Ranked Approval Voting with Run-off

[A Former User]

@cfrank STAR is quite a similar procedure in spirit (if not in implementation). You could also expand STAR to runoff with more candidates, for example you could take the Condorcet winner of the top few score candidates.

[cfrank]

@brozai yes I was considering that with a Condorcet extension to the top few approval winners, it could be interesting. I think STAR is an excellent system and it’s what I’m basing this concept off of, my main issue is the arbitrary nature of cardinal scores, which is why I’m trying to incorporate an approval/rank order system. I’m also trying to mess around with these:

N(c<d) := # voters formally marking c<d
N(c/d) := # voters formally approving c but not d

and since there are only two basic kinds of strategy (burial and reverse burial) I was thinking about these:

N*(c<d) := #voters honestly marking c<d
N*(c/d) := #voters honestly approving c but not d

If the top two approved candidates are a and b, I was trying to consider these functions:

G(a,b) := N(b<a)-N(a<b)
H(a,b) := N(a/b) -N(b/a)

J(a,b) := N(b<a)-N(b/a)
K(a,b) := N(a/b) -N(a<b)

Note that G(a,b)+H(a,b)=J(a,b)+K(a,b), and H(a,b) is also the difference in approval ratings between a and b.

If we let X be the set of voters who approved both a and b, and let Y be the set of voters who approved of only one of a or b, then

|X| = N(b<a)+N(a<b)
|Y| = N(a/b) +N(b/a)

An ordinary STAR-like runoff would do the following:
(1) If G(a,b)+H(a,b)>0, elect a.
(2) If G(a,b)+H(a,b)<0, elect b.

but this is potentially sensitive to burial and reverse burial. The first two steps in the algorithm I’m envisioning are a bit more restrictive:
(1) If G(a,b)>0 and H(a,b)>0, elect a.
(2) If G(a,b)<0 and H(a,b)<0, elect b.

But it is possible that G(a,b) and H(a,b) are of opposite signs, and then we can continue the algorithm. One could do the STAR runoff but then we have a system that is the same as if we did the STAR runoff to begin with. If we want a system that has some resistance to burial and reverse burial we have to do something different first. I was trying to think about whether anything could be gleaned about what tactic is more likely to have been employed based on the relative sizes of X and Y, or the probability of having committing burial given being an element of Y along with the probability of having committed reverse burial given being an element of X. If that could be done, some independence assumptions could be used to model the expected value of

N*(b<a)-N*(a<b)+N*(a/b)-N*(b/a)

[rob]

@cfrank said in Ranked Approval Voting with Run-off:

@rob my goal is to establish a system where the generally optimal strategy (that is not computationally prohibitive) is to simply approve of the candidates you find acceptable and then to rank them amongst themselves.

Yeah I have trouble parsing the concept of "acceptable" as a binary concept, as I tried to explain above. In my mind, it lies on a spectrum, and is always implicitly relative to the alternatives. (whether those be ones currently running, or candidates or office holders in the past)

This is closely related to why most utility theory rejects the concept of absolute utility. The word "acceptable" implies not only that there is absolute utility, but also that there is a well-understood threshold for what counts as acceptable.

Basically, I "accept" whoever wins an election (and wish everyone did *cough* Capitol Riot *cough*), but I am happier with some than others.

Obviously, I can use such words as "acceptable" and their equivalents in everyday life. It is mostly when trying to represent them in numbers and do math on them that I need to ask for more clarification and have higher expectations regarding rigorous definitions. (and I have certainly voted in primaries or local elections, where my everyday usage of the word "acceptable" applied to all or almost all candidates.... so, there's that)

I understand that is not how everyone looks at it. Most people I run into are far more comfortable than I with putting things into neat little categories. (or maybe I should say, they are less comfortable with the "everything lies on a spectrum" and "everything is relative" worldview that I tend to have)

[cfrank]

@rob when I say “acceptable” I do mean a voter’s individual, subjective assessment of the candidates they would be “content enough” or satisfied with winning the election relative to alternatives. Ideally, there would be candidates whose elections would be generally positive events for more than just most voters. If not then to me that seems like there is an issue with the candidate pool selection process.

