In the spirit of the Godel thread, here's another doozy. I brought this up in a OTB thread, but many of those I like to ramble with about this kind of stuff may not read that section.

Like Godel, Arrow's theorem takes the wind out of your sails. In short, it shows that any voting system (with 3 or more candidates or choices) can never be "fair" and consistent.

You specify a number of criteria, rules (axioms or postulates maybe in a loose sense, but rule is better) that any "fair and rational election" should meet, and it seems that any system that violated them would be unfair or irrtational. Well, Arrow proved that set of rules, those criteria are *INCONSISTENT*. No system can meet them all at the same time!

Arrow's Theorem is oft stated in very provocative terms to get your attention, such as "only a dictatorship can be consistent". That has nothing to do with a dictator in the most common sense, but means that one voter of the group always determines the outcome, all others notwithstanding. But, that of course leads to the more common notion of dictator.

This little ditty has a lot of Godel-like qualities. We wonder what would happen if PM were found inconsistent. Well, Arrow gives us a microcosm of that in voting theory. We find that our gut, emotional definition of "fair and rational" are indeed inconsistent, and we're going have to "remove axioms" so to speak if we want to make it consistent.

I can't formally state Arrow's Theorem, as I'd botch it, but if any are interested, just Google, and you'll find discussions about it and the voting criteria it finds to be inconsistent. It takes a while to for it to click, especially if you aren't familiar with the jargon of voting theory.

Arrow is most disconcerting to those with, shall I say, only a naive appreciation of "democratic processes" and voting theory.

But the big picture is simply that only individuals can be consistent in their choices. An individual is a dictator of his own conscience, of course. The notion of a group choice, a consensus, in the sense of the "public wants this" or "the public doesn't like this", etc, etc simply cannot be defined in a logically consistent way.

One of the roots of this can easily be seen. A majority of a group is not a single entity. Different members will be part of different majorities, and majorities can be constructed in different ways accordingly. So it should surprise you that "the majority" can easily contradict itself.

And finally, while no system can be completely fair and consistent, certain systems can be more unfair and more irrational that others. Another disconcerting result of voting theory, coming from some very simple analysis, is that of all the common methods, the simple plurality method is the worst of all.

Most general elections in the US are a simple plurality wins scheme (for reasons of expedience and cost, if nothing else), and many just want to cover their ears and run screaming from the room when confronted with the reality that it is just about the worst voting system to have. It can, with greater than two choices, pick a winner despised by a 2/3 majority of the electorate.

-Richard

Arrow's Theorem and many of the results of voting theory are not very well known. How many of the membership here have heard of it before? Voting theory is sort of a field little known outside of economics and political science and the subset of mathematics that support it.

The result of that ignorance is that people contine to be befuddled and incredulous when voting and ranking systems give paradoxical or insane results, and they don't understand why. But years ago Arrow and voting theory proved that is defect of all voting systems and is to be expected and there's nothing that can be done about it. Irrational results are going to happen.

Let's look at two of Arrow's criteria for a fair, rational system that are pretty big:

1) If every voter ranks A above B, the system will rank A > B

2) If the system ranks A above B, a change of the relative ranking of some third C with one or more voters should not change the system's ranking of A > B. [ETA: I worded that wrong, and corrected it. It is "If the system ranks A above B", not "If every voter ranks A above B"]

Guess what? Those two together are inconsistent (this is part of Arrow). Only a dictator, a single individual can sastify both.

There are some famous examples of violations of #2 causing uproars. One happened in the world figure skating championship back in '95, all of all places. Michelle Kwan finished her run and bumped herself up to 4th place. In doing so, she flipped the ranking of the original 2nd and 3rd place winners who had long since finished their routines.

Everyone was flabbergasted. How can that happen? Something is screwy in the ranking system. If A > B and are finished, how can a change in C result in B > A? It happened again in '97, and the international figure skating authority made changes to the rules to try to fix it.

But as any student of voting theory could've told them, while they may have prevented that "flip-flop paradox" under the specific conditions they analyzed, they didn't fix it in general. Just like the carpet with the pooched up air pocket, if you mash it down, it just pops up somewhere else.


