Approval Voting is better without runoffs
Most of my research deals with various voting systems and how they can be manipulated by insincere voters. For many reasons, I think approval voting is the best voting system for single-winner elections. That's why I have supported The Center for Election Science. Unfortunately, they've been inconsistent in their proposals for voting reform in different American cities.
Here's a simple example that shows why I don't recommend using runoffs with approval voting: Say 100 voters are voting in an approval election with a runoff, and they have the following sincere preferences over three candidates A, B and C:
- 45%: A > B > C
- 20%: B > C > A
- 35%: C > B > A
Given these preferences, B would win an A/B runoff, C would win an A/C runoff and B would win a B/C runoff. A would not win any runoff. So, if all voters are at least roughly aware of the other preferences and are strategic, we may reasonably assume the following:
- The A>B>C voters want an A/B or B/C runoff.
- The B>C>A voters want an A/B or B/C runoff.
- The C>B>A voters want an A/C runoff.
Therefore, the dominant approval strategies in the general election are
- A>B>C voters: B only
- B>C>A voters: B only
- C>B>A voters: A and C
Then, depending on the runoff rules, either B wins immediately or wins the runoff. (Voting in the runoff itself is straightforward; insincerity is never rewarded.) So the sincere Condorcet winner wins the election at equilibrium. What's wrong with this result? Well, the required strategy is complicated, which would amplify the power of the most sophisticated voters. But, more importantly, the best strategies for 80% of the voters turned out to be strongly insincere: They had to approve one candidate without also approving all candidates they preferred, destroying what is one of approval voting's most desirable properties.
Of course, this reasoning applies equally to range (or score) voting, which is why I don't support STAR voting, though it's certainly better than IRV (often now known as "RCV").
posted by Rob @ 18:47
11 comments
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11 Comments:
It was great chatting with you on Sass's weekly call ( https://democracydiscussion.com/ ). Per my comments there, I think that primaries are important for vetting candidates and pruning the list of candidates down to a manageable number for the general electorate. I think having a threshold (say 40%) approval rating for candidates and letting ALL candidates that exceed the threshold advance to the general election seems like a way of making it so that an approval primary doesn't distort the general election. A funny thing would have happened in St Louis in March 2025 if candidates were REQUIRED to get 40%: only Cara Spencer would advance. It seems like a MINIMUM of two candidates should advance to the general, but have a high threshold (e.g. 50%) for a third candidate to reach the general election. All candidates should strive to get a majority approving of them.
I don't see how a 40% threshold would ever work. What guarantee is there that _any_ candidate would meet the threshold?
A better way to prune the list of candidates would be to hold party primaries, whether open or closed. The main requirements are that voters can only participate in one party's primary, and that only one candidate from each party advances to the general election. The primaries and general election can all be conducted using approval voting, without the perverse strategy incentive shown above.
Bart: the way a 40% (or even 50%) threshold could work is if the law REQUIRES at least one or two candidates to advance. If the top candidate gets only 35% approval, and the runner-up gets 32%, one could have a law that states that the top candidate advances from the primary to the general, and then the 32% candidate would be like all also low-approval candidates, and could still be a write-in candidate in the general. Or, it could be that the law just requires a minimum of two candidates (regardless of approval level), and then advance all candidates that ALSO meet the threshold besides the top two.
The point of the threshold is to make it so that the candidates are more motivated to get a higher approval for themselves rather than mudslinging at the rest of the field. The primary motivation is for achieving the highest absolute approval level rather than relative approval level to other candidates.
Sure, you could exempt one or two (or more?) candidates from the threshold requirement. But I don't think it would work as intended. In particular, it worsens the "classic" collusion case, which I think goes all the way back to Brams and Fishburn (at least that's where I first saw it). By that I mean the case where two unrelated fringe candidates agree to support each other in the primary in order to eliminate a Condorcet candidate (or other sincere approval winner), so that they can compete directly in the general.
If the fringe candidates' plurality totals are each 30 percent, and the Condorcet candidate has 35, then the fringe candidates can potentially puff each other's primary totals up to 50 or 60 percent. If the Condorcet candidate is then eliminated by the 40 percent rule, then...
It seems to be impossible to reply to a reply with Google's/Blogger.com's weird threading system.
My example shows me that adding a runoff to a voting system can encourage counterintuitive and counterproductive voting strategies in the "primary", often giving runoff systems unintended and undesirable consequences. In practice, I wouldn't mind trying an Approval primary with a fixed threshold, though I'd prefer the threshold be as low as possible to eliminate fewer candidates. Ideally, none would be eliminated, turning the "primary" into an expensive official poll. I agree with Bart that primaries are best left to parties/blocs/groups; besides, an Approval general election would minimize the strategic advantage of nomination by primary.
I don't follow your reasoning on what "we may reasonably assume" about the preferences of strategic voters with respect to the runoffs. Could you please flesh that out a bit more? Specifically, you say that A voters would want either an A/B or a B/C runoff, but how can you be so sure that they would want their top candidate knocked out of the running with a B/C runoff, rather than keeping A in the running and mounting a resurgent campaign to get A elected in the general?
