Is Approval Voting Strategyproof?

Well yes, but actually no. Unless...

Introduction

Updated 12/7/2025.

Recently, I wrote a post about approval voting, and I would like to go more in depth here about one of its most appealing properties: strategyproofness. Which is to say, the property that a voter has one and only one optimal strategy, and that strategy is to vote honestly, with their heart. There is no benefit to trying to game the system by misrepresenting one’s preferences.

The question we are trying to answer is: is approval voting strategyproof? The short answer is:

well yes, but actually no

Approval voting is strategyproof under very specific (and arguably unrealistic) conditions. However, I aim to show that by reframing how we think about our voting goals, we can in fact regain strategyproofness from the perspective of an individual voter.

For this discussion, it’s worth being explicit up front about the definition of the voting systems we will be referencing: plurality (our current system, also called PV) and approval voting (AV).

While Ranked Choice Voting is also a popular alternative, it is still highly susceptible to strategy. The Gibbard-Satterthwaite theorem shows that no reasonable ranked voting system can be strategyproof. Indeed, RCV fails a number of other desirable properties like monotonicity, so beyond examples of strategic voting in RCV which we will get to shortly, we won’t dwell on that system too much. For a more in-depth discussion of RCV’s flaws (and why I believe AV to be strictly superior), see my post on approval voting.

Strategy Under Plurality

Let’s first get a sense for what it feels like to have to vote strategically under plurality voting.

Let’s consider an election with three candidates: \(A\), \(B\), and \(C\), and suppose you are a voter with the preference \(A>B>C\) (which we denote \(ABC\)). Now, let’s say that your favorite candidate, \(A\), has basically no chance of winning, and the real contest is between \(B\) and \(C\). How you should vote highly depends on what voting system is being used.

Under plurality, where you can only vote for one candidate, voting for \(A\) is effectively a wasted vote. Suppose that the final result before your vote is:

And \(C\) wins by one vote. If you vote sincerely for \(A\), the result is unchanged. You might as well have stayed home. However, if you vote strategically for \(B\), the result becomes:

And now \(B\) and \(C\) are tied! This is a strictly better outcome for you, since you prefer \(B\) over \(C\). Now, however the tie is broken, there’s still a chance that \(B\) could win. Whether it’s a coin flip (50% chance), or a runoff election is held between \(B\) and \(C\), where \(A\) can’t potentially spoil the election, your strategic vote has improved your expected outcome.

This is a problem, however. You as a voter had an honest preference which would have led to your least preferred candidate winning if expressed sincerely.

This is likely what Nader voters felt in the 2000 US presidential election. By voting for their favorite, Nader, instead of strategically voting for Gore, they allowed Bush to win by a mere 537 vote margin in Florida.

Strategy under Ranked Choice Voting

Frustrating scenarios can also occur under ranked choice voting (RCV). RCV is the most popular alternative voting system being discussed today, where voters rank candidates in order of preference, and candidates are eliminated (by lowest first-choice votes) one by one until one candidate has a majority. As a ranked system, RCV is subject to the Gibbard-Satterthwaite theorem, but this vulnerability particularly comes from the shortsightedness of eliminating arbitrarily by first-choice votes only. In general, elimination methods are highly susceptible to strategic voting, because it comes down to gaming the elimination order, as we shall soon see.

Suppose a situation with ranked ballots, where you still prefer \(ABC\), and the votes are as follows:

Before your vote, the final result depends on which candidate is determined to be eliminated first. If your favorite \(A\) is eliminated first, then your second choice \(B\) wins. However, if \(B\) is eliminated first, then \(C\) wins. Therefore, your vote is incredibly important. If you vote sincerely for \(A\) first, then you force \(B\) to be eliminated first, and \(C\) will win. You changed the outcome from a tie that could have gone to your second choice or worst choice, to your worst choice winning for sure.

However, if you vote strategically for \(B\) first, then \(B\) is safe from elimination, and you force your favorite to be eliminated, leading to \(B\) winning for sure. Again, strategically voting for your second choice has improved your outcome, at the cost of sincerity.