-Richard

Voting is all about letting people you wouldn't let walk your dog pick your leaders. The concept is based on 18th-century ideals about the Innate Nobility And Wisdom Of The Common Man that kinda fell apart once television and the internet showed us what jackasses most of us really are.

__________________
"If this were play'd upon a stage now, I could condemn it as an improbable fiction."
Shakespeare, Twelfth Night
"The Mayan symbol for "book" looks a lot like a triple hamburger, but I've never seen them claiming it as proof the Mayans had Big Macs." - KaiYeves
"Distance doesn’t matter much in space, where if you just start a thing off with the right kind of shove, sooner or later it will get where you want it to go." -Frederik Pohl, Mining the Oort

You may be wondering that, given Arrow, are there any voting systems which are somehow the best, in the sense of the least irrational and most fair.

Well, ironically but naturally obviously, there is no consensus amongst the experts on determining consensus. And consensus amongst those is probably the hardest to achieve of any.

There are two top contenders, the Borda count/method and the "approval method". That argument will get quite heated amongst voting theorists.

One of the top experts in this field, Dr. Donald Saari wrote a 50 page paper on this published in some Economic theory journal back around 2000.

http://en.wikipedia.org/wiki/Donald_G._Saari

Interestingly Saari also dabbles in celestial mechanics and the n-body problem and has done some significant work there.

His 2000 paper has been described as an impressive tour deforce in linear algebra applied to voting theory, a juggernaut of Principia Mathematica both framing and analyzing a problem.

Suppose you have N choices. There are then N! possible rankings of those choices. Saari constructs an N! dimensional vector space, with each permutation being a basis vector. The number of people holding that ranking, that opinion, is then the coordinate along that "direction". The group is then the vector you get from all components. The voting system, the group "choice function" is then one that constructs a single ranking from that vector.

And off he went. He was able to show that Arrow problem, the irrationality, comes about from a certain subspace there, which could be seen as a component orthogonal to the total. Or something like that.

Anyway, the result was the Borda count was the best, and he declares he proved it. No one disputes his math, no error there at all, they just disagree with his definition of "best". Depends on what you mean by best, IOW. Which sounds familiar......

-Richard

Well, as Winston Churchill once quipped, the greatest argument against democracy is a 5 minute conversation with the average voter.

But that is the wrong thing to take from Arrow and voting theory, actually. Yes, some people are idiots and I hate for idiots to vote. But that's not the issue here.

The issue here is while individuals can be rational, the collective cannot. There is no logically consistent way to define a collective opinion. To me, it's just one more feather in the individualist vs collectivist debate.

So, even with a set of rational, intelligent voters, good voters who simply disagree on important questions, any attempt to define a consensus can be irrational. In some cases, it can be rational, but it is not guaranteed.


-Richard

Sounds like Arrow was a man after my own heart.
As I said somewhere else,

Do you know of any country whose voting body are intelligent and rational? And if so do they accept immigrants?

__________________
"If this were play'd upon a stage now, I could condemn it as an improbable fiction."
Shakespeare, Twelfth Night
"The Mayan symbol for "book" looks a lot like a triple hamburger, but I've never seen them claiming it as proof the Mayans had Big Macs." - KaiYeves
"Distance doesn’t matter much in space, where if you just start a thing off with the right kind of shove, sooner or later it will get where you want it to go." -Frederik Pohl, Mining the Oort

__________________
Andre

"They did not know it was impossible, so they did it!"
Mark Twain

There were some very nice articles in the German issue of the Scientific American this year. Only thing I remeber without digging it up is that there was an election somewhere in Switzerland where they used a new method accounting for many of the problems you ususlly have with simple voting systems. Sorry don't remember the details at the moment

__________________
Andre

"They did not know it was impossible, so they did it!"
Mark Twain

It may be informative to look at just what the meaning of "rational" is in the context of voting theory.

We have three choices, three options, candidates, whatever and we dub them A, B, and C. Take your pick of how you rank them. Any permutation of those is possible for an invidual voter. A>B>C, or B>A>C, etc, etc.

