

Arrow's Theorem - infinity
http://plato.stanford.edu/entries/arrows-theorem/

======
YokoZar
Arrow's theorem has to be one of the most overhyped and overplayed theorems in
existence. Here, for instance, are three major things what it doesn't say: 1)
That all voting systems are equivalent in any sort of meaningful sense 2) That
the current voting system you are using is not awful 3) That a perfect class
of voting system doesn't exist if you are willing to accept that rock-paper-
scissors situations can happen among people's preferences.

I'm not being hyperbolic here. Number 1 and 2 are how people quickly use vague
citations to Arrow's Theorem to shut down talk about voting reform even when
the status quo consists of provably terrible systems like plurality voting.

Number 3 is the true result that if you relax the rather overly-strongly
defined IIA criteria with a much more-reasonable criteria -- that the winner
must remain among the top rock-paper-scissors loop of the voters -- then
Arrow's theorem simply doesn't apply. This is well known: that "top loop" is
the Smith Set and every Condorcet method of voting satisfies it.

There's also another interesting result: if voters have merely "single-peaked
preferences", such as opinions about where to set a volume knob, then Arrow's
theorem also doesn't apply since there will be no rock-paper-scissors set of
equally fair options.

~~~
aaron-lebo
Number 3 is a valid criticism, but criticizing the theorem on the basis of
people using it to support the status quo is like criticizing water because
some people drown.

His theorem says nothing about the way things should be, but rather about how
things are. I mentioned Poundstone's _Gaming the Vote_ in another post, but
that book in particular discusses the implications of the theorem and analyzes
better alternatives to current systems.

~~~
Retric
The problem is people assume it means more than it does. It's like someone
using Godel's incompleteness theorem to argue that math is pointless.

------
tunesmith
"Arrow's theorem says there are no such procedures whatsoever—none, anyway,
that satisfy certain apparently quite reasonable assumptions concerning the
autonomy of the people and the rationality of their preferences."

The trick there is the "apparently quite reasonable" part. Far too many people
take it as a given that those assumptions are sacred, which leads to a form of
worship about this theorem. The IIAC in particular is problematic, and if that
is relaxed, it's not quite so certain anymore that a "perfect" vote-counting
method is impossible.

~~~
baddox
I'm curious which of the assumptions you think are unreasonable.

~~~
tunesmith
If you have a Smith Set, you don't want to just arbitrarily remove one of its
candidate to resolve the "tie".

Similarly, if an election has a Condorcet Winner, and _adding_ a candidate
creates a Smith Set (thus potentially violating IIA if a certain candidate is
chosen in a "tiebreaker"), this is not necessarily a bad thing.

If that happens it means that people always legitimately had those views, and
were not able to properly express their views by choosing among the more
limited number of candidates.

~~~
baddox
Regarding your second sentence, who says that would be a bad thing? I'm a
little confused about the situation you're describing. Most analysis of voting
systems I've seen isn't concerned with the fact that valid candidates may be
left off the ballot. That's obviously an important consideration in real
elections, and if there are primary elections the choice of voting system
there obviously matters, but in the final election it doesn't seem
particularly relevant.

------
Kryptor
This theorem only applies to _ordinal_ voting systems. Cardinal voting
algorithms like score and approval voting escape this predicament.

[http://rangevoting.org/ArrowThm.html](http://rangevoting.org/ArrowThm.html)

~~~
tunesmith
Cardinal voting algorithms are only superior if people vote sincerely.
Unfortunately, cardinal voting methods offer significant incentives to vote
insincerely. Condorcet voting does not.

I always thought an interesting vote gathering technique would be something
that actually interviews/polls the voter. Ask them for their ordinal ranking.
The voter would know that if a Condorcet Winner exists, then that candidate
would be the winner, but also ask for various cardinal ranking numbers, along
with their approval line. That way there would still be the incentive to vote
sincerely, while the cardinal information could be used to break the Smith Set
loops. (The only downside here is that some people claim that the presence of
cardinal tiebreakers creates an incentive for people to vote insincerely to
_create_ a Smith Set...)

