Best posts made by cfrank

I think there are many of us here who prefer some voting system or another over approval voting. I also think there is room for improvement. However, approval voting has a huge advantage in its simplicity and potential for integration into existing infrastructure. This is totally besides the comparisons to make in terms of game theoretical stability with Condorcet methods and expressivity with Score or others.

My thought is that, if we are really going to make progress by consolidating our support behind a single voting system, then realistically, Approval voting fits the bill. That isn’t to say that it should be the final destination for voting reform, but it would absolutely be a major step forward. While IRV is something of a tokenism, Approval would be an actual game changer.

Any thoughts about this are welcome.

This is a minor attempt to modify Condorcet methods in a simple way to become more responsive to broader consensus and supermajority power. It’s sort of like the reverse of STAR and may already be a system that I don’t know the name of. In my opinion, the majority criterion is not necessarily a good thing in itself, since it enables tyrannical majorities to force highly divisive candidates to win elections, which is why I’ve been trying pretty actively to find some way to escape it.

For the moment I will assume that a Condorcet winner exists in every relevant case, and otherwise defer the replacement to another system.

First, find the Condorcet winner, which will be called the “primary” Condorcet winner. Next, find the “secondary” Condorcet winner, which is the Condorcet winner from the same ballots where the primary Condorcet winner is removed everywhere.

Define the Borda difference from B to A on a ballot as the signed difference in their ranks. For example, the Borda difference from B to A on the ballot A>B>C>D is +1, and on C>B>D>A is -2.

If A and B are the primary and secondary Condorcet winners, respectively, then we tally all of the Borda differences from B to A. If the difference is positive (or above some threshold), then A wins, and if it is negative or zero (or not above the threshold), then B wins.

For example, consider the following election:

A>B>C>D [30%]
A>B>D>C [21%]
C>B>D>A [40%]
D>B>C>A [9%]

In this case, A is a highly divisive majoritarian candidate and is the primary Condorcet winner. B is easily seen to be the secondary Condorcet winner. The net Borda difference from B to A is

(0.3+0.21)-2(0.4+0.09)<0

Therefore B would be chosen as the winner in this case.

Some notes about this method:
It certainly does not satisfy the Condorcet criterion, nor does it satisfy the majority criterion. These are both necessarily sacrificed in an attempt to prevent highly divisive candidates from winning the election. It does reduce to majority rule in the case of two candidates, and it does satisfy the Condorcet loser criterion, as well as monotonicity and is clearly polynomial time. It can also be modified to use some other metric in the runoff based on the ballot-wise Borda differences.

Continuing with the above example, suppose that the divisive majority attempts to bury B, which is the top competitor to A.
This will change the ballots to something like

A>C>D>B [30%]
A>D>C>B [21%]
C>B>D>A [40%]
D>B>C>A [9%]

And if the described mechanism is used in this case, we will find instead that C is elected. So burial has backfired if B is "honestly" preferred over C by the divisive majority, and they would have been better off indicating their honest preference and electing B.

And again, suppose that the divisive majority decides to bury the top two competitors to A, namely B and C, below D, keeping the order of honest preference between them. We will find

A>D>B>C [30%]
A>D>B>C [21%]
C>B>D>A [40%]
D>B>C>A [9%]

In this case, the secondary Condorcet winner is D, and the mechanism will in fact elect D, again a worse outcome for the tactical voters.

Finally, suppose that they swap the order of honest preference and vote as

A>D>C>B [30%]
A>D>C>B [21%]
C>B>D>A [40%]
D>B>C>A [9%]

Still this elects D.

As a general description, this method will elect the Condorcet winner unless they are too divisive, in which case it will elect the secondary Condorcet winner, which will necessarily be less divisive. I believe that choosing the runoff to be between the primary and secondary Condorcet winners should maintain much of the stability of Condorcet methods, while the Borda runoff punishes burial and simultaneously addresses highly divisive candidates.

This is a concept I had in mind which may already have been described, although not all of the logistics are necessarily hashed out and there may be issues with it. The idea is described below, but first I want to make a connection to “cake-cutting.” The standard cake-cutting problem is when two greedy agents are going to try to share a cake fairly without an external arbiter. An elegant solution is a simple procedure where one agent is allowed to cut the cake into two pieces, and the other agent is allowed to choose which piece to take for themselves. The first agent will have incentive to cut the cake as evenly as discernible, since the second agent will try to take whichever piece is larger. In the end, neither agent should have any misgivings about their piece of cake.