Now let's look at binary comparisons. If you favor A over B, and B over C, that implies you *should* favor A over C logically. That would be the ABC ranking. Just do that with your favorite candidates from across the spectrum and you'll starkly see this. But that doesn't have to be if you actually hold the pairwise elections. A voter who would vote for C and A after choosing A over B and B over C is therefore irrational, insane. Voting theory doesn't worry about that so much in the context of rationality of the group. If the individuals are irrational, then all bets are off for the whole. The only question is do you force rationality by you voting your scheme.

Note that individuals can shuffle A, B, and C around all they want and still be pairwise rational.

Now, let's look at a group. Suppose we have three voters and they form a little cyclic permutation of ABC. One voter is ABC, the second is BCA, and the third is CAB. Let's hold the pairwise election for the group.

A vs B:

Voter 1 choose A. Voter 2 chooses B, and voter 3 choose A. A wins this round 2 to 1.

Now B vs C:

Voter 1 chooses B, voter 2 chooses B, and voter 3 choose C. B wins this round 2 to 1.

Note the group has chosen A over B, and B over C.


Finally A vs C:

Voter 1 choose A, voter 2 chooses C, and voter three chooses C. C wins 2 to 1.

The group has chosen C over A! The group is not rational. But note that every individual voter voted rationally. But the group result was irrational. This little conundrum is know as Condorcet's Paradox, and is (one of the) roots of Arrow's evil here.

What would call the results of such an election? There is simply no consensus there. You can call it a "tie", 3-ways if you like, but it's sort of more than that, or less than that, depending on how you view it. Indeed, if you held a three-way vote, the initial ballot would be an exact 3-way tie, with no run off set possible for any majority method.

It turns out those cyclic permutations and those of the "backwards" sequence, ABC vs BAC, are the ones that cause all the problems, causing strange things to happen, even when the numbers of each or not equal. Various "flipping problems", such as the Michelle Kwan thing all come from these cyclic permutations.

-Richard

I used to think that Arrow's theorem is some terribly magic and clever theorem. Indeed the proof I was given of it, in a course of lectures given by Amartya Sen that I was terribly privileged to attend, didn't really illuminate quite how simple it is true.

But in fact it is terribly simple. It all comes down to the Independence of Irrelevant Alternatives (IIR), which is the condition Publius labelled as (2) above. What Publius calls (1) is known as Pareto Efficiency.

What IIR means is that in setting the order between A and B, all information about C, D, E, etc is irrelevant. So the only information we have is how many people prefer A to B, and how many B to A. (The formal version of the theorem also admits the possibility of indifference, ie, some people rate them the same, but let's forget that.) What choices do we have, when that is the only information to go on? If dictatorship is not allowed, then essentially the only option available to us boils down to a majority vote. That then leads us into Condorcet Paradox area, ie, if there are 3 options and we use majority voting to set them in order, we can violate transitivity A>B>C>A, as Publius mentions.

There is a proof that essentially follows that line of argument, but it isn't the simplest proof, as the hand-waving I employed above has to be made precise, which isn't so easy.

If you are interested in the proof, there are three of them in this paper. The last one is the easiest to follow as a proof, though it doesn't really illuminate anything, just succeeds in proving it very economically, only about a page, which I think the numerically literate layman could follow with some effort. The Condorcet route is also given in this paper.
http://web.archive.org/web/200608292...a/d1123-r3.pdf

Just to be clear, Arrow's Theorem only works if there are at least three options. It then says that there is no social welfare function (ie, a "voting function") that satisfies the following: (1) Non-dictatorship (2) Independence of Irrelevant Alternatives and (3) Pareto efficiency.

Arrow originally started looking for social welfare functions that had certain "good" properties, and had a rather longer list of "good" properties than this, which of course soon threw up the impossibility. His insight was in discarding the things that were unnecessary, in other words identifying what weak conditions impossibility arises. His original version of the theorem had a different condition called Monotonicity instead of the Pareto condition, but the Pareto condition is strictly weaker.

Publius also talks about good voting systems. As Arrow proves, in an entirely general situation there is no good voting system, if we desire it to handle any situation. But in the real world we can probably say that we don't need entirely general voting systems.