For example, if I asked a child to rank their top few favorite colors, I’m sure they would have no issue with that. There are certain colors that pop out as relative favorites, and then they have relative preferences among those favorites. I asked my 8 year old nephew to do this, and he ranked “white<blue<red.” I asked him if he wanted to include any other colors, and he said no.

But more relevant to your thinking, this approval/rank structure basically restricts possible methods of individual tactical voting to two: burial and reverse burial. If you don’t like that, the system can always just be looked at formally.

You can also consider this structure to be a restricted kind of weak ranking where only the bottom slot is allowed to indicate indifference.

[rob]

@cfrank I understand. Most people have intuitive notions of such concepts that don't tend to hold up under more rigorous scrutiny. A child would be expected to have a notion of like vs. dislike that is more simplistic (and typically more "absolute" and binary) than that of an economist, game theorist or neuroscientist.

What if you were being a cool uncle and buying an RC car for your nephew's birthday, and you didn't even know if red, white or blue was going to be available? Now he might be more willing to express his preference between purple, green and pink as well.

(my seven year old considers certain foods unacceptable. I've found that simply limiting her options over the long term -- e.g. less sugar and junk food -- changes her calculus on the matter considerably. She may think it isn't relative, but it obviously is)

You do mention "relative to alternatives", so we are at least somewhat on the same wavelength here. But once you start analyzing it as relative to alternatives, you start having to ask "but which ones count as alternatives?" All of them on the ballot? What about some fringe candidate on the ballot that you know has a snowball's chance in hell of being elected? If you are considering likeliness of being a front runner, now we get into guessing at that, a burden on voters that I personally prefer we try to eliminate [1] from voting methods.

I often come back to the idealized example of voting for a numerical value, such as a bunch of people who share an office space, and having a vote for the temperature to set the thermostat to. If I were in that office, it would be very hard for me to think in terms of "acceptable" or "not acceptable" in such a case. But I can certainly tell you, given any two temperatures, which I prefer. Or, I can simply tell you my ideal temperature. If we use a reasonable voting system [2], I shouldn't have to express my preference as a binary (acceptable or not), and I shouldn't have to give a moment's thought to how others are likely to vote.

@cfrank said in Ranked Approval Voting with Run-off:

But more relevant to your thinking, this approval/rank structure basically restricts possible methods of individual tactical voting to two: burial and reverse burial.

Ok. Is that good? Not sure I understand the advantage of that.

More to the point, can you describe the strategy a savvy voter would use under such a system? If their goal is to actually get the best outcome for themselves, what strategy should they use for determining who to include in their ranked list and who not to include? Wouldn't they be wise to go ahead and rank all of them?

1. recognizing that no method can eliminate it completely, but Condorcet methods, in my opinion, reduce it to being insignificant or nearly so

2. The ideal voting system in such a case, in my opinion, would be for each person to pick their preferred temperature and then select the median. However, if you simply list a whole bunch of options with sufficient granularity, and people rank them, I would expect a Condorcet method to converge on that median. (unless people rank them unexpectedly, such as preferring both 68 degrees and 70 degrees to 69 degrees) I think it is notable that in such a scenario, the concept of "majority" becomes quite meaningless as well, even if Condorcet methods technically use pairwise majorities under the hood.

Youtube Video

[cfrank]

@rob what I meant to say by restriction of tactical methods is actually that there is only one kind of burial, and that there is only one kind of reverse burial, which may make tactics in this system much easier to analyze. And while it does require voters to make a subjective indication of their relative approval, it also foregoes the need for every voter to rank every candidate.

The color example stands, the candidate pool is limited for the most part to the colors of the rainbow plus white, black, brown, and a few others. We in fact brainstormed colors together and that did not change his ranking.

[rob]

@cfrank said in [Ranked Approval Voting with Run-off](/forum/post/1000

And while it does require voters to make a subjective indication of their relative approval, it also foregoes the need for every voter to rank every candidate.

