1844: Voting Systems

Explain xkcd: It's 'cause you're dumb.
Jump to: navigation, search
Voting Systems
Kenneth Arrow hated me because the ordering of my preferences changes based on which voting systems have what level of support. But it tells me a lot about the people I'm going to be voting with!
Title text: Kenneth Arrow hated me because the ordering of my preferences changes based on which voting systems have what level of support. But it tells me a lot about the people I'm going to be voting with!


This comic is about types of single-winner voting systems:

  1. Approval voting has voters "approve" (i.e. select) any number of candidates. The winner is the most-approved candidate. It works with the same unranked ballot as plurality voting, but would allow a "compromise" candidate who is the second choice of a majority to defeat a candidate who is supported by a plurality but disliked by other groups.
  2. Instant-runoff voting (also known as Ranked Choice or Preferential Voting) has voters rank the candidates in order of preference. Ballots are initially counted for each elector's top choice. If a candidate secures more than half of these votes, that candidate wins. Otherwise, the candidate in last place is eliminated and removed from consideration. Ballots that had this candidate as the top choice now have the second preference as the top choice (this is the "instant runoff"). The top remaining choices on all the ballots are then counted again. This process repeats until one candidate is the top remaining choice of a majority of the voters or all but one candidate have been eliminated. IRV's proponents have successfully implemented it in a few places, such as the city of San Francisco, and Federal elections in Australia.
  3. A Condorcet method elects the candidate that would win a majority of the vote in all of the head-to-head elections against each of the other candidates. A candidate with this property is called the Condorcet winner. Due to the Condorcet paradox, an election with 3 or more candidates might not have a Condorcet winner, so Condorcet methods differ in the secondary set of rules used to handle that situation.
  4. Not directly mentioned in the comic, First-past-the-post voting (FPTP, aka plurality voting) is the method currently used in the US, UK, and several other countries. It only allows voters to choose a single candidate. Experts on voting methods agree there are multiple reasons why FPTP is not the best way to implement democracy,[1][2][3][4][5][6][7][citation needed] but they made little progress in replacing it in the United States for decades. However, this is changing; the state of Maine and numerous cities have adopted either IRV or Approval in recent years.

Arrow's impossibility theorem gives a list of criteria for ranked voting systems and states that no system can satisfy all of them at once, despite that for each of them it may seem "obvious" that an electoral system ought to satisfy it. Some voting theorists (such as Cueball) dislike IRV because it fails more of the criteria than Condorcet does.

The primary joke in the comic is the premise that people who are pedantic or knowledgeable enough to find Arrow's theorem to be relevant will self-fulfill the theorem by being inclined to disagree on any effort to change the voting system. This is illustrated by Cueball's voting system preference that is contingent on the preferences of other people, which defeats their effort to produce a community-wide ranking.

A secondary joke in the comic is that often voters don't pick their favorite choice in a vote. Instead, they vote for a less favorable, but more likely electable, person as a way to prevent their least favorite choice from being elected. This is commonly called "spoiler effect"; in Arrow's parlance it is a form of IIA criterion failure. Cueball's strategic vote switch implies that they may be using FPTP (which they dislike) to make the decision, as FPTP is the only system to involve a potential "spoiler effect" (note, however, that certain vote distributions in systems such as IRV can produce a similarly problematic and illogical effect on the outcome).

A third joke is the recursive self-referencing inherent in voting to choose a voting system.

The title text stipulates that Cueball has no fixed ranking of preference for human candidates, but makes this choice dependent on which voting system is favored by the group. This exceeds strategic voting considerations as the ranking should have full information, whom Cueball prefers in each situation. Therefore Arrow's impossibility theorem and the analysis behind it assume the ranked preferences of an individual voter as a fixed given. To make them dependent on the voting system makes assessing the efficacy of the voting systems absurd or at least much more complicated to do as a general assessment. That is given as the reason why Arrow would wholeheartedly hate him.


[White Hat and Ponytail are standing on either side of Cueball who is talking while lifting one hand.]
Cueball: I prefer approval voting, but if we're seriously considering instant runoff, then I'll argue for a Condorcet method instead.
[Caption beneath the panel:]
Strong Arrow's theorem: The people who find Arrow's theorem significant will never agree on anything anyway.

comment.png add a comment! ⋅ comment.png add a topic (use sparingly)! ⋅ Icons-mini-action refresh blue.gif refresh comments!