So this is my attempt to apply that kind of procedure to political parties and representatives. Forgive my lack of education regarding how political parties work:

• There should be a government body that registers political parties and demands the compliance of all political parties to its procedures in order for them to acquire seats for representation;
• (Eyebrow raising, but you might see why...) Every voter must register as a member of exactly one political party in order to cast a ballot (?);
• Each political party A is initially reserved a number of seats in proportion to the number of voters with membership in A; the fraction of seats reserved for A is P(A). however
• For each pair of political parties A and B (where possibly B=A), a fraction of seats totaling P(A~B):=P(A)P(B) will be reserved for candidates nominated by A, and elected by B; these seats will be called ambassador seats from A to B when B is different from A, and otherwise will be called the main platform seats for A;
• Let there be a support quota Q(A~B) for the number of votes needed to elect ambassadors from A to B, and call P(A~B) the ambassador quota of party A for B. If E(A~B) is the fraction of filled A-to-B ambassador seats (as a fraction of all seats), I.e. nominees from A who are actually elected by members of B, then A will only be allowed to elect P(A~A)*min{min{E(A~B)/P(A~B), E(B~A)/P(B~A)}: B not equal to A} of its own nominees. That is, the proportion of reserved main-platform seats that A will be allowed to fill is the least fraction of reserved ambassador seats it fills in relation to every other party, including both the ambassadors from A to other parties, and the ambassadors from other parties to A.

This procedure forces parties to also nominate candidates that compromise between different party platforms in order to obtain seats for any main-platform representatives. If a party fails to meet its quota for interparty compromises, it will lose representation. On the flip side, this set up will also establish high incentives for other parties to compromise with them in order to secure their own main-platform representation. In total, this system would give parties high incentives to compromise with each other and find candidates in the middle ground, which will serve as intermediaries between their main platforms.

Basically, here the outlines indicate seats open to be filled by candidates who are nominated by the corresponding party, and the fill color indicates seats open for election by the corresponding party:

Seats with outlines and fills of non-matching color are ambassador seats, and seats with matching outline and color are main platform seats. In terms of party A, by failing to nominate sufficiently-many candidates who would meet the support quota Q(A~B) to become elected as ambassadors from A to B, or by failing to elect enough ambassadors from B to A, party A restricts its own main platform representation and that of B simultaneously. By symmetry the reciprocal relationship holds from B to A. Therefore all parties are entangled in a dilemma: to secure main-platform representation, parties must nominate a proportional number of candidates who are acceptable enough to other parties to be elected as ambassadors.

To see that all needed seats are filled in the case of a stalemate, where parties refuse to nominate acceptable candidates to other parties and/or refuse to elect ambassadors, the election can be redone with the proportions being recalculated according to the party seats that were actually filled.

The support quotas collectively serve as a non-compensatory threshold to indicate sufficient levels of inter-party compromise. Ordinary PR is identical to PR with ambassador quotas but with all support quotas set to zero, whereby there is no incentive to nominate compromise candidates.

The purpose of this kind of procedure is twofold: firstly, it should significantly enhance the cognitive diversity of representatives, and secondly, it should significantly strengthen more moderate platforms (namely those of the ambassadors) that can serve as intermediaries for compromises between the main platforms of parties. Every party A has a natural “smooth route” from its main platform to the main platform of every other party: The main platform of A should naturally be in communication with ambassadors from A to B, who should naturally communicate with ambassadors from B to A, who should naturally communicate with the main platform of B.

Also, this procedure gives small parties significant bargaining power in securing representation. Large parties will have much more representation to lose than the small parties that are able to secure seats if the small parties refuse to elect any ambassadors, so rationally speaking, large parties should naturally concede to nominating sufficiently many potential ambassadors whose platforms are closer to the main platforms of those small parties. The same rationale holds for the potential ambassadors nominated by small parties, who also should tend to have platforms closer to the main platform of the small party.

Finally, this system creates significant incentives for voters to learn about the platforms of candidates from other parties who stand to reserve seats for representatives.