Ok, I am not aware of any ranked voting system where you must rank all candidates. You certainly don't have to here in SF. But the general criteria put forward for not ranking them all is generally that you don't think they are likely to be elected anyway, or you just don't care between them. It isn't whether they cross a subjective threshold of disapproval. (I did, however, post another thread where I propose a way of handling this, even if it seems unlikely to be adopted in any political election any time soon)

We in fact brainstormed colors together and that did not change his ranking.

Well, when voting we tend to have something substantial on the line, giving more incentive to rank all the candidates you actually have an opinion on. I'm not convinced that if he knew he might end up with one of these two bikes if his top three color choices weren't available, he wouldn't be interested in expressing a few more rankings. (*)

Whether or not you want to explore the issue further of using an "approval threshold" or what-not in the context of voting (and why I am uncomfortable with it), I'll go ahead and put this out there for anyone who does. It's an analogy, but I think a very good one.

You've mentioned being impressed by Vickrey and his auction concept, as am I. It is counterintuitive to many, since they think "why wouldn't you sell it for the amount of the highest bid, rather than the second highest?"

But clearly, doing the former is 1) not game-theoretically stable, and 2) it is not clearly defined what a bid actually "means." Does it mean "the mostthe bidder is willing to pay"? No, a smart bidder would shade it downward from that, based on a guess as to other bidder's valuations, so they don't end up paying the exact maximum they would be willing to pay, when they might have gotten it for less. So a bid becomes a much more ambiguous concept compared to one in a Vickrey auction. Some may define it vaguely (e.g. "what you think it is worth") and others may define it in strategic terms (e.g. "the most you think anyone else is likely to pay while also being less than the maximum you would pay").

But under a Vickrey auction, the most strategic bid disregards what the bidder thinks others might pay, and simply bids the highest amount the bidder would be willing to pay. Therefore, it is very easy to tell someone how to effectively bid: "state the most you would be willing to pay." There is no ambiguity in the explanation, and there is no need for anyone to try to guess how others will bid.

Certain voting methods work similarly to a Vickrey auction, or close to it. Voting for a numerical value by specifying your preferred value and then selecting the median is an ideal example. It is game theoretically stable, and the best strategy is simple to explain. There is no need to shade your answer based on how you think others will vote.

I would argue that Score and Approval are both far from this ideal.

* the left bike's color technically not being in the rainbow being a non-spectral hue, but I guess that's neither here nor there

[cfrank]

@rob I do understand your point about the bike colors, and you are right about voters having something on the line and that it very likely would change the colors present in a ranking, and this ties in to predictions about which candidates will be most highly approved.

One question I have is how computationally expensive it is to make such predictions with reasonable confidence. For example, if voters generally are not committing burial, it doesn’t seem easy to predict which candidates are going to be most approved, which would make it also difficult to commit reverse burial in an effective way.

Here also to be clear I am defining burial as the exclusion of a candidate from a ranking that a voter actually prefers to a candidate present in the ranking, and reverse burial as the inclusion of a candidate into a ranking while there are excluded candidates the voter would actually prefer.

Furthermore, if the approval paradigm is more-or-less palatable, then I see no reason why not to take in the additional information about the rankings of approved candidates. It only allows the voter to provide more information to the approval system.

[rob]

@cfrank said in Ranked Approval Voting with Run-off:

One question I have is how computationally expensive it is to make such predictions with reasonable confidence.

I'm not sure what you are referring to... the other post where I was suggesting a collaborative filtering mechanism to predict unstated lower rankings?

In that case, yes, it is probably a bit computationally intense but I'd think the bigger problem is that it just seems a bit too magicky for political elections.... at least now. If the whole world adopted ranked elections, I might well seriously propose we look into it further.

(on the other hand, if you are just worried about how hard the computers have to work to produce results.... I'd estimate such a thing might be calculated -- for a national election such as president -- with about the same amount of processing as to produce a single frame of some video games )

Is that what you are asking about? You refer to burial, which is known to be a theoretical issue with Condorcet elections, but which I don't think would have much if any effect in real world elections. But that seems to be straying a bit from the topic here.

