Arrow's Impossibility Theorem (The Problem of the Third Party)

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An explanation and proof of a simpler version of Kenneth Arrow's Impossibility Theorem, a statement of his full theorem, and an argument against third parties.
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Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy and more!
Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy and more!

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16 сен 2024

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Комментарии : 35

@_skysick_ 6 лет назад

Attributing this to the existence of a third party is going to grossly mislead people. Arrow's Impossibility Theorem doesn't just apply to "voting system" where people go and mark their preference on a ballot and then a winner is announced. If you phrase it that way, you're misleading people into believing that two party systems aren't suffering from the contradictions implied by the Theorem. The impossibility theorem applies to ANY method of social choice between more than two options. Even just restricting ourselves to the actual civic voting system, we've got primaries leading up to an actual election. This is a method of social choice between many options, and therefore the outcome and its relation to people's preferences is affected by Arrow's Impossibility. Arrow's impossibility is not "the problem of the third party". It is "the problem of reconciling differing preferences".

@JohnRandomness105 7 лет назад

In one of the videos, we discovered that "dictatorship" doesn't mean quite what we think it means. Perhaps some of the other criteria don't mean (or imply) what we think they mean (or imply). There is always the question, who the heck gave us these particular choices? Not necessarily Trump and Clinton, but rather, in a two-candidate system, who the heck gave us these two candidates? Is it really any kind of democracy, if the powers that be give us two candidates and tell us we only have those choices? That happened with Clinton in the Democratic Party. Trump was probably a rebellion against that in the Republican Party. In the two parties, what about the elections for candidates, if they aren't imposed top-down? Are we only allowed two factions in each party -- each faction with it's own sub-candidate? I think that there's serious trouble here.

@CarneadesOfCyrene 7 лет назад

The argument would be that it is more important to have more candidates than to fulfill one of the other criteria. In the final video of the series we look at some of the reasons we might abandon one criterion or the other.

@nienke7713 2 месяца назад

Is there any case in the world that truly has a two-party system? I know some countries, such as the US, have effectively a two party system due to the use of pluraliry/FPTP system being insanely prone to the spoiler effect, but it's not truly, more parties are allowed and they do exist, most people just vote strategically to not waste their vote. In order to truly have a two-party system, you need to artificially restrict range, because it's highly unrealistic that there will only be 2 options if more are allowed. So when you argue for a two party system, either you're arguing to have a multi-party system with plurality/FPTP (an absolutely terrible system), or you're arguing to resteict range. Now what's better ultimately depends on what you're choosing, but for an election of a president or other single role intended to represent a large amount of people, it is best to choose someone who is considered more acceptable than each other candidate, i.e. a condorcet winner; or in case none exists, then at least someone from the set that all consider more acceptable than those outside the set, i.e. someone from the Smith set. Determining a condorcet winner, ir the Smith set, should thus be a priority. If we found our Condorcet winner, great, we're done. If we found a Smith set larger than 1 (which I think is quite rare in realistic elections) then we need a way to pick someone from that set. Since we all desire to limit tactical voting and interference from the spoiler effect, and knowing that full IIA is impossible, it seems sensible to go for the option that satisfies all three of the weaker forms of IIA: LIIA, Smith-IIA, and Indepence of Clones, which all make strategic voting and spoiler effect more difficult and less likely. Thus, we should opt for the Ranked Pairs Condorcet method. If we want a legislature or something that represents the diversity of opinions that exist in society, then we shouldn't be looking at these methods at all, we should be looking at various methods of proportional representation; local representatives on a national/federal level is an absolutely ridiculous system and also assumes that voter preferences are large based on locality whilst completely ignoring other demographics: why local representatives rather than representatives based on demographics such as age, race, gender, sexuality, religion? For mamy people those are as important as, if not more portant than, theie locality for their voter preferences; local representatives is just a form of identity politics, favoured mostly by the people who whine the most about identity politics. Perhaps in the olden days it made sense because you only really had any connection to your local community with little to no clue to what was going on outside of it; but nowadays with the inteeconnectivity we have, it just doesn't seem that sensible anymore. Now if you just want to pick a movie to watch with your friends, you can go for something simple as approval voting; it's quick, easy, and as long as the stakes are low and no real reason to try to vote tactically, it'll be fine, and you can pick the option that most of you are okay with. Need to judge a bunch of performances in a music contest? Some score voting is probably thw right pick for that, as long as those voting don't have a stake in the results and thus no reason to tactically inflate/deflate their scores, that's a pretty good system to get a final ranking of the candidates. Want the middleground/compromise/moderate option out of the picture and its fence-sitting centrist supporters picking a side? Then the centre-squeezing Instant Runoff Vote (a.k.a. Alternative Vote) is the right choice for you

