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Voting Paradoxes [video] (exploratorium.edu)
67 points by butterfi on Nov 4, 2016 | hide | past | web | favorite | 57 comments

I found this video really interesting. I knew that the current US system, first past the post, has a lot of problems from watching CPG Grey's [1] videos on it. I didn't know that the alternative systems, like ranked voting, had all these other weird problems.

It's depressing, really.

[1] https://www.youtube.com/watch?v=s7tWHJfhiyo

While it's true that every voting system has paradoxes, I don't feel that this makes every system equally bad.

The US has what I would consider a big systemic problem, which is that the first-past-the-post system leads to spoiler effects, and the result is a two party system. When I've talked to some people about this, the response I got was "well all voting systems have problems so we can't fix it without introducing new problems".

But the monotonicity paradox for elimination voting doesn't seem quite as serious. It seems to only be likely to come up when the two major choices are close anyway. If all voting systems are evil, it's the lesser evil.

If the US could implement elimination voting, we could remove a big problem (the two party system) and replace it with a smaller one: an occasional wrong choice between the two major parties. But this can happen anyway, for other reasons, eg one candidate wins the popular vote and the other wins the electoral college.

I'm aware that the Democratic and Republican parties benefit from the two party system, so they might not want this, but it seems to me that this is what voters should want.

I'd like to add that changing the "front-end" of voting (having people rank choices rather than pick-one) enables a whole slew of potential better-algorithms, because the input data is fundamentally better.

This might interest you if you think that elimination voting solves the spoiler effect: https://www.youtube.com/watch?v=JtKAScORevQ

What I think would solve some issues is to have a last option on the ballot that says, "None of the above" and if that wins, then the current president if they reached their term limit, steps down, and the vice president takes over for the next term. IE pretend the head of state has passed away and move everyone up in succession.

Basically it removes the "lesser" of two evils from the equation. Yes one could argue, the party in power could stay in power if every year the party abstains from voting but I don't think that would be a likely outcome because, lets face it. The VP probably wouldn't be a leader everyone likes come the next election cycle. They could also abstain from running in the next election if they choose.

EDIT: Also puts a lot of emphasis on selecting a good VP for either party since they could possibly succeed after 2 terms.

I would prefer a "none of the above" option that is not actually implicitly choosing a specific person.

If "none of the above" wins, then the sensible thing to do would be to re-start the elections process from the primaries, wherein any person who was a candidate in the previous iteration(s) is ineligible to run again. Mid-November to January is more than enough time to re-do the entire election cycle.

This makes sense to me, excepting the timeline perhaps, only I'd also modify it to mean only that whichever candidates were the finalists in the previous iteration are ineligible to run again. In other words, my party may have picked a candidate that the nation as a whole rejected, but it does not mean that another candidate who ran in my party's primary would be rejected by the nation as a whole. Otherwise I think you may be throwing the baby out with the bathwater.

The public story of the primary races is that they select for the most desirable candidate of those running. If the electorate decides that the "best" of those candidates is not suitable, isn't it reasonable to assume that none of the rest could beat the "none of these" option, either? Do you really want to force vote after vote of people saying "no" before trying an entirely new crop of candidates?

You're not throwing the baby out with the bathwater. You're throwing out the unidentifiable goo, the dirt clod, the rotting fish, and the floating turd. The "none of these" result is a clear indicator that there is no baby worth saving in the entire tub full of bathwater.

If we had open primaries everywhere, I'd be more inclined to agree with you here. As someone who lives in a state with closed primaries, the way it looks to me is that the candidate selected as "most desirable" by a party's base is not at all necessarily the one who will be "most desirable" for the country as a whole--especially when combined with the much lower percentages of people who come out to vote in the primary elections, hence my hesitation to toss 'em all out. The ones most likely to succeed are the ones who pander the most to their hardcore base, to the distress/distaste of much of the rest of the population, especially those outside the party in question.

I am however on the record elsewhere as saying that a necessary part of the path to improving Washington is, in fact, to throw them all out, so I won't fight you too hard on this!

You start to run into issues if you don't allow the same candidates to run again. Issues being that you are suppressing the right to run for office. Now, if the party made the candidate ineligible then that's a different story. Also no clue why I was downvoted without a reason.