@rob especially if the state is a swing state, making it more difficult for the large parties to secure voters for their platform I think would be a significant influence forcing large parties and their candidates to more scrutinizingly determine the real interests of voters in those states. It may dilute the interests of less competitive states, but since the competitive states are crucial to obtaining the presidency, the large parties will still have to invest strongly in the interests of voters in those states in order to compete with alternatives (and obviously each other) for the crucial swing points. This may lead to something like an arms race of concessions, which happened in New Zealand in 1996 and led to the national adoption of a PR system, according to Arend Lijphart. Obviously that's quite a leap for the U.S., but maybe a less extreme analogue is not so far-fetched.

Maine is one of the thirteen most competitive states for elections according to a 2016 analysis (Wikipedia: Swing state), so I’m not sure their recent establishment is actually strategically foolish, although it’s possible that it wasn’t fully thought through. I agree it isn't clear.

I think it will definitely be interesting to observe how the current political apparatus responds to Maine--and apparently, more recently, and strangely, Alaska:

https://news.yahoo.com/alaska-is-about-to-try-something-completely-new-in-the-fall-election-193615285.html

Since Alaska is far from competitive, I do think this transition was in fact foolish for the reasoning you stated, but it remains to be seen. If we saw a state like Florida transition to a system like Maine's, it would be very interesting to study the relative differences between federal treatments of Florida, Maine, and Alaska as a case study for how "swingy-ness" might influence the effect of such voting system transitions. If Maine experiences an increase in federal power, it would be a good case for the remaining swing states to make a similar transition. If that occurred, the swing states would become a platform foothold for alternative parties to grow.

@k98kurz I don’t think there should be any uncertainty in the default for a voter’s ballot.

@lime the point of this post isn’t to argue that approval voting is superior to other methods or that modifications wouldn’t improve approval voting, it’s to point out that despite other methods being potentially superior, standard approval voting is probably the most realistic target for near future steps toward substantially reformed voting.

Unfortunately, more choices does mean the system is more complicated. You can observe that the addition of even a very simple, marginal modification as you suggest already raises questions. Every question about a method is an opportunity for distrust to be exploited, even if the method is ultimately better. Plurality is terrible, but almost nobody had questions about it, and that’s why it’s stuck around for so long. Do you see what I mean? I may be a bit jaded, but I’m hoping to be realistic.

I don’t mean to be a downer, but my point is a bit sad: in terms of what people would prefer, such as more choices or buttons, what we have to deal with is exactly the fact that people are having a hard time getting what they prefer. The political status quo is strongly opposed to voting reform, it will have to relinquish substantial power and accountability to the people under an effective voting system. There’s a reason only flawed tokenisms like IRV have passed through legislature in recent times. In fact, there is a history of voting reforms being enacted and then reversed.

@democrates I meant a Condorcet winner only among the front runners (for example, the candidates with the top K scores, here we are taking K=2). If there is no Condorcet winner among them, then we can choose the top scoring candidate.

In your case, if two candidates have the same second-greatest score, and the three front runners form a Condorcet cycle, then you can use the scores to break the tie. If this was used, then Jill Stein would have won the election.

That isn’t “the correct” solution (there is no such thing), but it is somewhat less arbitrary than flipping a coin or operating by alphabetical order, neither of which has anything to do with relevant information that is readily available on the ballots.

If we were being engineers about choosing a high quality candidate to win the election, we could even compute the distribution of scores, take the candidates whose scores exceed some elbow point, and find the Condorcet winner among those candidates with the top scoring candidate as the backup if no Condorcet winner exists. That’s basically a generalization of STAR with a dynamic front-runner selection method.

There are other ways to proceed. For example, we could remove Condorcet losers, then try to find the Condorcet winner of all remaining candidates, iteratively eliminating the lowest scoring candidate until a Condorcet winner emerges.

@toby-pereira yes you’re right, it was just a thought that occurred to me when I was thinking about how to discourage bullet approvals, but it has irreconcilable flaws that are now apparent.

@k98kurz mirroring @Lime, I think any advantage conferred to one candidate over any other in an election should be granted on an opt in basis. A voter shouldn’t have to opt out from conferring an advantage to a candidate.

