An election system

[mike-l]

It just occurred to me that in Qaanol and my simulations with 1000 voters, we are modelling even distribution among the population on all issues, and just varying the candidates. I suggest now that my original model was actually better, as by having fewer voting blocks, there's much higher population variance between runs. At the very least, we should be picking some population bias on the 3 issues instead of using symmetric distributions every time.

[Роберт]

I say that we should get rid of 'terms'. Just keep a running tally. People could change their vote at any time. If someone is doing a god job, why make them go out and campaign? If someone is doing a bad job, why wait 4 years to get rid of them?

That's actually a terrible idea. Let me ask you: why do we have longer terms for senate than we do the house?

[broken lader]

Range (real, honest range) is, as expected, the best system

That's because our metric is how well each system agrees with range voting.

Assuming yo're talking about Bayesian Regret, that's wrong. It's astonishing how often people make this error.

Bayesian Regret starts with a distribution of utilities or "welfare values" which represent how well off each voter would be depending on which candidate wins the election.

Voters then have an "estimate" of their utility, which is produced by distorting the real utility via "ignorance factors".

Voters using Score Voting then normalize those utility estimates to fit the scale, e.g. 0-10, 1-5, etc.

Tactical voters then maximize or minimize many of the intermediate scores.

So the final results are radically different than what you'd get if you merely added up the real utility values. So it is absolutely not a "given" or an obvious expectation or a "tautology" that Score Voting wins. It was actually a surprise to Warren Smith, who conducted the world's most extensive Bayesian Regret calculations.

And actually, there are "exotic" voting systems which outperform even Score Voting, but are much too complex to be viable for political elections.

And this assumes that these 'preference' numbers are actually meaningful things and comparable between voters, and that optimizing them is what's best for the group.

These really aren't assumptions. They're pretty well proven.
http://ScoreVoting.net/UtilFoundns.html

Which is a reasonable assumption for sure, but an assumption nonetheless, and one that a number of economists disagree with.

Those economists are nowadays basically dinosaurs. Their notions that e.g. utility is ordinal have been robustly disproved.

[broken lader]

It just occurred to me that in Qaanol and my simulations with 1000 voters, we are modelling even distribution among the population on all issues, and just varying the candidates. I suggest now that my original model was actually better, as by having fewer voting blocks, there's much higher population variance between runs. At the very least, we should be picking some population bias on the 3 issues instead of using symmetric distributions every time.

If you are running voting simulations, I highly suggest you join discussion forum for The Center for Election Science. It is possible you've made some significant errors, or missed out on some previous discoveries, either of which would likely be pointed out to you by people who've been doing this a long time. Or possibly you've discovered something new, in which case there could be a productive conversation there.

https://groups.google.com/forum/#!forum/electionscience

[broken lader]

I think that the ideal voting system is a preferential voting list, where each vote is a ranking of all possible candidates. So each person ranks the candidates in order of preference. In the first round, only the first picks are counted. Whichever candidate has the least votes is knocked off and all of his voters get their vote changed to their second preference. Then the second least popular candidate gets knocked off and all of his votes get swapped for the next preference and etc etc. This way you don't have to decide between voting for the party you want and voting strategically against who you don't want. If you want to vote Ralph Nader and he gets knocked off, your second pick and still go to the Democrats. I am sure that there is a name for this system but I can't find it right now.

This is Instant Runoff Voting, and it's quite terrible.

http://scorevoting.net/CompleteIdioticIRV.html
www.electology.org/approval-voting-vs-irv

[mike-l]

Ok, that triple post explained a few of my questions. The reason the Bayesian Regret of sincere range voting is non zero that you both normalize and distort preferences for ignorance.

The post you were replying to however, was comparing sincere range voting to sincere range voting, we had no distortions or normalizations in our simulations, and that was the metric we were using.

I will definitely check out the discussion forum you linked.

I do have one question though, are there any real world examples of approval voting being used on a large scale (at least 1000 voters) that hasn't degenerated back into most voters voting for just one candidate?

[nitePhyyre]

I say that we should get rid of 'terms'. Just keep a running tally. People could change their vote at any time. If someone is doing a god job, why make them go out and campaign? If someone is doing a bad job, why wait 4 years to get rid of them?

That's actually a terrible idea. Let me ask you: why do we have longer terms for senate than we do the house?

I'm Canadian, we have an appointed senate with life terms. (Forced retirement at 75) They are much less powerful than the lower house, and rarely do they not pass something that made it out of commons. There are plenty of plans for senate reform, with some of the strongest voices advocating for complete dissolution of the senate - the same thing that the provinces have already done. So my answer to you is this:

Because the founding fathers had a perverse fetish for 'checks and balances'. Outside of that, there is no reason for 2 houses, let alone different sets of term lengths.

Perhaps a more satisfying answer? From http://usgovinfo.about.com/od/uscongres ... senate.htm

All members of the House are up for election every two years. In effect, they are always running for election. This insures that members will maintain close personal contact with their local constituents, thus remaining constantly aware of their opinions and needs, and better able to act as their advocates in Washington. Elected for six-year terms, Senators remain somewhat more insulated from the people, thus less likely to be tempted to vote according to the short-term passions of public opinion.

In that light, I can see how always running for election powered by modern technology is a terrible idea when compared to always running for election constrained to the technology of the 1700s.

[Роберт]

In that light, I can see how always running for election powered by modern technology is a terrible idea when compared to always running for election constrained to the technology of the 1700s.

Exactly! My point wasn't that the U.S. system was great, my point was literally constantly running for election is terrible.

[broken lader]

I do have one question though, are there any real world examples of approval voting being used on a large scale (at least 1000 voters) that hasn't degenerated back into most voters voting for just one candidate?

Well, there are real world examples of PLURALITY voting being used, in which significant numbers of voters don't vote for their favorite candidate. E.g. exit polling in 2000 showed that about 90% of those who claimed to favor Nader voted for someone else, mostly Gore. So then what do you think those people would have done if given the option to vote for multiple candidates?

1) Still just vote for Gore.
2) Vote for Gore and Nader.
3) Switch their vote from Gore to Nader.

If you picked option 2, congratulations. You aren't crazy.