For example, in the UK it used to be approximately true that there was a party of the left (L), a party of the right (R) and a party of the centre (C).
Therefore, if people were rational, one might presume that the only preferences people could have would be L>C>R , R>L>C, C>L>R and C>R>L, in other words a Lefty would prefer a Centrist to Righty, etc. In this kind of system where peoples preferences can be limited and simplified, then one can devise voting systems that are consistent, and of course it would tend to favour the centre. But in reality it is not so simple. In practice many people in Britain do have preferences such as L>R>C and R>L>C, because the centre party is weak and not seen as a credible governing party by many people. And of course left, centre, right is an over-simplification these days, there are other parties in some regions, etc.

Could you be more specific? What you mean by that?

__________________
"A witty saying proves nothing" Voltaire.
"All your bias are belong to us" Ara Pacis.

It will be late tonight before I have the time, but here's a quick example (which I went through in another thread, but don't cheat ). This is a fairly common voting theory example.

15 people are going to a party and they need to decide what beverage to serve. There are three choices, tea, beer, and wine. They decide to vote:

There are three groups, factions if you will, that have following preferences:

Group A, 6 people: tea, wine, beer

Group B, 5 people: beer, wine, tea

Group C, 4 people: wine, beer, tea

These preferences are "love", "okay with", "hate". Group B and C hate tea. Anything but tea is their mantra.

Now, they hold a vote. Which one wins by plurality? What is the fraction of the total that *hates* that winner?

Now, go to some runoff, majority method. Who wins there? How many people hate the winner there?

Is there a choice there that nobody hates? Did the plurality or even majority method correctly find it?

-Richard

I think it should stressed that L>R>C or R>L>C are not irrational of themselves. An individual could hold that preference and maintain transitivity just fine. Taking the LRC, he would prefer L to R and R to C, L to C and be perfectly consistent.

It is only an external system that orders C in between L and R where that choice would seem irrational. And indeed, such an enforced order would eliminate much of the problem (I don't say all, because I'm not sure).

What's the saying about those who are neither hot nor cold but merely lukewarm? There is something to that -- if you stay in the middle of the road, you tend to get run over. Pick a direction and go. But then on the other hand, in some things lukewarm is what you want.

And yes, imposing a 1D L-C-R ranking is simplified. There are more dimensions that, and in that larger space, two candidates that seem far apart along one axis aren't necessarily far apart in the full space.

Me, I tend to *vote* according to that imposed ordering, but as far as character, who I'd rather hang out with so to speak, I prefer those who have made up their minds and have a clear philosophy, not the lukewarm who can go either way, contradicting themselves in the larger picture.

-Richard

Let's look at this. If you went though it, you'll find that our little drink election went the following:

Tea: 6; Beer 5; Wine 4.

By a plurality system, tea wins and so tea will be served. But look at the rankings. Groups B + C both *hate* tea. Together, they are 9 total, which is 60% of the total 15 people. Note we can say the plurality system choose Tea>Beer>Wine as the ordering of the group.

But it just picked a winner who is despised by a 60% majority of the voters. The anti-tea vote was simply split. This 60% raises a ruckus. They can't stand tea, they represent a huge majority and they're going to take their marbles and go home.

So, hey, let's go to a majority system and have runoff. There are all sorts of ways to this, but with just three choices, they all collapse to the same to the same thing. So we'll just say it's the exhaustive method, low man out each round.

With that, wine is the low one out, and it is elimated. The run off is between tea and beer. Group C throws their vote to beer, their second choice and so the result is:

Beer 9; Tea 6.

Beer wins with a 60% majority. The winner now has a 60% majority who either prefer or don't mind it. Beer must be the consensus, no? So, we can say our majority system said Beer>Tea>Wine was the group ordering, flipping beer and tea.

But note that while a 60% majority is at least okay with our winner, 40% still despise it.

Look at the actual rankings of the groups. Wine is actually the true consensus. All 15, 100% either prefer or are okay with wine! Nobody hates wine. Both systems, the plurality *and* the majority failed to find the true consensus of the group.

Is their a system that would correctly find wine? And the "correct" group ordering is Wine>Beer>Tea. Is their a system that would find that?

There is in this case. Do the Condorcet method, holding all pairwise elections, Wine vs Beer, Beer vs Tea, and Tea vs Wine. You will find that wine wins all of its pairings, Beer wins one but looses the other, and tea looses both. Wine is the "Condorcet winner", beer is in between, and tea is the "Condorcet loser".

