kevin_standlee | Range Voting

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Range Voting

The folks advocating Range Voting contacted WSFS (actually, the WSFS webmaster,

sfrose) lobbying WSFS to change its voting system from the Instant Runoff Voting system we currently use for site selection and the Hugo Awards. Sharon told them how our rules work and suggested that if they want to change them, they come to WSFS business meetings and propose and debate the changes there, like all other rule changes. The advocate's response, in my opinion, amounted to, "Our proposal is so obviously Right that we shouldn't have to do all that hard, expensive work. You should change your rules because we tell you to do so."

I often tell people who come to me with rules-change proposals, "If you think it's worthwhile, come and submit it yourself. I'll help you with all of the technicalities to the best of my ability, but you have to make your own case, lobby people yourself, and get the votes by convincing people." Most of the time, this discourages them -- democracy is hard work! But sometimes we get people who are willing to work and debate, and sometimes we even get workable changes and improvements.

WSFS rules are intentionally designed to be resistant to change; however, they can be changed if people work hard enough at it. But it's not enough to just lobby a Board of Directors or subvert the Chairman; you have to convince the members.

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Sure, make the switch process difficult, like how Congress has to have a 2/3 super-majority to overrule a veto. There's no reason why, once properly educated, 100% of your group shouldn't want to switch from IRV to Range Voting. It is considerably, objectively, better than IRV. Consider these voter satisfaction ratios (expressions of social utility efficiency):

Utility measurements: Group A: 5 candidates, 20 voters, random utilities; Each entry averages the results from 4,000,000 simulated elections. Group B: 5 candidates, 50 voters, utilities based on 2 issues, each entry averages the results from 2,222,222 simulated elections.
Voting system VSR A VSR B
Magically elect optimum winner 100.00% 100.00%
Range (honest voters) 96.71% 94.66%
Borda (honest voters) 91.31% 89.97%
Approval (honest voters) 86.30% 83.53%
Condorcet-LR (honest voters) 85.19% 85.43%
Range & Approval (strategic exaggerating voters) 78.99% 77.01%
IRV (honest voters) 78.49% 76.32%
Plurality (honest voters) 67.63% 62.29%
Borda (strategic exaggerating voters) 53.26% 51.78%
Condorcet-LR (strategic exaggerating voters) 42.56% 41.31%
IRV (strategic exaggerating voters) 39.07% 39.21%
Plurality (strategic voters) 39.07% 39.21%
Elect random winner 0.00% 0.00%

Well that would totally convince me to support Range Voting, if the voters were computer simulations rather than actual people. Figuring out how to express my opinions through voting is hard enough without adding additional unnecessary complexity. Especially since (correct me if I'm wrong here Kev) we use paper ballots. It will be impossible to tally by hand if we use Range Voting.

My second point is that if it actually works then we'll shortly have evidence of it working for other groups. Then we can decide based upon real world evidence rather than theoretical simulations.

Well that would totally convince me to support Range Voting, if the voters were computer simulations rather than actual people.

And if I trusted electronic voting machines. (Diebold common stock does not have a niche in my portfolio.)

Ron Rivest (the "R" in RSA) actually invented a method that is not all that complex, that allows us to use paper ballots, and have reasonable certainty that no fraud has occurred in counting them, yet not be able to prove to a third party (as in, another person) how we voted.

It can be easily applied to Range Voting, and even to plurality voting with a little work, but NOT to IRV.

See: http://rangevoting.org/Rivest3B.html

In a nutshell, suppose the range of your Range Voting ballot is from -5 to +5. You get three ballots, and the sum of your scores for any one candidate is your score for that candidate (so obviously the scores can't sum to be higher than 5 or lower than -5, and a dumb machine verfies that). Then you get a certified copy of ONE of those ballots, and drop your ballots into a bin. Later, the results of EVERY ballot are publicly posted. You check that your ballot (the one you got to keep) matches what is publicly posted. Just having 1/3 of your full ballot means you can't prove how you voted, so you can't sell your vote. But if fraudsters try to cheat the system, and edit a ballot, there's a 1/3 chance they edit a ballot-third that someone took home a copy of, meaning they get caught. If they alter TWO ballots, there's a 5/9 chance the fraud is detectable. After a certain point, say 10 ballots, the odds of fraud being detectable, and with PROOF in hand (Hello, Mr. Dan Rather? I have a ballot here which doesn't match the publicly posted result) approach the mid 90's. Go up to a more significant number, and forget it. Fraudsters can't get away with it.

