How Approving should Approval Voting Voters Be?

Ranked Choice and Approval Voting have emerged as the two leading alternative voting systems being advocated in the U.S. I’ve looked a bit at how Ranked Choice might perform, particularly in a polarized political climate, and found that, though it usually is better at picking broadly agreeable candidates than the Plurality system used in most elections, it doesn’t consistently perform better than other voting rules like the much simpler Plurality Runoff (often referred to as “Jungle Primaries”).

A lot of smart people (and also myself) expect that Approval Voting would be a better alternative to Plurality. The Center for Election Science, which promotes Approval Voting, has managed to help Approval Voting become used for elections in St. Louis and Fargo, with possibly more adopters coming in the near future. But Approval Voting works a bit differently than the other voting methods I’ve looked at so far, and we need to sort out a bit about how voters will vote in Approval elections before using simulated elections to compare Approval Voting to Ranked Choice and Plurality.

In Approval Voting Elections, voters can vote for however many candidates they please. But how many candidates should voters vote for to make Approval Voting work best? And how many candidates will voters vote for?

We’ll find that Approval Voting works best when voters are very willing to “approve” many candidates — far more than they seem to “approve” in actual Approval Voting elections –, using a simple model of approval. But Approval Voting still works decently well even if real voters are significantly less likely to approve of candidates than “ideal” Approval voters.

Voting in Approval Voting Elections

One of Approval Voting’s biggest selling points is its simplicity — just vote for however many candidates that you want. No need to rank candidates like in Ranked Choice Voting or give candidates scores like in STAR voting; simply give each candidate a thumbs up or thumbs down. But how many candidates should you vote for? If you want to be a good Approval voter, should you only vote for candidates you really like, or should you vote for any candidate you would be willing to tolerate?

Approval voting gives voters an additional degree of freedom that plurality elections don’t — in an election with 5 candidates, voters could vote for one, two, three, or four of them (voting for five has the same impact as not voting). If voters only ever vote for one candidate, Approval Voting is identical to Plurality — certainly not the outcome Approval Voting advocates want or expect.

But what does peak performance Approval Voting look like? Does Approval Voting work best if voters are picky, only voting for candidates they like a lot, or if voters are much more liberal with their approvals, voting for any candidate they can tolerate? And how much do real Approval elections look like ideal ones?

How “Approving” should voters be?

To get a picture of how approving voters should be for Approval Voting to perform best, we’ll need two things:

  1. A way of measuring how good an election outcome is.
  2. An adjustable way to model how choosy voters are (how likely they are to vote for more or fewer candidates).

The first may sound hard, but fortunately there’s a pretty straightforward way to measure how well election outcomes turned out, one I’ve used since my first post on comparing voting methods: The average utility voters assign to the winning candidate. Simply put, this is how happy voters are with the winners of elections on average. Ideally, a voting system maximizes how happy voters are with election outcomes.

The second is a little trickier, because there are a lot of reasons a voter might or might not vote for candidates. How much they like a candidate, how many other candidates they like more, how much they know about all the candidates, and how picky they are about candidates can all influence all influence what a voter’s approval ballot might look like. Making a model to capture all of these factors in a way that accurately reflects how they affect the behavior of real voters just isn’t something I can do at this point — approval voting just hasn’t been used widely enough yet to produce the kind of data one would need to get a robust idea of how these factors affect voter behavior.

So instead we’ll keep things simple for now. Voters will choose whether they want to vote for a candidate purely based on how much they like that candidate compared to how much they like the candidates on average. We can adjust how choosy voters are by adding or subtracting from the average utility.

The model is very simple — voters will vote for every candidate that is above a certain utility threshold (they like more than a certain amount) and won’t vote for any candidates below that threshold. The threshold is determined by the average utilities voters assign to candidates, µ, (how much voters like candidates on average), the average standard deviation of the utilities voters assign to candidates (a measure of how differently voters view the various candidates), and a parameter that tells us how choosy voters are, N. Voters will vote for a candidates if they like the candidate better than


otherwise, they won’t vote for the candidate. The higher N is, the more picky voters are (because the utility has to be higher for them to vote for a candidate); the lower N is, the more willing they are to vote for candidates.