It only allows the voter to provide more information to the approval system.

If I understand correctly what you are referring to, I respond in detail on another thread started by @culi :

https://www.votingtheory.org/forum/topic/180/approval-irv

Basically, I don't think providing approval info in addition to ranking is providing meaningful additional information. That all goes back to my position that utilities are inherently relative.

[cfrank]

@rob I mean that it may be difficult for voters to make reliable predictions about which two candidates will be most highly approved.

[rob]

@cfrank Oh you said "computationally expensive" so I assumed a computer computation.

Are you saying it is cognitively difficult? If so... yes, of course. It can also be a result of following the polls and such, but even that gets really challenging when people are deciding who to vote for based on how they predict others will vote... that's a feedback loop.

And yes, I am against that. That is my whole reason for not being a fan of methods that reward you for being a good guesser.... which is my main gripe with Score and Approval.

[cfrank]

@rob cognitive difficulty isn’t totally important once you have polling statistics and algorithms running in the background to make predictions. You can very well consider the determination of the front-runners to be analogous to a password-guessing problem. The question then becomes how much information do you want to spend time gathering in order to have whatever given strategic payoff, which is more or less computational expense.

I don’t think the “Hall of Mirrors” effect can be eliminated, as per Gibbard’s theorem. The only thing that can be done is to make effective strategic behavior too computationally expensive to execute.

[rob]

@cfrank said in Ranked Approval Voting with Run-off:

I don’t think the “Hall of Mirrors” effect can be eliminated, as per Gibbard’s theorem.

I think that is well understood, just as a mechanical engineer knows that you can't eliminate friction. But you can using bearings and lubrication and the like to reduce it as much as is practical.

The only thing that can be done is to make effective strategic behavior too computationally expensive to execute.

I guess that is one way of putting it, assuming you mean "computationally" to mean "cognitively" (i.e. happening in a brain, not in a man-made computing device).

However, I would suggest that it goes a bit beyond that in that it deals with things beyond what a single brain can calculate, unless they can read other voters' minds.

But yes, as I have said many times, I think a typical Condorcet method would make it impractical to even bother trying to guess how others will vote. I suspect, but am not as sure, that even an IRV system would make it impractical the vast majority of the time.

In 1992 (first election I voted in), my preferences for president were Perot > Clinton > Bush. No one really knew if people were going to vote for Perot (the actual first choice of a LOT of people), or would decide it was safer to vote for a major party candidate. It was as "hall of mirrors" as you can get. I honestly considered flipping a coin as to whether to vote for my first or second choice. I factored in both my estimate as to how likely it was that Perot would be a front runner (not very), how likely it was that Clinton would beat Bush if it came down to those two (very), and how much I cared about each candidate. (I was fairly "meh" about Clinton, but thought it would be awesome to see Perot win)

But any ranked system would have meant that very few people would bother trying to even guess. I would have simply ranked them honestly. (under approval or score, it would be better than FPTP, but it still wouldn't have been straightforward for me to decide how to cast my vote for Clinton)

[cfrank]

@rob that is the way that economists and computer scientists would put it. In general, when I say computationally, I mean computationally. I'm significantly more concerned about a concentrated, organized group with access to loads of computational resources rigging a system than I am about a diffuse collection of self-interested individuals with no common goal. A person who tries to use their brain alone to out-play a well-informed AI at its own game simply doesn't stand a chance.

[rob]

@cfrank yes, well I guess the problem can exist at a lot of different levels.

Individuals can certainly vote strategically, and effectively so, under FPTP. Under approval, they almost have to. (unless they truly do think in simplistic black and white "like" and "dislike" terms)

I am also concerned about forming parties and eliminating candidates through primaries (etc), which FPTP strongly incentivizes. To be honest, that is the biggest problem because it causes so much polarization.

I am less clear on how organizations could game it through computation, although I don't doubt it is possible. They certainly do that with gerrymandering, with a lot of sophistication.