This is what people who are obsessed with voting methods cook up for fun.

CLAY

The fact that you can't figure out how to apply it to IRV doesn't mean that nobody can. There are truly horrible ways (that take more than 3 ballots per voter) that can easily be shown to work for IRV; I haven't tried to minimize the excess.

A Princeton math Ph.D, who incidentally told Rivest how to use this method for plurality voting, says:

Interestingly, 3ballot turns out to work most naturally, securely, and simply, for approval voting and range voting. It still works – but less naturally, securely, and simply for plurality voting (the kind of voting currently most common throughout the USA and world) – and it essentially does not work at all for voting methods based on rank-order ballots such as instant runoff voting.

So, let's see you refute him please. Prove that it's possible. I'm pretty sure he's proven that it's not.

sethb has a Yale math/computer science Ph.D, he also holds a BA and MA in math from Harvard... BFD about the Princeton math Ph.D. One of my bosses in the Air Force was a Princeton Ph.D. in EE, who used to disappear at lunch hour to teach at Northrup University where he was the department head (and yes, he was also simultaneously an active duty lieutenant colonel in the USAF at LA Air Force Station).

Yeah, there are a lot of smart people out there. If Seth has the credentials you say he does, he should be able to analyze our claims and find flaws in them if those flaws exist. If he can find any methodological errors in our utility efficiency calculations, for example, is is welcome to point them out, or even work on our OPEN SOURCE utility efficiency calculation program, which presents the results in terms of Bayesian regret I should add.

Maybe he could solve some of our unsolved puzzles http://www.rangevoting.org/PuzzlePage.html

> Well that would totally convince me to support Range Voting, if the voters were computer simulations rather than actual people.

The computer simulations model real human behavior. 720 parameterized models were used, in which voters ranged from 100% strategic to 100% honest, and from 100% ignorant to 100% informed. Somewhere within that Range must be the same percentage of real people who vote strategically vs. honestly. Somewhere within that Range must be a level of ignorance that is approximately equal to the real amount of ignorance in society. Range Voting bested the other methods in ALL of the models.

Let's take a sample scenario. 3 voters, with the following utility values for candidates A, B, and C.

A, B, C
V1 -4, 1, 2
V2 -3, 3, 4
V3 -2, 5, -6
AVE: -3, 3, 0

Now in a real-life scenario, we KNOW what these voters would choose using plurality voting. V1 and V2 would choose candidate C, and V3 would choose candidate B. Candidate C would win. This would produce a utility score of 0, which when rescaled to a random-winner=0, ideal-winner=1 scale, would produce 0 / (3-0) = 0%. That's the social utility from this one simulation. Now do MILLIONS of them, and you start getting meaningful numbers. Calculating the values under honesty is a piece of cake. It's devising realistic strategies that gets complicated. That's why lots of strategies have been tried, in the effort to use strategies that model what humans would really do. In a plurality election, the best strategy is to vote for the person you like more among the two perceived front-runners. You can model that effect in digital elections pretty easily. In general, the same strategy applies to IRV as well, which is why IRV degrades to being as bad as plurality the more strategic voters get.

Now, if you aren't happy with the strategies we've employed, and think that the ENORMOUS disparity between Range Voting and IRV can be overcome by simply correcting some of our methodological errors, then please be my guest and cite our errors, and even get the credit you deserve for improving our simulation. But until you find methodological flaws, your objection amounts to nothing but incredulity. "I don't believe your simulations could model reality very well, therefore your results are wrong." Well PROVE it. Engage in science, not superstition.