Issues being that you are suppressing the right to run for office.

Their ability to run for office isn't being suppressed, they ran for office the first time and lost. They could possibly run again at the end of the next term. This is no more suppressing a right to run for office than the 22nd amendment suppresses a right to run for office.

You run the risk of the exact same outcome running every body again.

We have that risk now, yet it doesn't happen.

Personally, I would rather choose a random sample of all politicians within the country to put into power. Over time, this should tend towards a compromise for everyone.

Failing that, each party having a number of seats proportional to the number of votes they received is not that bad of a system.

This is a great introduction to voting theory, but there are several notable omissions.

The video doesn’t tell the whole story about what it presents as “ranked voting”, usually known as Condorcet voting. There are systems that always yield the Condorcet winner when one exists, and do a good job of resolving the unavoidable but rare preference cycles as consistently as possible. The most standard one is the Schulze method, which is used by the Debian, Ubuntu, and Gentoo projects, among many other organizations.


The video asserts that what it presents as “elimination voting”, usually known as instant-runoff voting, doesn’t suffer from the third-party spoiler effect. But that’s only true as long as the third party never gains enough support to have a real chance of winning.


(I wonder: can you think of a pair of powerful organizations who might want to ensure that third parties never gain enough support to have a real chance of winning? Hmmm.)

Finally, the video doesn’t talk about approval voting or score voting, which use a different ballot type to which Arrow’s impossibility theorem does not apply. Some game theorists argue that these systems actually do a better job of finding the ideal Condorcet winner than Condorcet systems do, in the presence of strategic voting.


That's not how ranked voting should work - why would you look at pairs individually? You can simply take the average rank of all flavours and be done (in which case Vanilla wins definitively).

Also, most party systems tend to clump together in ideology making the 'paradoxical' choices rare. For instance in Canada we have two leftist main parties and one conservative. Ranking the conservatives in the middle would be virtually unheard of.

I really wish they would have gone into more voting methods (and counting methods) than they did. The video as-is seems to be very incomplete.

I thought the same thing about party rankings in the US. If we had ranked voting, and the Green and Libertarian Parties ran serious campaigns, it seems that the vast majority would rank G/D and L/R together. I suppose if we had many parties, though, we could see a 'paradoxical' choice.

I haven't fully investigated it yet but at one point I played with the idea of a "Markov election" based on the fact that Arrow's impossibility theorem assumes determinism. Nondeterminism is not necessarily a bad thing, if people agree that it was fair! [1]

In this case your rankings for the candidates constitute a Markov matrix where you specify that your vote should "flow" from these candidates over to those candidates. For example someone voting for Alice over Bob over Carol might have a Markov matrix which states their vote is:

    1.0  1.0  0.5
    0.0  0.0  0.5
    0.0  0.0  0.0
saying "All of the A votes I make should ideally stay for A, all of my B votes I would like to flow to A, all of my C votes I would like to flow to B or A." Part of the reason that I haven't figured this out yet is that I don't know how one should pick these numbers, exactly, to be robust against Arrow's problems (like adding another candidate not affecting the relative preferences of two existing ones).

Anyway, based on the election, we average everyone's Markov matrices together! Then we start from a state where each candidate has an even share of the vote and everyone collectively determines the flow of the votes in successive cycles. This basically selects out the eigenvector of the Markov matrix with the highest eigenvalue, which represents some probability distribution among the candidates. Now to resolve the pesky problem of cyclic preferences, we use a fair random number generator to choose from the resulting probability distribution. Thus everyone has an even say in who they want to win, someone gets chosen at the end, and if someone really is strongly dominated by someone else then they will generally get eliminated by the flow of votes from the other to them.

[1] While I have you excited about randomness I am also a fan of choosing a leader at random out of the entire population, educating them during a transition period, then handing them the reins. If you happen to disbelieve in the Buddhist metaphysics then an example is the tulku system for obtaining leaders, where they follow somewhat-ambiguous instructions from the previous holder of the seat to seek out a child born at a specific place at a specific time who is believed to be the specific reincarnation of the previous holder of the seat. If you don't believe that metaphysics is right then the search essentially chooses a child at random to be the leader. Seems to usually work pretty well for them either way. There's another example from Muslim history where the leader of the dynasty did not go to one's child because one did not have children; instead it was transferred from eunuch slave-soldier to eunuch slave-solder.