Alternatively, if a Smith Set occurs, schedule a second round of voting for
only those candidates (like a Louisiana Runoff, but Condorcet style), so that
voters could better educate themselves on the remaining candidates.

~~~
YokoZar
I will note that a Smith Set Condorcet loop can occur even among 100%
rational, wholly informed voters. It's not an aberration due to irrationality,
it's just a fact of politics not being one-dimensional (quite literally -- if
voters rank candidates based on whomever is closest to them in n-dimensional
space, then for any n > 1 you can have a cycle).

You can draw your own example of this if you like. Draw an equilateral
triangle and its altitudes, creating 6 regions inside. Put some dots in the 6
regions in the middle (voters), but leave every other region blank. Now,
declare the vertices to be candidates (a 3-way race). If you compare any two
of them, and have voters vote for whomever they're closest to (based on which
side of the altitude they fall on), you'll end up with a rock-paper-scissors
situation.

~~~
tunesmith
It's interesting and it brings up the question of how a vote should actually
be interpreted if there is that kind of legitimate Smith Set. The only options
I can think of are either factoring in intensity of preference (see above), or
some kind of power-sharing agreement.

Also interesting to me is that I believe the IIAC can actually _uncover_ that
kind of completely-legitimate cycle. Meaning, while introducing an additional
candidate can never lead you from one Condorcet Winner to another, it can lead
you from a Condorcet Winner to a Smith Set. If people change their preferences
in that manner, then it means that they have found better choices for them. In
other words, if IIAC happens, it could be an indication that the original set
of candidates wasn't really appropriate for the voters in the first place.

Thinking about both at the same time is uncomfortable, because if Smith Sets
aren't an indication of voter-population confusion that can be resolved with
more education and communication, then it basically means that the more choice
you offer, the less likely there will be one candidate deserving of victory.
If that's true, then making the arbitrary choice (among most likely
candidates) might actually be the best outcome. Not exactly democratic though.

~~~
baddox
I think nondeterminism is fairly reasonable in a cyclical Smith set. In fact,
I think nondeterminism in voting systems has a worse reputation than it
deserves.

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theschwa
Is there a good data structure to represent Ordinal Data? I frequently see
pairwise count matrices used [1], but they don't seem to completely capture
all of the information. For example, the number of times a choice was first,
second, third, or last such as would be necessary for the Borda method [2].

[1]
[http://en.wikipedia.org/wiki/Condorcet_method#Pairwise_count...](http://en.wikipedia.org/wiki/Condorcet_method#Pairwise_counting_and_matrices)
[2]
[http://en.wikipedia.org/wiki/Borda_count#Voting_and_counting](http://en.wikipedia.org/wiki/Borda_count#Voting_and_counting)

~~~
rspeer
Well, Condorcet and Borda do fundamentally different things with people's
preferences.

There's no real way to simplify it beyond "here's a set containing the rank
order of every voter" if you don't know in advance what election method will
be used. In fact, you might argue that the rank order is insufficient, and you
also need to know whether the voter would approve of each option in an
approval vote, or 0-10 rankings for range voting, or whatever.

Condorcet cares most of all about people's preferences between pairs of
options, so that's why you'd summarize it as a matrix made out of those
pairwise preferences. Borda cares most of all how many times something is
"first place", "second place", and so on, and cares nothing about the pairwise
rankings; Borda advocates claim that pairwise rankings will just confuse the
issue.

(Opinion: I happen to believe that Borda is garbage for anything where
politics is involved. It might be okay for the kind of contests where it's in
use, like the baseball MVP or Eurovision, except those are probably secretly
political too.)