It could be even worse under RCV, however. Consider the situation:

Now, without your vote, either \(B\) or \(C\) could be eliminated first, leading to either \(A\) or \(C\) winning (in terms of your utility, this is an incredibly chaotic situation). Your sincere vote for \(A\) first would be completely inconsequential, because it only matters how the votes for \(B\) and \(C\) are distributed. However, if you vote strategically for \(B\) first, then you guarantee that \(C\) is eliminated first, leading to \(B\) winning for sure.

But if you had instead voted for your least favorite \(C\) first, then \(B\) would be eliminated first, leading to your favorite \(A\) winning for sure! In this case, voting for your least favorite candidate has led to your most preferred candidate winning!

Perhaps you think this is contrived, but this is actually a generic strategy in RCV. Take the 2022 Alaska special election. We had the Democrat Peltola, Begich the Republican (who was the Condorcet winner), and Palin (another Republican who was determined to likely be a Condorcet loser).

Peltola won the first round (expected, since it would be the two republicans who split the first choice votes). Now, as a Peltola voter, it might actually optimal to rank Palin > Peltola > Begich (or, not rank Begich at all), even though you likely hate Palin. You might deduce that Begich would likely beat your favorite, Peltola, in a head to head matchup. Therefore, by ranking the weaker candidate Palin first, you can contribute to the possibility that Begich is eliminated first, leading to Peltola winning in the final round against Palin. In fairness, the head to head results were not clear, so this would have been a very risky strategy. But, if any Peltola voters did, in fact, do this, then their high risk strategy worked, since Begich was eliminated first, and Peltola won.

That is, if your favorite \(A\) is ahead in first place, but not close to a majority, and there is a perceived Condorcet winner \(B\) and Condorcet loser \(C\) (or, at least a candidate \(A\) would likely beat head to head), then it can be optimal to rank \(CAB\), even if you hate \(C\), in order to get \(B\) eliminated first, leading to \(A\) winning.

Therefore, in both plurality and RCV, strategic voting can improve a voter’s outcome, at the cost of sincerity. A voter sometimes has to betray their honest preferences, or even vote for their least favorite candidate, in order to achieve their best possible outcome.

I would like to highlight how unfortunate it is that in both systems, the fundamental flaws in how they work result in causing voters distress about whether they should vote sincerely or strategically, and potentially being forced to vote for their least favorite candidate in order to achieve their most preferred outcome. This is a failure of the voting systems, not the voters. It does not have to be this way, and this is where approval voting comes in.

Approval to the Rescue?

Brams and Fishburn showed that approval voting satisfies a number of highly desirable properties. In particular, sincerity under two or three tiered preferences, and strategyproofness under dichotomous (two tiered) preferences. However, they also prove that once we leave two or three tiers, even approval voting is no longer strategyproof or necessarily sincere.

Let’s first define what these terms mean.

Note that a sincere strategy doesn’t need to include all candidates that the voter is indifferent to in the lowest included tier. For example, if a voter prefers \(A > (B=C) > D\), then the sincere strategies are: \(\{A\}\), \(\{A,B\}\), \(\{A,C\}\), and \(\{A,B,C\}\). Including \(B\) but not \(C\), or vice versa, is still sincere, since the voter is indifferent between \(B\) and \(C\). In this situation, the only candidate(s) that \(v\) prefers over \(B\) is \(A\), so if \(B\) is included, then we only require that \(A\) is also included for sincerity.

Finally, we say that a voting system is strategyproof if there is always only one optimal strategy for any voter in any scenario, and that strategy is sincere.

In their 1978 paper, Brams and Fishburn showed that

  1. In plurality voting, the only strategy that is never optimal is to vote for a voter’s least preferred candidate. Any other strategy can be optimal in some scenario. This can potentially include the optimal strategy being to vote for the voter’s second least preferred candidate.
  2. In approval voting, every optimal strategy includes voting for all candidates in the voter’s top tier of preferences and no candidates in the voter’s bottom tier of preferences. All strategy is always exclusively about how to handle candidates in the middle tiers.
  3. Therefore, if there are only two tiers of preferences (dichotomous), then there is one and only one optimal strategy under approval voting: vote for all candidates in the top tier, and no candidates in the other tier. That is, approval voting is strategyproof under dichotomous preferences.

