

Arrow's impossibility theorem - Gray0Ed
http://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem

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gjm11
A few remarks:

1\. Arrow's theorem concerns the situation where your election procedure needs
to deliver (not just a single winner, but) a _ranking_ of all the candidates.
You might hope that relaxing this condition will help, but ...

2\. There's a closely related theorem with the magnificent name of Gibbard-
Satterthwaite, which says that if you have more than two candidates, any
procedure that takes in ranked preferences and spits out a single winner must
(1) give all the power to one voter, or (2) leave at least one candidate
unable to win whatever the voters' preferences, or (3) be susceptible to
tactical voting, meaning that in some situations a voter does best to rank the
candidates in an order that doesn't match his or her actual preferences.

3\. However, there is a loophole "at the other end". For instance, if the
input consists not of _rankings_ but of _scores_ (e.g., from 0 to 100), then
the conditions of Arrow and Gibbard-Satterthwaite don't apply. And, in fact:

4\. If there are only three candidates then "range voting" or "score voting"
(each voter scores every candidate and the candidate with best average or
total score wins) has the desirable properties Gibbard & Satterthwaite forbid
for ranking-based voting systems. (Almost: sometimes optimal voting strategy
might require you to give two candidates the same score even though you have a
definite preference between them.) But, alas,

5\. With more than three candidates no score-based system has those properties
either.

(An interesting simplification of range voting is "approval voting", where the
only possible scores are 0 and 1.)

~~~
MichaelDickens
For #2, what if you relax the "no tactical voting" requirement to say that
voters cannot predict how to vote tactically unless they have an impractically
large quantity of information about other voters?

~~~
YokoZar
This is the insight. Proving "tactical voting is always theoretically
possible" sounds bad, until you realize that there exist systems where
tactical voting requires not only nearly perfect information about every other
voters exact preferences ahead of time, but also solving cryptographically
hard problems

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verteu
An amusing anecdote which illustrates why independence of irrelevant
alternatives is desirable:

After finishing dinner, Sidney Morgenbesser decides to order dessert. The
waitress tells him he has two choices: apple pie and blueberry pie. Sidney
orders the apple pie. After a few minutes the waitress returns and says that
they also have cherry pie at which point Morgenbesser says "In that case I'll
have the blueberry pie."

~~~
mcphage
When you say it like that, IIA does sound pretty obvious. But if you change
the terms a little bit, you can see why the IIA doesn't match up with how
people actually vote:

Say there's an election between a moderate democrat "blueberry pie" and a
third party liberal "apple pie". As a liberal, Sidney would rather vote for
the third party ("Sidney orders the apple pie"). However, if you introduce a
republican candidate "cherry pie", Sidney will probably vote for the democrat
(blueberry pie) instead of the third party candidate, because he'd be worried
about his vote costing the more moderate candidate the election.

IIA means that you won't vote differently than your preferences—but people do
that all the time. And sure, a voting system where that wasn't necessary would
be nice, but losing that condition isn't as nonsensical as it seems at first.

~~~
logicchop
In the context of voting, all IIA represents is the requirement that we only
take into account the information on the ballots..

~~~
dllthomas
I don't see this. Could you elaborate?

~~~
logicchop
I recall that Bordes and Tideman were a good source on this issue; I believe
the (relevant) paper is "Independence of Irrelevant Alternatives in the Theory
of Voting." \-- The upshot is that the condition known as IIA (I think it is
sometimes known as Sen's condition-alpha, and the condition that many people
harp on, including Michael Dummett) is in fact a stronger condition than is
needed for the result; roughly, that instead of needed a condition about
consistency among selections over possible ballots (like: if y were selected
in ballot [xyz], y should be selected among yz on ballot [yz]..) we simply
need a requirement that the selection only involves information on the ballots
(and that the relative rankings of candidates not on the ballot are irrelevant
to the selection).

~~~
dllthomas
Ah, I think your phrasing was confusing. "IIA" _is_ considering an attribute
of the ballots, but including IIA is stronger than necessary for the theorem
to hold - it can be replaced with the much weaker "nothing but the ballots".

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marcoperaza
The criteria considered by the theorem concern how _democratic_ a voting
system is. I'd argue that 'democraticness' is secondary to two other criteria:
accountability and government-effectiveness. Accountability means that bad
rulers can get voted out of office when enough people are displeased.
Government-effectiveness means that the resulting government can practice good
governance. When considering these criteria, I think plurality/first-past-the-
post voting systems are superior. Unpopular leaders are voted out and
elections usually result in single-party majorities that can govern
effectively.

------
Tloewald
It seems to me that concern over the dictatorship criterion is ill founded.
Its failure merely implies that for any given set of alternatives there will
always be one voter whose preferences, whatever they may be, determine the
outcome. Now if everyone's preferences were known and constant then in theory
you could find that person and they would be able to decide the outcome. In
practice preferences don't stay constant and no-one knows them all, so as long
as you have secret ballots the dictatorship criterion doesn't matter. So you
can have the other two criteria which are far more important.

------
c-slice
Can I get a simple wikipedia explanation? That was the most challenging
wikipedia article I've ever read.