But okay, I'll indulge you. Here's some Approval Voting election data from the Pirate Party in Germany. These are real contentious elections within a party that holds, for instance, 15 of the 148 seats in the parliament in the state of Berlin.
http://www.electology.org/pirate-elections-germany

Here's a more in depth look at the bullet voting issue you're talking about.
http://www.electology.org/bullet-voting

Also, here's proof that FairVote, the primary organization promoting Instant Runoff Voting, is astonishingly dishonest about this.
http://www.electology.org/fact-check

EDIT: And this is really cool. http://www.electology.org/german-approval-voting-polls

[mike-l]

But okay, I'll indulge you. Here's some Approval Voting election data from the Pirate Party in Germany. These are real contentious elections within a party that holds, for instance, 15 of the 148 seats in the parliament in the state of Berlin.
http://www.electology.org/pirate-elections-germany

This shows about 15% of people voting for BOTH candidates in a 2 candidate election. Presumably then, some people vote for all or almost all the candidates in bigger elections too, and if that number is anywhere near 15%, then the rest look to be voting for just one person. Really though, I'd like to see a breakdown of what % of people voted for N candidates, instead of just an average N. Nonetheless, thanks for the link.

EDIT: And this is really cool. http://www.electology.org/german-approval-voting-polls

Very neat. This data is from 2008, did they repeat this in 2009-2011? Did this lead to the adoption of it anywhere in Germany? (Maybe this is why the pirate party picked it up?)

Anyway, it's clear from this that when picking approval cutoffs in simulations, there needs to be a bias towards fewer candidates than more, with probably at least half of the people bullet voting unless there are 7+ choices. I'm running some new numbers tomorrow with this criteria and with inherent population biases. Will post when I have numbers.

Can you link any peer reviewed papers on the subject?

[Qaanol]

How do these sound for strategies some voters might use? Realistic?

Before the election, there is an opinion poll where a small random sample of the population gives an honest vote, perhaps FPTP-style or maybe Approval, for their favorite candidate(s). In the actual election, each voter looks at the result of the opinion poll and does as follows:

Of the candidates this voter approves of, whichever one did best in the opinion poll will be this voter’s top choice. Furthermore, the original honest rating for that candidate will be the voter’s new Approval threshold. Call this the “Rally around the leader” tactic.

Of the candidates this voter does not approve of (aside: if the above is used too, should this consider the original threshold or the new one?), whichever one did best in the opinion poll will be this voter’s bottom choice. Call this the “Sink the opponent” tactic.

[mike-l]

Seems reasonable... what is the average approval rate with this method and our previous criteria (ie what's the average number of candidates a voter approves)? The links that Clay provided suggest we should be looking for about 1.4-1.6 for 5 candidates.

[Qaanol]

I haven’t run any simulations recently, I’ve been working on implementing a native Cocoa application to run larger simulations faster.

I think the Rally and Sink strategies need to be somewhat different from how I described them above. Using the Bush/Gore/Nader terminology, some people who honestly approve of both Gore and Nader will tactically rank Gore ahead of Nader, but some of them still rank Nader ahead of Gore. Furthermore, some people who don’t really approve of Gore may tactically approve Gore because they strongly disapprove of Bush.

Perhaps something like three categories are in order when considering poll results: Approve, Neutral, and Disapprove. The Approves can only go up, the Disapproves can only go down, but the Neutrals could go up or down depending on whether they’re needed to defeat a Disapprove or whether they threaten to defeat an Approve.

I’m still not certain how best to translate from poll results to tactical decisions though. What if they poll shows many candidates neck and neck, and this voter also rates those candidates very close together? Then I think whatever tactics the voter uses, should be used on all of those candidates.

[Derek]

Wikipedia gives some voting strategies for approval voting. This one sounds pretty good and is close to what you were trying to do, I think:

Vote for any candidate that is more preferred than the expected winner and also vote for the expected winner if the expected winner is more preferred than the expected runner-up. This strategy coincides with the optimal strategy if there are three or fewer candidates or if the pivot probability for a tie between the expected winner and expected runner-up is sufficiently large compared to the other pivot probabilities.

[Qaanol]

Wikipedia gives some voting strategies for approval voting. This one sounds pretty good and is close to what you were trying to do, I think:

Vote for any candidate that is more preferred than the expected winner and also vote for the expected winner if the expected winner is more preferred than the expected runner-up. This strategy coincides with the optimal strategy if there are three or fewer candidates or if the pivot probability for a tie between the expected winner and expected runner-up is sufficiently large compared to the other pivot probabilities.

Wait, Wiki claims that strategy coincides with optimal (for the voter) when there are three or fewer candidates? What if there are three candidates who I rate as follows:
A = 0.91
B = 0.90
C = 0.10

And the results of the straw poll show:
A = 33.5%
C = 33.4%
B = 33.1%

I think it’s pretty obvious that any reasonable person is going to vote for A and B, because they really don’t want C to win. Any sensible definition of “optimal” for the voter needs to indicate that course of action for this scenario.

Also, to the point of most people approving a small number of candidates, usually 1 or 2, there is also the reverse strategy, choosing all but a small number of candidates, in order to essentially “vote against” 1 or 2 especially-disliked candidates.

[nitePhyyre]

In that light, I can see how always running for election powered by modern technology is a terrible idea when compared to always running for election constrained to the technology of the 1700s.

Exactly! My point wasn't that the U.S. system was great, my point was literally constantly running for election is terrible.

Ohh, ok then. Protip: In English, this type of phrase construction...

Let me ask you: why do we have longer terms for senate than we do the house?

...is used almost exclusively when someone is trying to make the point you weren't trying to make.

Additionally, unless the fact that the senate is much better at governing, that the laws the senate introduces are of a higher quality and better for the country than the laws in the House, that whenever the senate rejects something that made it out of the House it was a truly terrible piece of legislation, and that all these quality differences are a result of term lengths and not other factors such as body size, procedural rules, election process, minimum age requirements, etc. are all common knowledge... not only did you write the opposite of what you meant, you haven't even made the point you were trying to make.

@anyone: Is all the above true? I think I would have picked up on it, but I may have missed it.

In other words, I've seen better attempts at backtracking and covering your ass, I've seen worse.

[Qaanol]

The thought occurs to me that so-called Condorcet methods, whether they involve ranking candidates from best to worst, or filling out “Who would you vote for in this matchup?” for every head-to-head matchup, can in real life (and should be expected to sometimes) produce a winner other than the Condorcet winner. They can even elect a Condorcet loser, I believe.

Why? Because in a two-candidate race, there is no possible tactical voting: everyone votes honestly. Therefore the actual Condorcet winner—the candidate that would actually win every head-to-head matchup, if such a candidate exists—is a function of the true, honest voter opinions of candidates.

But in a multi-person race, it can be expected that many voters will tactically rally around the leading candidate of whom they approve, and sink the leading candidate of whom they disapprove. Therefore, whether ranking the candidates from best to worst, or listing one’s choice in a hypothetical head-to-head matchup, it can be expected that rational voters will not give their honest opinions.

As a result, there may be a candidate that the ballots indicate is a Condorcet winner, when in actual fact if a runoff were held some other candidate would beat that one. Indeed, it is conceivable that every other candidate would beat the alleged Condorcet winner if an actual runoff were held, on account of voters tactically misrepresenting their true preferences.