It seems to me your analysis is an oversimplification of real elections, ignoring such subtleties as:
1. In reality, the set of voters who are voting in the primary is smaller and more motivated/informed than the set of voters who vote in the general election. Thus, it's not guaranteed that the preference matrix will remain the same in the general.
[continued in replies, since my comment was too long otherwise]
2. In reality, campaigns are dynamic and shift resources depending on which other candidate(s) is(/are) viewed as the main competitor(s) and how much of a threat they're perceived to be. Thus, it's not reasonable to assume that strategic campaigns would expend all of their resources in the primary and then the general election would simply perfectly reflect the state of the world from the primary. Rather, if they're reasonably assured of gaining a spot in the runoff, A's campaign would likely conserve their resources during the primary and withhold any major "October surprise" type of bombshells and their most expensive ad campaigns until after the primary is over and they determine who they're up against and what the new lay of the land is. If it seemed like it might pay off, they might even spend some money in the primary trying to prop up B, knowing they're the more vulnerable candidate with fewer ardent supporters, so that they could knock B down more easily in the runoff. The A campaign might even explicitly tell some of their supporters to also vote for B, to ensure B's spot in the runoff. To assume that ardent A voters (and presumably the A campaign itself, if they're "reasonable"), would assess A's chances of winning based on an analysis of the preference matrix leading up to the primary and completely abandon their candidate who is leading in the polls strikes me as a profound misunderstanding of the psychology of most voters and all campaigns/candidates.
3. In reality, relative preference orders can map to very different absolute preferences, and each voter's sincere approval threshold may lie anywhere along that relative scale. The A>B>C voters can be subdivided into voters who would (sincerely) approve of no candidates, those who would approve of only A, those who would approve of A and B, and those who would approve of all three. Simply providing the preference matrix isn't enough for determining what strategy each of these sub-types of A voters would reasonably want to follow and which strategy they would actually be able to stomach. For example, you can ask me to rank Adolf Hitler, Josef Stalin, and Saddam Hussein, and I might be able to give you a ranking order, but I'm not sure I could bring myself to cast a vote "approving" of any of them, even if it would help my "favorite" win or keep my "least favorite" from winning. Similarly, you can ask me to rank Buddha, Jesus, and the Devil, and I would say B>J>D, but really I think B >> J >> D, and I would probably strategically approve of both Buddha and Jesus to try to keep the Devil out of the runoff at all costs, even if you gave me some convoluted story about how my approving of the Devil would somehow ultimately pay off. Finally, if there were three candidates who I had a very high esteem for, I might not bother voting at all if I felt the differences between them were small enough to deem it a waste of my time.
4. In reality, voters don't know how the totality of other voters will actually behave in the voting booth, and reasonable strategies have to take into account all possible strategies. If A>B>C voters think that "smart" C>B>A voters will vote for A and C, then shouldn't they continue to stick with their favorite A also and handily earn that runoff spot? In the absence of complete knowledge of what percentage of C>B>A voters might vote that way, is it better for A>B>C voters to stick with A or not stick with A? And if some percentage of A>B>C voters will likely stick with A because they know that some percentage of C>B>A voters will also vote for A, then surely B>C>A voters should vote for A also, to ensure that it's B and A in the runoff, rather than B and C, right?? So then A ends up with 100% approval in the primary but loses to B in the runoff! But then, knowing that they don't really need to help A get into the runoff, C>B>A voters decide to NOT approve of A, so they don't have to feel weird about approving the candidate they prefer the least, so they don't actually do it. But then, knowing that C>B>A voters decided not to approve of A, A>B>C voters decide they should be smart and only approve of B. And round and round we go...
Returning to your example, you assume that A>B>C voters would prefer to ditch A, rather than risking that A might lose to C in the runoff. That may be the case, or it may be the case that B and C are virtually identically awful to A voters, in which case, they'd vote for only A in protest and leave it up to the rest of the voters to choose their fate between two awful alternatives. You also assume "reasonable" C>B>A voters would cast an approval for A, but if they think C is just barely better than B and A is a potential fascist that would destroy democracy if given the chance, then that would be an insanely dangerous strategy.
In other words, determining "optimal" strategy is not trivial, and I strongly suspect that the vast majority voters will vote sincerely in the face of such uncertainty.
In the long run, real-world voters adapt to the voting system in use and become more sophisticated than we often give them credit for. It's true that most real-world voters still wouldn't be quite so sophisticated as in my example, but I prefer to make the pessimistic, game-theoretical assumption that all voters are thoroughly informed and willing to vote insincerely if it helps them selfishly to improve the outcome of the election. In my example, each voter has a dominant strategy regardless of what the other voters choose to do. Dylan, your point about a campaign gaining from "propping up" a rival in order to have an easier opponent in the runoff is a good one, and shows that insincere strategies can be used by both voters and campaigns when the voting system allows it. My point is just that, in the worst case, Approval with a top-two runoff can encourage widespread insincere voting, which can't be said for Approval without a runoff.
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