Condorcet correctly found the choice that nobody despised and found the correct ordering of the group.

But note how awful the plurality method was here. The actual preference of the group got the least number of votes, and the one hated the most by the group got the most! It turned it upside down.

Put that in your pipe and smoke it a while. This demonstrates why the plurality system is the worst of all the common methods.

-Richard

The system won't let me edit again, and I see I made a mess with "their" vs "there" (like I do with it's and its). That drives me nuts and I know better -- it's just my brain does something funny there getting the signals to the typing fingers.

-----------------------------------

But back to our subject.

As you can see from that drink election example, the plurality system, with 3 or more choices that are evenly divided can really screw up. Here, the issue was something inocuous, what to serve at a party, but for the really important things where passions are inflamed (and rightly so), it will easily lead to instability. I mean, if your winner is hated by large majorities, things just aren't going to go well.

And there is the root of something. In the real world, various sides get information about how the whole will probably vote. That information is certainly not complete, but it's there.

So groups B and C learn that A, the tea people will probably win. It is then advantageous for them to get together and agree to vote for a common to avoid splitting their vote. If they got together, and held a preliminary election to decide who they were all going to vote for in the actual election, beer would win 5 to 4, and so beer would win the final election 9 to 6. The effect would be the same as the majority run off.

And thus political parties are born. Preliminary election = primary. But note that fails to see the tea folks have something in common there, wine, which everyone would be fine with.....

It is oft stated the two party system (which pretty much forces a binary choice in the general election) is the reason for (US at least) stability. But note the argument is strong, very strong, that the root source of instability is the stinking plurality system in the first place, and the two party system is the stabilizing response to that.

The anti-tea crowd getting together like this is a simple example of what is called "tactical voting" in voting theory. This means, in a nutshell, voting in some way other way, dishonestly in some cases, that how you really want to in order to better you own chances.

In the beverage race, the anti-tea folks getting together is not a bad thing, as the result is winner with majority support. It still doesn't find the best result, but it beats a minority position winning.

Bad tactical voting is when one side uses it thwart the majority and get a minority candidate in. A very common example is cross-over primary voting. One party sneaks into the other's primary and voters for an extreme or otherwise weak candidate so their own candidate will win in the general election. There are many examples of this, the California governor's race several years back being a big example. That let a very unpopular governor win relection which ultimately led to that big recall mess.

All systems will have tactical voting vulnerabilies, although they get complex and subtle sometimes. It's Condorcet's paradox and IIR violation at work. If you know how to game IIR, you can pull some stunts.

Ironically, it's pre-election information that can be used to inform these tactical games. And that information can be good as well. But it can be bad. All in all, and for other reasons, that's why I'm against polls. I urge everyone to hold how they will vote close to the vest. If political operatives had no clue how things were going, well, you'd see them peeing their pants. It would be a hoot.

-Richard

In many countries the publishing of polls is not allowed on the days, sometimes even weeks, leading up to the election.
This is very hard to enforce though.
In the US there might even be constitutional issues.

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The whole business reminds me a lot of the principle of sailing into the wind. It seems "rational" that a wind must push a boat in a direction that is at less than a right angle to the direction of the wind, not more. But even a flat board nailed onto a boat can easily sail into the wind-- the trick is that you have two, not one, processes involved in determining the force. One is that the sail picks out the component of the wind that is perpendicular to the sail. By itself, that process could never yield an "into the wind" force. But then the boat has a keel, which picks out whatever component of the sail-force that is perpendicular to the keel. By doing this "perpendicular component" analysis twice, one gets a counterintuitive force into the wind. It's like doing two elections, A vs. B and B vs. C, and being surprised when one simply compares to A vs. C. But note there is nothing irrational in sailing, so that term may be the real source of the problem.