So, not only is the Condorcet criterion the wrong criterion to use—the true honest preference criterion is what actually measures the will of the people—but voting systems that identify a Condorcet winner from the ballots might in reality identify a candidate that would not actually satisfy the Condorcet condition, and could even be a Condorcet loser.

Now, my third paragraph in this post might make it sound like multi-person races are doomed, and we should stick to two-person races, but that is not really what I’m saying. Instead, I’m saying multi-person races where voters are asked to rank the candidates are doomed, and we should stick to election systems where the options are “Yes” and “No”. Namely, Approval voting.

[mike-l]

Yes, any ranking system is subject to tactical voting and may give 'incorrect' results because of that.

Do you think that's somehow unique to ranked systems? Approval voting is inherently tactical, as you HAVE to pick a cutoff. And different cutoffs yield different results. And that's just sincere voting. Let alone not approving 'rivals' while approving less appealing third parties

The question is, what are the chances that such things happen. Perhaps this week I'll try simulating some 'sink the biggest threat' tactical voting an post the results.


Here's my mathematical criteria

Spoiler:

I'm using the same same setup we originally had with 3 issues, one centralizing and one dividing. But im adding a population bias. For each issue I'm picking a central biased number ((rand + rand)/2) and after that is picked, each voter rolls rand for each issue and if it's above, then their value is on that issue is made to be 5 + abs(orig -5), otherwise it's 5 - abs(orig-5). Approval will use your rally method. FPTP, Condorcet and IRV will give the front runners +/- rand/2 In their rankings, depending on which they prefer.

I also still vehemently deny that there is a measurable 'individual value' function that voters can measure themselves and can be aggregated linearly into a 'social value' number. I'm only granting that voters assessment of such a function is a metric which is reasonable to look at (among others)

Aside from Yakks objections that such functions just don't exist, lets grant for a moment that such a function f
Does exist. It's still not reasonable that you can simultaneously have tha people will vote according to it and that the social value is the aggregate. This is because many voters will vote according to their estimate of the aggregate and not just their own value. For example, as someone with an upperclass background and no plans for a family, no medical issues, etc, its beneficial to me by most metrics to lower taxes and cut services, but I still vote exactly the opposite to that.

Edit: Looking at http://halshs.archives-ouvertes.fr/docs ... roval4.pdf right now. It claims that with adequate polling that strategic voting will elect condorcet winners in AV. Also, it claims that the unique best strategy is to put your cutoff at the polling leader, and to include them or not so that you have exactly one of the two frontrunners. Essentially your Rally if you prefer the leader, and Sink if you don't. There's some typos but they don't seem to matter to the actual paper. I do have some issues after skimming it though and I'll have to see how they are addressed on closer reading.

[Qaanol]

I also still vehemently deny that there is a measurable 'individual value' function that voters can measure themselves and can be aggregated linearly into a 'social value' number. I'm only granting that voters assessment of such a function is a metric which is reasonable to look at (among others)

Aside from Yakks objections that such functions just don't exist, lets grant for a moment that such a function f
Does exist. It's still not reasonable that you can simultaneously have tha people will vote according to it and that the social value is the aggregate. This is because many voters will vote according to their estimate of the aggregate and not just their own value. For example, as someone with an upperclass background and no plans for a family, no medical issues, etc, its beneficial to me by most metrics to lower taxes and cut services, but I still vote exactly the opposite to that.

Right, in all actual elections, rational voters will tactically vote for the candidate(s) that maximize their expected utility.

I am saying that, if an omniscient being were able to “see into the minds” of voters, that omniscient being could measure the percent agreement of each candidate with each voter, using a consistent metric. Then the sum of all the voters’ levels of agreement with a given candidate is the net agreeability of that candidate. The candidate with the highest value there is the “optimal” candidate that best represents the will of the people and “should” win.

I acknowledge that it is impossible to measure such a things in real life, but I still posit that the foundation is solid. Individuals do have levels of agreement with candidates, we just don’t have a way in real life to compare those between different voters on the same scale.

My point is, in running these simulations, we get to be the omniscient observer, so we can give each voter a level of preference for each candidate, with those drawn from the same scale in the first place. Having done so, we can now calculate the optimal will-of-the-people candidate by summing those values. Similarly, we can calculate the actual Condorcet winner using those honest preference levels, since rational voters are always honest in a two-person election.

Then we have to figure out what the results of an actual election would be using various voting systems. This is where all the tactical stuff comes into play, with voters acting in their own best interest rather than being honest to a fault. Once we come up with realistic strategies for voters to follow, we can calculate how they will behave in various elections. That lets us determine how often each of the proposed voting systems will elect the optimal will-of-the-people candidate, how often each will elect the actual Condorcet winner if one exists, and so forth.

Edit: Looking at http://halshs.archives-ouvertes.fr/docs ... roval4.pdf right now. It claims that with adequate polling that strategic voting will elect condorcet winners in AV. Also, it claims that the unique best strategy is to put your cutoff at the polling leader, and to include them or not so that you have exactly one of the two frontrunners. Essentially your Rally if you prefer the leader, and Sink if you don't. There's some typos but they don't seem to matter to the actual paper. I do have some issues after skimming it though and I'll have to see how they are addressed on closer reading.

I’ll have to look through that paper. On its face, that strategy does sound reasonable. However, I submit my previous counterexample (numbers modified slightly):

Some voter has the following honest opinions of candidates:
A = 0.91
B = 0.90
C = 0.10

The poll results come in like this:
A = 33.5%
C = 33.3%
B = 33.2%

That paper’s strategy would say to vote for A only. But any real person would vote for B as well. Essentially the problem is that there could be—once margins of error are accounted for—a virtual tie among more than two candidates.

Here’s what I see as a reasonable strategy for a voter:

Given a voter’s honest ratings, r₁, r₂, ⋯, r_N, and the poll results p₁, p₂, ⋯, p_N, let r_max be the highest rating and r_min be the lowest rating.

For each rating, let the shifted rating s_k = r_k - (r_min + r_max) / 2 be the result of subtracting the midpoint of the ratings. Now let the tactical rating be given by multiplying by the poll results, t_k = s_k * p_k. This means candidates that poll poorly are drawn toward the midpoint. Now scale and shifted those tactical ratings so the lowest is 0 and the highest is 1.

This provide a tactical ranking of all the candidates that can be used for all the voting systems, whether ranked or rated or what-have-you. For Approval Voting, I’d say look at all candidates that rate at or above 0.75 on the new scale. Find the one of those with the lowest honest rating. Use that honest rating as the threshold, and do vote for that threshold candidate.

I understand this is rather complicated, and no one actually does such a calculation in real life, but I think it provides a reasonable approximation for how people intuitively weigh the results of polls against their own opinions.

