Introducing standard deviation into Score Voting formula

[Eric Sanders]

Geeky, but something I've been wondering about...

Can anyone think of a good way to introduce standard deviation into a Score Voting calculation? For example, imagine two candidates being scored by two voters:

Candidate A: 0, 10 = 10 total, 5 avg.

Candidate B: 5, 5 = 10 total, 5 avg.

The totals and averages are identical, but Candidate A's scores were 0 and 10 while Candidate B's scores were 5 and 5. Do we really want to call this result a "tie"?

[Benjamin Grant]

I am very interested in responses to this questions – if they don’t go over my head.

 

-Benn Grant

eFix Computer Consulting

be...@4efix.com

603.283.6601

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[Aaron Hamlin]

I think the chances of a tie in Score Voting are extremely rare when you have a reasonably-sized electorate. If it weren't I could maybe see the case for selecting a candidate that has less variance in her/his score. But because it's so rare, I'm not sure that it matters.

[Carl S. Milsted, Jr.]

 

On Thu, 3 Oct 2013 08:50:14 -0700 (PDT), Eric Sanders wrote:

  Geeky, but something I've been wondering about...

  Can anyone think of a good way to introduce standard deviation into a Score Voting calculation? ...

 

--

I brought up this issue sometime back without acclaim. In my opinion, with honest votes the candidate with the higher standard deviation should lose.

 

My reasoning is as follows: when it comes to governance, strict utilitarianism is morally questionable. To some degree of natural rights should also figure into the equation. This is why we have a Bill of Rights and other constitutional restrictions on legislatures.

 

Governments violate rights. That's what they do. Governments exist to handle those cases where the sum of individual choices does not lead to a happy solution. Well behaved governments violate individual choice in order to handle public goods, externalities, etc. Alas, even when we can objectively determine what constitutes a public good or an externality, values of public goods and the costs of externalities are still subjective. How valuable is that public park? That wider highway? Just how bad is it to have a bit of air pollution? Or that noisy motorcycle?

 

This is why we have elections.

 

Governments cannot please everyone since we have different preferences for public goods and different dislikes of externalities. To violate individual liberty to enforce one set of preferences is violate the rights of those with different preference levels. A rights based voting system would then attempt not to maximize utility, but to minimize rights violation.

 

If people voted honestly, I'd elect the candidate who has the least squared negativity vs. the candidate with the highest mean. That is, on a 0-10 scale, a 9 vote is a negativity of 1, an 8 a negativity of 4,...a 5 a negativity of 25, etc. With such a count it pays not to violate the rights of minorities.

 

But people don't vote honestly, and I haven't figured out a voting system which approximates this criterion...yet.

[Jameson Quinn]

Coincidentally, I'm right this instant doing my statistics homework on means and standard deviations using Lebesgue integrals. So, taking a busman's holiday from the homework:

One easy-to-find parameter of interest is: for candidates A and B, what is the chance that a random sample of N voters, with replacement, from the actual electorate, will elect A? That would be called a bootstrap win %, and a candidate with a higher standard deviation will be more random, while one with lower STDEV will tend to get numbers more like 1 or 0. If you did that pairwise with N=1 a bunch of times, the most frequent winner would be a Condorcet winner; if you did it groupwise with N=1, you'd get plurality. If you do it pairwise with N=large, you'd get a median system (GMJ, I think); and if you do it groupwise with N=large you get Range. So this trick would let you get hybrids in between those 4 corners, by varying the average number of candidates you threw into the ring, and the average number of ballots you picked. This is computationally tractable; you could pretty easily prove which candidate had the best win probability analytically, if the empirical bootstrap percentages came out close to a tie.

Wow. That's awesome. 4 of the 5 "good" voting systems (except not approval), all seen as special cases of the same basic procedure. And approval, the fifth, is of course the special case of all those special cases.

If this isn't clear to someone... I don't have time to explain this instant, but I'll definitely come back and post more with a bigger exploration of this idea some time later.

Jameson

2013/10/3 Eric Sanders <er...@electology.org>

[Jameson Quinn]

Whoops. I certainly did not mean to claim that plurality was one of "the 5 good voting systems". Still, it's pretty cool. More later.