------
yomritoyj
Interestingly, the theorem is false if the number of voters is infinite
[http://blog.jyotirmoy.net/2013/10/arrows-impossibility-
theor...](http://blog.jyotirmoy.net/2013/10/arrows-impossibility-theorem-is-
false.html)

------
aaron-lebo
William Poundstone's _Gaming the Vote_ is a great read that covers this and
related topics.

------
chimeracoder
> Arrow's theorem says there are no such procedures whatsoever—none, anyway,
> that satisfy certain apparently quite reasonable assumptions concerning the
> autonomy of the people and the rationality of their preferences

The article alludes to one very important corollary, though IMHO it doesn't
explain it very well: Arrow's theroem is like the CAP theorem - it seems to be
a much stronger restriction than it is. In other words, if you're willing to
make just a few very straightforward assumptions (ie, compromises), you _can_
create a system that appears to "satisfy... rationality of their preferences".

Nobel laureate Amartya Sen[0] has demonstrated that if you assume that there
are certain rankings of preferences that are rare enough to be ignored
altogether, then instant-runoff voting[1] will in fact satisfy all the
constraints of the Impossibility Theorem[2].

Let's use the 2000 US Presidential election as an example. There were three
main candidates in Florida (Bush, Gore, Nader), for a total of 6 rankings.
While Nader played a spoiler role, Nader and Gore shared more of a platform
than Nader and Bush did. So it is very reasonable to assume that there are few
people who would have voted for Nader over Bush, but Bush over Gore. This
reasoning allows us to eliminate a number of those 6 rankings - and more
importantly, enough rankings that the criteria of the Impossibility Theorem
are likely to hold.

[0]
[https://en.wikipedia.org/wiki/Amartya_Sen](https://en.wikipedia.org/wiki/Amartya_Sen)

[1] [https://en.wikipedia.org/wiki/Instant-
runoff_voting](https://en.wikipedia.org/wiki/Instant-runoff_voting)

[2] The article does mention Sen, and alludes to this finding, but I don't
think it explains it very clearly.

~~~
tunesmith
This is probably just being pedantic, but the Nader->Bush->Gore ranking was
commonly preferred by Nader supporters, not because they preferred Bush's
policies to Gore's, but because they thought it would teach society the error
of their ways in the long run or something. Also because they hated Nader
being labeled a spoiler candidate, so it was probably something of a spiteful
response.

------
jordanpg
A fascinating result. I am interested in understanding how this can apply to
decisions about consuming news.

Given some objective, like "stay informed" or "make the best possible choice
when I vote", and given certain problems like "news is corrupted by
advertising" or "news is dumbed down", is it at all plausible to achieve any
of those goals in 1 hour of news consumption each day?

A naive application of this theorem suggests not... "none, anyway, that
satisfy certain apparently quite reasonable assumptions concerning the
autonomy of the people and the rationality of their preferences."

In other words, might it be the case that there is little intrinsic value in
consuming the news piecemeal, in the way that most of us do?

~~~
yongjik
NOPE.

It just means that those " _apparently_ quite reasonable assumptions" weren't
reasonable after all and must be relaxed a bit according to a precise
definition.

You just took a mathematical theorem and went into a completely unrelated
tangent. It's as if someone says there's no point in keeping development
schedule because temporal order is relative to the observer.

Please don't do that.

~~~
jordanpg
"Say there are some alternatives to choose among. They could be policies,
public projects, candidates in an election, distributions of income and labour
requirements among the members of a society, or just about anything else.
There are some people whose preferences will inform this choice, and the
question is: which procedures are there for deriving, from what is known or
can be found out about their preferences, a collective or “social” ordering of
the alternatives from better to worse?"

Keep reading. This one is more than just math.

------
Symmetry
Always interesting, but given the median voter theorem is I'm not sure it's a
problem in practice.

[1][http://en.wikipedia.org/wiki/Median_voter_theorem](http://en.wikipedia.org/wiki/Median_voter_theorem)

~~~
eru
That only applies for one dimensional politics, and, if plurality voting is
used, a two party system.