All of this, however, is predicated on individuals knowing their own levels of agreement with each candidate. That is fine for a well-informed populace, but in real life many people don’t know much about most candidates. I’m not sure how or if we should account for voter ignorance.

[mike-l]

Right, in all actual elections, rational voters will tactically vote for the candidate(s) that maximize their expected utility.

I just said that I personally vote to maximize (in my opinion) social utility, not my own. Maybe that's not rational, but then that suggests that looking only at rational voters doesn't reflect reality, as I'm sure I'm not alone.

I am saying that, if an omniscient being were able to “see into the minds” of voters, that omniscient being could measure the percent agreement of each candidate with each voter, using a consistent metric. Then the sum of all the voters’ levels of agreement with a given candidate is the net agreeability of that candidate. The candidate with the highest value there is the “optimal” candidate that best represents the will of the people and “should” win.

And I disagree that the 'net agreeability' in any way necessarily reflects the social value of any particular choice. Ie net agreeability has no reason why it should be a better metric than anything else. There are plenty of things I have very strong opinions on that don't affect me directly, so they would contribute to my individual 'agreeability' but if we were trying to count a 'social agreeability' number, those issues should not affect my portion of the score.

[Qaanol]

A political election is a popularity contest. As such, it is right and proper that the most popular candidate should win. That’s what is measured by the honest level of preference. Given the premise that voters are choosing who they “want” or “would like” to have elected, and in that way the democratic process reflects the will of the people, it follows that election systems should be compared with regard to how well they select the best candidate by that metric.

If you want to argue that an election “should” be something different from a popularity contest, perhaps with the goal of maximizing utility, that’s a different discussion.

[mike-l]

A political election is a popularity contest. As such, it is right and proper that the most popular candidate should win.

Wherever did you get that idea?

[Qaanol]

I just said that I personally vote to maximize (in my opinion) social utility, not my own. Maybe that's not rational, but then that suggests that looking only at rational voters doesn't reflect reality, as I'm sure I'm not alone.

That’s why we’re doing all our models based on internal levels of preference. It doesn’t matter whether you like a candidate because they’ll give a tax break, or because they’ll build wind turbines, or because their middle name makes you giggle. All that matters is how much you like the candidate. That’s what “level of preference” measures, and that’s what determines who you’re going to vote.

A political election is a popularity contest. As such, it is right and proper that the most popular candidate should win.

Wherever did you get that idea?

Reality.

The fundamental premise behind democratic elections is that the voters decide who wins. The winner is the person whom the voters support the most. The voters are the people. The population. The populace. Their collective opinions are the definition of “popularity”.

[mike-l]

Right, in all actual elections, rational voters will tactically vote for the candidate(s) that maximize their expected utility.

That’s why we’re doing all our models based on internal levels of preference.

Which one is it?

The winner is the person whom the voters support the most.

So, popularity depends on what system we're using?

[Qaanol]

Right, in all actual elections, rational voters will tactically vote for the candidate(s) that maximize their expected utility.

That’s why we’re doing all our models based on internal levels of preference.

Which one is it?

Those are both simultaneously true statements, and they do not contradict. Furthermore, they are not even referring to the same things. Look at what each was in response to. Rational voters will vote to maximize their own expected utility. That’s the definition of rational behavior.

We’re basing our models on levels of preference because that’s what voters base their decisions on.

The winner is the person whom the voters support the most.

So, popularity depends on what system we're using?

Voting systems are different ways of attempting to measure popularity, and none of them do it perfectly. We are trying to find which one does it best.

[mike-l]

Right, in all actual elections, rational voters will tactically vote for the candidate(s) that maximize their expected utility.

That’s why we’re doing all our models based on internal levels of preference.

Which one is it?

Those are both simultaneously true statements, and they do not contradict. Furthermore, they are not even referring to the same things. Look at what each was in response to. Rational voters will vote to maximize their own expected utility. That’s the definition of rational behavior.

We’re basing our models on levels of preference because that’s what voters base their decisions on.

What? The second quote was in response to my response to the first quote! That we either need to have a system where rationality allows us to vote for our expected social utility, and not just our own, or abandon the idea of rational voting being a useful modeling tool.

The winner is the person whom the voters support the most.

So, popularity depends on what system we're using?

Voting systems are different ways of attempting to measure popularity, and none of them do it perfectly. We are trying to find which one does it best.

But you are claiming that aggregate internal preference is the best measure of that, and your proof is that this is what voting systems do (or are trying to do?). In reality, almost nobody uses range or approval voting, suggesting this is actually not what they are trying to do.

[broken lader]

This shows about 15% of people voting for BOTH candidates in a 2 candidate election. Presumably then, some people vote for all or almost all the candidates in bigger elections too, and if that number is anywhere near 15%, then the rest look to be voting for just one person.

I would doubt that. It's conceivable that 14% of voters could "approve of" both candidates in a two-candidate race. But as you add candidates, the probability that any given voter approves all the candidates goes down rapidly.

Really though, I'd like to see a breakdown of what % of people voted for N candidates, instead of just an average N.

You and me both! Here are some results from a somewhat contentious/meaningful Score Voting election that a San Francisco running group recently held.
http://www.electology.org/sf-frontrunners-2012

I know it's not a political election, but it's interesting.

This data is from 2008, did they repeat this in 2009-2011? Did this lead to the adoption of it anywhere in Germany? (Maybe this is why the pirate party picked it up?)

I don't know the answer to any of these questions, except I'm pretty sure that Approval Voting is not used in any government elections. It was used for experimental (democracy was experimental at that point) government elections in some parts of the USSR, right before its collapse. It was "disapproval" voting, so psychologically a little different. But as for Germany, the North Rhine-Westphalia (one of 16 German states) Pirate Party has on their platform that they'd like to see mayors elected via Approval Voting. Also there's a German democracy reform organization called Mehr Demokratie (More Democracy) which promotes it.
http://nrw.mehr-demokratie.de/buergermeisterwahl.html

it's clear from this that when picking approval cutoffs in simulations, there needs to be a bias towards fewer candidates than more, with probably at least half of the people bullet voting unless there are 7+ choices.

I can't say I agree with that exact assessment. It's very complicated. For instance, say you have two clear frontrunners, and 90% of the voters favor one of those two. Then you could conceivably have 90% of the electorate bullet voting. But that does not mean that they are "bullet voters" as a rule. If their favorite candidate isn't a frontrunner, maybe the vast majority of those voters would vote for more than one candidate. We don't know from those German results whether most voters are bullet voting as a rule, or just because the preference distributions they have just happen to lead to bullet voting. Further, these elections seem to be different from ordinary political elections in that there's much less pre-election polling, and so voters have a less clear picture of who the frontrunners even are. Now, rationally/logically/statistically speaking, they should then just approve all candidates they like better than average — that's the best strategy. But it could be that in a zero-info (or low-info) scenario, psychology causes people to be optimistic that their favorite candidate has a chance, and so they are more likely to bullet vote. These results could certainly be incorporated into making our simulations more realistic, but it's not clear what parameters we'd want to tune differently.

