Arrow's Impossibility Theorem

[Owen Densmore]

During the last Friam, we got talking about voting and Arrow's  
Impossibility Theorem came up.  It basically discusses anomalies in  
voting when there are more than two choices being voted upon.

The result depends strongly on how the votes are tallied.  So for  
example, in our last election, due to having three candidates, we  
entered the Arrow regime.  But "spoilers" like Ralph are not the only  
weirdness.

The html references below have interesting examples, and the pdf  
reference is a paper by SFI's John Geanakoplos who gave a public  
lecture last year.

"Fair voting" schemes are getting some air-time now a-days.  There are  
several forms, but the most popular I think is that you basically rank  
your candidates in order of preference, the "top-most" being your  
current vote. There are several run-offs which eliminate the poorest  
performer and let you vote again, now with the highest of your ranks  
still available.  This insures you always have a vote if you want one.  
This would have won the election here for Gore, for example, presuming  
the Nader votes would favor Gore.

Various web pages with examples:
   http://www.udel.edu/johnmack/frec444/444voting.html
   http://econ.gsia.cmu.edu/Freshman_Seminar/notes_on_arrow.htm
   
http://www.personal.psu.edu/staff/m/j/mjd1/ 
arrowimpossibilitytheorem.htm
   http://www.sjsu.edu/faculty/watkins/arrow.htm
Three proofs by John Geanakoplos
   http://cowles.econ.yale.edu/P/cd/d11a/d1123-r.pdf

Owen Densmore          908 Camino Santander       Santa Fe, NM 87505
[hidden email]    Cell: 505-570-0168         Home: 505-988-3787
AIM:owendensmore   http://complexityworkshop.com  http://backspaces.net

[Robert Holmes]

I'm having issues with this notion of transitivity (if aPb and bPc then
aPc). I can see how it seems all logical but it just doesn't reflect the way
I think about my preferences. It certainly doesn't bear any relation to how
I can express those preferences in a first-past-the-post electoral system.

Here's the gedanken conversation I'm having with a psephologist. We're
talking about the UK system - Blair (Lab), Howard (Con) and Kennedy
(Libdems):

P: So who would you prefer, Blair or Howard?
R: Easy - Blair
P: Blair or Kennedy?
R: Also easy - I'll take Blair, please
P: Finally, who would you prefer, Kennedy or Howard?
R: Errr.. neither of them. I prefer Blair.
P: No, you don't understand, I'm trying to construct your social welfare
function. Now who would you prefer, Howard or Kennedy?
R: Look, I don't like either of them. And anyway, I've only got one vote.
Even if I had a preference it's not as if I could express it. So I prefer
Blair.
P: No no no no. That's not a valid way of thinking. You can't have that sort
of preference. You've got to be transitive.
R: I apologise. I didn't realise.


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[Robert Holmes]

A party political broadcast on behalf of the SDP-Liberal alliance in the mid
1980s. John Cleese was a big supporter of theirs. Makes a good case for P.R.

http://play.rbn.com/?url=realguide/showbox/g2demand/wto/johncleese.rm&pr
oto=rtsp

[Roger Critchlow-2]

Hmm, you're trying to get a total ordering where only a partial ordering
is required.

Blair > Howard and Blair > Kennedy establishes no need to have an opinion
about Howard vs. Kennedy, Howard <> Kennedy is an acceptable answer.

A utility function on the candidates would establish a total ordering of
candidates.  But a transitive preference relationship only establishes a
partial ordering.  And a partial ordering is enough foundation for game
theory.

Which isn't to say you don't have real issues with transitivity, but you'll
need another gedanken conversation to illustrate them.

-- rec --

Robert Holmes wrote:

>I'm having issues with this notion of transitivity (if aPb and bPc then
>aPc). I can see how it seems all logical but it just doesn't reflect the way
>I think about my preferences. It certainly doesn't bear any relation to how
>I can express those preferences in a first-past-the-post electoral system.
>
>Here's the gedanken conversation I'm having with a psephologist. We're
>talking about the UK system - Blair (Lab), Howard (Con) and Kennedy
>(Libdems):
>
>P: So who would you prefer, Blair or Howard?
>R: Easy - Blair
>P: Blair or Kennedy?
>R: Also easy - I'll take Blair, please
>P: Finally, who would you prefer, Kennedy or Howard?
>R: Errr.. neither of them. I prefer Blair.
>P: No, you don't understand, I'm trying to construct your social welfare
>function. Now who would you prefer, Howard or Kennedy?
>R: Look, I don't like either of them. And anyway, I've only got one vote.
>Even if I had a preference it's not as if I could express it. So I prefer
>Blair.
>P: No no no no. That's not a valid way of thinking. You can't have that sort
>of preference. You've got to be transitive.
>R: I apologise. I didn't realise.
>
>
>
>  
>
>------------------------------------------------------------------------
>
>============================================================
>FRIAM Applied Complexity Group listserv
>Meets Fridays 9AM @ Jane's Cafe
>Lecture schedule, archives, unsubscribe, etc.:
>http://www.friam.org
>

