Voting is a Sham Mathematically Speaking

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by eibrahim • 14 years ago

95 74 14 years ago

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mtgx• 14 years ago

Approval Voting mostly solves the "strategic voting" part that almost forces you to choose the "most likely to win" candidate, or if you hate that one, the one closest to him, while eliminating the spoiler effect, and giving 3rd party candidates a much higher chance of winning than with current traditional voting systems.

http://www.electology.org/approval-voting

http://en.wikipedia.org/wiki/Approval_voting

DennisP• 14 years ago

Plus it doesn't run into trouble with Arrow's theorem, since it's not a "rank-order voting system," unlike plurality, instant runoff, and various others.

Range voting has the same advantage. In computer simulations measuring how well the election result matches voter preferences, either range or approval is as much an improvement over plurality as plurality is over picking someone at random (or, if you like, monarchy). http://rangevoting.org/BayRegsFig.html

gus_massa2• 14 years ago

I'm not sure about the definitions, but if the Arrow theorem doesn't apply to the Approval Voting sistem them I think that it must not be applicable to the "majority rules" criterion.

In this two system the idea is that you get very little information from the voters (best candidate / a set of candidates) and don't know all the information about the order of preference and the relative strength. So I don't understand why having less information is better (theoreticaly).

jivatmanx3• 14 years ago

Don't know if anyone is going to read this, but I was reading about ordinal vs. cardinal utility, which are interesting because they seem to correspond to Ranked Choice vs. Approval voting, with the various voting systems simply being a different algorithm of calculating the winner from these.

Anyway, apparently, with "Inter-temporal choice", or, different utility of the same thing measured in different time periods, Cardinal Utility needs to be used.

I wonder if the same applies to grouping utility of different people together in order to get an aggregate. If so, it's an argument for the use of Approval/Score voting as opposed to Ranked choice.

The quote is "Models of utility with several periods, in which people discount future values of utility, need to employ cardinalism in order to have well-behaved utility functions." an appears in the cardinal utility article.

bhauer2• 14 years ago

Score voting is actually slightly superior to Approval Voting (which is Score Voting with a range of 0 to 1). To my understanding, the Center for Election Science ("electology.org") is the preeminent organization championing Approval and Score Voting.

http://www.electology.org/score-voting

Eric Sanders of that same organization has written a few articles on the subject, including the following:

http://bigthink.com/action-in-action/why-the-republicans-los...

I've personally been trying to campaign for the adoption of score voting:

https://usa.brianstaskforce.com/task/362/score-voting-for-na...

jivatmanx3• 14 years ago

I think that, over time, people would drift towards using only 0 and 10 and none of the middle scores, similar to how, over time a plurality voting system drifts towards only having two political parties (Duverger's Law). 0 And 10 are the "strategic votes" of score voting as 2-major-parties are the "strategic votes" of plurality voting.

A possible counter-argument to this is that score voting doesn't have a reinforcing feedback loop (each election, the minor parties get weaker until they remain at around 1%)

bradbeattie3• 14 years ago

Let's say there are two head-line candidates: A and B. I hate A, am neutral towards B, and love the less popular of the three candidate C.

So do I vote 0/0/1 or 0/1/1? Well, that depends on how popular the three are and is very much an example of strategic voting.

When proposing an alternative, one should probably be up-front with its flaws.

jivatmanx3• 14 years ago

That's not strategic voting according to the definition. In both cases you're still voting for C, the candidate you like the most.

"In voting systems, tactical voting (or strategic voting or sophisticated voting or insincere voting) occurs, in elections with more than two candidates, when a voter supports a candidate other than his or her sincere preference in order to prevent an undesirable outcome.[1]"

If you mean it in the larger sense of using strategy when voting, Gibbard–Satterthwaite theorem shows that no voting system is strategy free unless it's either dictatorial or nondeterministic.

pc86• 14 years ago

Would it be better to have something like 0/1/2 which is essentially IRV, or some other method of point distribution among candidates?