I'm also a fan of Monte-Carlo sampling and would like to see it implemented more in choosing a government, but fully random people wouldn't have legitimacy to their rule. Social order only works if people believe in it and I don't see how we can make people believe in fully random choice. That person could only have power if they fully control the army and why would the army let themselves be controlled rather than kill the random person and put someone else in charge, who will promise them the cushy life.

You did mention giving the person training. The problem with that is this will result in a fully self-perpetuating system that finds ways to not give power to people who disagree with it.

It still sounds better than the current system, so I'm all for it.

> I am also a fan of choosing a leader at random out of the entire population, educating them during a transition period, then handing them the reins.

> If you don't believe that metaphysics is right then the search essentially chooses a child at random to be the leader. Seems to usually work pretty well for them either way.

A cynical take on this might be that those educating the next ruler get a chance to shape him/her to their specific desires.

That is really interesting, but I don't see many American voters supporting non-deterministic elections. I've had trouble convincing some that anything is wrong with the current system. (Although perhaps that's my fault.)

I'd be interesting in how you would explain this system to an average voter. I didn't really follow it myself, but I've at least heard all the words before (except probably tulku).

that is part of the reason why ranked choice is better for the voter even if they lean strongly to one direction. It allows they to show their disapproval for their parties candidate without spoiling the vote and giving election to someone they really dont want

Right, but in the parliamentary system you also end up with parties like the Greens. They are not fiscally liberal, even though socially they are. Many people would rank them differently depending on their own values. Given the possibility of votes "counting" more in Canada, other Green-like parties may evolve, since now people may actually vote for them and have it matter.

Also, interesting point about averages and ranked voting. By the values, Strawberry would have won out, however as stated in the video, a majority does prefer chocolate to strawberry—so how would that be dealt with?

I look forward to the discussions around our [Canada's] new electoral/voting system this December.

Personally I would be very happy if Canada's parliament realizes that cooperative minority governance should be the norm and is the healthiest option.

I am not sure which part of the video you were looking at where Strawberry would have won, but it seemed by my quick averaging, that the 'right' choice always wins under simple averaging (unless there is a tie which is fine in Canada, not so much in the US).

I too hope that cooperative minority governance becomes the norm, unfortunately we've been fed for far too long that EU style cooperative governance is "weak" and undesirable.

I was referring to where In ranked voting where the paradox was cyclic preferences, Paul mentions that strawberry would win on averages, but yet the majority of voters would have preferred chocolate over strawberry.

There are many ranked voting systems and that one is called a "Borda count." It's really cool but it has its own flaws, as you'd expect from Arrow's impossibility theorem.

Here's the most basic problem: the video that we're looking at made an implicit assumption when it proposed ranked voting: it said that "here are peoples' preferences!" and then it copied those preferences over to the ballots. Given the way that they're doing ranked voting, this equivocation makes sense, of course you're going to rank in your actual preference order. But with the Borda count this assumption is totally wrong and a lot of the confusion from first-past-the-post systems reoccurs: there is no reason to vote your actual preferences. Consider the 2016 US presidential election as done via Borda count, if we assume peoples' present polling is honest. Think about someone who supports Jill Stein; their preferences are typically Jill > Hillary > Gary > Donald. Will Jill lead their ticket? Given the very lackluster showing of Jill, they will probably narrow down the race mentally to "Hillary or Donald" and turn in the ballot "Hillary > Jill > Gary > Donald," to try to avoid a Trump presidency. It's the same spoiler effect as we had before.

It's not just an isolated problem like that suggests. Suppose an election where we have three candidates, Alice, Bob, and Carol. There are nine voters with the preferences,

    4xABC, Alice > Bob > Carol
    1xACB, Alice > Carol > Bob
    2xBAC, Bob > Alice > Carol
    2xCBA, Carol > Bob > Alice.
Alice is the majority's first choice and only 2/9 of the peoples' last choice, so you'd figure she'd be a shoe-in to win the election. Indeed, she'd win a ranked-runoff election and first-past-the-post election, while the pairwise election has the following ballot counts:

    A beats B 5 - 4
    A beats C 7 - 2
    B beats C 6 - 3
The pairwise election therefore also decides Alice > Bob > Carol. But while these other systems all have Alice winning for relatively easy-to-understand reasons, if Alice's supporters turn in honest ballots in the Borda election, then under a Borda election, Alice LOSES!