Looks like 2 of us added explanations at the same time. Someone else want to consolidate them and produce a concise explanation? ~blackhat (talk) (please sign your comments with ~~~~)

I tried merging our explanations, so there is a small improvement, but there is still some duplicated information. Plus I'm not a native english speaker, so a consolidation by a third editor would be welcome. (talk) (please sign your comments with ~~~~)

Something I don't understand about the Arrow Impossibility Theorem: In the example given, the result of the election is obviously a 3-way tie, where each candidate got exactly equal support. Surely the Arrow Impossibility Theorem doesn't complain about voting system's inability to intuitively break an exact tie? (talk) (please sign your comments with ~~~~)

I think there is another layer of explanation here. When Cueball is discussing this - he's talking about voting for which voting system is to be chosen. The choice is Approval versus Instant Runoff - but isn't Cueball arguing about using a Condorcet method to decide WHICH voting method to choose? This is emphasised by the mouse-over text which talks about him dynamically changing his choice of ultimate candidate based on the election system chosen - which is exactly the Condorset paradox, but when applied to the selection of which voting system you want rather than the choice of candidate. Again reinforced by the discussion of "Strong Arrows theorem" which at that same meta-voting level. 15:40, 31 May 2017 (UTC)
The "exact tie" only exists because ranked-choice ballots destroy any information about strength of preference. It likely wouldn't be an exact tie with a Score voting ballot, for instance. 00:02, 2 June 2017 (UTC)

Generally the idea behind Arrow's Theorem is that you would get different results if you did a vote where the choices were just A or B, B or C, C or A, thus no option wins head to head against the others (Condorset Paradox). An example I recently read was economic policy, and how the options being presented can cause policy to fluctuate wildly in a democracy as the outcome depends on the options compared. -- 16:01, 31 May 2017 (UTC)

Neither Arrow's Theorem nor the joke makes any reference to Condorcet's paradox. Rather, the joke is that it shows an individual voter who apparently fails to satisfy independence of irrelevant alternatives. This is one of the criteria in Arrow's theorem, and it is normally always regarded as being true of any individual's opinions, just not necessarily of the outcome of an election. Zmatt (talk) 18:38, 31 May 2017 (UTC)
For reference: both instant run-off voting (IRV) and every concorcet method fail independence of irrelevant alternatives. Some (most?) condorcet systems satisfy all other criteria of Arrow's theorem, while IRV also fails monotonicity. Approval voting satisfies both, but it is outside the scope of Arrow's theorem as it is not a ranked voting system. Zmatt (talk) 18:47, 31 May 2017 (UTC)

"Arrow's impossibility theorem states that when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide ranking." Arrow's theorem does not say that. Arrow's impossibility theorem says "When voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide ranking that is complete, transitive, Pareto efficient, have universal domain, has no dictator, and independent of irrelevant alternatives." The conditions matter, and the non-dictatorship condition in particular is horrible misnamed.

"The theorem may be interpreted in a way suggesting that no matter what voting electoral system is implemented in a democracy, the resulting democratic choices are equally imperfect". No. Perfection is an absolute so things are either perfect or they are not. "Equally imperfect" is a tautology. If you are going to throw in "equally" some voting methods are manifestly closer to perfection than others, some voting methods satisfy all but one of Arrow's conditions, while others satisfy none of them. 18:05, 31 May 2017 (UTC)

Quite true. Monotonicity is not desirable because it enables the kind of strategies which make Condorcet systems almost as unstable in practice as FPTP. Arrow's Theorem can be disposed of by the realization that nonmonotonicity is what makes IRV impervious to strategy. 07:16, 5 June 2017 (UTC)
Arrow's Theorem is based on a fundamentally flawed approach in the first place, which he realized later in life. Using ordered rankings to estimate utility is not a very good plan. Voting systems based around estimating utility directly (rated rather than ranked) are much better. It was based on economist dogma that utility can't be compared meaningfully between individuals, but interpersonal comparisons of preference are even less valid. 00:02, 2 June 2017 (UTC)