[Frank Wimberly]

A couple of weeks ago I recommended a book to Robert on the Bayesian
approach.  It was "Good Thinking" by I. N. Good.  Robert said it was
philosophical and not mathematical enough so I said I would recommend
another.  Owen asked if I would send that recommendation to the group.  The
other book is "The Foundations of Statistics" by Leonard J. Savage.

For balance you may also want to ready my "boss's" essay "Why I am not a
Bayesian" in Clark Glymour, Theory and Evidence (Princeton: Princeton
University Press, 1980).

Also, I mentioned a scaling method used by psychologists in which subjects
are required to rank pairs of items.  The choices are then rated in a way
which assigns a value to each despite inconsistencies so that a linear order
can be constructed.  Owen asked for a reference.  It's called "the method of
paired comparisons" and the relevant citation is:  "Theory and Methods of
Scaling" by Torgerson (John Wiley and Sons, 1958).

Frank

[Frank Wimberly]

Sorry for responding to my own mail.

There is another interesting paper by R. Rosenkrantz with the title "Why
Glymour IS a Bayesian" in J. Earman, ed., Testing Scientific Theories, vol.
10, Minnesota Studies in the Philosophy of Science (Minneapolis: University
of Minnesota Press, 1983).

----- Original Message -----
From: "Frank Wimberly" <[hidden email]>
To: "The Friday Morning Complexity Coffee Group" <[hidden email]>
Sent: Friday, December 19, 2003 12:10 PM
Subject: [FRIAM] Follow-up to This Mornings Discussions


> A couple of weeks ago I recommended a book to Robert on the Bayesian
> approach.  It was "Good Thinking" by I. N. Good.  Robert said it was
> philosophical and not mathematical enough so I said I would recommend
> another.  Owen asked if I would send that recommendation to the group.
The
> other book is "The Foundations of Statistics" by Leonard J. Savage.
>
> For balance you may also want to ready my "boss's" essay "Why I am not a
> Bayesian" in Clark Glymour, Theory and Evidence (Princeton: Princeton
> University Press, 1980).
>
> Also, I mentioned a scaling method used by psychologists in which subjects
> are required to rank pairs of items.  The choices are then rated in a way
> which assigns a value to each despite inconsistencies so that a linear
order
> can be constructed.  Owen asked for a reference.  It's called "the method
of

> paired comparisons" and the relevant citation is:  "Theory and Methods of
> Scaling" by Torgerson (John Wiley and Sons, 1958).
>
> Frank
>
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9AM @ Jane's Cafe
> Lecture schedule, archives, unsubscribe, etc.:
> http://www.friam.org

[Andy Wuensche]

Dear Friam-ites
Announcing the release of multi-value Discrete Dynamics Lab
-----------------------------------------------------------
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from Cellular Automata to Random Boolean Networks and beyond.

This is a major update, including..
* Generalizing all aspects of DDLab for multi-value logic (up to 8
  values or colors), instead of just Boolean logic (0,1).
* Methods for finding multi-value "life"-like dynamics in 2d and 3d,
  with a variety of complex interacting/propagating
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Multi-value DDLab includes many other new features, improvements and
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DOS. Irix and MacOSX versions are in the pipeline.  The code is
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For further details and download see
http://www.ddlab.com 
or
http://www.cogs.susx.ac.uk/users/andywu/ddlab.html

Please forward this to other individuals or groups who might be interested.
regards
Andy Wuensche

[Owen Densmore]

Sorry for the (old) repeat, but Arrow's Impossibility Theorem came up  
at last Friam, so I thought I'd resend the past post.

The conversation relating to AIT included noting that the US two  
party scheme does not escape the issue, due to the primaries  
basically being a tournament run off amongst 3 or more candidates,  
and that the parties themselves are built from several coalitions,  
thus are greater than a 2-choice vote/game.