This is because the remaining four voters see it as down to an Alice-vs-Bob election and want Bob to win, so they turn in ballots saying BCA. The resulting total of points (3 for first place, 2 for second, 1 for third) is:

    (ballots: 4x ABC, 1x ACB, 4x BCA)
    Alice: 4 * 3  +  1 * 3  +  4 * 1 = 19 points
    Bob:   4 * 2  +  1 * 1  +  4 * 3 = 21 points
    Carol: 4 * 1  +  1 * 2  +  4 * 2 = 14 points
Alice however is a savvy politician and convinces her own supporters to rally against Bob to get Carol to look much better than Bob, so that they hand in ACB ballots. This is a measured tactic to turn it into an Alice-vs-Carol race which gets the BAC voters back on her side, since now Bob is no longer a strong contender and they vote Alice. It gets complicated!

    (ballots: 5x ACB, 2x ABC, 2x CBA)
    Alice: 5 * 3  +  2 * 3  +  2 * 1 = 23 points
    Bob:   5 * 1  +  2 * 2  +  2 * 2 = 13 points
    Carol: 5 * 2  +  2 * 1  +  2 * 3 = 18 points
Except Carol has a trick up her sleeve. She sees that Alice is doing this and so runs with her friends Dwayne and Erica, who agree with her on everything but are slightly less preferred by the voters in general. Since they are so similar our original setup is 4xABCDE, 1xACDEB, 2xBACDE, 2xCDEBA. Now if the ballots are cast with the smear campaign against Bob in full force you have:

    (ballots: 5x ACDEB, 2x ABCDE, 2x CDEBA)
    Alice:  5 * 5  +  2 * 5  +  2 * 1 = 37 points
    Bob:    5 * 1  +  2 * 4  +  2 * 2 = 17 points
    Carol:  5 * 4  +  2 * 3  +  2 * 5 = 42 points
    Dwayne: 5 * 3  +  2 * 2  +  2 * 4 = 27 points
    Erica:  5 * 2  +  2 * 1  +  2 * 3 = 18 points
Suddenly Carol wins! Or, if Alice smears Carol almost as much as she's smeared Bob, then Dwayne just takes Carol's place and Dwayne wins; remember, they're largely identical! (You might wonder whether the effect of adding Dwayne and Erica to the voting has meant that Alice doesn't need to smear Bob anymore and whether Carol becomes her enemy #1; the answer is that if we just substitute C -> CDE in the above "Bob wins" tally then Bob gets 34, Carol gets 32, and Alice gets 29, at which point we need to re-evaluate the strategic voting because you KNOW that after that mock-poll comes out, the news will classify Alice as a 3rd-place finisher and her own majority voting block might rebalance around the assumption that it's largely a Bob-vs-Carol race.)

Great comment! Borda certainly has flaws, but it seems there are variants that can address some of them (I like the Nanson method). I am Canadian, where a ranked system is more ideal (but not implemented yet).

With Nanson it seems that the second to last scenarios (Carol wins) is cleansed of the strategic voting leading to the 'correct' Alice win.

The presentation is extremely nice and detailed oriented. This also co-relates to how messed up 2016 US election cycle is and in general how complicated elections are in USA.

1) You have few candidates on each side Democratic & Republics and eventually folks get eliminated

2) Then you can have independent candidates that can jump in

3) People finally vote for the president

- But we have winner takes it all by state

- The number of electoral votes is decided by the population distribution on the state

- Another complication each state can have a different way of solving ties

I still maintain one of the best voting systems has you vote successively for a majority winner.

- Many candidates run - If someone has a majority (more than 50% of vote), that person wins - Otherwise, top x number of candidates move on to next round - repeat

This way, changes in availability are addressed, so people can make fair choices between one item and the next.