And read this: http://rangevoting.org/WhyNoHumans.html

> Figuring out how to express my opinions through voting is hard enough without adding additional unnecessary complexity. Especially since (correct me if I'm wrong here Kev) we use paper ballots. It will be impossible to tally by hand if we use Range Voting.

Range Voting is orders of magnitude easier to count, since you just sum up the votes in a single round, where precinct totals can be sent up. It is also consistent, meaning that if two precincts both elect A the winner, A will still be the winner when you merge their ballots. No the case with IRV. With IRV, A can win in two states, yet when you sum the ballots together, A can lose. Sounds nuts, huh? Yeah, IRV _is_ nuts.

And with Range Voting, you can use ordinary dumb plurality voting machines, instead of having to get special multi-round IRV machines, or (ugh) do IRV's multiple rounds by HAND.

See: http://rangevoting.org/Complexity.html

Get this nonsense that Range Voting is complicated out of your head. It's way simpler than IRV.

>My second point is that if it actually works then we'll shortly have evidence of it working for other groups. Then we can decide based upon real world evidence rather than theoretical simulations.

The theoretical simulations prove it VASTLY better than any real world simulations, since you can't read people's minds, but you CAN read the minds of digital voters. There will NEVER be a real world test of voting systems that can compare to the quality of computer simulations, at least for calculating social utility efficiency. As for judging the easy of implementing them, and having real world voters use them, that does require real world tests. Millions of people score things online all the time. That's a great science experiment that proves our point. Not to mention my exit polling in Texas. http://RangeVoting.org/Beaumont.html

What the computer simulations do not (so far as I can tell) do is model the fact that in reality, people's preferences are correlated in strange and wondrous and extremely non-random ways.

Well, they are ALL random. It is RANDOM that I got the genes I got that shaped my personality the way they did. It is RANDOM that I happened to have the upbringing I got. Everything that affects my preferences is random.

Also, understand that utilities in these simulations were based both on randomness, AND on random positions on ISSUES. That is, take a plane (two-dimensional issue space) and put some dots on it to represent candidates, as well as lots of dots to represent voters of various stripes (one axis can be stance on abortion, and perhaps another can be position on taxes...the possibilities are endless, and you can use as many dimensions as you want). Now assign the utilities based on ideological difference (or "distance" in the graphical sense). What we find is that the results from using extremely realistic distributions of ideology, even based on real results of large-scale politics quizzes (with tens of thousands of participants) don't vary much from just picking random utilities. Hence your point has been fairly soundly refuted. But just to be on the safe side, these utility calculations were done with both random utilities, and utilities based on positions on various issues.

So your comment doesn't really make any sense upon closer inspection.

But I welcome any more articulate criticism you may have.

IRV contradicts itself (best is worst?) and other disasters
#voters their vote
9 B>C>A
8 A>B>C
7 C>A>B

In this 24-voter IRV election, A wins after C is dropped. But now suppose every voter reverses his preference order (i.e. they are now attempting to choose the worst candidate rather than the best). In that case A still wins after B is eliminated. I.e. IRV contradicts itself; IRV's unambiguously "best" candidate A is here the same as its "worst"! (Such embarrassing winner=loser reversal failures are, however, never exhibited in range, approval, Borda, plurality, Tideman Ranked Pairs, or Schulze beatpath voting, all of which are logically self-consistent in this respect.)

This failure is really just a mere symptom of the flawed logic underlying IRV.

This example also illustrates a bizarre kind of strategic voting. Suppose 3 of the B>C>A voters reverse their votes to A>C>B. In that case B is eliminated whereupon C wins 13-to-11 over A. The reversal's raising of A from bottom-to-top in their vote caused A to lose – and voting maximally dishonestly as though they were suicidally trying to elect the worst candidate, was actually optimal strategy that caused the election result to improve from their point of view.