2013/10/3 Jameson Quinn <jameso...@gmail.com>

[Warren D Smith]

On 10/3/13, Jameson Quinn <jameso...@gmail.com> wrote:
> Coincidentally, I'm right this instant doing my statistics homework on
> means and standard deviations using Lebesgue integrals. So, taking a
> busman's holiday from the homework:
>
> One easy-to-find parameter of interest is: for candidates A and B, what is
> the chance that a random sample of N voters, with replacement, from the
> actual electorate, will elect A? That would be called a bootstrap win %,
> and a candidate with a higher standard deviation will be more random, while
> one with lower STDEV will tend to get numbers more like 1 or 0. If you did
> that pairwise with N=1 a bunch of times, the most frequent winner would be
> a Condorcet winner; if you did it groupwise with N=1, you'd get plurality.

--agree re Condorcet & plurality.

> If you do it pairwise with N=large, you'd get a median system (GMJ, I
> think);

--disagree. Counterexample election:
A's scores 0,0,0,0,0,5,6,6,6,6,6,6
B's scores: 4,4,4,4,4,4,9,9,9,9,9,9
A has greater median with 5. B has greater average. I wrote this example
for six below-median scores and six above, but imagine that "six" is
"very large."
Now upon taking a random sample of N scores, N large, the one with the higher
average will almost surely win (i.e. will have higher sample average).

>and if you do it groupwise with N=large you get Range.
> So this
> trick would let you get hybrids in between those 4 corners, by varying the
> average number of candidates you threw into the ring, and the average
> number of ballots you picked.

--careful, above error indicates this is not quite what you think it is.

> This is computationally tractable; you could
> pretty easily prove which candidate had the best win probability
> analytically, if the empirical bootstrap percentages came out close to a
> tie.

--More precisely: Given V range-style ballots, we wish to determine
"who would have
the greatest win-probability, if we selected N ballots at random &
ignored the rest?"
(0<N<=V.)

This is an interesting algorithm-design problem. (Actually there
could be two problem-variants -- sampling with & without replacement.)
However, at present, I am not seeing any efficient algorithmic answer
to either problem variant. The obvious
algorithm is to exhaustively enumerate every possible sample. Without
replacement there are binomial(V,N) samples. With replacement there
are V^N.
These numbers are exponentially large. (Consider V=2000000, N=1000000.)
So the obvious algorithm is infeasible.

You could use Monte Carlo experiments to get a probabilistic answer
(that comes with
a statistical confidence) but in close elections you'd be screwed and
I'm not seeing
an efficient "analytical" way to resolve near-ties.

Is this merely because I am too stupid to see a good algorithm (which
nevertheless exists) or is it because this problem really is hard?

[By the way, I actually used this technique in the past to examine a
poll by me, Greene & Quintal for 2004 USA presidential election. The
point was by subsampling our poll & doing Monte Carlo bootstrapping
you get confidence intervals for various claims about
the election. Balinski & Laraki also did a similar study another
time, thought of the same kind of stat-analysis technique, which by
the way is not due to us, this is called the "bootstrap" idea and has
been around for a while.]

Anyhow. I think this is a good algorithm problem. I would initially guess
it is #P-hard or NP-hard, probably "weakly" so (like the NUMBER
PARTITION problem,
read book by Garey & Johnson) in which case this whole idea is,
although nice in some
mathematical sense, useless practically.

That "weakness" insight pays off -- there is a "pseudopolynomial time algorithm"
in the event the allowed scores are selected from a fixed finite set
such as (to be concrete) the 10-element set {0,1,2,3,4,5,6,7,8,9}.
It works as follows. You compute the entire probability distribution
for the C score-totals of the N-voter sample (C=#candidates) with N=1.
Then, you add one more voter to the sample and recompute the entire
probability distrib.
You keep doing that, adding 1 more, then recompute entire probdist.
The point is
that with that fixed finite score-set, the entire probdist can be represented as
a table with at most (10*N)^C entries, each entry is a probability value.
Since C and 10 are constants, this is a polynomial function of N.

If however, "10" is instead exponentially large then this is no longer
polynomial
(that is the meaning of the terminology "pseudo"polynomial time).
Also if C is allowed
to be unbounded, ditto.