It is interesting that some voting systems, especially those associated with sports (not normally considered a haven of rational thought), have found good solutions. For example, when coaches vote for who they think are the top teams, they are asked to rank all the teams, and each team is given some number of points based on that ranking. That's quite logical to do, rather than counting plurality. After all, the most absurd of all possible systems would be to take the most frequent ranking as the chosen one, because in any situation where the possible rankings are far more numerous than the voters, it is likely that most rankings will appear but once, so if by pure chance any are repeated, no matter how irrational they seem compared to the rest, that could be the most frequent ranking. Coaches know that would be a bad way to go, and they are hardly number theorists! Would the results of primaries in the U.S., for example, be different if voters were asked to rank their choices the way the coaches do? It can occasionally happen (though I'm not sure I recall it) that the number one ranked team does not have the plurality of number one votes.

Sounds like Instant Runoff Voting, Ken G. Something I'm quite in favor of. I'd like to hear how badly it does, in terms of the problems publius described above...

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"What do you care what other people think?" -- Richard Feynman
"For a successful technology, reality must take precedence over public relations, for nature cannot be fooled." -- Feynman, at the conclusion of his Challenger report

As I understand pluralities as it works in Europe, when parties tally their totals between however many groups there are, some level of coalition building is done to achieve actual majority and control.

While it has drawbacks, that forced consensus can prevent runaway freight trains that have happened in the US when one of two parties achieves legislative majority and executive control.

__________________
The last time I felt a warm fuzzy feeling, I was informed by my doctor that it was just gas.

I like instant runoff as well (for general elections). And I'll point that what is called instant runoff is just one way among many to do a "find the majority" process. Instant run off usually means and exhaustive, low man out process. Tally up the first place votes, then eliminate the low man and distribute the second place votes amongst the remaining set, then tally again. Repeat until a majority is found. Ties and how to handle them get interesting there.

However, as you'll see in the drink election above, the instant runoff still not find the Condorcet winner. But it did better than a straight plurality.

The Borda method is another way to do it. I think, but am not certain that it will always return the Condorcet winner if it exists.

THe Bordo count is basically have voters rank the candidates or options. If there are N candidates, you give the the first choice N-1 points, second N - 2, etc. Now, just add up all the points. Borda winner has the most points. There are different ways to do the points, but they are all equivalent. Question is what to do for voters who don't want to rank all of them. Tactical voting comes into play there, as well as voters who just don't care at all for some candidates.

Anyway, the instant runoff and other majority methods are basically "find the majority" if it is there. And which you majority you find depends on how you do the counting. But you do find a majority.

Borda and Condorcet are about greatest consensus/least discontent. IOW, it finds the candidate least despised so to speak, not a candidate who simply has a majority.

That is good for things like what drink to serve at a party. It's not so great for solutions to tough problems, as it tends to favor sort "do nothing and hope the problem goes away".

There are diseases where the medicine is rough. You can have two factions who are willing to take the bitter pill, but disagree on which one to take. And then there is the muddled middle.

In that case, the majority methods will tend to find a majority for doing something at some point, rather that just sitting around and doing nothing.

-Richard

That's certainly one example where the plurality rule gives cause to ponder, but I still don't understand why you say that it's the worst voting method of all. Under all circumstances?

__________________
"A witty saying proves nothing" Voltaire.
"All your bias are belong to us" Ara Pacis.

The winner of a plurality election is enfranchised to power by a relative minority of the population. In a democratic system established on the premise of majority rule, this is a clear contradiction.

__________________
The last time I felt a warm fuzzy feeling, I was informed by my doctor that it was just gas.

All the other minorities were smaller.

__________________
"A witty saying proves nothing" Voltaire.
"All your bias are belong to us" Ara Pacis.

Well, it's not just me. If you'll allow me a little appeal to authority, while voting theorists certainly argue about the best possible system (in light of Arrow), they all agree the plurality system is the worst, and they certainly look at all sorts of circumstances.

All systems can pull some strange tricks, and if you get into voting theory, you'll see complex examples involving all sorts of voter profiles that will make your head spin trying to keep track of it.

Ah, there is the problem, and it's a common one. A minority is a minority. More are for something else that are for it. The best way to state a majority is "more for that against".

With only two choices, a plurality is a majority and no problem. However, with three or more (and the more, the worse), top choice alone is a poor measure indeed. Yes, more people have a given top choice than any other, but no matter how you slice it, *more people are against that choice than for*.