The issue is that real voter preferences are rarely dichotomous. Most voters have more nuanced preferences, with at least three tiers of preference (ex. love, like, lukewarm, weak dislike, strong dislike). Brams and Fishburn showed that under three tiers, approval voting is sincere but not strategyproof. This is because a voter should always include all candidates in their top tier, and exclude all candidates in their bottom tier, but any combination of candidates in the middle tier would still technically be sincere. Therefore, there can be multiple optimal strategies, but all of them are sincere.

Strategy in Approval Voting

Under four or more tiers of preferences, however, approval voting cannot be guaranteed to be sincere (and therefore not strategyproof). For example, consider a voter with preferences \(A > B > C > D\). Voting for \(A\) and \(C\) is not sincere, since the voter is missing \(B\), which they prefer over \(C\). However, this strategy can be optimal in some scenarios. For example, suppose that polls indicate two possible election results:

or

Perhaps, there are three voters who are deciding between voting for \(A\) and \(B\), and voting for \(C\) and \(D\). Here, the optimal vote would be for \(A\) and \(C\), since if \(A\) and \(B\) are tied for first, then voting for \(A\) and not \(B\) causes \(A\) to win (better than an \(A\) and \(B\) tie). And if \(C\) and \(D\) are tied for first, then voting for \(C\) causes \(C\) to win over tying with your least favorite \(D\). Therefore, voting for \(A\) and \(C\) is optimal, but not sincere, since the voter is missing \(B\), which they prefer over \(C\).

Issues with Dichotomous Preferences

We have seen that under plurality and RCV, strategic voting can improve a voter’s outcome, at the cost of sincerity. We have also seen examples where voting insincerely can improve a voter’s outcome even under approval voting, when the voter’s preferences are not dichotomous.

In reality, voters have nuanced preferences that are rarely dichotomous. The idea that a voter can be indifferent between all candidates that they approve of versus all candidates that they disapprove of is unrealistic. Most voters have at least three tiers of preference, if not more.

But what if there was an alternate framing that could make voter preferences dichotomous?

The Game

let's play a game

Consider a hypothetical election with three or more candidates. Perhaps we’re talking about a crowded primary of 10 or more candidates. As a voter, you probably have a favorite, a few candidates you like, a few you dislike, and a few you really dislike. However, with so many options, it’s unlikely that your favorite candidate can be guaranteed to win. Under approval voting, voting for them only would not be strategically prudent, as you could be throwing away your vote when it comes to the real contest between more viable candidates.

We might reframe the situation away from thinking about a strict ranking of 10+ candidates, and choosing among the thousand possible approval votes to get one’s most favorite viable candidate to win, and instead think about which candidates we consider “acceptable” versus “unacceptable”.

Now, this isn’t much different from regular approval voting, where we necessarily have to project our nuanced preferences onto a dichotomous split of “checked” versus “not checked”. However, instead of thinking about two distinct checks as “this would get me this much utility if they win which is more than the utility I would get from this other candidate winning” (i.e. though both simply get checked, one check has more “value” than another), we can frame it under a simpler game.

Suppose we instead choose our goal to be “I just want any acceptable candidate to win, and I want to avoid any unacceptable candidate winning”. While the individual utility between candidates may vary, under our goal, all candidates in each tier are truly equivalent. If anyone we find acceptable wins, then we “win”. Otherwise, we “lose”. Under this framing, our preferences are truly dichotomous.

Definition:

We call a voter’s goal a dichotomous goal if their only concern is having any candidate in an arbitrarily chosen non-empty strict subset \(S\subsetneq \mathcal{C}\) win, and avoiding any candidate outside of \(S\) winning. We can say that any subset \(S\) of candidates induces a dichotomous goal for a voter.

Intuitively, this is equivalent to a voter constructing a dichotomous preference where all candidates in \(S\) are in the top tier, and all candidates outside of \(S\) are in the bottom tier. This allows us to say:

Theorem (AV Goal-Relative Strategyproofness):

Consider an approval voting election with candidates \(\mathcal{C}\). If a voter \(v\) has the goal of having any candidate in any arbitrarily chosen non-empty subset \(S\subsetneq \mathcal{C}\) win, and avoiding any candidate outside of \(S\) winning, then the only optimal strategy for voter \(v\), under any scenario, is to vote for all candidates in \(S\), and no candidates outside of \(S\): \(S_v = S\)

This theorem follows directly from the strategyproofness of approval voting under dichotomous preferences. By changing our goal to be dichotomous, we are essentially changing our utility function to represent a dichotomous preference for which AV is strategyproof, as shown by Brams and Fishburn.