The most important thing I see in those results is that Approval Voting absolutely does not degenerate into Plurality Voting, as people like Rob Richie of FairVote (a total and complete shill for IRV) have asserted constantly through years.
I'm running some new numbers tomorrow with this criteria and with inherent population biases. Will post when I have numbers.

Can you link any peer reviewed papers on the subject?[/quote]

[broken lader]

How do these sound for strategies some voters might use? Realistic?

We generally expect tactical voters to vote for the same person they would with Plurality Voting, plus everyone they like even more. And then sincere Plurality Voters (e.g. Nader voters who like jousting windmills) would probably be split roughly evenly between those who would still bullet vote for Nader, and those who would go ahead and tack on a tactical vote for e.g. Gore to their sincere vote for Nader.

But if you want to know what's actually the ideal strategy, that's explained here:
www.electology.org/threshold
ScoreVoting.net/RVstrat3.html

[broken lader]

I haven’t run any simulations recently, I’ve been working on implementing a native Cocoa application to run larger simulations faster.

Umm... have you looked at Warren Smith's simulation code? I hope you're not doing too much reinventing of the wheel. See paper 56 here:
http://scorevoting.net/WarrenSmithPages ... works.html

Even "little" things can get you, like the fact that most generic random number generator libs fail horribly to meet basic statistical checks. They are good enough for video games, but not for simulations.

Using the Bush/Gore/Nader terminology, some people who honestly approve of both Gore and Nader will tactically rank Gore ahead of Nader, but some of them still rank Nader ahead of Gore.

Depends on whether they are honest or tactical voters. I assume you have a simple boolean flag which indicates this for each voter.

Furthermore, some people who don’t really approve of Gore may tactically approve Gore because they strongly disapprove of Bush.

I don't know what you mean "really approve". There's no such thing as "approval", there's just utility. Approving a candidate is a mark you make on your ballot. It's pretty arbitrary what you count as a "sincere" Approval Voting ballot. Smith's simulations consider this to be, I believe, approving all candidates you like better than average, regardless of their probabilities of winning.

Perhaps something like three categories are in order when considering poll results: Approve, Neutral, and Disapprove.

Oh, so you're getting pretty sophisticated with your pre-election polling. Interesting.

I’m still not certain how best to translate from poll results to tactical decisions though. What if they poll shows many candidates neck and neck, and this voter also rates those candidates very close together? Then I think whatever tactics the voter uses, should be used on all of those candidates.

The strategy is, approve everyone whose utility is higher than your expected utility — meaning, the utility of each candidate times his probability of winning, all summed up. But then how can you determine "win probabilities", given that you know exactly who the winner is, based on the pre-election polls? Perhaps a simple model would be to just make the polls 100% accurate/clairvoyant. Even then you'd have some error, if some voters are tactical in the real election, but not in the polls, or if ignorance factors change between polling time and election time (i.e. voters become more or less educated about the candidates, causing their estimates of their utilities to change — remember, voters vote based on their ESTIMATES of their utilities, not based on their real utilities, because they aren't omniscient).

I'm sure if you had talked this over with Warren before getting into the code, he could have helped you out a lot. You should join our discussion group and bounce some ideas off him.

[broken lader]

What if there are three candidates who I rate as follows:
A = 0.91
B = 0.90
C = 0.10

And the results of the straw poll show:
A = 33.5%
C = 33.4%
B = 33.1%

So the expected utility is (.335*.91)+(.334*.9)+(.331*.1), or about .64. So you'd approve of A and B, because you prefer them both to your expected utility.

[broken lader]

But in a multi-person race, it can be expected that many voters will tactically rally around the leading candidate of whom they approve, and sink the leading candidate of whom they disapprove. Therefore, whether ranking the candidates from best to worst, or listing one’s choice in a hypothetical head-to-head matchup, it can be expected that rational voters will not give their honest opinions.

Exactly. Leading to horrendous consequences like this:

http://ScoreVoting.net/DH3.html
http://ScoreVoting.net/CondBurial.html

And regardless of what is actually "rationally" tactical, most voters using ranked systems will just use the "Naive Exaggeration Strategy", which causes ranked systems to degenerate toward Plurality Voting:
http://ScoreVoting.net/NESD.html

This is also why Score Voting and Approval Voting may, in practice, be better Condorcet methods than real Condorcet methods.
http://ScoreVoting.net/AppCW.html

So, not only is the Condorcet criterion the wrong criterion to use—the true honest preference criterion is what actually measures the will of the people—but voting systems that identify a Condorcet winner from the ballots might in reality identify a candidate that would not actually satisfy the Condorcet condition, and could even be a Condorcet loser.

I think I love you.

Now, my third paragraph in this post might make it sound like multi-person races are doomed, and we should stick to two-person races, but that is not really what I’m saying. Instead, I’m saying multi-person races where voters are asked to rank the candidates are doomed, and we should stick to election systems where the options are “Yes” and “No”. Namely, Approval voting.

Or Score Voting in general. Approval Voting is Score Voting on a 0-1 scale, but a 0-10 scale is much better. E.g.
http://ScoreVoting.net/ShExpRes.html (Shentrup Experiment Result)

[broken lader]

Yes, any ranking system is subject to tactical voting and may give 'incorrect' results because of that.

Do you think that's somehow unique to ranked systems? Approval voting is inherently tactical, as you HAVE to pick a cutoff.

But that causes VASTLY less severe consequences, because the PERCEPTION of not being a frontrunner isn't a self-fulfilling prophecy. Which is the focus of this:
http://ScoreVoting.net/NESD.html

A core issue here is the Favorite Betrayal Criterion. Cardinal methods never punish a voter for supporting his favorite candidate.
http://scorevoting.net/FBCsurvey.html

And different cutoffs yield different results. And that's just sincere voting. Let alone not approving 'rivals' while approving less appealing third parties

The question is, what are the chances that such things happen. Perhaps this week I'll try simulating some 'sink the biggest threat' tactical voting an post the results.

I'm using the same same setup we originally had with 3 issues, one centralizing and one dividing. But im adding a population bias. For each issue I'm picking a central biased number ((rand + rand)/2) and after that is picked, each voter rolls rand for each issue and if it's above, then their value is on that issue is made to be 5 + abs(orig -5), otherwise it's 5 - abs(orig-5). Approval will use your rally method. FPTP, Condorcet and IRV will give the front runners +/- rand/2 In their rankings, depending on which they prefer.

