# Arrow's Paradox (Arrow's Impossibility Theorem)

Page 1 of 2
I just learned about this today actually. In my "Making Sense of Politics" class we had a full discussion on a thing called "Arrow's Paradox" which basically shows how flawed our electoral system is.

Say you have a 3 person society

Person 1

Person 2

Person 3

And they try to make a government

A, B, C stand for who they vote for in order of preference

1 A B C

2 B C A

3 C A B

So obviously no one is going to win, but say you pit 2 of them together at a time

So say we have A v B

A wins based on preference

So now we have A v C

C wins based on preference

Now let's try another one

A v C

C wins based on preference

Now we have C v B

B wins based on Preference

Now for C v B

B wins based on preference

Now B v A

A wins based on preference.

Do you see that no matter who wins, it's not a majority who wants that person? The person wins based on the design of the electoral system.

Here's another one (again in order of preference)

1| A B C

2| A B C

3| A B C

4| B C A

5| B C A

6| C B A

7| C B A

Based off one vote A would win with 3 votes

BUT, if you look farther, you'll see that 4 people would have picked A as their last candidate.

Weird isn't it? I thought that it was supposed to be based off of what the majority wants. And Obviously the majority didn't want A to win.

See if you based it off of a point system

A - 3 + 3 + 3 + 1 + 1 + 1 + 1 = 13

B - 2 + 2 + 2 + 3 + 3 + 2 + 2 = 16

C - 1 + 1 + 1 + 2 + 2 + 3 + 3 = 13

B SHOULD have been the winner.

Kind of funny how that works isn't it?

Discuss.

P.S. Sorry about the long post, it just really seemed interesting to me when I found this out. I would put a TL;DR but really can't with this.
Austin Les Paul
Alvarez Acoustic

Marshall JCM 2000 DSL 100
Peavey 5150 2x12 60watt combo
Orange Micro Terror
Jet City Made in USA 2x12
Last edited by BeastModeEngage at Oct 21, 2011,
wat

I didn't get any of that.
pinga
using XYZ would've made more sense than ABC
ggg1 ggg3

.
This is why there are dictatorships in the world
Good thing the're at least 3 people in our society....
Quote by behind_you
using XYZ would've made more sense than ABC

It's what he used in class

Quote by bambi_slaughter
Good thing the're at least 3 people in our society....

The probability of the majority getting someone they don't want actually increases with the more people you have, and the more people there are running for that office
Austin Les Paul
Alvarez Acoustic

Marshall JCM 2000 DSL 100
Peavey 5150 2x12 60watt combo
Orange Micro Terror
Jet City Made in USA 2x12
Last edited by BeastModeEngage at Oct 21, 2011,
Who you would pick last doesn't mean shit. There's no third place.
It makes sense, and could be extrapolated to any number of people in a society. It isn't a problem of three, or 5, or any number issue.
Quote by Spud Spudly
Who you would pick last doesn't mean shit. There's no third place.

But there is a "definitely not that guy" place.
Kill person 1 and 2. 3 wins by default.
Quote by SkepsisMetal
This is why there are dictatorships in the world

It all makes sense now.
I get what he's saying, I think.

Pretty much, only 49% of the population may vote for candidate 1.

Other candidates could receive, in one instance, 11%, 25%, and 15%.

Most people didn't vote for candidate 1, but he still won.

Anyways, I'm not sure how one could fix this issue. I suppose they could have a second election between the top two candidates, but I'd bet in the majority of cases, the original winner would win the reelection as well.
Quote by High&Mighty
It all makes sense now.

ggg1 ggg3

.
Quote by Spud Spudly
Who you would pick last doesn't mean shit. There's no third place.

It does though.

Think a little bit. Say 60% of the people voted on someone that wasn't Obama.

But 40% voted for Obama

But now think that even though those 60% of the people didn't vote for Obama, they have there vote split up so much that they can't get one other person over that 40% marker.

So Obama wins, and 60% of the public doesn't actually want him there.

Does that make sense?

Quote by Kevin19
I get what he's saying, I think.

Pretty much, only 49% of the population may vote for candidate 1.

Other candidates could receive, in one instance, 11%, 25%, and 15%.

Most people didn't vote for candidate 1, but he still won.

Anyways, I'm not sure how one could fix this issue. I suppose they could have a second election between the top two candidates, but I'd bet in the majority of cases, the original winner would win the reelection as well.

^ +1

This guy's got it.