That would basically be instant runoff voting. The main issue with that is that in a polarized election you can have a moderate compromise option eliminated early. For example, suppose the election is between a far-left, moderate, and far-right candidate. 40% prefer the far-left candidate, 20% prefer the moderate candidate, and 40% prefer the far-right candidate. In this case 60% of the population prefers the moderate candidate to the far-right candidate, and 60% prefer the moderate candidate to the far-left candidate and so the moderate candidate should win the election. But in an IRV-type system, the moderate candidate gets eliminated immediately and the election is then between the extreme candidates.

Also the people who vote earlier in the process their vote essentially becomes useless if the candidate drops after a month.

Not every state is winner-take-all either. And representation is not exactly proportional by population. You get one vote for each senator and each representative your state has in congress, which favors smaller states.

What I was trying to say is if you come from a small state like Hawaii or Rhode Island, your votes becomes part of this small group which does not deliver a bigger impact like New York or California. I was just trying to highlight list of complications like the presenter did with 3 small choices.

I guess this is more of a problem in states that have a large majority one way or the other - you feel like the "extra" votes in that direction are wasted. But I'm from NH. We only have 4 votes but because we are evenly divided (and highly over-represented for our population), we regularly make "tipping-point" lists and each vote has a lot more power than it does in other states.

I am from Jersey, I agree the distribution is not at all even and that is one reason why few state decide the entire election.

Could we conclude that it's better to choose individually in as many circumstances as possible, since choosing in groups has so many flaws? It seems the more choices are made by some for all the more people are dissatisfied.

How do you get rid of collective decision?

By liberalism.

You literally allow both choices, and each person gets to make their.

That obviously does not work for everything. But it's always important to keep in mind, and if you look at any modern government by those lenses you'll see plenty of superfluous regulation that does nothing but put people down.

Isn't this tautological?

Easy enough: (1) You choose any method, (2) if you hit the paradox you declare the result as invalid, (3) there is only ice cream after a valid vote, and (4) if there's no ice cream yet goto step 2.

Now the worst case is not that you make a few people unhappy (the majority), but the worst case is that people start killing each other in the hopes to change the result. But if you think about it that is also not a bad result, since then there is more ice cream for you.

I like your optimism! Unfortunately, when Arrow got the nobel prize for proving that there is no way to avoid this, it's possible that your solution was considered...

For example: there is no upper bound on the loop. Why would anyone change their vote when you retry? I.o.w. you might be voting and getting the same result forever.


There is an upper bound to the loop. People die. People lose interest (thereby changing the population). People get annoyed and thereby more willing to do stuff to change. Not mathematically provable, but works in most situations because humans are adaptable.

*edit: I feel a little disappointed that people don't seem get the humor in my post. Maybe dying because people voted chocolate ice cream but you wanted vanilla is a reasonable scenario for some.

In the last method, how do you detect if you've hit a paradox given only a single election? It only shows up if you change the options and make everybody fill out a new ballot.

it is always visible he just used the change to illustrate a normal case and a paradox case. If the results of the second vote had been the only vote you would be able to see the paradox was there by examining peoples rankings

We don't always have to vote on one outcome. Multiparty systems often split representatives by party percentage. This has its own problems, but seems to allow more party mobility.

There's no decent voting system for a body of government this large. Every kind has critical flaws. The best solution is not to have a nation as gigantic as the U.S., but obviously that's not gonna happen any time soon. It used to be a flawed option too, before WMDs, but now all nations are on roughly the same playing field. Plus there's still something to be said for owning as much land as you can, for things like oil more so than farms.

I feel like the world would be better if some nation(s) were legislatively stream-lined for experiments in secession (perhaps with trial periods and other conditions). Would allow for governance innovation without the calamity of conflict that's required for carving out space today.

EDIT: Perhaps it could involve a hypothesis of sorts before secession, with baked-in options to re-join the parent state if the parent considers adopting some learnings after a successful experiment. Lots of possibilities. Would likely involve re-gearing the concepts of how quasi-independent states (child + parent) would keep relations open, so as not to polarize populations unnecessarily. I don't normally think of diplomacy as exciting, but learning how to make such a system work effectively would be pretty interesting

How can anyone vote for Chocolate. Chocolate is clearly CORRUPT. He's openly buying votes! This election is rigged! I just watched a video on youtube explaining how the ELITES plan to trigger an election paradox! Investigate the WILLY WONKA FOUNDATION. Vote Vanilla or we won't have a COUNTRY in 5 years!!!