@JohanDanielsson8802 8 дней назад

I agree with much of what you say here, but ranking a long list of candidates would, realistically, be a bit too tedious to many voters. While electing someone for one position, I would prefer to first let the voters elect 4-6 candidates out of many through score voting, and then hold a second election round where the voters get to rank those 4-6, preferrably with the Ranked Pairs Condorcet method. As someone who live in a country with proportional representation, I would say it has both pros and cons. I think it is a superior way to mirror the people´s opinions compared to local representatives, but at the same time, the MP´s (or city council members or whatever) do not answer individually to their voters, but rather to their parties. Thus, if you want a system where the politicians can be held accountable by their voters, I would argue that proportional representation have less potential (although many systems with local representatives out there also fail rather miserably in this regard). I would argue for a two chamber parliament: one chamber elected through proportional representation, which get to make any decisions which are too small to settle though referendums. One chamber of local representatives, elected through the ranked pairs Condorcet method, which also can be ousted by their constituents. The chamber of local representatives would get to decide which decisions should be settled through referendums, and which should be entrusted to the proportional-representation-chamber (which in theory should mirror the people´s opinions, but can not be properly controlled by the people). The fact that people living on the same place do not have that much in common, would (I think) be an advantage rather than a drawback, in the roles of these local representatives. After all, their role would not be to push their own political agendas, but rather to be guardians of democracy, and let the voters get what the voters want. Thus, them being elected by a heterogeneous group of people, in a way that does not produce center squeeze, would (hopefully) keep them from pushing agendas in the way the proportional-representation-chamber is there to do. I would also give different local representatives different voting power, depending on how many people they represent. That way, they could each represent a naturally cohesive, actual administrative area, and it would be much more difficult to come up with any excuses for gerrymandering. As a side note, if people elected their political representatives based on race, gender, sexuality or religion, the government would have to actually ask all citizens about these things, and register it. I would not be entirely comfortable with that. It would also be possible (probably even easy) to simply lie about one or more of these characteristics, in order to get to vote within that group.

@TheMikeshmel 7 лет назад

welcome back!

@ulmeydasmile 7 лет назад

@Joeviocoe 7 лет назад

It seems that all of these criteria are still only calculable based on ranked voting. Approval voting, with no ranking but more than just two parties,... makes it impossible to measure with these criteria, yet is desirable and fair.

@buggaby9 7 лет назад

Ya, I found that on the Wiki article for Arrow's Impossibility Theorem. From that article: 'Arrow originally rejected cardinal utility as a meaningful tool for expressing social welfare, and so focused his theorem on preference rankings, but later stated that a cardinal score system with three or four classes "is probably the best".'

@nienke7713 4 года назад

I'd think approval voting would suffer from something similar to the spoiler effect in FPTP; if you know your preferred candidate has a low chance of winning, and you dislike/disprove of the two larger ones that have a chance of winning, there's a good chance you'd give an approval of the big one you dislike/disapprove of the least because you really don't want the other big one to win. Meanwhile, supporters of the big ones would have no incentive to approve of the smaller ones even if they are kind of okay with them, because they prefer their big one who has a good chance of winning, and approving of smaller ones would only risk the smaller ones becoming a legitimate competitor to their favourite.

@skyhistory6602 2 месяца назад

@@nienke7713no they aren't same because in aprroval voting, if everyone vote honestly without tactical voting, the introduction of new party that lose can't cause winning party to lose because that party still be count in aprroval vote, in fact approval voting can be consider as score voting that only have range only 0 and 1

@nienke7713 2 месяца назад

@@skyhistory6602 "if everyone vote honestly without tactical voting" that's precisely the issue though, people will vote tactically, and that's fully within their rights, if you don't, then you're duping yourself

@skyhistory6602 2 месяца назад

@@nienke7713 In approval voting their are no spiler affect if seat contest between A and B with 60:40 vote under normal fptp if C are introduce and have political stance close to A result in aplit vote vetween A and C that make result become for example A:B:C is 35:40:25 make shift A from winner to B become winner by introduce C this is what called spoiler affect but under Approval Voting, Party C suppotter prefer C Over A and Over B know if only vote for C will make B become winner So also vote for A too like as if C party didn't exist by assume that party C supporter consider A as in acceptable range and assume that another B party are not gain vote from this system because B political stance is very different from A and C make vote result for A:B:C is 60:40:25 as seen spoiler affect had be orevented by allowed voter to express more than 1 choice but another tactical voting such as Chicken Dilemma still possible

@billyte1265 7 лет назад

Like other people have mentioned, the Impossibility Theorem is only valid for ranked-order systems. Cardinal systems like Range Voting satisfy all of Arrow's criteria. Read more here: governology.wordpress.com/2017/09/05/kenneth-arrow-is-a-dick/

@nienke7713 Год назад

It would be seriously susceptible to tactical rating. If my honest rating (ranging from 0 to 10) would be A: 8 B: 5 C: 2 I would have incentive to exaggerate my preferences: A: 10 B: 5 C: 0 And if the race seems like it's going to be mainly between A and B, I might just lower my rating of B as well, or if it seems it's mainly going to be between B and C, I might increase my rating of B. There's also no guarantee that it would satisfy the Condorcet condition, and keeping in mind how people are likely to exaggerate and manipulate their scoring, we cannot really rely on the scores to say how well liked or divisive a candidate truly is (which seems to be the main argument to justify this lack of meeting the Condorcet criterium). Meanwhile Condorcet has a tendency to select candidates who are considered more moderate, as moderate candidates will always get support from moderate voters along with support from one extreme opposing the other extreme when matched against the other extreme, and vice versa, which is key in order to win every match. We should also not pretend like our option is either 2-party or rating. We can choose a ranked choice system and accept that lack of IIA for example, just like choosing a rating system would accept the lack of Condorcet and the exaggeration and manipulation of ratings.

