Monday, July 19, 2010

Three Is More Than Two (But Less Than Infinity)

Why are approval voting and score voting such good voting methods? This certainly wasn't the expected result when Dr. Smith's voting simulation was first run, and a lot of work has gone into trying to make sense of the result.

I've Got a Theorem...

When anyone first starts to dig into the vagaries of electoral systems, they will quickly hit upon the Nobel-winning work of Kenneth Arrow and his impossibility theorem. But there was another very important--perhaps more important--theorem that came along just a few years later. It goes by the mouthful-of-a-name of The Gibbard-Satterwaite theorem. You do have to admit "Arrow" is a much catchier, and certainly shorter, name; so we'll call this "G-S" for short.

What G-S proves is that, under any electoral system using rank-order ballots, if there are at least three candidates, there will always be situations where a voter who knew how every other voter was voting, would be better off by voting strategically. Always. Even if you allow for equal-rank preferences (which some Condorcet methods can handle) you still run into this problem. So the conjecture is that no single-winner, rank-order-based method could ever possibly support three strong parties, since any candidate would eventually run headfirst into this problem and the perfectly-informed voter will have to choose between honesty and strategy.

But approval voting and score voting don't have this problem with a third candidate. It is still always in your best interest to rate your true-favorite highest, and your true-hated lowest, and doing so will never cause the election outcome to be worse than any other outcome you could achieve. In short, there's no incentive to vote strategically.

Preemptive Counter-Counter-Arguments

Now, there are a couple counter-arguments that people will bring up at this point. One of them is that, since G-S is about a "perfectly-informed" voter, they claim that this means approval (and score) only work this well if there is perfect polling for an election. Which is clearly impossible, so we should obviously use INSERT_FAVORITE_METHOD instead. The logic here is entirely unsound. First, even if G-S doesn't guarantee the effectiveness of ratings-based methods, it certainly does guarantee the ineffectiveness of all ranking-based methods; to stump for a known-bad over an unknown-but-potentially-good, seems blindingly counter-productive. Secondly, the fact that even the best-informed voters may have to strategize to avoid a bad outcome will tend to cause less-than-perfectly-omniscient voters to hedge their bets, and strategically go with the lesser of two evils out of fear of the greater evil.

But a more bizarre (or perhaps just more brazen) argument is that, since approval and score can't pass G-S with four or more candidates, then they are clearly insufficient. Which is an absolutely mindboggling argument, since even school children know that three is still more than two, regardless of the fact that three is less than infinity. Are these methods perfect? No, they aren't. Are they able to deliver an outcome that it is impossible for any ranking-based method to deliver? Yes, they are.

Too Infinity!

The fact that these methods have no perverse incentives in three-candidate elections is probably a large contributor to their improved performance in three-candidate elections, even if they still aren't perfect. And we know that all methods do better when everyone is honest, so having nothing to gain from being dishonest probably accounts for something. And perhaps this improved performance with three candidates somehow carries over and provide better results with four-or-more candidate races too, since performance drops at a noticeably slower rate than the rank-order methods do.

Perhaps knowing why something works isn't as important as knowing that it works; but being able to explain why may help convince some people who refuse to believe that.


  1. Regarding dishonesty (switching order) with >3 candidates , perhaps it should be mentioned how rare this is (1 in a 1000 voters with negligible effect). See bottom for conclusion:

    If this dishonest ordering with more than 3 candidates somehow created concern, there could be an alternative. First, one could cut the losses of whatever tactics would be used in a primary. So, one has an open primary (this allows more representative candidates).

    Then, the top three rated candidates would advance to the general election. And now there's no issue with order tactics. Your advancing candidates would also be the most centrist/representative candidates.

  2. Excellent point (I'm sure I'd read that link before, but couldn't find it while writing this piece.) It also handles the less-than-perfect information case quite well.

    Yeah, "top three runoff" might not be too shabby; but I wonder, do you always have a runoff if there are more than 3 candidates, or is there some reasonable (non-arbitrary) threshold that could be set to avoid runoffs? I don't believe there is, unfortunately (50% more approvals than 2nd place, perhaps? That would likely be so infrequent as to not ever avoid a runoff. 20% more? 15%? I don't know.)

  3. I can't imagine exceptions to allowing more than three for a range general election. The reason I say that is that then you'd have that possibility for order switching.

    I know I said that you'd be cutting your losses against order switching in the primary election, but I'm second guessing that. First, consider how a top-two-plurality runoff (TTPR) allows one to look within the top three front-runners for a pick. This contrasts with regular plurality which demands you look at only the top two. (unless it's really close)

    See here for a visual of range voting tactics: The example here doesn't actually show order switching, but provides a visual for how it could happen.

