How many candidates should you vote for in an Approval voting election?

Approval Voting offers a simple voting reform: instead of voting for just one candidate in an election, voters can vote for however many they like. This is a pretty obvious fix to a fairly common voting problem; you don’t have to choose between voting for the candidate you really love who has little chance of winning and the candidate you kind of like who is a real contender if you can just vote for both.

Approval Voting would have allowed voters to voter for this guy and Al Gore.

But this newfound freedom comes with a new question: how willing to vote for multiple candidates should you be? Should you only vote for candidates you really love, or should you vote for any candidate you’d be willing to tolerate?

Previously, I looked at the conditions in which Approval Voting works best and found that Approval Voting consistently gives the best results (elects winning candidates that all voters are most favorable toward, on average) when all voters are very willing to approve of candidates. Approval Voting tends to give the best outcome for all voters when everyone approves of all but their most despised candidates.

But you aren’t “all voters”. You have your own desires about who wins. How many candidates should you vote for to maximize your ability to get a winning candidate that you like? Should you be pickier than other voters, only voting for candidates that you like best? Should you be more willing to vote for more candidates than other voters? Or does your best strategy change based on the behavior of the other voters? We’ll look at a model that shows that there is an ideal level of “pickiness” that you should adopt to maximize your ability to get what you want from an Approval election, and real voters happen to look about as picky as strategic voters, at least in the few Approval Voting elections that have recently occurred in the US.

How can I be a smart Approval Voting voter?

For (almost) any voting system, there are a cases where voting cleverly can help get you a better outcome for yourself. Approval Voting is no different. We’ll look at one simple strategy for Approval Voting — adjust how picky you are when choosing whether to approve candidates — and get an idea for how willing to approve of candidates you should be if you want to vote as effectively as possible.

How many candidates to approve?

The model we’ll look at for how voters decide whether to vote for a candidate is quite simple: Voters vote for every candidate they like more than a certain amount and don’t vote for any candidates they like less than that.

The threshold determining whether a voter votes for a candidate is based on how much voters like the candidates on average (µ, the average utility voters assign to candidates), how differently voters view different candidates (σ, the average standard deviation of the utility voters assign to candidates), and a constant that lets us adjust how picky voters are (N). Voters vote for any candidate they value more than

µ+Nσ

and don’t vote for any candidate they value less than this threshold. Increasing N makes voters pickier, and decreasing N makes voters more willing to vote for candidates.

One ideological cohort of voters will attempt to steer elections in their favor by using a different value for N than the rest of the voters. For these simulations, we’ll have all the voters in Quadrant I use a different N in their approval threshold than all the other voters (the voters in Quadrants II, III, and IV) to try to improve their ability to get winning candidates that are further into Quadrant I ideologically.

We’ll look at elections where voters and candidates are uniformly distributed in a two dimensional ideological space (they are equally likely to have any ideological views). I looked at a few different values for ß, which determines whether voters mostly care about how ideologically similar candidates are to themselves (ß near 1) or mostly care how far candidates would push governance in the ideological direction they preferred (ß near 0). Essentially, if ß is near 1, voters like candidates who they agree with the most, and if ß is near 0, voters like candidates that are more extreme than themselves as long as they’re going to push politics in the direction they like. (For the full story and why ß should be between 1 and 0 for real voters, see the book on the topic.)

The success of Quadrant I’s strategy is measured by taking the average dot product of the winning candidate’s position in ideological space and the vector (1,1). I explain this metric in more detail in my look at strategic voting in Borda Count, but the gist is that, the higher this value, the better Quadrant I’s strategy worked, the lower (more negative) this value, the more counterproductive Quadrant I’s strategy was, and a value near 0 tells us the strategy didn’t have any consistent effect.

Smart Approval voters are equally picky

The results were essentially the same in all the conditions I looked at: Your best strategy is to be a little picky, but not too picky, and deviating from this ideal “pickiness” is not a good strategy on average, no matter how everyone else is voting.

Quadrant I voters’ best strategy is to adopt an N in the range of 0 to .75, depending slightly on the value of ß, regardless of what threshold voters in Quadrants II-IV use. Likewise, voters in Quadrants II-IV can respond by adopting the same threshold and eliminating any advantage that Quadrant I could gain.

The graph of how effective Quadrant I’s strategy is vs. the value of N they and the other Quadrants’ voters used looks similar no regardless of the number of candidates running. Changing ß to be smaller slightly shifts the graph so that the ideal strategic approval threshold is less picky (the best N to choose is about 0 when ß=0), but this doesn’t change the graph’s shape.

