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A better way of voting

by causalset
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Mech_Engineer
#19
Mar5-12, 02:49 PM
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I'm starting to see the arguments for the weighted system, but let's be honest I'm not sure the added complexity and confusion would really bring enough added value.

I think when it comes down to it politicians try so hard to be ambiguous and general these days! if you have two candidtaes they each get about half the vote, if there are three candidtaed they each get about a third of the vote... People vote for whomever they like best on TV regardless!
cmb
#20
Mar5-12, 02:55 PM
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Sounds like you'd achieve the same result as some existing variations of the Single Transferable Vote scheme.

Try looking up some of these voting schemes, see if you like them;

Gregory method,
Meeks method,
Warren method,
Hare-Clarke method,
Wright method,

and if you are doing internet searches, combine these terms with the words 'Droop quota', and that should pull up the links to these [and other] schemes.
Gokul43201
#21
Mar5-12, 07:09 PM
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Here's another test of the Causalset method:

Consider the case of 3 candidates: A and B are essentially identical moderates, liked equally by a large fraction (nearly 2/3rd) of the electorate, while C is an extremist with a significant, though smaller group of supporters (a little over 1/3rd of the electorate).


Current Method:

A and B split votes from their supporters, while C takes the rest.
Fraction of total vote that A gets, [itex]V(A,total) = 1/3 - \Delta[/itex]
Fraction of total vote that B gets, [itex]V(B,total) = 1/3 - \Delta[/itex]
Fraction of total vote that C gets, [itex]V(C,total) = 1/3 + 2\Delta[/itex]

C wins the election for arbitrarily small positive [itex]\Delta[/itex]. For instance, C wins if [itex]V(A,total) = V(B,total) = 0.333,~ V(C,total) = 0.334[/itex]

That doesn't sound terribly fair to me. There are 2 candidates that appeal to a large majority of the voters (how often does that happen in real life?), yet they both lose to the one extremist that appeals to a much smaller plurality.


Causalset Method:

One plausible scenario: Supporters of A and B (let's call them 'group ab', and say they make up the same fraction of the electorate as above, namely [itex]2/3 - 2\Delta[/itex]) give each of their equally favored candidates a score of +1 (normalized to +1/3) and give C a score of -1 (normalized to -1/3). Supporters of C (call them 'group c', making up [itex]1/3 + 2\Delta[/itex] of the electorate) give C a score of +1 (normalized to +1/3) and give each of A and B a score of -1 (normalized to -1/3)

[itex]V(A,total) = V(A,ab) + V(A,c) = (2/3 - 2\Delta)(+1/3) + (1/3 + 2\Delta)(-1/3) = 1/9 - 4\Delta/3[/itex], where [itex]V(X,y)[/itex] is the share of X's total vote-fraction coming from 'group y'.

Similarly, [itex]V(B,total) = V(A,total) = 1/9 - \Delta[/itex]

And [itex]V(C,total) = V(C,ab) + V(C,c) = (2/3 - 2\Delta)(-1/3) + (1/3 + 2\Delta)(+1/3) = -1/9 + 4\Delta/3[/itex]

For C to win this election, it would require that [itex]-1/9 + 4\Delta/3 > 1/9 - 4\Delta/3[/itex] or [itex]\Delta > 1/12[/itex]. So, C would need to have the support of at least half (1/3 + 2/12) of the population in order to win.

Sweet! I like that. Note that I didn't try to optimize strategies for 'ab' or 'c' - I just did the lazy thing and guessed what might be close to an optimal strategy. I wonder if my guess was indeed optimal. An alternative guess would be for 'group ab' to vote (A=0,B=0,C=-1), and maybe 'group c' goes with (A=0,B=0,C=+1). Someone with a little free time might crunch those numbers, or see if 'ab' and 'c' can independently optimize their votes, and figure out what does.

Or better still, does someone have a nice situation where the Causalset Method produces a less desirable result than the Current Method?
causalset
#22
Mar6-12, 03:32 PM
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Here is more general pattern. Suppose we have extreme republican, mild republican, mild democrate and extreme democrate. If people could only vote for one person, the democrates will split EVENLY between mild democrate and extreme democrate. Likewise, republicans will split EVENLY between mild republican and extreme republican. If votes can't be split, all democratic votes would go exclusively to democrates, thus ''mild republican'' and ''extreme republican'' would be equally likely to win. Likewise, since all republican votes will be exclusively republican, mild democrate and extreme democrate would be equally likely to win, as well. On the other hand, if the split votes are allowed, then democrates would give ''split vote'' to mild republican in order to prevent extreme republican from winning, and republicans will give ''split vote'' to mild democrate just to prevent extreme democrate from winning. Thus, my way of voting will ''shift'' the elected candidates ''even closer'' to the middle and ''further away'' from extreme left or extreme right.
Office_Shredder
#23
Mar6-12, 03:54 PM
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Gokul, the voters for C totally blew it in your scenario. It's more efficient for them to just put all their vote in for C (If they ding both candidates A and B, each C voter is only worth 2/3rds of a vote for C essentially)

Let's consider that situation - the voters for A and B split their vote three ways and the voters for C go all in for C:

[tex] V(A) = (1/3-\Delta)*2/3 = 2/9-2\Delta/3 = V(B)[/tex]

[tex] V(C) = 1/3+2\Delta-(1/3-\Delta)*2/3 = 1/9+4/3\Delta[/tex]

so C will win as long as [tex] \Delta > 1/18[/tex] in this situation.