> Wow. That's awesome. 4 of the 5 "good" voting systems (except not
> approval), all seen as special cases of the same basic procedure. And
> approval, the fifth, is of course the special case of all those special
> cases.
>
> If this isn't clear to someone... I don't have time to explain this
> instant, but I'll definitely come back and post more with a bigger
> exploration of this idea some time later.
>
> Jameson

--yeah, well, it looks like your idea was somewhat half-baked.
However, there is certainly an interesting problem in there.
As I said I would guess that this core problem can be proven to be
#P-hard or NP-hard.
But that would be the interesting research -- to prove (or disprove) that.
Can you? I do not currently see a solution, but feel there is a
decent chance this can be accomplished.



--
Warren D. Smith
http://RangeVoting.org <-- add your endorsement (by clicking
"endorse" as 1st step)

[Jameson Quinn]

Hmm... too tired right this instant to figure this out. Re the "analytical" thing, I think you are right; my "analytical" intuition was based on bounded C and R. But as for the rest of it... better sleep on it.

2013/10/3 Warren D Smith <warre...@gmail.com>

[Eric Sanders]

"But people don't vote honestly..."

You raise an interesting question, Carl, and this is a bit off-topic re: the original thread, but here's my view: Voting is inherently strategic. Therefore, it is not a problem that people "don't vote honestly," because the goal of a voting method is not to encourage people to vote "honestly" but to give them the broadest ability to express their strategic votes/scores and then calculate these strategic votes/scores in a manner that produces a result with maximal utility (= broadest overall satisfaction).

The reason that voting is inherently strategic is simple: When you vote (including Score Voting) you are comparing options to each other. You must use strategy in order to determine how much you want one option vs. the others. It is simply not an option to not think strategically, since you have a specific goal in mind--to help elect the candidate(s) you most prefer.

Conversely, when you rate a product on Amazon.com, you are rating it against an abstract, idealized version of that product. You have no vested interest in the outcome and therefore have no reason to think strategically about how to rate the product; in this case, you could be said to be "rating honestly."

The term "vote honestly" is a bit of a misnomer, as far as I'm concerned. All voting is strategic--and not only is this not a problem, it's built into the very notion of "voting" itself. It's unavoidable and we should embrace this fact when evaluating voting methods.

[Carl S. Milsted, Jr.]

On Thu, 3 Oct 2013 13:57:52 -0700 (PDT), Eric Sanders wrote:

...

  You raise an interesting question, Carl, and this is a bit off-topic re: the original thread, but here's my view: Voting is inherently strategic.

 

I agree that voting is inherently strategic. I'm just saying how I'd handle standard deviation if voting wasn't.

 

My ultimate point is that minimizing Bayesian Regret is not my optimal criteria for a voting system. The distribution of the regret matters, since government is by nature an imposition -- the majority must pay some penalty for governing in a way far at odds with the minority.

 

When I get the time -- sometime after the sun cools down at bit at the current rate -- I'd like to run some of Warren's simulations with my distributed regret scoring system.

[Clay Shentrup]

On Thursday, October 3, 2013 9:22:41 AM UTC-7, Carl Milsted wrote:

some degree of natural rights should also figure into the equation.

"Rights" are just "utility increasers", like "not having cancer" or "being born into a wealthy family".

[Bruce Gilson]

On Thu, Oct 3, 2013 at 6:25 PM, Carl S. Milsted, Jr. <cmil...@quiz2d.com> wrote:

[...]

 

My ultimate point is that

minimizing Bayesian Regret is not my optimal criteria for a voting system. The distribution of the regret matters, since government is by nature an imposition -- the majority must pay some penalty for governing in a way far at odds with the minority.

 

This of course (that minimizing Bayesian Regret is not my optimal criteri[on] for a voting system) puts you at odds with Clay. But I think I'm more inclined to agree with your position than his.

[Eric Sanders]

Carl, I am super interested in where you're going with this. I look forward to seeing how your simulations compare with Warren's!

On Thu, Oct 3, 2013 at 6:25 PM, Carl S. Milsted, Jr. <cmil...@quiz2d.com> wrote:

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[Carl S. Milsted, Jr.]

On Thu, 3 Oct 2013 15:30:01 -0700 (PDT), Clay Shentrup wrote:

  "Rights" are just "utility increasers", like "not having cancer" or "being born into a wealthy family".