Steko• 14 years ago

Approval voting suffers the largest number of "Especially Intolerable Failures" in the comparison here:

http://en.wikipedia.org/wiki/Comparison_of_instant_runoff_vo...

Empact3• 14 years ago

If you examine these supposed failures of Approval voting, you'll find that the bulk of them can be attributed to the divide between a "Majoritarian" or a "Consensus-driven" perspective. That is, many of the criteria suppose that the most-preferred candidate of the majority should win, but should that be the case? Imagine there are 3 candidates, one who is adored by the super-majority, a second who is very well-liked by that super-majority and a third who is hated. Meanwhile the minority hates the first and third candidates, but well-likes the second.

We can visualize this like so:

  #   Approval | Disapproval
  80 |-1--2----|--------------3-|
  20 |-2-------|-----------3--1-|

So while the first candidate is the favorite candidate of the super-majority, the second candidate has the unanimous approval of population. Which is preferable? I would say the second candidate.

So you see, the criteria themselves incorporate an ideological perspective - a majoritarian bias - rather than articulating some absolute objective truth.

Also I have to take issue with the basis for the "Especially Intolerable Failures" explanation - does it make sense to reject a system for being theoretically capable of producing a result, "regardless of the probability that these paradoxes may occur"? Given 2 systems, one which exhibits several flaws often, and another which rarely exhibits flaws but is theoretically capable of exhibiting a certain offensive flaw, does it make sense to reject the better-behaving system because of a theoretical result? I think not.

streptomycin• 14 years ago

Also, there's this: http://papers.nber.org/papers/w15220

saraid2167• 14 years ago

> Voting is a method that a group of people use to pick the “best choice” out of a set of candidates. It’s pretty obvious, right?

And like many other pieces of "common sense", this isn't correct.

Wikipedia says, "Voting is a method for a group such as a meeting or an electorate to make a decision or express an opinion—often following discussions, debates, or election campaigns. Democracies elect holders of high office by voting." This is very different from "picking the best choice".

I realize that the American public has been indoctrinated for the past few decades that voting is the only way you make yourself heard, but this isn't true and never has been. I recently learned about Wellstone Action ( http://en.wikipedia.org/wiki/Wellstone_Action ); I encourage everyone to look into enrolling. (I haven't done so myself yet. I probably will at some point, though.)

> On one hand, this seems to be an endorsement of the two-party political system we have in the United States.

Actually, what it's an endorsement of is all of our other voting systems where the choice is between APPROVE and REJECT. You have to endorse the existence of political parties in the first place before you can endorse a two-party system, and Arrow's theorem goes nowhere near that.

derleth3• 14 years ago

> I realize that the American public has been indoctrinated for the past few decades that voting is the only way you make yourself heard

This has never been true. Americans have always protested, signed petitions, and even joined special interest groups like the ACLU and the NRA. Americans have a very long tradition of things like that.

saraid2167• 14 years ago

Tell that to the next person who complains that their vote doesn't count.

bo10242• 14 years ago

> Wikipedia says, "Voting is a method for a group such as a meeting or an electorate to make a decision or express an opinion—often following discussions, debates, or election campaigns. Democracies elect holders of high office by voting." This is very different from "picking the best choice".

I don't see how they are different. The author never said that candidates have to be people. Substitute the word "alternatives" if you prefer.

saraid2167• 14 years ago

> I don't see how they are different.

"I think this is the way we should proceed" is qualitatively different from "I think this is the best choice".

> The author never said that candidates have to be people. Substitute the word "alternatives" if you prefer.

Substitute it in place of what? Where did I require that the candidates must be people?

basseq• 14 years ago

> A voting system can only, at times, choose the most preferred of the options given. But it doesn't necessarily present us with the best candidates to choose from in the first place.