Finally, this election also illustrates a "no show paradox": If those three B>C>A voters had simply refused to vote, then C would have won, an improvement in their view. Different way of saying the same thing: these three voters' decision to cast an honest A-last vote caused A to win.

That's an argument for Condorcet. Condorcet solves that problem without changing the voting method. Range voting introduces vastly uncontrollable new variables.

Well, no, it's an argument _against_ IRV. Both Condorcet and Range Voting solve problems like this, so you can't just say that it's an argument "for Condorcet". And Condorcet cyclic ambiguities show that Condorcet is deeply flawed. For instance, consider this scenario:

35 A > B > C
33 B > C > A
32 C > A > B

Who wins this election? Perhaps your Condorcet method of choice picks A. Now, if B drops out of the race, but not a single voter's preferences change, C beats A by a huge 65-35 margin. Hence the utter broken-ness of the "majoritarian" idea. We propose something better, the _utilitarian_ idea. The most good for the most people. And if you know even the most basic economics, you want the highest expected value. Range Voting gives that to you. Condorcet, with such paradoxical behaviors, often doesn't. And under strategy, it falls apart, whereas Range Voting degrades, under total strategy, approximately to Condorcet. That is, Range Voting at its WORST is about the same as Condorcet at its best.

See the DH3 pathology: http://www.rangevoting.org/DH3.html

Why Range Voting is better than Condorcet:
http://www.rangevoting.org/CondorcetExec.html

Range Voting kind of IS a Condorcet method
http://rangevoting.org/AppCW.html

I began my trip into the election methods debate as a Condorcet enthusiast. It didn't take long before I saw the light. I encourage you to study the facts objectively, and make the same move.

Clay

An argument against one system is meaningless unless it's an argument for a better one. (Remember Churchill saying that "Democracy is the worst form of government, except for all the others"?)

You acknowledge that both Condorcet and range voting solve the particular problem you described earlier. Therefore your criticism of IRV is an argument for Condorcet for the reason I stated: that it solves the problem without introducing new variables which might create new problems. And the uncontrolled point system of range voting emphatically does.

So, Range Voting and Condorcet best IRV. Then if we compare Range Voting to Condorcet, it beats Condorcet, so...

If range voting is kind of a Condorcet method - and I can see that - then I definitely prefer Condorcet.

By the way: you write,

I began my trip into the election methods debate as a Condorcet enthusiast. It didn't take long before I saw the light.

Do you have any idea how much that makes you sound like a Scientologist? We in science fiction fandom have good and long reason to be wary of that kind of starry-eyed insistence that perfection has been found, and that accounts for a lot of the immediate bristling your posts have been generating. (Have you ever heard of a fellow named Claude Degler?)

> If range voting is kind of a Condorcet method - and I can see that - then I definitely prefer Condorcet.

Well, you actually do not prefer Condorcet. You'll be less happy with the results of elections if you use Condorcet. There is of course the off chance that you'll say, "Man, I loved voting with Condorcet so much more than voting with Range Voting, that it's worth getting a crappy president", but then you'd be pretty abnormal. If you just want a voting method to be fun, regardless of its quality, then I assure you, I can think of some fun ways to vote. Throwing darts at balloons on a wall, with the names of candidates inside is fun. You should vote like that.

> Do you have any idea how much that makes you sound like a Scientologist?

How ironic, considering how much time I've spent heckling Scientologists, and going into their little tents and causing a scene. I actually included Xenu in my Range Voting poll for the 2008 US Presidential election, and he's coming in last place. http://zohopolls.com/us/pres

> We in science fiction fandom have good and long reason to be wary of that kind of starry-eyed insistence that perfection has been found

I specifically showed an example scenario of an election where Range Voting would produce a very poor winner. So clearly I know it's not perfect. It's just a huge improvement over IRV. You are making a straw man argument here.