As for why this theorem is true, you can get the sense intuitively as follows:

Therefore, the only optimal strategy is to vote for all candidates in \(S\), and no candidates outside of \(S\): \(S_v = S\). Any deviation could lead to you contributing to a failure to achieve your goal, and no deviation can ever improve your chances of achieving your goal.

Note that monotonicity certainly helps here. Unlike in Ranked Choice Voting, where a sincere vote can hurt your favorites, voting for candidates in \(S\) cannot hurt their chances of winning, and, in fact, strictly helps them. And also unlike in plurality, you need not worry about which candidate in \(S\) has the best chance of winning. The fact you can vote for all of them simultaneously is what simplifies the strategy so significantly (to essentially no strategy at all).

Note that this theorem says nothing about

  1. How the voter feels about candidates within \(S\). They might actually hate them! The point is that under a dichotomous goal, there is one and only one optimal strategy, regardless of how the voter feels about candidates within \(S\).
  2. How the voter chose the set \(S\). They could have used a random number generator to pick the candidates they put in \(S\). Or this could be a very honest reflection of what they feel, in their heart, that “acceptable” means. Either way, there’s only one optimal strategy under this goal.
  3. What any other voters are doing. Even if every candidate in \(S\) has absolutely no chance of winning, voting for them is still the only optimal and rational strategy under this goal. If multiple candidates in \(S\) have a chance, there’s still no reason to exclude any of them from your ballot, because what if there’s an upset, or a polling miss, and a withheld vote causes a candidate outside of \(S\) to win?

However, if this dichotomous goal is a rational projection of a voter’s nuanced preferences (i.e. if the voter’s set \(S\) defines a sincere strategy for them), then we can in fact say

Corollary (AV Strategyproofness under Dichotomous Goals):

If a voter has a sincere strategy \(S_v\), then choosing the dichotomous goal induced by \(S_v\) leads to approval voting being strategyproof for that voter under that goal.

To be explicitly clear: this is not saying that approval voting is properly strategyproof in general. This is essentially a newly invented concept of being “strategyproof” but only under a specific goal, chosen by a voter. To “win” under this goal, there is only one optimal strategy. But this strategy is also sincere for the voter. That makes it strategyproof for them under this goal. There could be an alternate strategy that could be optimal under a different goal (such as trying to get one’s absolute favorite candidate to win), but that strategy would also necessarily open the voter up to risk of not achieving the dichotomous goal.

Note that, again, this says nothing about how what any other voter is doing. If you, as a voter, choose your approval set \(S_v\) sincerely, and decide to adopt the dichotomous goal induced by \(S_v\), then there is no scenario where voting for \(S_v\) is not optimal for you. We have essentially eliminated all strategy from your perspective (once we have a dichotomous goal). The only ballot it is rational to cast is a sincere expression of this preference.

Where did the strategy go?

So where did the strategy go? The sleight of hand here is in how we constructed the sincere strategy \(S_v\) for which our dichotomous goal is induced. If a voter has preference \(A > B > C > D > E\), then there are four possible sincere approval strategies depending on where the voter draws their line:

  1. \[\{A\}\]
  2. \[\{A,B\}\]
  3. \[\{A,B,C\}\]
  4. \[\{A,B,C,D\}\]

And it gets more complex if it’s not a strict ranking. If the voter is actually indifferent between \(B\) and \(C\), then \(\{A,C\}\) is also a sincere strategy. Each of these sincere strategies induces a different dichotomous goal. Therefore, the strategy has not disappeared, it has just been moved to a much simpler decision: where to draw the line between acceptable and unacceptable candidates.

The key is that once a voter has drawn that line, then choosing a dichotomous goal induced by that line makes approval voting strategyproof for them. There is no longer any incentive to try to game the system by including or excluding candidates in the middle tiers of their preferences. The only rational choice is to vote sincerely according to their chosen line.