Reminds me of HHGTTG: "Anyone who is capable of getting themselves made President should on no account be allowed to do the job."

aprescott• 14 years ago

In this case, Hanselman is the clear winner with three votes, whereas the other two candidates each have two votes. This is how our elections are held today.

This is dependent on the exact election taking place. With the US presidential elections, my understanding is that a plurality of electoral college votes is not enough to win, you need an actual majority. In the event of a simple plurality win with no majority, the result is decided by the House of Representatives (which may itself be tied).

rjzzleep• 14 years ago

is it? the actual candidates already get preselected.

even if youre right, which is likely, it doesn't matter, because any majority was previously generated by pluarility.

wam• 14 years ago

Learning about Arrow's theorem definitely changed the way I think about elections in the US. It also changed the way I think about election news coverage. I used to be an ardent "horse race news" hater. I still am, in terms of how utterly it dominates election news, but now I see some utility in it as well.

Arrow and these others have focused how I look at the game-theoretic underpinnings of elections and the importance of being up to speed on exactly how candidates and interested parties are crafting strategies around the complexities built into the game. When people conflate the "message" of the candidate with the strategy (which is always) I still get irritated. I have a tendency toward partisanship and that kind of thing clouds my judgment. But the day-in day-out workings of the campaigns and PACs are more interesting to me now, because they shed light on what's fundamentally "broken" (from my point of view) in the underlying system, as opposed to what I simply find distasteful or disappointing.

Math!

saraid2167• 14 years ago

Social choice theory is One Of Those Things which everyone (myself included) needs to spend more time learning about.

hcarvalhoalves• 14 years ago

If you look at Brazil, which has multiple parties and plurality voting, the problems are pretty clear.

In this year's elections, the candidate with 28% of the votes was elected mayor in my city.

Noughmad• 14 years ago

I find it interesting that in all those discussions about voting systems, which are mostly focused on USA president elections, nobody mentions two-round voting, also known as run-off voting.

This is what we have in Slovenia for electing our president. In the first round, there are many candidates, and each voter can vote for one. If any candidate gets at least 50% of votes, he automatically wins.

If, on the other hand, there is no majority winner, the two best candidates compete head-to-head in the second round.

Such a system allows you to always vote for your favourite candidate in the first round, and if your candidate doesn't make it into the second round, you can vote for the fallback one.

Details: https://en.wikipedia.org/wiki/Two-round_system

kscaldef• 14 years ago

I don't believe this satisfies the Condercet criterion either. Consider these rankings of preferences:

    20% A ...
    20% B ...
    15% C ...
    15% D C ...
    15% E C ...
    15% F C ...

In a two-round run-off, one of A or B will be elected, despite the fact that 60% of voters prefer C over either A or B.

im3w1l• 14 years ago

And in the real world, people would second guess this, and enough people would tactic vote for C that it would't be a problem.

"But then they can't vote for their prefered candidate which was the whole point"

Well, some people can. D, E, F could still get a few percentage points. More importantly, I don't think we would see convergence to a 2-party system.

Unless I am missing something, it looks like at least 3 parties could be sustained.

gradstudent• 14 years ago

Preferential voting solves all these problems. You vote by ranking the candidates on order of preference. If your top candidate does not win the vote goes to next guy down the line until eventually it ends up for one of two candidates.

Pinckney2• 14 years ago

Preferential voting does not satisfy the Condorcet criterion.

http://en.wikipedia.org/wiki/Instant-runoff_voting#Voting_sy...

bradbeattie3• 14 years ago

To demonstrate this, consider the following.

80 people: A, C, B

50 people: B, C, A

35 people: C, B, A

IRV eliminates C (as it has the fewest first-place votes) and elects B. But voters on the whole prefer C over B (115 to 50). This is the failure that Pinckney refers to.

finnw• 14 years ago

You mean IRV does not satisfy the Condorcet criterion. IRV is not the only preferential voting system. There are others, e.g. ranked pairs[1] which do.