Quadrant I’s strategy is more effective where the graph is higher along the z-axis (more red) and counterproductive where the graph is lower along the z-axis (more blue) Quadrant I voters’ strategy is more effective when they use a threshold with N closer to .5 than the threshold voters in Quadrants II-IV use. Quadrant I voter’s strategy is counterproductive when they use a threshold with N farther from .5 than the threshold Quadrant II-IV voters use.

Crucially, when both Quadrant I voters and Quadrant II-IV voters pick the ideal value for N (in the range of 0 to .75; the exact number depends on the value of ß), neither group can get better results by picking a different value. This suggests there is a Nash Equilibrium (a concept I’ve described before in the quite different context of determining what Pokemon type is objectively the best) where no ideological cohort can expect to get better results by choosing a different approval threshold than others. Smart approval voters are equally picky.

This is perhaps easiest to see when we map out what happens for each combination of N values for Quadrant I and Quadrants II-IV.

  • The more red, the more effective Quadrant I was at getting candidates further into their quadrant elected
  • The more blue, the more counter-productive Quadrant I’s strategy was, and thus the further away from Quadrant I the average winning candidate was.
  • White areas show where Quadrant I’s strategy was ineffective.

As you would guess, picking the same N as all the other voters is an ineffective strategy. Quadrant I voters’ best strategy is picking an N around .5, but Quadrant II-IV’s voters can neutralize their strategy by picking the same N. Crucially, neither set of voters can respond again by changing the N they use to get a better outcome for themselves — they’re both stuck getting the best outcome they can. Both sets of voters are always best off choosing the same approval threshold (in this case, the one that uses N=.5).

This is good sign for approval voting for a few reasons:

  1. Very simple strategies like “vote for more candidates than other voters” or “only vote for your favorite candidate” aren’t going to work. FairVote (an organization that advocates for Ranked Choice Voting) has contended that Approval voters should approve of only one candidate, but this isn’t generally the case.
  2. Any voting strategy that involves adjusting how willing you are to approve of candidates can be completely neutralized by other voters.
  3. Voters using the strategically smart approval threshold might not vote all that differently from how real voters vote in most approval elections. St. Louis’ mayoral approval primary results were similar to what one would expect from voters with an N around .6 (albeit in a slightly different pick-2 primary context) and Fargo’s mayoral primary results look like what one would expect from voters with an N just under 1.

There are some less great signs for Approval voting

  1. Strategically smart Approval voters are way pickier than the ideal voters who would make Approval Voting perform best. The approval threshold voters would choose to make Approval Voting as good at maximizing utility as possible is quite low. Utility-maximizing voters approve of way more candidates than strategically-smart voters.
  2. Approval elections where voters pick more strategically viable approval thresholds might not look as impressive when compared to elections using other voting methods. Average utility from Approval elections starts to fall off between the N=0 to N=1 range, which is precisely where voters should be operating if they’re being strategically selective. Precisely how bad this is for approval voting depends enormously on the precise values of N and ß and the exact distribution of voters and candidates, meaning that this could either be a significant problem or no problem at all.
  3. In Approval elections with lots of candidates, real voters might not vote for as many candidates as strategic voters do. More data is needed to determine how serious this worry is, but I suspect that elections with a lot of candidates are more likely to include “junk” fringe or vanity candidates that shouldn’t be treated as real candidates, and this isn’t likely too much of a problem.

Overall, I’d say the positives takeaways for Approval voting are on more solid ground than the negatives, but the negatives are worth exploring further to get a better idea of how much (and in what conditions) they actually matter.

How do smart approval voter votes?

What does ideal strategic Approval voting actually look like? We can use the average number of approved candidates when N=.75 and N=0 to get a rough range (I used ß=1 for these numbers). This won’t work for every election (some elections might have sets of extremely similar candidates or a bunch of “junk” fringe weirdo candidates that might as well not count as candidates), but it gives a rough idea of what strategically smart Approving might look like.

Candidates N=.75 approvals
per voter
N=0 approvals
per voter
31.0 1.7 
41.3 2.3 
51.7 2.9 
62.0 3.5 
72.3 4.1 
82.6 4.7 
92.9 5.3 
103.3 5.9 
113.6 6.5 
123.9 7.1 
134.2 7.6 
144.5 8.2 
154.9 8.8 
165.2 9.4 

Overall, the ranges seem pretty similar to what we’ve seen from the few Approval voting elections that have been run in the US, particularly the ones that have been run with fewer candidates. Again, elections with lots of candidates probably tend to have more irrelevant candidates, so the fact that real Approval Voting voters seem to vote for fewer candidates than this range would suggest they should might not actually mean that you (a hypothetical Approval voter) can take advantage by voting for more candidates.

If you want to see more of my results or join me in exploring Approval Voting further, feel free to leave a comment or email me at timrschmitz @ gmail . com.

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