 

I beg to differ, Clay. Though I actually come out more of a consequentialist than a rights based political theorist, I do think the concept of natural rights is operatively useful.

 

Rights are simply what you are allowed to have/do without interference. I have the legal right to do certain things in my back yard. My ancestors had the legal right to own slaves.

 

Though there be multiple definitions of "natural rights" floating about, I use the term to mean rights which would be granted by a state of nature. In a state of nature I could roam about at will, hunt any available animals, exploit any natural resources, take any herbal intoxicants, wear what I would, etc. I would not have the right of childcare, healthcare, own slaves, etc. Some of these latter rights are indeed rights in some welfare states, but they are not natural rights. They require an imposition on someone in order to provide them.

 

Contra most libertarians, property rights also compromise natural rights. My right to private property in land interferes with the rights of others to trespass upon that land, use said land, etc. It is impossible to avoid truncating natural rights once you add in a society of significant population. The question is how we divide up the available natural rights to 6 billion people.

[Eric Sanders]

How do we divide up "natural rights" with 2 people, let alone 6 billion? And do you factor other animals into your calculation, or only humans? Seriously wondering, not trying to be snarky...

--

[Carl S. Milsted, Jr.]

On Fri, 4 Oct 2013 02:37:44 -0400, Eric Sanders wrote:

  How do we divide up "natural rights" with 2 people, let alone 6 billion? And do you factor other animals into your calculation, or only humans? Seriously wondering, not trying to be snarky...

 

With two people it is pretty easy to divide up the earth such that each has more land than they can use without interfering with each other. Both can live whatever hunter gatherer lifestyle they like. As we ramp up the population, however, it is impossible for every to live a leisurely hunter-gatherer lifestyle in an ideal climate. Contra Ayn Rand, warfare is more the nature of Man than production -- unless you believe in a recent creation in which the first humans were tending a certain garden... To live the idyllic hunter-gatherer lifestyle means culling the population of neighboring tribes.

 

Relative peace and productivity have put quite a squeeze on nature. Culling the population through warfare is an unpleasant process and would probably destroy what's left of much of nature given modern weapons. So I am interested in peaceable means to divide what we have.

 

How to divide up land rights with a dense population is an interesting challenge. Henry George advocated taxing "ground rents". The idea predates George. Thomas Paine and Adam Smith discussed the idea. It got traction under George because the frontier had just closed. Taxing natural resource use and land vs. taxing labor would be a good start. Keeping some lands public, including roads and parks, preserves some of the old right to roam (sorry libertarians). In Norway, you can take a walk through someone else's land without it being trespass as long as you stay out of view of their home. I'm not sure I entirely like the idea, but it is an interesting compromise.

 

Some people argue that the taxes thus collected should be given out as a dividend to all. This makes those who own large amounts of quality land pay rent to those who use less natural rights directly. One problem with this arrangement is that you can collect more rent for your family by having lots of children. The ancient Law of Moses handled the problem by allocating land to families equally, and mandating that the land pass through inheritance only. You could rent out your share, but you could not sell your descendants birthright. Conversely, if you have more children than your land can support, some of them need to go out and get jobs.

 

As for animals, we can look at it from two sides. From the human side we clearly see that the libertarian ideal of non-initiation of force is not a natural right. Wild animals steal from humans. Predators eat humans. Only non-initiation of force from humans would exist in this theoretical construct. So I have mellowed out and am willing to be a bit fuzzy on the concept of natural rights. But I still think it a reasonable point to approximate what constitutes a taking.

 

From the animal point of view, I would say that animal rights advocates who demand veganism are off base. Predation is natural. Penning animals into concentration camp conditions, however, is not. I am willing to pay extra for free range meats, and it wouldn't bother me if the animal rights people successfully outlawed some modern meat farming practices (which are actually subsidized by the way we subsidize corn).

 

 

[Thomas]

I'd rather require qualified majority of zero (definite disapproval) vs.
non-zero (nuances of approval) scores, so that very controversial
candidates can be eliminated.

[Bruce Gilson]

What is this attachment you put on. I don't recognize it as any kind of file I've ever seen before.

[Thomas]

It's an industry standard S/MIME signature.
Most major MUAs (Outlook, Apple Mail, Thunderbird ...) have built-in
support for S/MIME encryption.