The problem is you really CAN'T fix it.
Austin Les Paul
Alvarez Acoustic

Marshall JCM 2000 DSL 100
Peavey 5150 2x12 60watt combo
Orange Micro Terror
Jet City Made in USA 2x12
Last edited by BeastModeEngage at Oct 21, 2011,
You essentially just explained Canadian politics.
Quote by Godsmack_IV
You essentially just explained Canadian politics.

No, if we had a Parliamentary system like Canada, or the UK, we would honestly be better off.

Our presidential system sucks.
Austin Les Paul
Alvarez Acoustic

Marshall JCM 2000 DSL 100
Peavey 5150 2x12 60watt combo
Orange Micro Terror
Jet City Made in USA 2x12
the best possible government would be a dictatorship where the dictator is actually a good person and uses their ultimate power to benefit the people.

however, absolute power corrupts absolutely
Quote by jimRH7
my cat used to have a penchant for rubber bands, if that counts.

edit: totally doesn't count.

Quote by JacobTheMe
It looks like you are an asshole.

Sorry, kid.

Quote by red157
So, basically faster. And not so Jamaican-like.
Elections are determined by who's most popular, not who's most unpopular. This is why I get so annoyed at 3rd party voters who are making some sort of political statement.
Check out my band Disturbed

Genius!
HILT!

Where's Waldo?

Quote by BeastModeEngage
It does though.

Think a little bit. Say 60% of the people voted on someone that wasn't Obama.

But 40% voted for Obama

But now think that even though those 60% of the people didn't vote for Obama, they have there vote split up so much that they can't get one other person over that 40% marker.

So Obama wins, and 60% of the public doesn't actually want him there.

Does that make sense?

But he was still the majority choice of the candidates.
Quote by Dirge Humani
But he was still the majority choice of the candidates.

If he was the real majority he'd have 51% of the vote. Your logic is flawed.

He just has the most votes of the candidates.

Quote by Godsmack_IV
With an Alternative(run-off) voting system B would have won in your last example.

See the only problem is, you have to take into account EVERYONE who was in the race at one point.

Say there were 50 people that started the presidential race, but 48 dropped out.

Now, you still have to take into account everyone in the population who wanted those other 48 members to be president.

See why it's hard to fix?
Austin Les Paul
Alvarez Acoustic

Marshall JCM 2000 DSL 100
Peavey 5150 2x12 60watt combo
Orange Micro Terror
Jet City Made in USA 2x12
Last edited by BeastModeEngage at Oct 21, 2011,
Quote by BeastModeEngage
If he was the real majority he'd have 51% of the vote. Your logic is flawed.

He just has the most votes of anyone else.

Note "of the candidates"
Yeah. I didn't understand that one.
My Gear:
Yamaha FGX720SCA
Quote by Dirge Humani
Note "of the candidates"

Right, sorry.
Austin Les Paul
Alvarez Acoustic

Marshall JCM 2000 DSL 100
Peavey 5150 2x12 60watt combo
Orange Micro Terror
Jet City Made in USA 2x12
Quote by BeastModeEngage
No, if we had a Parliamentary system like Canada, or the UK, we would honestly be better off.

Our presidential system sucks.

Envision the most recent Canadian election. The Conservatives won a majority (50%+ of seats in parliament) with only 40% of the vote. NDP had 30% and Liberals had 19%. It is not unlikely that a majority of Canadians were opposed to a conservative government, nevertheless a majority conservative government.
Quote by BeastModeEngage

Here's another one (again in order of preference)

1| A B C

2| A B C

3| A B C

4| B C A

5| B C A

6| C B A

7| C B A

The majority would always show a minority? This is stupid. Why couldn't the preference order for the second person be A C B? Or change 7 to CAB.

This makes no sense at all.
Current Stage Gear
HD500 Pedal
Bugera Vintage 22

Quote by metaldood91
Hi. Can someone tell me which guitars are real 24 fret guitars and which are just 22 fret guitars with 2 extra frets added on?
Quote by Godsmack_IV
Envision the most recent Canadian election. The Conservatives won a majority (50%+ of seats in parliament) with only 40% of the vote. NDP had 30% and Liberals had 19%. It is not unlikely that a majority of Canadians were opposed to a conservative government, nevertheless a majority conservative government.

Okay point taken. I forgot that some Parliamentary systems have more than 2 parties.

But see, this is also where Arrow's Paradox takes place.