    It appears to me that having a top-three-range runoff provides a buffer against order switching in similar manner as a TTPR allows voters to consider the person in third as their selection. My thinking is that for the range primary election a top three runoff would prevent tactical order switching for at least those polling in the top four. So there's a slight improvement to this minimal problem.

    Another perk here is that given there is a good quantity selection in the primary, you'd have some great candidates in the general election. And three is a nice number to manage information gathering and organizing debates for.

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  5. Hey Dale, if you're going to allude to our dialogues, I think I shd get a link or two?

    My point is that the analysis used to demonstrate AV or SV as the best is based on assumptions that are demonstrably false and therefore the debate continues.

    Seems like InfoHedon is suggesting something similar to my "top 3 IRV" system that uses a limited AV approach to pick the top 3 out of 7 candidates and then IRV for the rest.

    The fact some might have an incentive to vote strategically in a perfect info situation doesn't matter that much when they're without perfect info and if the final three are relatively centrist the "loss" from getting the wrong winner would be less.

    And it does pay to keep things simple...


  6. I'd really love to see a BR comparison of using limited AV vote to pick the final 3 out of 7 candidates and then either IRV or AV for the final 3.

    I'm guessing they'd be close, but AV wd be higher because of the fact they use more info and there may be some incentive for some folks to vote strategically still with IRV and 3 candidates.

    But what if "top 3 IRV" ends up being a gateway for AV? After all, folks can get used to AV in the primaries first and then the vote for 3 of the candidates restriction could be lowered and the primary/general election distinction collapsed, perhaps?

    I think we might be evolving towards AV or a SV (with 4 or more options), but we gotta focus on intermediaries first to prep voters for such. We can't all go from where we are now to where we shd be in one bound.


  7. "My point is that the analysis used to demonstrate AV or SV as the best is based on assumptions that are demonstrably false and therefore the debate continues."

    I have yet to see any of the assumptions be shown to be demonstrably false.

    I've seen a lot of you raising issues with them, that were then explained to you how there had already been efforts to incorporate your concerns, and the results did not appreciably change, and then you re-iterating the same concerns as if nothing had happened, with no reference to the counter-arguments that had been brought to your attention.

    Broken Ladder has the patience of a saint, by the way.

    What InfoHedon is suggesting is similar to your suggestion only in that it is a top-3 runoff; the theory behind his suggestion is quite strong, since top-2 runoffs do lead to different (often better) results than instant runoffs, and also because the runoff approval round can take advantage of the fact that approval has no perverse strategic incentives in the 3-candidate case (similar to how *no* election method has perverse strategic incentives in the 2-candidate case; hence this entire post and its title).

    YOUR suggestion is to use IRV for the 3-way runoff; which is in direct opposition to BOTH elements of the theory supporting an approval-with-top-3-approval-runoff.

    Using approval in your first round is a great start (although how you plan to limit round 1 to 7 entrants is still an unexplained restriction and uses an explained selection process). Your insistence on IRV for the runoff is mindboggling.

    That even the argument in your last paragraph in your first post of this very thread FAILS TO ACKNOWLEDGE the presence of the counter-argument I presented IN THE ORIGINAL POST that (attempts to) refute it, is another example of what I referred to in my second paragraph of this comment.

    If it will help knock some sense into you, yes, I suppose a BR sim of a 7-candidate approval-with-top-3-IRV-runoff could be done. Intuitively and by theory, I suspect the performance will not be better than approval-with-top-3-approval-runoff; although I don't know if it would be worse than a single-round approval election or better. But no one had a theory or suspicion for straight approval being as great as it is, so I suppose anything is possible?

    I believe BL has, elsewhere, pointed you to the IEVS sourcecode. Since you're the one with the point to prove, I think it'd be appropriate for you to do the legwork; do you have enough programming experience to do that?

    As for evolving towards better systems: obviously I have no model for this either, but I don't see "top 3 irv" being any sort of a stepping stone toward approval; some I'm not going to put my effort into it (sorry), but good luck.

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  9. Thanks. I didn't have the patience to reply after seeing the persistent comments on the "As It Ought To Be" blog.

  10. How do you define "demonstrably false"?
    Can I demonstrate that we don't have cardinal (not ordinal) utility rankings of all candidates in an election deep, deep down in our heart of hearts? Or isn't that almost self-evidently a heroic assumption (borrowed from neo-classical economics utility functions) that throws out lots of the messiness of reality for heuristic reasons?