Totally unrelated to the discussion, but interesting that Cueball has moved from being between a black hat and a black haired women in 1842: Anti-Drone Eagles to being between a White Hat and a white haired woman, two comics later, where he starts speaking in both comics. :-) --Kynde (talk) 18:09, 31 May 2017 (UTC)

For deep (but simply explained) insight into voting systems, (and why the American first past the pole system sucks), see this playlist of youtube videos by CGP Grey --Kynde (talk) 18:16, 31 May 2017 (UTC)

Unfortunately he repeats incorrect statements like "IRV eliminates the spoiler effect" and obviously hasn't done honest research on it. 00:02, 2 June 2017 (UTC)


Cueball almost perfectly matches my views on voting. I think Approval is far and away the best (due to ease of implementation and low chance of paradox). Condorcet & IRV use the same ballot design, but IRV is mathematically inferior, so I don't get why anyone likes it, other than bandwagon effects. The only situation where I'd support IRV is if it were the only viable option to replace FPTP, which is unfortunately the case in many places. - Frankie (talk) 22:45, 31 May 2017 (UTC)

Frankie, the two established parties Democrats and Republicans both favor IRV over Condorcet precisely because of its mathematical biases. The 'deficiencies' of IRV tend to eliminate centrist moderates early in the process and leave the established parties in political power. IRV represents a slower change to the political status quo. Barrackar (talk) 07:35, 2 June 2017 (UTC)
Addendum: after reading about recently-developed voting methods, I find that Majority judgement is excellent. I still approve of Approval, because it offers a lot of improvement for almost no cost, but if you're going to replace all the voting machines, rated is better than ranked. - Frankie (talk) 11:53, 10 June 2017 (UTC)

The larger the democracy, the less a single vote matters, regardless of the voting system. I, for one, support a return to the system of democratic city-states with annual elections. If a sizeable focal minority don't agree with their government, they can just break off and declare their area a separate city-state. Of course, this could eventually create a loose alliance of house-states or even people-states each with their individual laws and foreign policy. --Nialpxe, 2017. (Arguments welcome) 02:44, 2 June 2017 (UTC)

Obvious counter-argument: voluntarily replacing our existing structure of nations with city-states is so much less likely to happen than replacing FPTP, that it's really not worth discussing as a plausible option at this point in history. OTOH, if Trump starts WW3, all bets are off after the apocalypse. - Frankie (talk) 13:26, 4 June 2017 (UTC)

I think this might be the first xkcd, in over 1,800 comics, that I understood literally nothing on my own. Wow. Except that this was something about voting, caught the word voting, LOL! I usually get at least a few things, and come here to fill in any gaps. Guess discussing these 4 things is particularly American, I've never heard of any of them (as a Canadian, and on an iPad where I can only see the title text here).! - NiceGuy1 03:36, 2 June 2017 (UTC) [Hey, someone replied in the middle of my comment block! LOL! Copying my "signature"/time stamp here in the hopes of making it complete again as two separate blocks] - NiceGuy1 04:12, 7 June 2017 (UTC) I finally signed up! These two comments are mine. NiceGuy1 (talk) 06:22, 13 June 2017 (UTC)

The same with me. After reading the comic, explanation AND comments, I can't even find the joke, let alone understand it.These Are Not The Comments You Are Looking For (talk) 03:26, 4 June 2017 (UTC)
The topic of voting systems is particularly relevant for Canadians under the current administration, because one of the major planks of their campaign platform was "ensuring that 2015 will be the last federal election conducted under the first-past-the-post voting system" (https://www.liberal.ca/realchange/electoral-reform/). Some of us consider it one of the top two or three priorities for the current term actually! Jkshapiro (talk) 04:13, 4 June 2017 (UTC)
Ah, yes, I should have said "As a Quebecer". :) We don't have the luxury of voting our beliefs, we have to vote defensively to ensure we continue BEING Canadian. Any discussion of voting I hear is about THAT. :) - NiceGuy1 04:12, 7 June 2017 (UTC) Also my comment! NiceGuy1 (talk) 06:22, 13 June 2017 (UTC)
Actually I find the explanation worked for me, just that there's not much joke here. As I understand it, Arrow's Theorem means there's no clear Best System, that there's no agreement or something (sorry, I didn't re-read the explanation, so I'm working from my memory or reading 2-3 days ago, LOL!). This is saying anyone who knows enough about Arrow's Theorem to embrace it will automatically be a part of it, and magically likewise fail to agree with each other (which would take embracing the theory to a ridiculous level). - NiceGuy1 04:15, 7 June 2017 (UTC) So's this! NiceGuy1 (talk) 06:22, 13 June 2017 (UTC)