The best case scenario for the voters for A and B, if they really don't care, is to just give a -1 to candidate C. Then candidate C can never win unless he has an outright majority, and the winner of A vs B will be decided by whatever pitiful fractions of votes the non-rational voters toss their way.

So candidates A and B, knowing that they want one of them to win, could run on a campaign of: give C a -9 vote and give your favorite candidate a +1 vote. C will be guaranteed to lose but A or B will win depending on who has more support amongst the non-C voters
Gokul43201
#24
Mar6-12, 07:02 PM
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Quote Quote by Office_Shredder View Post
Gokul, the voters for C totally blew it in your scenario.
That was my suspicion. Thanks for doing the legwork.

It's more efficient for them to just put all their vote in for C
...
The best case scenario for the voters for A and B, if they really don't care, is to just give a -1 to candidate C.
I suspected (as I was finishing up the last post) that this might in fact be the case. Which is why I suggested it in passing, at the end of my last post.
Quote Quote by Gokul43201 View Post
An alternative guess would be for 'group ab' to vote (A=0,B=0,C=-1), and maybe 'group c' goes with (A=0,B=0,C=+1).
Now, with a pen and paper I see that this is indeed their optimal solution - lucky that it turns out to be quite easy to optimize their strategy. And I find it nice that with both groups voting optimally a majority is required for C to win.
Vanadium 50
#25
Mar6-12, 07:38 PM
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There is something about voting systems that causes everyone to think there is no need to read up on what previous work has been done. I recommend everyone take a look at something called Arrow's Theorem, which proves that an ideal voting system cannot exist. Given that, the discussion should be centered around the faults of the proposed replacement being less severe than the faults of the baseline.
cmb
#26
Mar7-12, 02:33 AM
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Quote Quote by Vanadium 50 View Post
I recommend everyone take a look at something called Arrow's Theorem
Thanks for reminding me of that name. I've been trying to recall it recently in conversation when describing these proofs.

which proves that an ideal voting system cannot exist.
Well, let's be specific, it proves an ideal voting system cannot exist, given the axioms he used.

...and I believe there is a quite simple way to change the axioms, given the advent of modern electronic voting systems:

A scheme to break Arrow's axioms I tend to propose (when in discussion on this topic) is to have a system whereby [electronic] voting begins a week before the vote is counted. When you go to place a vote, you are presented with the votes cast to date. Then you vote.

Some will vote early, who have already decided firm who they want. Their advantage in doing so would be to show others that a minor candidate that others might not feel would get any support is getting it. That would encourage the group who would prefer that choice but would otherwise tend to make another choice to avoid their least favoured candidate.

Those who favour some head-strong character might want to try to sneak in at the last minute because if the running result was all biased to some extreme character the next voter in the booth might then choose to vote tactically against him.

I'm not at all sure how this would work out in practice, but having a voting scheme with a running total breaks the axiom of a fixed set of 'decision criteria' in the theorem, because instead you end up with a practically limitless number of non-independent decision criteria according to that of the voter and also of the running state of the vote.
Gokul43201
#27
Mar7-12, 07:47 AM
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Quote Quote by cmb View Post
Their advantage in doing so would be to show others that a minor candidate that others might not feel would get any support is getting it.
How would they show this? The only possibility I can think of is exit polling. Right now, it's possible for say CNN to get good exit polling numbers because they need to hang out at polling stations for only a day. I wonder if they'd still want to do it if they had to spend a whole week camping outside polling places, talking to a much slower trickle of people. I suppose a candidate with a vested interest in these numbers might pay for the service but that could then raise doubts about bias.
cmb
#28
Mar7-12, 08:24 AM
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Quote Quote by Gokul43201 View Post
How would they show this?
It is an idea dependent (actually, inspired by) electronic voting. When the voter is presented with a touch-screen to vote on (on on-line selection), next to each candidate would be the running total to date.
Office_Shredder
#29
Mar7-12, 08:37 AM
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People overrate Arrow's theorem. Consider the idealized one dimensional political axis, where everyone's political position is a number between 0 and 1, and everyone votes for the politician who is closest to them on the scale, and there are two candidates, the median voter is a dictator. Does that mean that majority wins when having two candidates face off is unfair or not ideal? I would guess most people say no
cmb
#30
Mar7-12, 08:40 AM
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Quote Quote by Office_Shredder View Post
People overrate Arrow's theorem. Consider the idealized one dimensional political axis, where everyone's political position is a number between 0 and 1, and everyone votes for the politician who is closest to them on the scale, and there are two candidates...
Then Arrow's theorem doesn't apply. It relates to voting for 3 or more options.
Office_Shredder
#31
Mar7-12, 08:42 AM
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The statement of Arrow's theorem is: if blah blah blah about voting systems, then there is a dictator. People gasp. Oh my god, a dictator!!! They scream. But the system I described has a dictator, and nobody thinks that's a terrible thing.
cmb
#32
Mar7-12, 08:44 AM
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Quote Quote by Office_Shredder View Post
The statement of Arrow's theorem is: if blah blah blah about voting systems, then there is a dictator. People gasp. Oh my god, a dictator!!! They scream. But the system I described has a dictator, and nobody thinks that's a terrible thing.
I think the terrible thing about this is your understanding of Arrow's theorem!


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