One thought I had on the matter was:
   - Let everyone vote in both primaries
   - Hold all primaries on the same day

BTW: Since this was written, Robert Holmes discussed with several of  
us one of the fair voting schemes in the UK.  I forget the details,  
but the aim was to insure the individual voters maximized their input  
into the vote.  Robert -- do you know where that scheme fits into AIT?

I would like to add that AIT is germane to ABM: may of the models  
have agents "voting" amongst each other for access to resources, and  
similarly, voting within themselves for behavior rules, often with  
knowledge of the community's preferences.  Both are within AIT, I'd  
guess.

We may want to put our heads together at an upcoming wedtech to see  
if we understand this, and its impact on our work.

     -- Owen


Begin forwarded message:

> From: Owen Densmore <owen at backspaces.net>
> Date: December 18, 2003 10:20:05 AM MST
> To: The Friday Morning Complexity Coffee Group <friam at redfish.com>
> Subject: [FRIAM] Arrow's Impossibility Theorem
> Reply-To: The Friday Morning Complexity Coffee Group  
> <Friam at redfish.com>
>
> During the last Friam, we got talking about voting and Arrow's  
> Impossibility Theorem came up.  It basically discusses anomalies in  
> voting when there are more than two choices being voted upon.
>
> The result depends strongly on how the votes are tallied.  So for  
> example, in our last election, due to having three candidates, we  
> entered the Arrow regime.  But "spoilers" like Ralph are not the  
> only weirdness.
>
> The html references below have interesting examples, and the pdf  
> reference is a paper by SFI's John Geanakoplos who gave a public  
> lecture last year.
>
> "Fair voting" schemes are getting some air-time now a-days.  There  
> are several forms, but the most popular I think is that you  
> basically rank your candidates in order of preference, the "top-
> most" being your current vote. There are several run-offs which  
> eliminate the poorest performer and let you vote again, now with  
> the highest of your ranks still available.  This insures you always  
> have a vote if you want one.  This would have won the election here  
> for Gore, for example, presuming the Nader votes would favor Gore.
>
> Various web pages with examples:
>   http://www.udel.edu/johnmack/frec444/444voting.html
>   https://econ.gsia.cmu.edu/Freshman_Seminar/notes_on_arrow.htm
>   http://www.personal.psu.edu/staff/m/j/mjd1/ 
> arrowimpossibilitytheorem.htm
>   http://www.sjsu.edu/faculty/watkins/arrow.htm
> Three proofs by John Geanakoplos
>   http://cowles.econ.yale.edu/P/cd/d11a/d1123-r.pdf
>
> Owen Densmore          908 Camino Santander       Santa Fe, NM 87505
> owen at backspaces.net    Cell: 505-570-0168         Home: 505-988-3787
> AIM:owendensmore   http://complexityworkshop.com  http://
> backspaces.net
>
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9AM @ Jane's Cafe
> Lecture schedule, archives, unsubscribe, etc.:
> http://www.friam.org

[Robert Holmes]

Errr.... I've forgotten the scheme too. I'll look up what I wrote when I get
back from the cradle of democracy (I'm heading out there to visit family
tomorrow). I think this is a great idea for a WedTech - just don't schedule
it for next week as I'll be up a hill in Yorkshire in the rain.

R

On 3/27/07, Owen Densmore <owen at backspaces.net> wrote:

>
> Sorry for the (old) repeat, but Arrow's Impossibility Theorem came up
> at last Friam, so I thought I'd resend the past post.
>
> The conversation relating to AIT included noting that the US two
> party scheme does not escape the issue, due to the primaries
> basically being a tournament run off amongst 3 or more candidates,
> and that the parties themselves are built from several coalitions,
> thus are greater than a 2-choice vote/game.
>
> One thought I had on the matter was:
>    - Let everyone vote in both primaries
>    - Hold all primaries on the same day
>
> BTW: Since this was written, Robert Holmes discussed with several of
> us one of the fair voting schemes in the UK.  I forget the details,
> but the aim was to insure the individual voters maximized their input
> into the vote.  Robert -- do you know where that scheme fits into AIT?
>
> I would like to add that AIT is germane to ABM: may of the models
> have agents "voting" amongst each other for access to resources, and
> similarly, voting within themselves for behavior rules, often with
> knowledge of the community's preferences.  Both are within AIT, I'd
> guess.
>
> We may want to put our heads together at an upcoming wedtech to see
> if we understand this, and its impact on our work.
>
>      -- Owen
>
>
>

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