One thing I don't get: Why Condorcet can't be used on 3 or more candidates. I read a bit of the Wikipedia link about the Condorcet Paradox, okay, I see the POTENTIAL paradox, but it's not necessarily so. Sure it MAY be that 3 candidates get equal support in this way, but numerically this is so horribly unlikely I'm suprised it's not only being considered, but given such significant weight as to say it can't be used! As I understand it, using last year's election, it works like this: Trump, Hillary, and let's throw in Bernie Sanders as the third. As I'm understanding the explanation of the Condorcet Method, if a hypothetical election between Bernie and Trump would have Bernie winning (based on support? Sounds like no actual voting taking place), and a hypothetical election between Bernie and Hillary would also have Bernie winning, then Bernie is the winner. But that's 3 people, what doesn't work? And if Condorcet only works with 2 candidates, how is that not just a normal vote? The Paradox seems to say if exactly a third of voters rank Bernie over Hillary over Trump, one third says Hillary over Trump over Bernie, and the final third has Trump over Hillary over Bernie, then THAT'S the Condorcet Paradox. But that's SO specific, it's unlikely! - NiceGuy1 03:36, 2 June 2017 (UTC) My comment too! NiceGuy1 (talk) 06:22, 13 June 2017 (UTC)

I agree. Who cares about the Condorcet winner when there is the Smith set? Barrackar (talk) 07:35, 2 June 2017 (UTC)

To the Canadian commenter: have you followed the elections of the Conservative party? It looks to me like a recent large-scale use of an "non-traditional" voting system. I've heard it criticised for its complexity, but no discussion on why it was chosen. Description here 15:31, 2 June 2017 (UTC) anothercanadiancommenter

Nope. In addition to being a Canadian I also live in Quebec. All my political involvement is about remaining Canadian, I know nothing beyond that, LOL! We don't have the luxury of voting our beliefs (other than that one), so It seems pointless to look any further. All we can do is hope that the strongest party against separation behaves. - NiceGuy1 04:12, 7 June 2017 (UTC) Commenting a lot here, LOL! NiceGuy1 (talk) 06:22, 13 June 2017 (UTC)

Barrackar, any voting system can be used on any number of candidates. However, there are a lot of voting system criteria, and no voting system will be able to satisfy all of them. Arrow's theorem implies that any system based on rankings will fail at least one of 3 important criteria, and one criteria that can never be satisfied by a ranking system is immunity from irrelevant choices (IIC). However, Approval Voting (or any general cardinal rating method) is not a ranking method, per se, and so it isn't necessarily subject to the constraints of Arrow's theorem. But choosing between different voting systems is, in itself, a form of choice, and the comic uses this to point out that the implicit ranking of systems leads to lack of immunity from irrelevant choices -- by introducing IRV, Cueball's choice changes from Approval to Condorcet (which fails IIC). Note that Approval does satisfy IIC and another important criterion, Participation (adding another vote for your favorite doesn't cause your favorite to lose), but it does fail the Majority Criterion (MC) -- it is possible that by Approving all your approved candidates, including your compromise, a candidate who is in fact preferred by a majority won't win, but will be beaten by a candidate who would lose to that candidate in a direct pairwise comparison. IRV does satisfy MC, but it fails Participation and Immunity from Irrelevant Choice, is not summable (you can't do counts in separate precincts and sum the results centrally -- you have to do a central count overall to decide which candidate to eliminate next), and its monotonicity failures can lead to unpredictably unstable results. Personally, I prefer a ratings-based method, Majority Judgment, which is effectively a special kind of median rating that is highly resistant to strategic manipulation. But MJ can still fail Participation, so I think it would benefit from being the first stage in a 3-2-1 voting style approach -- use MJ with an A,B,C,D,E,F rating system, with A,B,C ratings approved and D,E,F disapproved, then take the top 3 MJ candidates as the 3-2-1 semi-finalists. Drop the least approved candidate from those 3 to get the top two semifinalists, and finally, choose the candidate who wins pairwise as the winner. There could be situations where MJ fails participation, but the participation loser would likely still be in the top three and would win both the "2" and final pairwise comparison. Araucaria (talk) 17:57, 2 June 2017 (UTC)