Under ranked pairs, each pair of candidates is considered individually, largest majorities first.

In bradbeattie's example we have:

    C > B: 115 votes
    C > A:  85 votes
    B > A:  85 votes

So the group preference is C > B > A and C wins.

Ranked pairs fails the participation criterion[2], but it is still possible to vote tactically in a way that guarantees you will not harm your first-choice candidate by participating (since if your first choice is A you can rank A first and all the others joint second. Similarly if your goal is to prevent C from winning, you can rank C second and all others joint first.)

[1] http://en.wikipedia.org/wiki/Ranked_pairs [2]:http://en.wikipedia.org/wiki/Participation_criterion

bo10242• 14 years ago

This is a very nice summary of/intro to the classic/standard mathematical approach to voting and Arrow's Theorem.

frozenport• 14 years ago

This is why we have a 2 party system :-)

maratd3• 14 years ago

This is off. Voting is not a system for selecting the best candidate. Voting, in whatever forms it exists, is a system to avoid violence and conflict.

Each side feels it had a fair shot, regardless of outcome. The purpose of voting is to leave you with that feeling, to avoid unpleasant behavior from the losing party. The best person for the job almost never gets it. That's why we don't vote people in when hiring somebody at a company.

diego• 14 years ago

Yes, and in addition it's also a system to ensure that elected officials and their parties have accountability. If you win an election and then "betray" your voters, you won't be reelected. If you can't be reelected, you could damage the chances of your party. In essence, democracy in its current form is not so much about choosing the right/best candidate. It's more about making sure the winner cannot become a despot.

bad_user• 14 years ago

In addition, people do not even try voting for the best candidate. People usually vote for the candidate whose interests best align with their own interests and if we're talking about that, then we can't avoid talking about compromises.

Which is why it's easy to see why preferences between candidates may be non-transitive, as the space we're talking about is no longer one-dimensional so candidates don't have a natural order.

ontheotherhand2• 14 years ago

"If you win an election and then "betray" your voters, you won't be reelected."

That ain't accountability, that's a joke. If you pay me 5000$ to do a job, and I don't lift a finger, and your "punishment" is that we won't do that a second time, then I have free money and you're a fool. It doesn't hurt the party one bit either; that's the whole point of corporatism, you can swap out individuals while the "brand" rolls on, saying "whoops, bad apple" every 5 seconds, with people forgetting after 2.

"It's more about making sure the winner cannot become a despot."

Correction: it replaces a single despotic individual with a tag team of people who basically can do whatever they want - within boundaries, sure, they at least have to be somewhat slick about it, and know how to make a puppy face, too; but certainly not within boundaries defined by the actual will of the people who handed their power as souvereigns over to their representatives. Having despotic entities control you is not one iota better than despotic humans, not in the long run.

Despotism is marked by the control going only down, accountability going only up -- period. Not by angry men on podiums necessarily, and not by bloodshed. (Not that there isn't plenty bloodshed, but that's besides the point) If you seriously see a huge difference or improvement there, you've fallen for it I'm afraid.

anigbrowl• 14 years ago

Voting, in whatever forms it exists, is a system to avoid violence and conflict.

If only someone had explained this to your namesake :-)

http://en.wikipedia.org/wiki/Jean-Paul_Marat

saraid2167• 14 years ago

Not really. Voting only avoids violence and conflict when the possible combatants may actually participate in the vote.

fleitz• 14 years ago

Actually some workplaces do elect people, and they find equally baffling results that voting people in increases productivity and reduces conflict.

As for a more common example of voting with in a company, look at a board of directors.