I have no idea what in god's name you're talking about. You want to start by just giving the voters utilities for each candidate. That can be done with a regular random Gaussian distribution. Or you can pick random positions on an n-dimensional issue space, and then assign utilities as 1-Ln distance or something of that sort. But it turns out that the more dimensions you use (trying to make it more realistic) the closer it gets to random distributions.

I have no idea what you're trying to accomplish with this "each voter rolls rand for each issue" part. You want to just assign utilities and estimated utilities (the latter being a transformation of the former, through ignorance factors), and then you want the voters to rate/rank based on their estimated utilities, plus whether they are sincere or tactical voters.

I also still vehemently deny that there is a measurable 'individual value' function that voters can measure themselves and can be aggregated linearly into a 'social value' number.

Well, you're wrong. The neurotransmitters that make us feel pleasure are discrete material things that have exact "numeric" states. You can absolutely add them up, both because of the underlying physics, and because additive utility is the only possible right model. This is mathematically proven.
http://ScoreVoting.net/UtilFoundns.html

lets grant for a moment that such a function f
Does exist. It's still not reasonable that you can simultaneously have tha people will vote according to it and that the social value is the aggregate. This is because many voters will vote according to their estimate of the aggregate and not just their own value.

Voters, BY DEFINITION, vote in order to maximize their estimated utility. If you enjoy altruism, then this simply means that you have a lot of personal utility for results which happen to benefit NET utility as well. But you are still ALWAYS acting to maximize YOUR OWN (estimated) utility.

Edit: Looking at http://halshs.archives-ouvertes.fr/docs ... roval4.pdf right now. It claims that with adequate polling that strategic voting will elect condorcet winners in AV.

Pointed out here: http://ScoreVoting.net/AppCW.html

Also, it claims that the unique best strategy is to put your cutoff at the polling leader

Utterly false. The strategy is to approve all candidates for whom your utility estimate is greater than your estimated expected utility (probabilities of each candidate winning, times their estimated utility).
http://www.electology.org/threshold
http://ScoreVoting.net/RVstrat3.html

[broken lader]

Right, in all actual elections, rational voters will tactically vote for the candidate(s) that maximize their expected utility.

That’s why we’re doing all our models based on internal levels of preference.

Which one is it?

Those are both simultaneously true statements, and they do not contradict. Furthermore, they are not even referring to the same things. Look at what each was in response to. Rational voters will vote to maximize their own expected utility. That’s the definition of rational behavior.

We’re basing our models on levels of preference because that’s what voters base their decisions on.

It seems like Qaanol gets it quite precisely.

The winner is the person whom the voters support the most.

So, popularity depends on what system we're using?

Voting systems are different ways of attempting to measure popularity, and none of them do it perfectly. We are trying to find which one does it best.

Yes. You are trying to find the system which maximizes net utility, or "minimizes Bayesian Regret".

[mike-l]

I haven't had time to read your septuple post yet, but I do get a kick out of you calling someone from FairVote a shill, while only linking your own two sites in every post. BTW, a shill is someone who is disingenuous about his connection to what he's criticizing/praising, which, fortunately for you, doesn't apply to either you or Rob Richie. But you're both shamelessly supporting one cause by linking your own website over and over. You fail the crank test pretty hard. Again, fortunately for you, this doesn't cast judgement on either Approval Voting or Instant Runoff.

Anyway

The question is, what are the chances that such things happen. Perhaps this week I'll try simulating some 'sink the biggest threat' tactical voting an post the results.

I'm using the same same setup we originally had with 3 issues, one centralizing and one dividing. But im adding a population bias. For each issue I'm picking a central biased number ((rand + rand)/2) and after that is picked, each voter rolls rand for each issue and if it's above, then their value is on that issue is made to be 5 + abs(orig -5), otherwise it's 5 - abs(orig-5). Approval will use your rally method. FPTP, Condorcet and IRV will give the front runners +/- rand/2 In their rankings, depending on which they prefer.

I have no idea what in god's name you're talking about. You want to start by just giving the voters utilities for each candidate. That can be done with a regular random Gaussian distribution. Or you can pick random positions on an n-dimensional issue space, and then assign utilities as 1-Ln distance or something of that sort. But it turns out that the more dimensions you use (trying to make it more realistic) the closer it gets to random distributions.

I have no idea what you're trying to accomplish with this "each voter rolls rand for each issue" part. You want to just assign utilities and estimated utilities (the latter being a transformation of the former, through ignorance factors), and then you want the voters to rate/rank based on their estimated utilities, plus whether they are sincere or tactical voters.

What I'm trying to accomplish is reflecting that in an average popultion, the distribution on any given issue is probably NOT centred at the same place for each issue, but our current distributions force that, so I'm adding a bias on each issue to one side of centre or the other.

[Qaanol]

We generally expect tactical voters to vote for the same person they would with Plurality Voting, plus everyone they like even more.

I think it’s obvious they will vote for those candidates. They might also vote for additional candidates too though. Think “Nader supporter who voted Nader in FPTP, but would vote both Nader and Gore in Approval.”

Umm... have you looked at Warren Smith's simulation code? I hope you're not doing too much reinventing of the wheel.

I’m a mathematician by training, with a specialty of modeling real-world human behavior. I intend to run my simulations the “right” way to the best of my ability. After I’m done, then I may look at what other simulations have done, and compare.

I hope I *am* reinventing the wheel, because that will mean you guys were doing things the right way too. But in case you guys made any mistakes or incorrect assumptions, I’d rather make my model without knowing what your model does.

Even "little" things can get you, like the fact that most generic random number generator libs fail horribly to meet basic statistical checks. They are good enough for video games, but not for simulations.

I’m using arc4random_uniform. I’m pretty sure I know what I’m doing on the math side of things here.

Using the Bush/Gore/Nader terminology, some people who honestly approve of both Gore and Nader will tactically rank Gore ahead of Nader, but some of them still rank Nader ahead of Gore.

Depends on whether they are honest or tactical voters. I assume you have a simple boolean flag which indicates this for each voter.

I have an int flag to distinguish among the many different types of tactics my simulated voters can employ.

Furthermore, some people who don’t really approve of Gore may tactically approve Gore because they strongly disapprove of Bush.

I don't know what you mean "really approve". There's no such thing as "approval", there's just utility.

I mean who they would or would not approve in the absence of any opinion polls. And there’s a lot more than just utility. In fact, most people can be assumed *not* to behave rationally.

Perhaps something like three categories are in order when considering poll results: Approve, Neutral, and Disapprove.

Oh, so you're getting pretty sophisticated with your pre-election polling. Interesting.