I don't understand the example provided in the description. In what election would Sanders, Clinton, and Trump be on the same ballot? Jkshapiro (talk) 04:13, 4 June 2017 (UTC)

Sounds like you're talking about what I said. This is why I worded it "let's throw in Bernie Sanders as the third", I needed a third candidate to explain what I was talking about, and he's the only other presidential hopeful whose name I know off the top of my head. :) I don't know WHY Bernie can't be on the same ballot - I suspect he's the same party as either Hillary or Trump, so he was competing with one of them to be the party's candidate - but his early disappearance from things last year led me to grasp that he couldn't be. (I should probably point out once again that I'm Canadian, therefore not my shindig, plus I'm proudly very politically unaware. See above comments for why). - NiceGuy1 04:12, 7 June 2017 (UTC) I finally signed up! This comment is mine. NiceGuy1 (talk) 06:22, 13 June 2017 (UTC)
If you want a specific 2016 candidate likely to be on the same ballot as Clinton and Trump and likely to beat either in a one-on-one contest, I recommend Evan McMullin (notably in Utah). - January First-of-May 10:38, 7 June 2017 (UTC)
It seems important to me to only use names I'm familiar with, from hearing them a lot. Thanks anyway. :) I suspect he's from Vermont, it and Northern New York are the parts of the States closest to where I live, to the point that the American network stations our TV providers use are from Burlington and Plattsburgh, so that might be why I hear about him a lot especially during election times. LOL! NiceGuy1 04:23, 9 June 2017 (UTC) Also my comment! NiceGuy1 (talk) 06:22, 13 June 2017 (UTC)
@NiceGuy1, sorry, no, I should have been more clear. This was in reference to an example in the explanation itself which I've since deleted. Clinton and Sanders were competing to be the Democratic candidate, and the winner, Clinton, then competed against Trump in the presidential contest. But I think the explanation works fine without any example. Jkshapiro (talk) 12:50, 7 June 2017 (UTC)
Ah. I don't recall reading that, and as you say you deleted it now. Seeing as it's such an odd pairing that even _I_ as an outsider realize it's odd, I wonder if someone added that to the explanation inspired by my comment, LOL! (Which would mean it was part of the explanation between times I read it). And thanks for clarifying. I may not get to participate in American elections, but I think this might have been the third election where I got the impression that the general opinion was "Well, of course Sanders should be president", where he wasn't elected and I think didn't even win being the candidate. Hence my using him for my example. :) NiceGuy1 04:23, 9 June 2017 (UTC) Another one of mine! NiceGuy1 (talk) 06:22, 13 June 2017 (UTC)

I had a discussion about "voting for a voting system" years ago on the election-methods mailing list. My proposal was to count the votes using *all* of the methods under consideration. Then, based on the premise that the best election method will choose the best election method, choose a method that chooses itself. 05:46, 10 June 2017 (UTC)

What does "people who study voting methods agree that FPTP voting is not ideal" mean anyway? Which "people"? What proportion of them? There could be 100 people, and 3 of them don't like this method, and this statement is still perfectly factual. It seems to imply that "there is a broad consensus", but it doesn't say that, and this exactly the kind of language people use to mislead other people by not stating things clearly. It could also be read as meaning that "all the people who study voting methods agree that this is a bad method", which is almost certainly not true. There are bound to be "people who study voting methods" who "agree" that it's just fine the way it is. When I read something like this, I think "someone with an agenda is trying to influence other people into agreeing with them by making it sound as if 'experts all say' [whatever they want you to believe]". 05:40, 21 January 2018 (UTC)

Is it better now? - Frankie (talk) 20:40, 31 May 2018 (UTC)
Thanks Frankie, I'm not the rule writer here but it's more common to embed the links into the floating text rather than having a references section. This Wiki is not Wikipedia. --Dgbrt (talk) 21:12, 31 May 2018 (UTC)