@christiancatalan2943 4 года назад

Finally shorter, and simple explanation. (online class is killing me)

@leandrosanchez1212 5 лет назад

Ranked voting to pick the condorcet winner in case of the condorcet paradox run IRV until there is 2 candidates and have a second round.

@menotyou135 7 лет назад

What would be the response to someone arguing that having 3rd parties is more desirable than the desirability conditions due to the easy exploitation of 2 party systems by power in modern democracies? This analysis doesn't seem to address the practical concerns of a 2 party system. How are desirability conditions justified, or are they assumptions?

@CarneadesOfCyrene 7 лет назад

I completely agree that I do not look into the practical implications of these theories. We will look into justification of the desirability conditions in the next video. The conditions do more to prevent voters from being able to manipulate the system by changing their preferences to not represent what they actually want, but what will benefit their candidate of choice more.

@menotyou135 7 лет назад

Thanks for the response! So desirability conditions are important to prevent strategic voting rather than voting based on honest opinion? In that case it seems pretty clear that having only 2 choices would prevent that because it makes your choice binary. It prevents strategic voting by giving you a single option to choose to get the desired outcome. The strategy and the honest opinion vote can't diverge because the choice is limited. I guess my main question toward this would be, is a desirability condition actually desirable if it's desirability is based on it's limiting aspects of the voter's agency?

@hadronoftheseus8829 3 года назад

"What would be the response to someone arguing that having 3rd parties is more desirable..." There is quite literally no argument. Any argument against ranked choice voting applies _a fortiori_ to first past the post. First past the post has exactly zero defenders who are simultaneously sincere and intelligent. They simply don't exist. This is _not_ hyperbole.

@lockvirtompson5287 7 лет назад

Is the independence of irrelevant alternatives really that important(considering biting the bullet)? For my change i vote to matter the new winner must have been close to winning before the change. So it only matters in a close race. This means that the general preference of the voters is intact. Because it is one of the majority candidates. Compared to having voting districts were your vote only counts as a subset of votes(voting for some one who then votes for you). Compared to that is't independence of irrelevant alternatives more fair?

@CarneadesOfCyrene 7 лет назад

I agree that Independence of Irrelevant Alternatives is probably the best place to put some pressure. It does allow third parties to act as "spoilers" in an election, which can mean that a majority of people with similar views who split between two similar candidates, can lose out to a minority of people with drastically different views who back a single candidate.

@_skysick_ 6 лет назад

Making that change is inventing a new voting system, which is also affected by Arrow's Impossibility Theorem.

@f_f_f_8142 5 лет назад

If the number of candidates is 0 or 1 the election is trivial and pointless, if the number is 2 the election is trivial but not pointless and if the number is greater than 2 the election is not trivial anymore. Just like solving k-CNF...

@ashnur 7 лет назад

I think the Independence of Irrelevant Alternatives is based on false assumptions and it's irrealistic. And I do not see why would anyone want it either. Can someone please explain?

@CarneadesOfCyrene 7 лет назад

So that third party candidates cannot act as spoilers, and split a majority group and elect a candidate with a minority of support.

@ashnur 7 лет назад

Thank you, I also went back and watched the video you made about it again. These things get confusing really fast. Based on this video it seems to me that there there is some kind of circular dependency which this rule breaks down. I would be happier with a more formal explanation than all this text and verbal reasoning :D.

@BlueLightningSky 7 лет назад

What concepts of set theory is missing to prove the theorem?

@CarneadesOfCyrene 7 лет назад

There's a number of things that we can mean by proof, most "proofs" of the theorem that you can find are informal proofs, which generally explain why something works. I like to do proofs that go line by line, and clearly demonstrate logical validity (like the ones that you can see in my series on logic). So in order to prove these we need axioms and rules dealing with simple things like membership and subsets, but, in all likelihood some on unions, intersections and more. When I started to make the video on it, it ended up being 90% about set theory, and only 10% about voting theory. We will do a video with a full proof. Just not yet.

@ashnur 5 лет назад

@@CarneadesOfCyrene do you think the proof could work with multisets instead of sets?

@josgibbons6777 Год назад

As explained here, some voting systems avoid the "one of these bad things conclusion must happen" consequent by collecting a different kind of data from voters from that assumed in the antecedent of Arrow's theorem: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-e3GFG0sXIig.html tl;dw The trick is to replace no-ties-allowed only-ranking-conveyed data with ties-allowed consecutive-preferences-gap-size-variable data.