The question is, how much against it are they compared to the others when their top choice is eliminated. The Condorcet method is the way to see this.

Consider if you had a very large field of candidates, say you just turned loose all the primary contenders from each party and anyone else that wanted to join in into the general election.

You might have 8 candidates or more running. Say you had 10 just to keep it simple. A candidate with just over 10% of the vote could win. And that candidate could be a extremist despised by nearly 90% of the rest of the electorate.

That has happened, France being an example a number of years ago. They have a top-two runoff. That demands a majority but picks the top two from the initial ballot, no matter if they total a majority between them or not. That let a extremist with 17% of the vote get into the final.

A plurality system alone will lead to chaos for this reason. The two party system comes about because of it. It is a natural response to prevent candidates whom a majority despises, hates, from being elected.


-Richard

I know very little about voting theory (a fascinating topic), but the little I remembered was that all the basic methods had more or less the same amount of qualities and flaws, and none of them was optimal...

You're going to have to explain that last remark about the two-party system!

__________________
"A witty saying proves nothing" Voltaire.
"All your bias are belong to us" Ara Pacis.

In light of Arrow, there is no perfect system, no perfect "decision alogorithm" possible. And, in the strict sense they suffer from the same problems, such as IIR violation.

However, how *badly* they can foul up, and what it the circumstances and probablities for those circumstances in which they can screw up is a different question indeed. By that measure, the plurality system just stinks.

You could come up with something even worse if you worked at it, I suppose (pick the Condorcet loser always, maybe ), but of all the ones commonly used, plurality is the worst.

Now, about the two-party system. Consider a general electrion free for all, I mentioned above. Everbody is in there, extremists, single-issue types, mainstreamers of all stripes, and just plain nutjobs and loose cannons. They're all on the ballot and the one with most votes is going to win.

Of that gaggle, there are five serious candidates you could support, and all the rest are opposite to you, or just plain wackos. Which one do you vote for?


There are bunch of other people who generally be pretty close to you, and would have a set of five themselves just about the same as yours. But if you all split your vote amongst them, one of the others may win.

And there is generally another bunch of people with another five of different polarity. The others are the extremists or nutjobs, with the single issue one-note Johnnies in there as well.

It would be chaos. And so those two bunches of people get together, decide which of the five they will all support so as not to split their vote.

That is a political party. And the result is to force only two choices in the general election, and make the plurality produce a majority (or close to it).

Any third party is seen by both sides as a potential spoiler, taking votes away from their side and allowing the other to win. So, through the levers of power when they get it, they make it as hard as possible for any spoilers to get on the ballot.

Adopt a (good) runoff system, and you take the spoiler pressure away. A "third way" will not be a threat, and it could actually flourish, growing over time.

-Richard

And anyone in the U.S. will recall that the presence of Ralph Nader on the ballot in 2000 is the entire reason that Bush has been president for 8 years, pure and simple. A runoff system and it's Gore-- no Iraq, more responsible carbon emissions policies, and no Nobel prize. Choose your poison, but instant runoffs are looking good to me on balance.

By the way, similar issues come up in other kinds of sports settings-- the tiebreakers when more than two teams end up tied in the standings, for example. If there are just two teams, it is simple to look at "head to head" results, but if there are three, you can get weird results. Often you might look at the sum total of all the scores involving the three teams, and whoever comes out the best wins. But that can put you in a situation where team A beat both teams B and C, but B really clobbered C so ends up ahead in the total score. So usually you only look at the records between the three teams and only go to the total score if the records are all even, but what if two teams still end up tied that didn't play each other at all? There's never a perfect system.

Also, I'm not sure there is really an "individual vs. group" distinction that matters all that much, in terms of identifying what is considered "rational". I see these problems as being fundamental, and not restricted to groups-- there are similar issues that happen for an individual who is weighing multiple criteria for their vote. Also, it's not clear that there's any irrationality involved, it's just the nature of the beast.