I suggest you read a book by Carl Sagan called "Demon-Haunted World: Using Science as a Candle in the Dark", if you like science as much as science fiction. I take Sagan's view (that seems to sort of be shared by my personal hero, Richard Dawkins) that irrational beliefs aren't just sort of mildly annoying, but dangerous...pernicious. I therefore care less about whether you actually USE Range Voting, and more about whether you are educated about it, such that you make the correct conclusion, that it is the best common voting method (I say "common" because we are excluding insanely complicated methods like CTT voting, that some genius economists created).

It's like when I was in 6th grade, and my teacher taught us there was an "Antarctic Ocean" in addition the four "real" oceans. Now, of course the concept of distinct oceans is arbitrary, since they have no physical boundaries. But I knew that the mainstream consensus was, Actic, Pacific, Atlantic, Indian. FOUR oceans. I told my teacher he was wrong. Everyone mocked me. I could have avoided that reaction by just not correcting him. But no way was I going to let that go, because then he'd just go around believing that, carrying around that false meme in his head. So I came the next day with the encyclopedia and proved him wrong. I fixed the broken-ness. It's the same reason I'm trying to get all my friends to use vorbis instead of mp3, and the same reason I use dvorak, and the same reason I'm trying to get you to use a better voting method. To fix things that are incorrect. Maybe you're not neurotic enough to relate. Shrug.

Sir, I will get a crappy president if a plain majority of the voters prefer a crappy president, no matter what the voting method. Getting a crappy president has nothing to do with the matter.

Nor has making the voting method "fun" have anything to do with the matter. Making it comprehensible and gaining the trust of the voters does. If we have nothing better than your arguments to offer, we will never gain the trust of these particular set of voters with range voting.

You seem to be mistaking science fiction fans for the kinds of soulless automatons who would apply scientific principles to every area of their lives. Convention running is not a science, but a combination of science and art. We must often balance what is theoretically most preferable with what is practically feasible to do, with what people want, and what they find easiest to do and to understand. That was the first lesson taught me when I went to work for a political campaign, and it applies just as well here. The theoretical, and if present the quite marginal, greater utility of range voting does not overcome these other factors, and it most emphatically does not overcome practical concerns about its utility.

This is especially true if, as you are saying, range voting is not perfect. And my point that you are calling "straw man" is not that you literally said it was perfect, it's that you are talking as if you think it is perfect.

Sir, I will get a crappy president if a plain majority of the voters prefer a crappy president, no matter what the voting method. Getting a crappy president has nothing to do with the matter.

Say I offered you a 1 in 50 chance of getting 200 dollars, or a 1/500 chance of getting 4000$...which would you take? With the former, your expected value is 4$. With the latter it's 8$. Either way, that's barely enough to buy you a jumbo soda for lunch. Both options are crappy. But one is LESS crappy.

Nor has making the voting method "fun" have anything to do with the matter. Making it comprehensible and gaining the trust of the voters does.

Whether voters trust it or not has nothing to do with whether it is actually the best voting method. You might try to explain to someone with Down's syndrome that his expected value with one game at the casino was much higher than with another. He might not understand your proof of that, but you'd still be right. He'd still be better off playing the game with the higher expected value, whether he had confidence in that or not.

If we have nothing better than your arguments to offer, we will never gain the trust of these particular set of voters with range voting.

My arguments are substantial. RV is monotonic; IRV is not. RV passes the independence of irrelevant alternatives; IRV does not. RV passes the favorite-betrayal-lesser-evil criterion (an honest vote for Nader is NEVER a vote for Bush); NO ordinal system passes this. I'm at a loss for what you mean when you say, "If that's all you have to offer".

You seem to be mistaking science fiction fans for the kinds of soulless automatons who would apply scientific principles to every area of their lives.

Why do you have to be soul-less in order to be rational in every aspect of your life? I pour my soul into my music, yet I always adhere to rationalism. Why must we ever substitute superstition for science in any endeavor?

Convention running is not a science, but a combination of science and art.

If your goal is to please the attendees, then all the art really is science. It's the application of your data about what pleases humans, in order to accomplish the goal of creating the greatest utility for them and/or for yourselves.