The practical upshot for you as a voter is that once you draw your line, then you no longer need to worry about strategy. If you are extremely principled, and you only approve of your absolute favorite candidate, then so be it. Choosing the dichotomous goal induced by that strategy, where it’s your favorite or bust, means there’s still no reason to second guess yourself (even if that goal is unlikely to be achieved, bullet voting is still the only optimal strategy under that goal).

If you are more pragmatic, you can choose to be strategic and might be more inclusive, which effectively strengthens your vote against the candidates you omit from your strategy, or less inclusive if you feel your favorites have a better chance (however, the chance that being slightly more inclusive will directly result in your single vote causing a favorite to lose against the candidate you just included is essentially zero). Either way, once you have drawn your line, there is no further strategy to consider under this goal.

Strategy in Approval Voting Revisited

Recall the example of strategy in approval voting we saw earlier, where a voter with preferences \(A > B > C > D\) and the two possible election results:

We saw that the voter found it optimal to vote for \(A\) and \(C\) (insincerely). We can reframe the situation as follows, depending on the voter’s goal:

Note that we can definitely see the other insincere strategies we considered earlier (like voting for \(A\) and \(C\) only) would be optimal under the strict, four-tiered preference we know the voter has. However, it appears that the most prudent dichotomous goal is to include \(A\), \(B\), and \(C\) in the acceptable set, since that maximizes the voter’s chances of achieving their goal. While bullet voting for \(A\) only is certainly an option, and is optimal under the dichotomous goal of only wanting \(A\) to win, by drawing the line more generously to include \(B\) and \(C\), the voter can increase their chances of achieving their goal to 100%.

This example, in fact, gives us a lesson in how to strategically choose our dichotomous goal. While I certainly don’t want to tell you how to vote (though, if you’re interested, I explain my approach to strategy below), I can say that it might be prudent to draw your line to at least include the most agreeable frontrunner. For example, if I was a Nader voter in 2000, I can say that the Nader and Gore dichotomous goal appears more pragmatic than the Nader-only goal, since it increases my chances of achieving my goal from 0% to something greater than 0%. While the Nader only vote would still be optimal under the Nader-only dichotomous goal, mathematically, the Nader and Gore goal is more likely to be achieved.

A Strategy Comparison

I would like to compare and contrast the types of strategy a voter has to employ under the three systems we have discussed: plurality, RCV, and approval voting under dichotomous goals.

For plurality, regardless of the size of their dichotomous goal, the strategy is the same. They have to find the candidate in that set with the best chance of winning, and vote for them only. The dichotomous goal is, in general, very weak under plurality. Really, the only optimal strategy in plurality voting is to vote for the most preferred of the top two viable candidates. However, under a dichotomous goal induced by a single candidate, the optimal strategy is indeed to vote for that candidate only. But the success rate of achieving that goal is very low, unless that candidate is a frontrunner.

For RCV, the strategy is much more complex. While the voter could naively vote sincerely according to their full ranking, this could backfire spectacularly, as we saw earlier. In terms of a dichotomous goal, the voter would think that it’s obvious they want to rank all in their goal above all outside of it (or not rank them at all). However, we saw that sometimes the voter has to rank their least favorite candidate first, in order to get their most preferred candidate to win. So it’s not as simple as just ranking all acceptable candidates strategically, and not ranking anyone else.

They must consider who is likely to be eliminated, who their votes are likely to transfer to, and then play the game again. They might have to play three eliminations ahead! And the elimination order itself can have huge ramifications on the state of the election just a few rounds in. The strategy gets incredibly complex very fast, in terms of playing optimally. In reality, the average voter will not likely get this right in a complex election, particularly with many candidates and many necessary eliminations. But, as we saw, the paradoxes appear even with just three candidates.

Thus, arguably, RCV makes it simpler by telling the voter “don’t worry, this election is way too complex to strategize about. Just vote sincerely and hope there wasn’t a paradox that would lead to your ballot actually leading to a worse outcome”.

And, in that respect, fair enough. But I can say, personally, that I don’t much like the idea of squashing strategy by intimidating the voter through unnecessary complexity. It feels like a cop-out.

In comparison, approval voting under dichotomous goals is much simpler, effective, and straightforward. Again, the strategy is delegated to the line drawing, but this is also optional. By monotonicity, just like voting sincerely in plurality is technically the optimal strategy under a dichotomous goal induced by your favorite candidate only, approval voting gives you that same safety. The difference is that under approval voting, this sincere choice to bullet vote is a choice, and not “the only way to express a sincere preference”.