It can mess with any electoral system

Quote by bluekirby7
The majority would always show a minority? This is stupid. Why couldn't the preference order for the second person be A C B? Or change 7 to CAB.

This makes no sense at all.

It's an example of how things CAN happen, not how they WILL happen.

It's a theorem. Not a law. But it can happen
Austin Les Paul
Alvarez Acoustic

Marshall JCM 2000 DSL 100
Peavey 5150 2x12 60watt combo
Orange Micro Terror
Jet City Made in USA 2x12
Quote by BeastModeEngage
See the only problem is, you have to take into account EVERYONE who was in the race at one point.

Say there were 50 people that started the presidential race, but 48 dropped out.

Now, you still have to take into account everyone in the population who wanted those other 48 members to be president.

See why it's hard to fix?

It isn't necessary to whittle out candidates until there are only two.
Quote by Godsmack_IV
It isn't necessary to whittle out candidates until there are only two.

No it's not, but, if there were four candidates it would be just as hard.
Austin Les Paul
Alvarez Acoustic

Marshall JCM 2000 DSL 100
Peavey 5150 2x12 60watt combo
Orange Micro Terror
Jet City Made in USA 2x12
I get what you're saying, but here's where I see the flaw. You start by showing us that depending on the order of the voting, A, B, or C could all potentially win. This seems wrong to me, because the first possibility would mean more people want A than C, and your third possibility implies more would want C than A.

IMO, Our system is fine if we got more than 2 parties into popularity and if we got rid of the electoral college bullshit.

EDIT: Here is why I'm fine with us narrowing it down to a few people. People say that votes for Nader made Gore lose to Bush. I don't know if that's true or not, I'm not looking at the numbers, but for the sake of this example, assume it is.

Now, if Nader would've been a Democrat, thusly getting voted out instead of being a third option, then Gore, whom the most people wanted, would've won, instead of Bush whom people didn't.
Last edited by Macabre_Turtle at Oct 21, 2011,
Yeah but you only vote for your favorite candidate, not for the one who you would like the least.

When parties form governments they compromise and thereby including each other in the legislative process.

When to parties who lets say have achived 25% and 35% of the votes form government together they represent 55%% of the population...

Quote by BeastModeEngage
No it's not, but, if there were four candidates it would be just as hard.

Are you saying you wouldn't be able to keep track of the interests and difference in agendas of more than 2 parties?
Quote by Robertoodavid
Yeah but you only vote for your favorite candidate, not for the one who you would like the least.

When parties form governments they compromise and thereby including each other in the legislative process.

When to parties who lets say have achived 25% and 35% of the votes form government together they represent 55%% of the population...

25 + 35 = 55
We have a new paradox to contend with.
Quote by SomeoneYouKnew
You should be careful what you say. Some asshole will probably sig it.

Quote by Axelfox
Yup, a girl went up to me in my fursuit one time.

Quote by Xiaoxi
I can fap to this. Keep going.
You can't vote against someone, only the votes FOR the candidate count.
I mean, If there was 3 candidates as in an earlier example, one can get 40% of the votes and still win despite the fact that 60% didn't vote for that person.
But hey, the 2 other ones got equal amount of votes, so do you think a candidate with 30% votes should win?
This makes perfect sense, since you can't give -1 or anything like that. The one that's the most popular win.
Last edited by JB95 at Oct 21, 2011,
Quote by JB95
You can't vote against someone, only the votes FOR the candidate count.
I mean, If there was 3 candidates as in an earlier example, one can get 40% of the votes and still win despite the fact that 60% didn't vote for that person.
But hey, the 2 other ones got equal amount of votes, so do you think a candidate with 20% votes should win?
This makes perfect sense, since you can't give -1 or anything like that. The one that's the most popular win.

No but (and this really applies to 3+ party systems) if you are given the option to order your choices for representation then you can attempt to avoid your perceived worst outcome if your ideal candidate does not get elected. In other words, even if the candidate that most closely represents your interests does not get elected then at least you can opt for a different candidate who can somewhat represent you instead of being completely unrepresented. Democracy is all about freedom of representation - more freedom of representation means more democracy.
Hell, if we could vote for an order of options we could start getting third parties into popularity because people won't worry about "throwing away their vote."
I shall grant you three wishes.

None of which will work.

Does the above post enrage, offend or confuse you?

Good.

I like my women how I like my guitars. Curvy and like it when I finger them.
What is this I simply do not even