What do “ideal” approval voters look like?

To answer this, we’ll look at several different electoral situations and see Approval Voting fairs when voters are more or less picky (as determined by different values of N). Quite conveniently, there was a consistent pattern for all the situations I tested: Approval voting fairs best when most voters vote for most candidates (not very picky at all!), and it gradually performs worse as voters become more stingy with their votes.

No matter the number of candidates in an election, whether voters care more about candidates being ideologically similar to them or pushing the polity in their preferred direction as much as possible being ideologically similar to them or pushing the polity in their preferred direction as much as possible (measured by ß, as described in this link), or whether voters and candidates are evenly distributed among all ideological positions or are polarized into two opposing camps, approval voting always did best when N was around -1.75.

After 1000 simulations of 100 voter elections with different values of N, we see a similar pattern across a host of electoral conditions: once voters are just picky enough to start weeding out candidates from their ballots, approval voting quickly peaks. It very gradually performs worse as voters become more choosy about what candidates they put on their ballots.

Graphing the average utility voters assign to the winner (how happy they are with the election result) against N (how picky voters were for voting for candidates) for different electoral conditions makes the pattern obvious. The average utility quickly spikes as voters become less generously approving of candidates (N grows), then it gradually falls as voters become stingier with their approvals.

This pattern holds regardless of whether voters and candidates are uniformly spread among ideological positions or are polarized into two opposed clumps, and whether they prefer candidates that are the most ideologically similar to themselves (ß near 1) or that are most likely to push the status quo the hardest in the direction they want the status quo pushed (ß near 0):

This means that Approval Voting did best when voters were very willing to vote for multiple candidates, sometimes voting for around 90% of candidates. When voters are very liberal with their approvals, broadly agreeable candidates tend to get votes from the majority of voters while less agreeable candidates lose by dropping off the ballots of politically opposed voters.

Thankfully for Approval Voting fans and potential Approval Voting beneficiaries, Approval Voting performs pretty similarly until voters become much less willing to vote for candidates. But how closely actual voters match the highly-approving “ideal” ones could effect how well Approval Voting stacks up in comparison to other voting methods.

Do real Approval voting elections look like “ideal” ones?

These “Ideal”, N=-1.75ish approval voters are the Golden Retrievers of politics, loving virtually everyone. Are real Approval Voting voters anything like this?

Not at all.

But how much does this matter?

Real Approval election voters are almost certainly much pickier than “ideal” ones that our simulations found, but Approval voting with more realistic voters can still perform well compared to other voting methods.

I haven’t found a lot of data from real approval voting elections (if you know of any treasure troves of Approval Voting data, definitely let me know), but we can use St. Louis’ 2021 Mayoral primary to ground ourselves in some real results. This election featured four candidates, and on average a voter voted for 1.56 candidates on their ballot. Using our model for voter selectiveness, this is roughly what we would expect if N=.6 (if candidates and voters are uniformly distributed ideologically, and ß=1), where our voters approve candidates if and only if they evaluate those candidates as better than µ+.6σ.

If we look at all the March 2021 St Louis primary results, this simple model does a surprisingly good job predicting how many candidates voters will vote for. For elections with at least three candidates running (so elections where Approval Voting is meaningfully different from Plurality), the actual number of approvals per voter differed from the N=.6 estimate by .156 on average, thanks primarily to a larger miss on the sole six candidate race.

I’m not going to oversell this result for this model. Predicting what Approval ballots look like in several three candidate races while missing fairly badly in the one race with more than 4 candidates (the Alderman WD21 race, whose results suggest voters with N=1) isn’t some stunning success. Voters might ignore most candidates in a many-candidate race, as these are often full of mostly irrelevant candidates. Voters could act especially picky in low-level elections with lots of candidates simply because most voters aren’t going to have a strong opinion on the fifth person running in a local Alderman race. And there might simply have been enough voters who weren’t familiar with the Approval System enough to consider multiple candidates.