@wolftune this is a well-known Condorcet method due in spirit to Tideman and called “Bottom N Runoff” where N=2 (hence “Bottom Two Runoff,” I.e. BTR or B2R). Generally speaking, these methods use some kind of absolute criterion (like least number of first place votes, lowest score, lowest approval, etc.) to decide which “bottom” candidates to subject to an elimination round, eliminates a Condorcet loser among them, and iterates until the desired number of winners remain. You are right, they’re pretty good methods. I like them.

@sarawolk you mean for us to use RCV instead of IRV? What do we call non-IRV ranked choice voting methods? I get that the onus is on us to make the alteration, but it’s actually both strange and quite annoying that IRV has presumptuously taken that title, as if other ranked choice voting methods don’t exist.

https://www.forbes.com/sites/steveforbes/2022/09/23/weird-undemocratic-voting-system-is-taking-hold-in-us/?sh=3e3faf2b2714

This is a very unfortunate depiction of IRV and rank choice voting in general, it’s the kind of terse and poorly-informed (or simply pernicious) dogmatism that upholds the current status quo.

But who knows, maybe it’ll backfire! Getting people to say, “What? What is this thing?” and to actually investigate may be a good thing in the end. As Condorcet wrote, “truth alone will obtain a lasting victory.”

Since I started learning about voting theory I've struggled with accepting the Condorcet criterion. I think my position on it has been tempered a bit and softened, but I still have some criticisms I wanted to express.

In my opinion one of the strongest argument for the Condorcet criterion is very simple and as follows: It should not be that the winner of an election should have any other candidate preferred over them by a majority in a head-to-head election between the two. This is totally practical, since in reality a large majority could perhaps in principle simply overthrow the election results, for example. Unfortunately this is not possible to satisfy in general, as demonstrated by the Condorcet paradox. Hence, the condition is qualified to be upheld `"whenever possible," and one arrives at the Condorcet criterion. If one applies Condorcet's Jury Theorem, and assumes that in any head-to-head election there is an objectively ``better" choice, and simultaneously that each voter is more likely than not to indicate that choice as their preference in each head-to-head match, then we concoct some level of information-theoretic corroboration for the criterion.

There are issues with this argument, however, and also with the conditions for applying the Jury Theorem. Most substantially, and as recognized by Condorcet himself, the concept of a "better" choice is inherently subjective---that is, each voter has a different conception of what might constitute a good choice, hence the notion of a socially shared objective ideal is questionable at best. Even in that case, it does not seem theoretically appropriate to treat voters simply as independent, identically distributed Bernoulli random variables. More or less, this is equivalent to saying that each voter is already biased to make the ``correct" choice, or that already more voters than not will almost certainly make that choice. By that reasoning the result of the theorem is almost circular. At the same time there is no possible empirical support for the conditions supporting its conclusion in the context where individual preferences rather than the subjective identifications of a single socially shared objective is being indicated.

There are valid applications of the Jury Theorem: for example, in a jury. If one assumes that the jurors are competent, and that each has a higher chance than not of identifying the guilt of a defendant, then majority rule of the jury is likely to produce the correct verdict. The issue with extending the scope of the Jury Theorem is that, at least in theory, the guilt versus innocence of a defendant is an objective circumstance with the potential of being identified, whereas the ``best candidate" in an election simply is not, because it is by nature an opinion.

My second issue with the Condorcet criterion is the rather obvious observation that the majorities that elect the winners of each pairwise face-off can differ from each other substantially. This may not seem important on its face, but in fact, this is the root cause of the Condorcet paradox. A Condorcet cycle must have links that are supported by distinct majorities, one crowd shouting something here and another crowd shouting something else there. The social choice theorist listening to these different voices as though they come from one mouth will naturally be twisted into confused circles. It is true that there is necessarily an overlap between any two majorities, but there needn't be any overlap at all shared by any three of them. As such, while the premise of a Condorcet system seems positively-motivated, a Condorcet winner appears to be a figure that rises from the ashen remains of a chaotic competition, not necessarily a figure that represents a broad consensus of the electorate, although this may be the case.