But yes, in general voting is used to provide legitimacy and credibility.

jQueryIsAwesome2• 14 years ago

> Each side feels it had a fair shot, regardless of outcome

Yeah, but the calm fades away when a big enought amount of people realizes that they are not actually part of neither side and they get angry about it.

justincpollard• 14 years ago

Whether or not voting is a system to avoid violence and conflict or create accountability, it's dominated by a system that allows for those outcomes (assuming they are binary) and a third: a better candidate is elected to the position. As long as the process is generally known and makes sense (whatever that means), I'd rather have all three outcomes than just two. I think the author's point is a simple and important one.

philwelch• 14 years ago

There's also the useful property that if everyone votes based on their own selfish interests, the winner will serve the interests of more people than any other candidate.

glenra• 14 years ago

I have two problems with that statement. First off, in practice very few people actually do vote based on their own selfish interests unless you're using a really broad and unusual definition of selfish. People largely vote as a form of signaling - they vote to send a message about how much they care about stuff and to have something in common with their group. (see also: _The Myth of the Rational Voter_)

The bigger problem is that there's a logical disconnect between your two clauses. The fact that people vote based on their own selfish interests does NOT actually guarantee the winner will serve their interests. It only would do so if people were really good judges of character and politicians were bad at lying and there were some reliable way to hold politicians to account for what they do.

Politicians lie about what they want to do, and we vote for them because we like the lies they tell. When everyone votes for the liar who promises to help everyone's own selfish interest, we get an unusually good liar who actually pursues his own interests when elected.

In the Bush v. Gore debate, Bush claimed to be firmly opposed to "nation-building". People who voted for Bush on that basis did NOT get a winner who was serving their interests.

mmphosis• 14 years ago

Mixed Member Proportional (MMP) Representation solves some of the problems. http://www.youtube.com/watch?v=QT0I-sdoSXU&feature=relmf...

The problem with MMP is when the parties choose the ranking of their list of representatives. I think it would be even better if rather than use a party generated list, instead the representatives are determined by people's votes.

nikatwork• 14 years ago

I've always thought New Zealand's mixed-member proportional (MMP) voting system [1] is the least bad solution currently in use.

I'm not from NZ so I'd be interested to hear what the locals think.

[1] http://en.wikipedia.org/wiki/New_Zealand_voting_system_refer...

lmkg2• 14 years ago

Not a local, but... Proportional voting systems suffer from the problem that voting "power" is not proportional to representation.

Consider a parliament with 100 members and 3 parties. Suppose the breakdown is: A has 49 members, B has 48 members, and C has 3 members. Guess what... A, B, and C all have equal voting power! Any two parties are enough to reach a majority of 51 votes, and any one party is not. Despite A having, in theory, over 16 times the representation of C, it does not have any voting advantage.

Keep in mind that any voting system based on parties will tend to have very partisan voting blocs. Representatives in the US are more independent and likely to break with the party because they are elected in geographically isolated elections. Representatives elected directly by a party generally have about as much independence as the Electors in the Electoral College.

NickNameNick• 14 years ago

I'm from NZ...

To form a government, the party with the most votes, or a coalition of parties which collectively holds a majority petition the governor general.

For a single party this is quite straightforward.

To form a coalition the member parties agree on a 'Confidence and supply agreement" This is basically a statement that in the event of a vote of no-confidence, all of the coalitions members will support the coalition, and also a broad agreement on the budget. Getting an agreement on confidence usually involves a certain amount of horse trading about ministerial and vice ministerial positions. Likewise, the agreement on supply will probably involve some intense budget and joint-policy negotiations.

If you had a parliament of 101 seats, split into an opposition of 50 seats, and a government of 51, itself made up of a large party (48 seats) and a small party (3 seats) what you will probably see is the small party only has the tiniest influence on the coalition agreement. They probably traded everything else to get their senior member a ministerial position.

gabemart• 14 years ago

I found this article quite frustrating.

>Condorcet formalized the idea that group preferences are also non-transitive. If people prefer Hanselman to me. And they prefer me to Guthrie. It does not necessarily mean they will prefer Hanselman to Guthrie. It could be that Guthrie would pull a surprise upset when faced head to head with Hanselman.

I found this by far the most interesting assertion, but the examples under "Historical Examples" don't demonstrate this phenomenon at all.

