategic voting. It’s very easy to design a metric that imagines voters coordinating in perfect unison or has everyone with a particular preference use some strategy. Such metrics cannot distinguish coordinated strategies from saturating or uniformly advantageous ones. This doesn’t mean they’re worthless, but it’s a big limitation.