For example, an individual might be trying to decide what movie to go see, among movies A, B, and C. It would not be hard to come up with a scenario where if movie C was not showing then he would pick A over B, if B were not showing he'd pick C over A, and if A were not showing he'd pick B over C. It all depends on the criteria. Let's say there are three criteria: cost, showtime, and desire. Maybe in terms of which movie he wants to see the most, it goes A, B, C. But he doesn't want to stay out late, and the order of earliest showtimes are B, C, A. Then the cheapest goes C, A, B. He must weigh all these factors, and there's no clear winner. But if A wasn't showing, he says B wins two out of three criteria so goes to B over C. If B wasn't showing, he notes C wins two out of three so goes to C over A. If C wasn't showing, he notes that A wins two out of three so goes to A over B. So it's just like the group result, but it's all in the mind of one individual, and there's nothing irrational about it.

Indeed, I might imagine a martial arts movie on this premise. You have three masters vying to be the last survivor in some martial arts battle. The three key factors in personal victory are skill, speed, and strength, let's say. By skill, the ranking is A, B, C. By speed, it's B, C, A. And by strength, it's C, A, B. Now let's say A knows that his great skill and reasonable strength will win out over B's speed advantage, so he can beat B, but his skill advantage will not be enough to triumph over C's better speed and greater strength. So he connives to insure that C will first face B, whereupon B's skill and speed advantage will triumph over C's great strength-- leaving B as easy pickings for A. Irrational? Not necessarily.

I was just reading some of this stuff again. The Borda count is *NOT* Condorcet compliant, as I said I thought was the case above. Borda favors "broad consensus" (if it's strong enough) over the Condorcet winner. In many cases Borda will return Condorcet, but not all. There is a variation of Borda that makes it choose the Condorcet winner if one exists, but otherwise returns a normal Borda winner (if it exists). And then variations that do something else. The "do something else" is where all the fun is.

For those interested in this, and I hope many of you are, I would strongly encourage you to acquaint yourself with voting theory. It can get quite involved (it's a been a while since I was into it, but I remember it) and quite fun. If you have a mathematical bent, it can be really rewarding. Framing it can be as a big an issue and the analysis itself.

Anyway, you can weigh all the various methods in your own balances, seeing all the convoluted circumstances that make them do different things, and decide which craziness you're willing to accept and which you're not.

And with all of them, sometimes there can be hard deadlocks for which no resolution is really possible. It can truly boil down to "flip a coin" as the only fair option.

But at any rate, here is the rub. Once you chosen a voting system, you want to execute it automatically. No chance for any monkey business or second guessing if things didn't go the way you hoped. Ponder it, argue over it all you want, but once you make a choice, go with it (and of course, change it later if you don't like it for future elections).

As Doris Day said, "Que sara, sara. Whatever will be, will be. The future's not ours to see. Que sara, sara. What will be, will be."

As we've seen of late, in a 50-50 country, when the margin of victory is within the margin of error, things go to hell. So, as a practical matter, make that margin of error go away by simply defining it away, and have hard procedures decided on beforehand and executed without question. Electronic voting can do that by forcing a voter to make a valid ballot (note he could still screw up and not vote the way he really wanted to, but you wouldn't know it and thus give lawyers wiggle room after the fact). But then we've got people complaining about that, fearing some conspiracy to hack the results by nefarious forces.... :sigh:

-Richard

Ken,

You'd make a good voting theorist. Condorcet's cyclic conundrum is the one that gets your attention.

Here's how I see this. You applied the "IIR" thing to an individual's ranking. Remove an "irrelevant" C and change the ranking of A and B. But how did you do it. By giving three criteria, weighted equally. And same thing with the martial arts masters. Which order you do the contest eliminates one of the contenders -- the three are equal in the whole.

What you've done with the movie example is create a deadlock between A, B, and C. By the criteria, A, B and C are all equal. The individual has no preference by his own criteria. Take one away though, and that collapses. But given all three, it's "flip a coin". Or "make up your mind". In doing the latter, you will either do the mental equivalent of flipping a coin, or change the relative weight of time, cost, and story preference.

And that's what it is with a group in the cyclic deadlock. There is no preference. Granted, the group could flip a coin, or minds could be changed, too. But they'd have to agree to do that (how ).

So maybe "irrational" is not the correct pedagogy as you like to say. *Deadlock*, no real preference, go any way depending on which way the wind blows, maybe. But an individual is the dictator of which way his own wind blows to break it. If he wants to.


-Richard