The third issue I take with the Condorcet criterion is that the pairwise face-offs operate fundamentally within a majoritarian paradigm. May's Theorem only applies to situations where each voter can make only one of three indications---the first candidate, the second candidate, or neutral. This scope does not encompass general range voting systems, for example, so that it is not generally clear that a majority rules principle is the optimal method of social choice even between just two alternatives. The standard thought experiment might be called the ``pizza topping" problem, where three friends wish to contribute equally to purchase a single-topping pizza, but two friends prefer a topping that the third is allergic to. A majority rule decision is simply anti-social in this instance.

My final issue is that rank order systems put a large burden on voters, and it may just be impractical to expect each voter to indicate a full ranking of candidates unless the candidate pool is fairly small.

Anyway, I know that people like the Condorcet criterion, and I see its merits as well. I just wanted to know whether supporters (more or less) of it would be able to address those points in its defense, or raise any other criticisms that might point to better alternatives.

Thank you!

The lab I'm working for has decided to organize a vote for a new lab logo! It's exciting because we can test a variety of voting systems out and do a comparative analysis of results. The deadline for submissions for my lab mates is February 21, if anybody wants to help do the analysis this is a great opportunity!

@democrates what should happen is that a Condorcet winner should be identified among the frontrunners if one exists. If one doesn’t exist, who knows.

@k98kurz the qualitative experience of a voter has only little to do with the tactics they will employ to get what they want most. We also have to consider what can be easily implemented. While I don’t disagree with you about the suggestion that certain improvements could be made, I do believe (in agreement now with @SaraWolk) that this kind of argument is actually counterproductive to getting voting reform off the ground in the current phase. Adjustments can be made later. Right now we need to get something decent through to replace choose one voting, and that should be urgent priority #1. I strongly believe that the best way we can do that is by pushing with a concerted effort for basic, ordinary approval voting. My opinion is that we need to shift away from the “anything other than choose one” attitude and toward a specific replacement, and my estimation is that approval voting—while definitely not perfect—is the least contentiously agreed upon improvement.

@rob I tried to send him an email Maybe an open letter is fine

“Hello Mr. Forbes,

My name is Connor Frankston. I recently read this very short article of yours, and I wanted you to know that the sentiments you express there are hindrances to necessary voting reform. While instant runoff voting is not by any means perfect, your article does absolutely nothing to address any of the positive qualities of instant runoff voting, and likewise does nothing at all to address any of the negative qualities of the present plurality voting system.

There are other ranked choice voting systems that many (including myself) consider generally superior to both methods, and to write off ranked choice voting altogether based on this specific, somewhat pathological instance, seems to reflect a pernicious sort of cognitive bias.

Otherwise, I would be interested to hear a less terse rendition of your arguments to the contrary.

Thank you,

Connor”

Here is a proof of Arrow's Impossibility Theorem:
Youtube Video

I can’t shake the feeling that there is something wrong with this proof, and that there is some kind of subtle logical fallacy. This isn’t because I feel like IIA and Pareto Efficiency are so important, in fact I think they are both somewhat peculiar to begin with, as is the requirement of an output that is a strict total ordering, and I’m not trying to “save” them per se. I have not read Arrow’s own formal proof (yet) or other formal proofs of his result on paper, but if the result follows the same logic as the video, then I have some doubts, and I’m trying to articulate what those doubts are about here.

As explained on the Wikipedia page on Kenneth Arrow's paper "Social Choice and Individual Values," where his impossibility theorem was proved, "Arrow himself expresses hope at the end of his Nobel prize lecture that, though the philosophical and distributive implications of the paradox of social choice were 'still not clear,' others would 'take this paradox as a challenge rather than as a discouraging barrier.'"

In line with this observation, I believe actually that there may be a fundamental issue with the establishment of the identity of the dictator as formally proved by Arrow. If my reasoning is correct, then the formal impossibility theorem has been misinterpreted, and whether a voting system is a "dictatorship" or "non-dictatorship” may actually be inconsequential. In particular, the so-called “dictator” as formally defined by Arrow may very well exist, but this does not actually imply that the “dictator” is in control of the election. Rather, it might just as well be that the election is in control of the “dictator.” Here is a narrative to illustrate the paradox:

Suppose that four children named Alice, Bob, Charlie and Dale have a game night every Thursday at Alice’s house, where they play one game each of Chess, Yahtzee, and Poker. They use a deterministic ranked-order voting system to decide the ordering of the games, and vote at the beginning of each game night, since each week, each of them is more eager to play their more preferred games. They all have agreed on a particular voting system, and are pleased to know for a fact that the system they have chosen to use satisfies the independence of irrelevant alternatives and Pareto efficiency criteria, and will return a strict ordering of the games. They use this system many times, and accumulate an extensive record of their votes and the outcomes of each election. The way they do this is in a specific order: firstly (1) they submit their ballots; secondly (2) they observe the result of the system; and then thirdly (3) they record the result and the corresponding set of ballots in their record book.