For instance, the author asserts that the Nader spoiler effect demonstrates nontransitive preference relationships. But from my reading, it wasn't the case that that group as a whole preferred (Gore over Nader) and (Nader over Bush) but (Bush over Gore). It was simply that due to the structure of the election, they happened to elect Bush. While this ties into the author's point about the "unfairness" of elections, it doesn't demonstrate nontransitive relationships in group preferences.

Could someone post an example of a group preference configuration in which the group prefers (A over B) and (B over C) but (C over A)?

I understand the concept of nontransitive relationships in general, but in the specific domain of fitness for office, I can't work out how this would come to be.

jsharpe• 14 years ago

I am wondering that too. He also gave the example of Ford beating Reagan (in the Republican primary), Carter beating Ford in the general election in 1976, and then Reagan beating Carter in 1980, but this is also not a great example since the Republican primary group is very different than the general electorate.

Dove• 14 years ago

Could someone post an example of a group preference configuration in which the group prefers (A over B) and (B over C) but (C over A)?

Sure, that's easy to construct.

    Peter's preferences: A, B, C
    Paul's: B, C, A
    Mary's: C, A, B

The group prefers A over B, by a 2-1 vote. Likewise B over C, and C over A.

Nursie• 14 years ago

It's late and I'm tired but I think in the following situation -

49% vote Bush, 42% vote Gore, 9% vote Nader. All Bush voters prefer Nader over Gore (unlikely!). All Nader voters prefer Gore over Bush. Half of Gore voters prefer Bush over Nader.

  - If the election was Gore:Bush, Gore would win 51:49
  - If the election was Bush:Nader, Bush would win 70:30
  - If the election was Gore:Nader, Nader would win 42:58

The group prefers Gore over Bush (A over B) and Bush over Nader (B over C) but Nader over Gore (C over A).

That's my reading anyway.

jQueryIsAwesome2• 14 years ago

Some of you are forgetting something; that even if you had some form of "stadistical fairness" (whatever that may be); you still have the biggest problem of most democracies: Uneducated people; people who think an atheist shouldn't be president, people who like to reinforce their biases more than they like to have deep discussions about the nation's issues, people who were never taught to do critical thinking... and without doing exceptions for their government, their parents, their religion and the law.

bluedanieru• 14 years ago

That's not really in scope for choosing a voting system that best represents the people. But yes, point taken.

Tloewald• 14 years ago

The article confuses two issues, one illustrated by Arrow's Theorem which is more relevant to parliamentary procedures (where any set of more than two choices has to be resolved as a series of binary choices, and the voting population is small and its preferences well understood) and first past the post electoral systems which are completely hopeless, especially when tiered, as in the US.

Most of the article is essentially discussing an example of Arrow's Theorem where if you know people's preferences and can present them with binary options in an order of your choosing you can obtain any outcome except the least popular option. This is very artificial and not a real flaw of preferential and proportional electoral systems where (a) individual preferences are not known and (b) the entire vote is done in one step, not in a carefully chosen series of binary options. Great for gaming a committee, lousy for elections.

As others have observed, the chief purpose of voting is allowing government transitions without violence and with the appearance of procedural fairness, but the fact remains voting works just fine when the population has a clear cut preference ("throw the bastards out").

Well, modulo corrupt redistricting.

Americans who want to talk about voting really need to understand that there are other voting systems than the horse and buggy system used in the US and UK.

stretchwithme• 14 years ago

Winner-take-all elections, no matter how they operate, leave many people without the representation they prefer. Proportional representation is much less likely to do this.

Proportional representation can used in the executive branch too. Switzerland does it.

fluxon• 14 years ago

Wasn't this issue addressed rather well in a recent hackernews-linked item which mathematically showed both that voting is not a sham, but that the Electoral College system is more fair than it has been represented? (sorry can't find the link!)

Voting is a Sham Mathematically Speaking

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