I have seen a few too many voting-system presentations that all seem to have a few of the same common arguments, arguments that I believe are flawed.

Here I can isolate it to these statements. First, "The whole point of the election was to force the group to make a choice about an ice cream flavor." This is false. The election's point was to interpret the preference of the electorate. We then use that preference to come to a choice. In the cyclic cases, this is no failure of the voting system. This is an accurate portrayal that the group is not able or ready to come to a decision.

"In its inability to identify a clear winner, ranked voter has failed in its primary task." This is also not true. It was not its primary task, and didn't fail at all. It succeeded wildly in identifying a confusion in the electorate itself.

Of course any attempt to "break that tie" will be flawed and invalid.

For those who don't know, the video up to that point explored Condorcet Voting, and the Condorcet Winner criteria. The existence of a cycle is known as a Smith Set or Schwartz Set (subtly different, but very similar). Tiebreaking methods to elect single winners from Schwartz Sets are all flawed, but that does not mean the Condorcet Winner method is flawed.

Elimination Voting is more commonly known as Instant Runoff voting. In his example at 7:04, this is again a Schwartz Set. So it's improper to declare that one of the three flavors is a winner, because the electorate is inherently undecided. The gold coins scenario changes nothing; it is still a cycle, and the electorate is still undecided. So the tiebreaker chooses a different winner - it doesn't really matter, because it is already invalid to pick a winner in that scenario.

But the bigger problem with IRV (elimination voting) is that it will sometimes pick the wrong winner even when there ISN'T a cycle. This is awful.

Finally, the presenter glosses over his description of Arrow's Theorem. You can practically hear the parentheses when he says, "given certain assumptions". In truth, not all voting criteria are the same. All Arrow's Theorem does is say you can't simultaneously meet four criteria, that he chose, in all cases. It does not prove that those four criteria are required. One of those criteria, the "Independence of Irrelevant Alternatives" criteria, is particularly problematic - and if I recall correctly, it can only move an election from a Condorcet Winner to a Schwartz Set. It cannot itself select a new winner entirely. And in my view, that is not a problem, because if adding a new candidate creates a Schwartz Set, then it only means that the electorate was probably not given enough choice from the outset. Although, there is a counterpoint that says that if you give voters more and more choice, cycles will inevitably occur.

So really when we're talking about voting theory, the entire problem with the subject is that we are conflating two needs. The first need is to measure the public's preference for a choice. But the other need is for the public to come to a choice. Often times, the process of the voting system is what motivates the public to do so. But it is not always sufficient. And at the same time, the fact that the voting system is not sufficient for this, is not a failing of the voting system itself.

I find it instructive to mull two questions, because I find that these two questions drive a lot of opinions on who feels what voting systems are superior.

1) If a candidate would beat all other candidates in a multi-candidate single-winner election head-to-head, should that candidate be the winner?

2) If, in a two-candidate election, one candidate narrowly defeats the other, but the other candidate's supporters are clearly more passionate, should the first candidate still win?

(EDIT: Calmed some language)

My main grip with these discussions is that nobody seems to have addressed how they affect realistic scenarios. Finding an intricate scenario in which a system fails is entertaining, but doesn't help in the real world.

I'm curious about what paradoxes emerge with score voting.

If you mean that the top candidate earns N points and the Nth candidate earns 1 point, I answered that in this comment:


No, that’s the Borda count. In score voting, each voter assigns a score from some range to each candidate independently (https://en.wikipedia.org/wiki/Score_voting).

Ah, what I know as "figure-skating voting". I don't know very much about these except that Arrow's theorem can't apply directly to them because they have a different ballot structure; however given that Borda is something of a special case of score voting (in the same way as in crypto; random permutations are special cases of random functions meeting different needs), presumably there are some regimes where you can push it so that it has similar behavior.

Borda could be thought of as a ballot-restricted version of score voting, but so could plurality; you can’t expect ballot-restricted versions of reasonable systems to remain reasonable. Generally, there’s no reason for voters to cast Borda-type ballots unless you force them to.

They could start with voting for witch flavor to remove.

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