On the day following one game night, another child named Kenny moves into the neighborhood. He is invited to Alice’s house that day and introduced to the other children, who mention their voting system and their record book. They also invite Kenny to join them on the game night of the following week. Kenny is intrigued by their voting system and the record they have kept of it, and asks to borrow the record for the rest of the week.

At home, he examines the record. However, he notices that, peculiarly, the other children have not indicated whose ballot is whose! Rather, the ballots are simply ordered from left to right. Tacitly, he assumes that the ballots have been arranged in alphabetical order, and using the exact logic in the proof in the video of Arrow's Impossibility Theorem and the record of the ballots, he concludes that Charlie is a dictator—that is, his conclusion is that their voting system simply ignores every ballot except for Charlie’s.

Before voting on the following game night, Kenny remarks that he thinks the children's "voting system" joke was very funny, since he has concluded that their "democracy" is in fact just a dictatorship controlled by Charlie. The children seem puzzled though, and thinking they are stringing him along, Kenny takes out the record book and demonstrates his proof that Charlie is a dictator. Charlie still seems puzzled, however, and states,

"But Kenny, we actually didn't put the ballots in alphabetical order. In fact, since the order we put the ballots in doesn't change the result of the election, we agreed to record our ballots in a totally random order from left to right each week!"

Kenny is now puzzled. "But... then what have I shown? Is it that the voter who is placed third from the left in this record is a dictator?"

Alice answers, "Well, I think that would be rather absurd, wouldn't it? The 'third from the left' voter does not even exist until after we have already seen the result of our system, and then the ‘third from the left’ voter will be chosen from among us totally at random. Are you suggesting that, somehow, our voting system can predict the future?”

Do you agree that it seems like something is wrong somewhere? That “something” could just be me making some kind of logical error, but I’m not having an easy time finding it if it’s there, and in any case I’m not having an easy time reconciling this narrative. What I mean to illustrate by this narrative is that there seems perhaps to be a causal misinterpretation of the formalism of Arrow’s theorem. Each time the ballots change, it is actually not clear that the voter identities “behind” each ballot have not changed as well. It is therefore not necessarily possible to “fix” the “dictator’s” ballot while simultaneously altering every other ballot, since the identity of the dictatorial ballot is potentially contingent on all of the ballots. If that’s true, then the specific identity of the dictator may not be predetermined by the system, but rather may be contingent on the election results.

I understand that the formality would have us associate the “first” ballot with “voter 1,” and the “second” ballot with “voter 2,” but again, voters 1 and 2 are indistinguishable except for their ballots and the arbitrary numbering. What is stopping us from applying the theorem to conclude that the “third from the left” voter is a dictator?

Just to be clear, I am quite confident that I fully understand all of the logic in the proof of the theorem. It is a theorem and the proof is correct. My skepticism is about how that formal theorem is being interpreted.

@toby-pereira I didn’t get to check this in detail but I find it interesting. It also makes sense, although I wonder if it should be stated a bit more generally, saying maybe that there is some fixed positive constant C such that for all epsilon>0, there is some k such that for all k’>=k, C*Var(l)/k’<epsilon.

Is there a reason for choosing the normalized variance Var(l)/k’ rather than the normalized standard deviation sqrt(Var(l))/k’? Or even expressing in terms of sqrt(Var(l))/E(l)?

@bmjacobs my intuition is that it will take a lot of education to convince the mainstream public that nondeterminism is a viable solution to certain voting theoretical problems. Putting that aside, what would your position be regarding auditing and transparency for a nondeterministic system, and whether this kind of system could be feasibly implemented without worries about rigging?