    At issue is realism. The solutions proffered by BL and you still work within the same general Bayesian Regret framework and, as such, are damn near impossible to verify as to whether they truly capture the problem in "reality". Thus, we are still left with a debate and lack of closure as to what is the "best" single-member election rule for various sorts of political elections.

    dsh:YOUR suggestion is to use IRV for the 3-way runoff; which is in direct opposition to BOTH elements of the theory supporting an approval-with-top-3-approval-runoff.

    dlw: my suggestion was to see how a 7-candidate election with a 3-way final run-off would compare in terms of BR depending on whether IRV or AV were used in the final round. My guess was that they'd be very close (more so without strategic voting). You point out that IRV would still have incentives for some to vote strategically. But it doesn't account for how many folks would have this incentive or how strong the incentive would be if the 3 candidates are in fact relatively centrist due to the limited form of AV was used to select them.

    And so, my point was to argue that if the BR diff were minor that my approach would be preferable on the basis of other criteria. Most notably, how voters, for better or worse, are more accustomed to rankings, like with the olympics or world cup soccer...

    As for how the 7 would be selected: local rules could be used to foster a learning-by-doing in this regard. But I think a good approach would be to give the incumbent a pass if (s)he wants to run again, and then have the other six chosen by their ability to get the most distinct signatures from potential voters, above a certain low thresh-hold. This would be a first (six or seven) pass-the-post approach. And so my proposal for single-member state senate elections would include each of the basic types of elections. It would advance awareness of Approval Voting, while not requiring a lot of voter-education.

    Politics is the art of the possible and that requires accommodating ourselves to low-info voters and their received habits of thinking about voting. If they get rankings better than approvals then it's better to start with a simple approach to approvals (making them approve of 3 of 7 candidates)

  11. The general non-certainty inherent in using utility is a recognized problem for many branches of economics. Pointing this out is not a new observation; I pointed it out last may.

    But the fact that they are an imperfect tool doesn't mean they are a useless tool, especially since in many cases they are the only tool we've got. The surprisingly strong showing of approval and range in utility-based Bayesian regret simulations is something that suggest these methods are certainly worth trying in the real world.

    Step one was the utility based simulations.

    Step two is real-world use.

    We're currently working on making step two happen.

    I'm confused by your "rankings, like the Olympics and world cup" claim. In the Olympics, for judged events, each of several judges score the competitor [the top and bottom score are thrown out]; highest average score wins.

    (A few sports have used rankings, which has provided some excellent examples of IRV-failure. Imagine: you're currently winning the gold. The last competitor is judged; they are universally judged worse than you, so you... take home the silver?)

    And the world cup uses so many systems: top-two score-based group rounds, and then an one-on-one elimination... sports tournaments make pretty terrible election templates; which is fine, they're suppose to produce excitement (and sell ads), not choose a representative leader of the fan-base. (Those usually use the Borda count, but that's an argument for another day...)

  12. I lost an earlier reply.

    Here's the gist.

    I'm all for using AV or SV more in real life, but I believe that there are common-sense tools to guide us in how to use them. IRV or rankings are easier to get for many voters because it's pretty much a recursive use of FPP. Who's the best? Okay, if that candidate were removed (for some reason), who'd be the best of the remaining? And so on...

    I like a limited AV for primaries, as noted before. The limitation of forcing everyone to approve of 3 of 7 candidates keeps the shopping costs of learning about the candidates down and it prevents later resentments from some voters getting more influence than most voters, who may have only approved one or two of the candidates. Once voters get used to approval-voting in this way, it'll be easier to use AV in other settings, with a fuller set of options for voters.

    As for score-voting, in line with the olympics metaphor. It should be used by "professionals" who have public accountability for their scores. The best example I can think for this is if the state representatives were given back the right to elect their state's US senators. Let's say that there are 3 or 5 candidates remaining. Then, the state representatives would get to give scores of 0-10 for all of the candidates. It could then be mandated for them to pick 5 criteria to use for all of the candidates and to give the candidates scores of 0-2 on each of the criteria. The criterion chosen and the scores, plus explanations, would then be made public information for the constituencies of the state representatives.

    This would be as close to the Olympics judge metaphor in political elections as I can think of at this point and the only sort of setting I would want to use an Approval Voting election rule. It would be very inappropriate for most voters to get to score candidates, since they'd have no incentive to do their homework or accountability to justify their score-votes.

    I do believe that unpredicticability or excitement matters for political elections. Most PR elections, with many seats, are very predictable and not very exciting. American Proportional Representation is not as predictable for the 3rd seat and potentially exciting since the candidate elected to the third seat would have a chance to decide which major party is in power (in the state assembly or city council).