In comparison to an elimination/runoff or ranked method, the line drawing is quite straightforward (and, again, optional). No consideration needs to be made about elimination order, vote transfers, or paradoxes. Nor which candidate in the acceptable set is optimal to take your one vote, as under plurality. You just vote for all of them!

For a more pragmatic voter, who perhaps always includes at least one viable candidate in their approval set, then the success rate of achieving their goal is much higher, and again, there is no further strategy to consider once the line is drawn.

The choice not to be strategic with your line drawing and just voting with your gut also has no risk of backfiring, as it does under RCV. Granted, the success rate of achieving your goal might be low if no one in your approval set is likely to win, but least you can be sure that your sincere vote won’t hurt any of them directly.

In summary, the primary risk of approval voting under dichotomous goals is that you may have been too stingy or generous with your line drawing. But, I will say that I think it’s far less risky to be too generous (approving too many candidates) than too stingy (approving too few candidates). The chance that being too generous is actually going to cause a less preferred, but still acceptable, candidate to win over a more preferred acceptable candidate by one single vote (yours) is extremely low (effectively zero), particularly in larger elections.

The Taylor Guide to Approval Voting Strategy

I will posit my personal approach to approval voting strategy, which is as follows:

  1. Identify your top favorite candidates. You never want to exclude them from your approval set, and it will never be optimal to do so.
  2. Identify any candidates you absolutely cannot stand. You never want to include them in your approval set, and it will never be optimal to do so.
  3. Among the remaining candidates, identify which ones you find genuinely acceptable. These are candidates you would be okay with winning, even if they are not your favorite. Let this be your initial approval set.
  4. Optional: consider the viability of candidates in your initial approval set.
    • If none of the candidates in your initial approval set seem viable, consider expanding your approval set to include at least one viable candidate that you find somewhat acceptable. This increases your chances of getting a candidate you at least don’t hate into office
    • On the other hand, if your initial approval set includes one or more viable candidates, then great! You can choose to keep this as your final approval set, if you want to maximize your chances of getting any acceptable outcome.
    • Alternatively, you can choose to narrow it down to your favorites among the viable candidates. Just be aware that narrowing your approval set too much could risk none of them winning.
    • You could also expand your approval set further if you want to strengthen your vote against the candidates you really dislike (ex. an “anyone but” strategy). Just be aware that expanding your approval set too much could risk an acceptable but less preferred candidate winning over your favorites. This can be especially effective for making a statement, where you approve a non-viable candidate you’re not even that crazy about, just to make candidates you dislike do worse.
  5. Once you have finalized your approval set, that’s exactly your optimal ballot. Vote for all candidates in your approval set, and no candidates outside of it. This ensures that those in your approval set are maximally helped by your vote, and those outside of it are maximally hurt by your vote, in alignment with your dichotomous goal.

Hopefully spelling it out like this shows that strategy under dichotomous goals in approval voting is exceptionally straightforward. The only real strategy is in step 4, which is also entirely optional. You can choose to be as strategic or as sincere as you like in drawing your line. But once that line is drawn, there is no further strategy to consider.

The Diluted Vote Myth

One criticism of approval voting is that the nature of giving equal votes to candidate you don’t feel equally about “dilutes” your vote. For example, if you approve of three candidates, but you really only love one of them (your favorite), then your vote is “weaker” than if you had only approved of your favorite candidate. However, this is not what the game theory shows us, and is also completely invalidated under the dichotomous goal framing.

In particular, note that your single choice to include a less preferred acceptable candidate (say, B) as well as your favorite (A) has the following effects:

In none of these scenarios do we cross from A winning to B winning strictly over A. At worst, we preserve ties or outcomes. Only in scenarios 3 and 4 would voting only for A technically be strictly better for a voter who prefers A over B (which leaves the Dichotomous goal framing if you chose to put both in your approval set). However, taking a step out of the dichotomous goal framing for a moment, if these outcomes are even possible, then:

  1. If B is unacceptable, then the voter should not have included B in their approval set in the first place. There is no incentive to extending your approval set from your favorite who is clearly viable or already in first place to an unacceptable candidate who is clearly viable and in second place. This is completely irrational.
  2. If B is not unacceptable, then B is acceptable. Perhaps you didn’t realize how viable B was, and that’s why you included both. But if B was a candidate you were even considering for your approval set, then you necessarily have to find them at minimum acceptable (or else, see the previous point).