We must often balance what is theoretically most preferable with what is practically feasible to do, with what people want, and what they find easiest to do and to understand.

I absolutely agree. A great reason to switch to RV would be the greater simplicity. Voters can even abstain from casting votes for unknown options, diminishing the harm done by ignorance, and making the voting process shorter and simpler on a per-voter basis.

That was the first lesson taught me when I went to work for a political campaign, and it applies just as well here. The theoretical, and if present the quite marginal, greater utility of range voting does not overcome these other factors, and it most emphatically does not overcome practical concerns about its utility.

You say "theoretical" as if our calculations are loose guesses. On the contrary, they were derived using quite rigorous modeling. Before discounting them, I'd look over the code, and tell us what methodological errors you find. Hand waving is not a scientific argument. And the utility differences are NOT marginal; they are huge. The benefit you get by going from IRV to Range Voting is almost as much as you'd get by going from RANDOM SELECTION to IRV in the first place. If I suggested you pick the winners by drawing names out of a hat, you'd think I was crazy, right? Well, just a bit crazier than one would be to use IRV instead of switching to Range Voting. It's also nice that RV is simpler than IRV to use and tabulate. It's a win-win-win..

This is especially true if, as you are saying, range voting is not perfect.

Your case is refuted by RV's being much better. It needn't be perfect.

And my point that you are calling "straw man" is not that you literally said it was perfect, it's that you are talking as if you think it is perfect.

No, I'm talking about it as though it is the best known voting system (that is practical to implement). It is.

Regards,
Clay

Clay

Whether voters trust it or not has nothing to do with whether it is actually the best voting method.

The fact that you say that with a straight face shows that You Don't Get It.

Whether the voters have confidence in the election system has a huge amount to do whether something is best. Let us presume that for a moment you were suddenly the King of WSFS and you imposed your voting system upon us. I expect that voter turnout -- already less than I would like to see -- would drop precipitously, and the SF community would almost certainly lose faith in the legitimacy of the results.

WSFS currently has a moderately-complicated election system, but it's been around long enough with few enough changes, and enough people in the field whose opinions matter agree with using it, that the results are widely perceived as legitimate. (Yes, anyone who thinks a system is Evil if it doesn't return his/her first preference isn't going to be convinced. I'm mainly interested in reasonable objections, not sour grapes.)

Voters can even abstain from casting votes for unknown options...

They can do that already. Indeed, we encourage people not to vote lower preferences if they have no preference. (This isn't the Australian parliament. On principle I object to any system that requires you to vote, because IMO the right to vote includes the right to abstain).

On the Site Selection ballot (not the Hugos), there is even an explicit "No Preference" selection, and any choices numbered after No Preference are ignored.

Re: Supermajority vote?

kalimac.livejournal.com - 2007-01-24 06:11 (UTC) - Expand

Re: Supermajority vote?

thebrokenladder.livejournal.com - 2007-01-24 11:41 (UTC) - Expand

That chart is really baffling. What do those percentages mean? It's surely not the percentage of voters who get their preferred candidate. Is it the opinion poll results for satisfaction with the voting method? That doesn't look right either.

I left a link to where it is described.

http://RangeVoting.org/vsr.html

Here's a quick example to explain it to you in a nutshell. We have voters V1 - V6, and candidates C1 - C5. Below is a table of the utility values from each voter for each candidate. Think of utility as "desire" or "happiness/satisfaction one would get by having that thing". These values are in "happiness units" of arbitrary magnitude. Don't worry that we can't objectively quantify that unit, because later we divide, which drops the terms out and leaves us with a unitless value - just a ratio.