Pre-election polls—or other methods to determine expected outcome—are the only things that enable tactical voting whatsoever.

So the expected utility is (.335*.91)+(.334*.9)+(.331*.1), or about .64. So you'd approve of A and B, because you prefer them both to your expected utility.

This works if the voter in question believes the results of the opinion poll can be interpreted as probability of winning the general election. It is not at all clear to me that voters actually believe such a thing. Furthermore, in my attempts to be realistic, I always assume that different voters have different strategies, interpretations, and reactions.

Or Score Voting in general. Approval Voting is Score Voting on a 0-1 scale, but a 0-10 scale is much better. E.g.
http://ScoreVoting.net/ShExpRes.html (Shentrup Experiment Result)

Range voting with more scores in the range is not, pragmatically, a priori better than Approval Voting. For starters, it makes the ballots, the voting procedure, and the tallying, much more complicated. Furthermore, the primary effect of choosing anything other than the highest or lowest possible rating is to make your own vote count for less than it could.

The question is, what are the chances that such things happen. Perhaps this week I'll try simulating some 'sink the biggest threat' tactical voting an post the results.

I'm using the same same setup we originally had with 3 issues, one centralizing and one dividing. But im adding a population bias. For each issue I'm picking a central biased number ((rand + rand)/2) and after that is picked, each voter rolls rand for each issue and if it's above, then their value is on that issue is made to be 5 + abs(orig -5), otherwise it's 5 - abs(orig-5). Approval will use your rally method. FPTP, Condorcet and IRV will give the front runners +/- rand/2 In their rankings, depending on which they prefer.

I have no idea what in god's name you're talking about. You want to start by just giving the voters utilities for each candidate. That can be done with a regular random Gaussian distribution. Or you can pick random positions on an n-dimensional issue space, and then assign utilities as 1-Ln distance or something of that sort. But it turns out that the more dimensions you use (trying to make it more realistic) the closer it gets to random distributions.

I have no idea what you're trying to accomplish with this "each voter rolls rand for each issue" part. You want to just assign utilities and estimated utilities (the latter being a transformation of the former, through ignorance factors), and then you want the voters to rate/rank based on their estimated utilities, plus whether they are sincere or tactical voters.

What I'm trying to accomplish is reflecting that in an average popultion, the distribution on any given issue is probably NOT centred at the same place for each issue, but our current distributions force that, so I'm adding a bias on each issue to one side of centre or the other.

Yeah, on this one we’re definitely doing it right. Choosing voter opinions of candidates randomly is completely unrealistic. Having a non-uniform issues-space is exactly how things work in real life.

I’ve been updating my model to use a system where the distribution of people for each issue has a random number N different peaks. The peaks are simple triangle bumps, as (left end of peak) + (width of peak)*(rand + rand)/2, where the width is (rand) and the left endpoint is (rand)*(1 - width). Each time a person is drawn from that distribution, first one of the N peaks is chosen (each peak has its own probability of being chosen, which sum to unity) and then the person’s opinion is drawn from that peak’s distribution.

So at the start I create the number of issues in the issue space, and for each issue I create a random number of peaks and assign each peak a width and left end-point. Then every time I make a person I pick their opinion from those issue distributions.

But you are claiming that aggregate internal preference is the best measure of that, and your proof is that this is what voting systems do (or are trying to do?). In reality, almost nobody uses range or approval voting, suggesting this is actually not what they are trying to do.

Perhaps I was not as clear as I could have been. I am using the following three axioms:

Axiom 1: The goal of a single-winner democratic election is to pick the candidate that best reflects the will of the people.
Axiom 2: The will of the people is defined as their aggregate internal preference.
Axiom 3: The quality of a single-winner voting system is defined as the average aggregate internal preference for the winners it picks in realistic scenarios over the course many elections.

[mike-l]

So the expected utility is (.335*.91)+(.334*.9)+(.331*.1), or about .64. So you'd approve of A and B, because you prefer them both to your expected utility.

This works if the voter in question believes the results of the opinion poll can be interpreted as probability of winning the general election. It is not at all clear to me that voters actually believe such a thing. Furthermore, in my attempts to be realistic, I always assume that different voters have different strategies, interpretations, and reactions.

Indeed. With sufficient polling data, the chances the leader will win approaches 1, which turns the posted method into the 'utterly false' method in the paper I linked.

But you are claiming that aggregate internal preference is the best measure of that, and your proof is that this is what voting systems do (or are trying to do?). In reality, almost nobody uses range or approval voting, suggesting this is actually not what they are trying to do.

Perhaps I was not as clear as I could have been. I am using the following three axioms:

Axiom 1: The goal of a single-winner democratic election is to pick the candidate that best reflects the will of the people.
Axiom 2: The will of the people is defined as their aggregate internal preference.
Axiom 3: The quality of a single-winner voting system is defined as the average aggregate internal preference for the winners it picks in realistic scenarios over the course many elections.

Yeah, I don't agree with axiom 2, and the general lack of any voting system in use that allows you to qualify how much you vote for someone (eg range voting) suggests that it isn't the primary goal of most people either.

[broken lader]

I haven't had time to read your septuple post yet, but I do get a kick out of you calling someone from FairVote a shill, while only linking your own two sites in every post.

I obviously don't mean that he's literally a shill. He and most of his associates are simply relentless supporters of IRV, who absolutely and completely disregard any data that refutes them. They are trying to get the data to fit their conclusion, rather than trying to make their conclusion fit the data. We have numerous examples of them just flat out lying, or being highly misleading at best. E.g.
www.electology.org/fact-check

Whereas we've conducted all kinds of research and mathematical modeling aimed at getting a better understanding of the facts.

But you're both shamelessly supporting one cause by linking your own website over and over.

I have no idea where "shame" would come into play. Our aim is to be the best and most accurate source for scientific analysis of voting systems, so of course I would link to our site. Our site does link to external sources in many cases, such as these:
http://www.electology.org/german-approval-voting-polls

But for the vast majority of analysis we've done, there are simply no other gifted mathematical minds out there doing anything comparable to the analysis that Warren has done. Even the extremely basic stuff, like demonstrating that the group can prefer X even when a majority of its members prefer Y, seems not to be found elsewhere online. It's as if there is just very little interest in doing mathematical analysis of voting methods. And most of the analysis we find is horrendously flawed. E.g. see the work of Saari and Tideman.
http://scorevoting.net/DonSaari.html
http://scorevoting.net/TidemanRev.html

You fail the crank test pretty hard.

That's a pretty meaningless and baseless accusation. Especially in light of the numerous basic novice errors you're making here.

What I'm trying to accomplish is reflecting that in an average popultion, the distribution on any given issue is probably NOT centred at the same place for each issue, but our current distributions force that, so I'm adding a bias on each issue to one side of centre or the other.