I have heard arguments that approving of both A and B and causing (or, rather, preserving) B winning over A would make the voter furious. Again, this is still leaving the dichotomous goal framing, but even so this is completely incoherent. If you would be furious at B winning over A, then you must find B unacceptable, and therefore should not have included them in your approval set in the first place. Case closed.

A Remark on Coalitions

I have also heard claims that RCV is better for candidates trying to form coalitions. This is provably false. The concept of a coalition is precisely a dichotomous goal. “I want any candidate in this coalition to win, and I want any candidate outside of this coalition to lose”. Based on the results outlined above, approval voting is basically the perfect system for coalitions. The nature of an approval vote gives the absolute most robust way to support a coalition of candidates, without worrying about vote splitting, or ranking order, or elimination order.

We have already seen that to achieve an optimal result under RCV, you may need to rank an unacceptable candidate over an acceptable candidate, in order to get an acceptable candidate to win. Thus, the idea of a coalition under RCV is not guaranteed to be optimal for its members. You cannot simply say “rank us and no one else” and expect that to be optimal. You may still need to game the elimination order by ranking unacceptable candidates (perhaps over acceptable ones).

In RCV, a coalition necessarily asks candidates to choose different rankings of the coalition members, which can lead to splitting votes. Sufficient splitting will likely lead to the elimination of members of the coalition. You must hope that the elimination order works out in your favor, and that the strongest candidate in the coalition is not eliminated first if a weaker one might lose in a later round (ex. if they’re a Condorcet loser). This necessarily creates genuine and unavoidable risk unless the coalition is supported by a sufficient number of voters (such that the coalition members are guaranteed to be the last ones standing, won’t accidentally boost another outside candidate to victory, or won’t end up in a situation where the last coalition member standing is a Condorcet loser).

As an example for the above, suppose that Palin and Begich formed a coalition in Alaska in 2022. The original result had that the Republicans split the vote to eliminate the Condorcet winner Begich first, leading to Palin (the Condorcet loser) to go up against the Democrat and lose. The Republican voters would have needed either more Begich voters to rank Palin above the Democrat, such that Palin can win against the Democrat in the final round, or just have prioritized Begich first more strongly, such that Begich doesn’t get eliminated first and could have beat the Democrat. But the nature of ranking in the coalition necessarily creates risk of vote splitting and elimination order and viability concerns.

There is no similar risk under approval voting, no matter how widely or narrowly supported a coalition is. Any voter who wants to support a coalition is necessarily adopting a dichotomous goal, and thus the only optimal strategy is to approve of all coalition members, and no non-coalition members. No risk whatsoever, besides the risk that none of the coalition members are viable. But you’re still not hurting anyone in the coalition, or helping anyone outside of it.

You can argue that viable candidates will be wary to form coalitions under approval voting. But the result is the same under RCV. If your favorite wins in RCV, then there’s nothing interesting. If your favorite is eliminated because of the coalition and then another coalition member wins, it’s identical to voting in a way that supports that other member. You’re doing the same thing as you would be under approval voting, but with extra steps that open you up to accidentally hurting your favorite because ranks are jank as hell. Sure, it feels better to rank candidates in a coalition, but it objectively acts as a weaker support mechanism for that coalition than you would have under approval voting.

If you aren’t convinced, think of it this way: in RCV, the only part of your vote that is ever looked at, at a given time, is your highest ranked candidate who is still in the race. Therefore, the fact you supported a coalition by ranking multiple members of it is irrelevant until your highest ranked member is eliminated. The Alaska republican voters who ranked Palin first and Begich second were interpreted as “we want Palin”, rather than “we want Palin, but Begich is also acceptable”. Therefore, when Begich was eliminated first, and Palin lost to Peltola, the coalition support was irrelevant. Such a voter’s ballot was not “for Republicans”, it was “for Palin”, and led to a Democratic victory because Palin was the weaker Republican.