Even if this model correctly captures the structure of how people choose how many candidates to vote for in Approval elections (it may not), the value of N that best captures voter behavior might depend on how many candidates are in the race, the prominence of the race, how knowledgeable and motivated the electorate is, how negative the campaigns are, and who knows what else.

All I can say now is that the idea that voters will voter for candidates they like more than a certain amount is pretty intuitive, and the fact that we can adjust N might can let us fit data we get from future Approval Voting elections, albeit in a post hoc way.

How does Approval voting compare to Ranked Choice, Plurality, and others?

This question deserves its own article (or rather, several articles), but it’s worth a rough pass to put the importance of how approving Approval Voters are in perspective. The short take:

  • When voters are more likely to approve of multiple candidates (N is -1.75), Approval Voting does very well, producing election winners with higher average utilities than Plurality, Ranked Choice, and Plurality Runoff (and on par with simulations of Borda Count and Copeland Method).
  • When voters are about as likely to approve of candidates as St. Louis voters were in their mayoral primary (N is .6), Approval Voting typically outperforms simulations of Ranked Choice, Plurality, and Plurality Runoff.
  • When voters are less likely to approve of multiple candidates (N is 1), Approval Voting fares worse. , typically performing worse or comparably to Ranked Choice. Worth noting is that these simulations of Ranked Choice assume voters fully rank all candidates, which is not realistic for races with many candidates and may affect Ranked Choice’s performance in races with lots of candidates.

Below are some results showing average utilities voters voters assign to election winners (y-axis) for Plurality, Plurality Runoff, Ranked Choice, and Approval Voting (with N=-1.75, N=.6, and N=1) elections with different numbers of candidates (x-axis). These span different electoral climates (uniformly distributed or polarized candidates and voters and different values of ß) and were found using 1000 simulations of 100 voter elections for each set of conditions.

These are great results for Approval Voting with ideally approving voters. It requires less from voters than systems like Ranked Choice which asks voters to rank candidates in order of preference, but it performs far better in simulations.

However, how good of a system Approval Voting is can have a lot to do with the voters who are using it. There’s still a lot to do to see how much this matters for real Approval elections, but at least for now we have some reason to think that Approval Voting might be more effective if voters become more willing to vote for more candidates than they currently seem to be.


This is a first pass at modeling Approval Voting, so there are a lot of aspects of this model of Approval Voting that we should be cautious about and that might be worth exploring more. Here are a few of them:

I have assumed that all voters use the same utility threshold, whereas some real voters might be more or less likely to approve of candidates than other voters. Changing this won’t do much besides add some noise to the results unless certain kinds of voters are more choosy than others. If, say, ideologically fringe voters are pickier than others or right-leaning voters are pickier than left-leaning voters, this could be a relevant factor.

What I found doesn’t necessarily tell you to vote for as many candidates you can tolerate if you’re voting in an Approval election! What we’ve looked at so far assumes all voters are voting the same way; it doesn’t tell you whether you should be pickier or less picky than other voters if you want to maximize the chance that the election goes your way, and it doesn’t say anything about how voters and candidates might learn how to game the system. I’ll look into whether you should be more or less picky than other voters in a future article.

The utility threshold I used is only based on how voters view the existing candidates in elections. This makes finding a value of N that fits election data pretty easy, because a value of N will consistently result in roughly the same number of average approvals per voter. This means N must be adjusted to fit the idea that some elections might be full of popular candidates that voters are more likely to vote for and other elections might be full of broadly unpopular candidates that most voters won’t want to vote for. This makes finding a particular value of N that works for all Approval Elections less likely.

I’ve been assuming that voters are voting sincerely. Real humans are clever and can use things like polling data and their own sense of how a political race is going to vote strategically.

3 thoughts on “How Approving should Approval Voting Voters Be?”

Leave a Reply