C1 C2 C3 C4 C5
V1 2 30 -16 52 56
V2 24 30 10 60 30
V3 40 -8 -30 12 -18
V4 37 36 8 -28 -37
V5 -26 -28 -11 -28 8
V6 45 58 22 -4 55
UTILITY AVG. (U) 20.33 19.67 -2.83 10.67 15.67
BAYSIAN REGRET 0.00 0.67 23.17 9.67 4.67
VSR 100.00% 91.27% -203.49% -26.64% 38.86%
VSR = (U - average(U)) / (max(U) - average(U))

In this example, C1 produces the greatest happiness for the greatest number, with an average utility of 20.33. C3, on the other hand, produces a an average utility of -2.83, the lowest. But these numbers are like cubits, only comparable to each other, but not to numbers from other scenarios. To resolve that, we rescale the utilities, such that the voter satisfaction ratio of picking the ideal winner is a 100%, and the VSR of picking a random winner (just the average of all the candidates' utilities) is 0%. The formula is mentioned below the table. For the highest utility value, U = max(U) so we get 100% social utility efficiency.

If we test honest plurality voting here, C1 and C5 tie, 2-2. But say we use strategic plurality, where each voter strategically votes for his favorite between the two front-runners. Then C5 wins 4-2.

Now say we try range voting, using the standard definition of "honest" range voting, where each voter simply rescales his utilies to the specified range (0-10, 0-99, whatever). Then we get:

C1 C2 C3 C4 C5
2.50 6.39 0.00 9.44 10.00
2.80 4.00 0.00 10.00 4.00
10.00 3.14 0.00 6.00 1.71
10.00 9.86 6.08 1.22 0.00
0.56 0.00 4.72 0.00 10.00
7.90 10.00 4.19 0.00 9.52
5.63 5.57 2.50 4.44 5.87

C5 just barely beats C1, even though C1 produces a tremendously higher voter satisfaction ratio. Of course that's partly because I tweaked this example (starting with random values that I then tuned) to show examples where Range Voting can indeed pick a pretty poor winner. (note that some candidates have negative values, because they produce less satisfaction than what would be expected from picking a RANDOM candidate - a voting method that picked them regularly would be WORSE than drawing a name out of a hat, instead of better, defeating the purpose of democracy).

Now say I go in and randomly plug a bunch of numbers in for the honest utility values again.

C1 C2 C3 C4 C5
V1 25 55 7 20 45
V2 12 25 1 18 19
V3 16 16 8 17 6
V4 35 40 0 34 41
V5 2 21 11 19 35
V6 8 12 15 5 7
UTILITY SUM (U) 16.33 28.17 7.00 18.83 25.50
BAYSIAN REGRET 11.83 0.00 21.17 9.33 2.67
VSR -31.48% 100.00% -135.19% -3.70% 70.37%

I dispensed with negatives this time, just because it's easier to type a bunch of random positive numbers in quickly without putting thought into it.

Now let's see which winner HONEST plurality would pick.

C2 and C5 tie, 2-2. So if voters were strategic with plurality (voting their favorite between just those TWO) they'd get C2, by a 4-2 vote. Producing a VSR of 100%! What about "honest" (normalized) Range Voting?

C1 C2 C3 C4 C5
3.75 10.00 0.00 2.71 7.92
4.58 10.00 0.00 7.08 7.50
9.09 9.09 1.82 10.00 0.00
8.54 9.76 0.00 8.29 10.00
0.00 5.76 2.73 5.15 10.00
3.00 7.00 10.00 0.00 2.00
4.83 8.60 2.42 5.54 6.24 <-- Averages

It would also pick C2, by a hefty margin. And if C2 dropped out, it would elect C5. So, here strategic plurality has actually worked pretty well, and so has Range Voting. Now imagine taking these simplified spreadsheet examples, and using something like 20 different voting methods, and MILLIONS of elections, with 720 different combinations of election parameters (strategic vs. honest, ignorant vs. informed voters). That's how we got the results I present to you.

Clay

Next time you post a chart, could you put the preformatted text HTML tag around it? (the letters PRE in angle brackets) As it is they're all flush left and impossible to read. Thanks.

Oh, sorry. Check this out:

http://spreadsheets.google.com/pub?key=pXPf6D8HwIWncwYJKKb4CcQ

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