Common Objections Addressed

Q: “But doesn’t approving multiple candidates dilute your vote?” A: No. See the section above for the mathematical proof. Your vote for A and B never helps B beat A (or vice versa) when you approve both. At the absolute worst case, you preserve an existing outcome between them. At best, you strengthen your vote against unacceptable candidates.

Q: “In AV, won’t my vote help my second choice beat my first choice?” A: No. This is mathematically impossible. See the diluted vote section for the full proof, but briefly: if you approve both A (favorite) and B (second choice), your vote gives each exactly one vote. You’re not taking votes away from A or giving extra votes to B. The only way B beats A is if other voters already find B more acceptable than A, meaning B has broader support, which is exactly what the system should reward. The chance that your single vote could have made A win over B by withholding your approval from B is effectively zero in any realistic election, and would require both being tied for first or within one vote of each other ahead of all other candidates beforehand. And, if that were the case, and you found B unacceptable, then you shouldn’t have approved B in the first place.

Q: “Won’t voters be upset if their second choice wins?” A: If you’d be upset with B winning, don’t approve B. The same logic applies to RCV. If B winning would genuinely upset you, don’t rank B second. In approval voting, just don’t approve of B. Both systems let you indicate second choices; the difference is that approval voting does it more directly.

Q: “Isn’t RCV better for coalitions?” A: No. See the section above. RCV coalitions face vote-splitting and elimination order risk. Approval voting coalitions face neither. A coalition is precisely a dichotomous goal, and AV is strategyproof under such goals.

Q: “What if I can’t decide where to draw the line?” A: That’s the only strategic decision you need to make, and it’s entirely optional (see step 4 in the guide). If you don’t feel you have the information to decide which candidates are viable, just go with your gut and ask “would I find this candidate acceptable if they won?” for each candidate and approve accordingly. This is the most sincere dichotomous goal and you do not have to worry about the optimal way to express it. Once you’ve drawn the line, voting is trivial: approve everyone above it, no one below it. You will strictly help all candidates you find acceptable, and strictly hurt all others. This is far simpler than strategic ranking under RCV or strategic compromise under plurality.

Q: “But ranking FEELS better than approving!” A: If you are okay with the rank-jank, the chances for paradoxes, the risk of your sincere ballot leading to a worse outcome, the failure to elect the Condorcet winner (which leads to frighteningly successful repeal efforts), and the cognitive load of strategizing about elimination orders, then sure. As long as you’re aware of the risks. But you could also, you know, not have to deal with all those structural issues with approval voting. Just saying.

Conclusion

The purpose of this post is not to argue that approval voting is actually strategyproof in practice. Rather, it is a suggestion for you as a voter. I am offering you a mental model. A goal for you to frame your approval vote, which allows you to cast a sincere ballot without worrying about strategy.

I know many people who feel very strongly about a few favorites. For them, the strategy of drawing that line would also be something they would cast aside. If you feel similarly, then this framing could greatly simplify your voting experience. You can just pick your favorites, and not worry about the rest.

A more pragmatic voter might want to think more carefully about where to draw their line. Perhaps consulting polls, or considering electability. But once that line is drawn, there is no further strategy to consider. Voting for all candidates in your acceptable set, and no candidates outside of it, is the only optimal strategy under that goal.

Personally, this is how I approach approval voting. I am generally less strict about my approval set. I would be generous with my approvals, particularly in a crowded field, and that makes AV very robust and appealing to me. Sure, I could construct a ranking. I could try to game the system and betray a candidate I like in order to help my favorite win. But as someone who is probably close to the median voter, I’m not that hard to please, and I also got other stuff going on, man. I don’t have to think too much about if I approve or disapprove a candidate, I just feel it in my gut. And I’d rather just vote my conscience, and feel good knowing that my voice will be heard:

I simply cannot say the same under plurality or RCV.

The hope, too, is that approval voting will encourage candidates who attempt to appeal to a broader base, since voters can approve of multiple candidates without fear of “wasting” their vote. Thus, ideally, it would become much easier to just vibe out an approval set based on genuine feelings, such that the required strategy of drawing the line itself, too, ebbs away.

If you find approval voting appealing, I highly recommend you check out The Center for Election Science, which is fighting the good fight for approval voting.

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