What do you mean "centered at the same place"? If you add some offset to each exit (e.g. X=3, Y=-2, etc.) then you just move the origin. What good does that do you, since it's relative position, not absolute position, that matters?

[broken lader]

I think it’s obvious they will vote for those candidates. They might also vote for additional candidates too though. Think “Nader supporter who voted Nader in FPTP, but would vote both Nader and Gore in Approval.”

Yes. I seem to remember having made this same point.

I mean who they would or would not approve in the absence of any opinion polls. And there’s a lot more than just utility. In fact, most people can be assumed *not* to behave rationally.

We would argue that the way you model irrationality is through ignorance factors. People have an imperfect estimate of their own utility. Tactical voters act rationally with regard to maximizing that estimate. That model is good because it doesn't care why their estimate is wrong. It could be irrational behavior, or just miseducation from lying TV ads. Ignorance is ignorance.

Pre-election polls—or other methods to determine expected outcome—are the only things that enable tactical voting whatsoever.

That's quite incorrect. For instance, say you "approve of" all of the candidates, in the sense that your utility for all of them is above a threshold you think of as your approval threshold. But obviously your best tactic is to approve only the ones you prefer to your average utility for all of the candidates.

That may sound a little contrived, since there's a question of what honestly "approving" of a candidate really means. But with Score Voting, where you can consider a sincere ballot just a normalization of utilities, it becomes even more obvious that the zero-info strategy is not the same as a sincere vote.

So the expected utility is (.335*.91)+(.334*.9)+(.331*.1), or about .64. So you'd approve of A and B, because you prefer them both to your expected utility.

This works if the voter in question believes the results of the opinion poll can be interpreted as probability of winning the general election. It is not at all clear to me that voters actually believe such a thing.

Okay, then they get their probabilities from some place else, like political betting markets. The calculation is agnostic as to how they get those probabilities.

Furthermore, in my attempts to be realistic, I always assume that different voters have different strategies, interpretations, and reactions.

Different strategies is realistic. I don't know what "interpretations and reactions" means. I assume interpretation means there's some additional "ignorance factor" applied to the poll results. So one voter could take "35% chance Romney wins" to mean "29% chance Romney wins", and another could read that differently. But then, different "reactions"?

Range voting with more scores in the range is not, pragmatically, a priori better than Approval Voting. For starters, it makes the ballots, the voting procedure, and the tallying, much more complicated. Furthermore, the primary effect of choosing anything other than the highest or lowest possible rating is to make your own vote count for less than it could.

The added tabulation complexity pales in comparison to the net utility increase seen e.g. here:
http://ScoreVoting.net/BayRegsFig.html

And Score Voting is much simpler to administer than e.g. Instant Runoff Voting, which we're using here in San Francisco, and other bay area cities. So the added complexity is pretty small.

And I'm well aware that sincere voting diminishes the strength of your vote. But a lot of voters will do it anyway, because they enjoy the expressiveness so much. Just like a lot of voters voted for Nader. So,

1) Score Voting is better for expressive ("sincere") voters, because they can express more.
2) Score Voting is better for tactical voters, since the expressive voters cede some ballot power to them.
3) Score Voting is better for the group overall, since the utility gained by the tactical voters is greater than what is lost by the expressive voters.

Choosing voter opinions of candidates randomly is completely unrealistic.

Warren Smith found that the more realistic you tried to make utility distributions (e.g. adding more dimensions) the more the utilities just looked like random numbers.

Having a non-uniform issues-space is exactly how things work in real life.

Can you be more clear about what that means? In the simplest case, I'm assigning issue-space coordinates on two dimensions. I pick X=5, Y=-2. If you offset that, you just move the origin, which does nothing. If you "compress" one dimension, relative to another (e.g. odds of X=4 are the same as odds Y=2), I'm not even sure that would have any effect in terms of relative outcomes among the different voting systems. I'd be curious to see.

I’ve been updating my model to use a system where the distribution of people for each issue has a random number N different peaks. The peaks are simple triangle bumps, as (left end of peak) + (width of peak)*(rand + rand)/2, where the width is (rand) and the left endpoint is (rand)*(1 - width). Each time a person is drawn from that distribution, first one of the N peaks is chosen (each peak has its own probability of being chosen, which sum to unity) and then the person’s opinion is drawn from that peak’s distribution.

I just simply cannot understand why in God's name you'd do this. It sounds totally arbitrary, not a way of modeling any real-world phenomenon.

If you look at actual ideological positions on a Nolan chart (e.g. the aggregate results of the OK Cupid politics quiz), you see what looks pretty much like a Gaussian distribution. If you're trying to be realistic, it would seem you want to find examples of real preference distributions, and try to create a utility generator which mimics those realistic distributions. It's not clear to me how your idea above has anything to do with modeling reality.

But you are claiming that aggregate internal preference is the best measure of that, and your proof is that this is what voting systems do (or are trying to do?). In reality, almost nobody uses range or approval voting, suggesting this is actually not what they are trying to do.

It is mathematically proven that the social utility function is just the sum of the individual utilities.
http://ScoreVoting.net/UtilFoundns.html

Of course that is not what most voting methods are trying to maximize, because most voting methods were created by people who DID NOT UNDERSTAND THIS. For instance, when I was a naive idiot who first started thinking about voting methods back in the summer of 2006, I independently invented the Condorcet system (although I didn't choose a cycle resolution algorithm). I was sooo sure that the only logically sensible thing to do was to elect the candidate who would beat all others by a majority. It just seemed right.

But then after weeks of arguing with Warren Smith, I realized that not only was I wrong due to theorems like those discussed in my previous link. But also, when I accepted utilitarianism, all the weird idiosyncrasies of election theory melted away.

Gone was Arrow's Theorem (Independence of Irrelevant Alternatives was consistently upheld)
http://ScoreVoting.net/ArrowThm.html

Gone was non-monotonicity, or the no-show paradox, or reversal symmetry, or any of the other self-consistency-based criteria. Because when you aggregate preferences the right way, there's no way to produce contradictions.

[broken lader]

Qaanol,

Are you going to express the results as Bayesian Regret, or "voter satisfaction index"/"social utility efficiency"?

The obvious benefit of Bayesian Regret is that it allows you to see a difference in results with the same parameters except for, say, different numbers of candidates.

[Yakk]

It is mathematically proven that the social utility function is just the sum of the individual utilities.
http://ScoreVoting.net/UtilFoundns.html

That page does not say what you think it says.

It says "it would be nice for social utility to be the average/sum of individual utilities. What do we have to assume to make that the case?" It also lists various problems with and objections to the assumptions. It does not say "it is mathematically proven that social utility is just the same of individual utilities" by any stretch of the imagination.