~~~
schmit
In short:

For a voting system (ranking of some candidates based on preferences of
voters), it would be nice if:

\- A single voter cannot determine the ranking (as a dictator) \- For every
possible set of voter preferences, there is an outcome (not random) \- If
everyone likes candidate A over candidate B, then in the final ranking
candidate A should be ranked higher than candidate B \- If one prefers A over
B when comparing just A and B, then one should also prefer A over B when an
additional option C is offered

Sounds like some reasonable properties for a voting system, right?

Well, the theorem states that if there are more than 2 candidates, then there
is no voting system that has all 4 properties above.

~~~
jonahx
Are the counterexamples offered by the theorem pathological, in the sense that
they are unlikely to occur in practice but are theoretically possible? Or
would they arise in practice frequently using standard rank voting systems?

~~~
YokoZar
As long as politics is not one-dimensional, there are completely reasonable
cases where you get a rock-paper-scissors situation between 3 top candidates
if voters select whomever is closest to them. That in turn violates the
criteria that says the election result can't change if you add a non-winning
candiate (rock beats scissors, but when paper enters the race now scissors is
computed the winner).

This criteria is called "independence of irrelevant alternatives". A common
criticism of Arrow's theorems usefulness is that it is a bit of a stretch to
call paper an "irrelevant alternative" when rock-paper-scissors forms a cycle
of preferences like that.

If politics is one-dimensional ("single-peaked preferences" in the article),
then you never get this situation.

It's also important to note that a much more reasonable criteria exists called
"local independence of irrelevant alternatives". This is the idea that total
losers joining the race don't affect the result, however someone who is in the
top rock-paper-scissors cycle (the "Smith Set") can still affect the result
even if they don't win themselves. This is far more reasonable, as it's fairly
arbitrary which of the candidates in that top cycle should win.

When there is no top cycle, the Smith Set is a single person (the "Condorcet
Winner"). When there is a Smith Set, most reasonable voting systems will pick
their winner somewhat arbitrarily as a member of the Smith Set, since everyone
in the Smith Set beats everyone outside the Smith Set.

------
aaron-lebo
A great popular read that covers this and similar topics is William
Poundstone's _Gaming the Vote_.

[http://www.amazon.com/Gaming-Vote-Elections-Arent-
About/dp/0...](http://www.amazon.com/Gaming-Vote-Elections-Arent-
About/dp/0809048922)

------
FLengyel
I wrote about a linear programming proof of Arrow's Impossibility Theorem due
to Rakesh Vohra and his collaborators in a series of posts, starting with
[http://deniallogic.blogspot.com/2015/04/transitivity.html](http://deniallogic.blogspot.com/2015/04/transitivity.html)
and ending with a proof of Arrow's theorem in
[http://deniallogic.blogspot.com/2015/05/arrows-
theorem.html](http://deniallogic.blogspot.com/2015/05/arrows-theorem.html).
The point was to fill in enough details that, for me, were missing from the
paper [1].

1\. Jay Sethuraman, Teo Chung Piaw, and Rakesh V. Vohra. Integer Programming
and Arrovian Social Welfare Functions. Mathematics of Operations Research Vol.
28, No. 2, May 2003, pp. 309–326.

------
VanillaCafe
Maybe I don't quite appreciate the significance. The Informal Proof section
seems to boil down to: If the vote is tied and there is one vote left, then
that last vote determines the outcome. The existence of a swing vote in this
circumstance doesn't seem very surprising.

~~~
verteu
"Part One" of the informal proof is trivial, for the reason you've described.
But "Part Two" is not: It shows that if the "pivotal voter" from Part One
votes B>C, then B will beat C in the election, even if _everyone else_ votes
C>B.

------
CurtMonash
I never published them, but I proved various extensions to the theorem back in
the day. Giving the ordinal voters more options doesn't help in the slightest.
And if you have some ordinal and some cardinal voters, the cardinal voters
taken together wind up as a dictator.

Where it really gets interesting is when you reinterpret the result into other
contexts. For example, suppose you're trying to reconcile several different
decision-making systems -- e.g., different moral codes. Those are like
different voters in Arrow's system, and hence there may be no "rational" way
to reconcile them other than simply adopting one of them (which would be the
"dictator" in the theorem's terms).

------
pixelcort
How does Quadratic Voting fare in meeting these criteria?

[http://ericposner.com/quadratic-voting/](http://ericposner.com/quadratic-
voting/)

~~~
rbkillea
I'm pretty sure that Quadratic Voting is a way to assign prices to votes,
whereas this theorem is about the features of a system in which the results
are determined by vote. So the results of quadratic voting will meet this
criteria because after the votes are bought they can be thought of as
emanating from discrete voters who align their preferences with those of the
buyer.

That is to say that the issues solved by Quadratic Voting and those presented
in Arrow's theorem are orthogonal.

------
gerty
Few theorems in economics, or political science in this case, are as
disappointing as this one. Another one is Sonnenschein-Mantel-Debreu.

[https://en.wikipedia.org/wiki/Sonnenschein%E2%80%93Mantel%E2...](https://en.wikipedia.org/wiki/Sonnenschein%E2%80%93Mantel%E2%80%93Debreu_theorem)

SMD: rational individuals do not sum up to a rational aggregate.

------
JDDunn9
[http://en.wikipedia.org/wiki/Voting_system](http://en.wikipedia.org/wiki/Voting_system)
has a nice comparison chart half way down.

------
kijin
Some people interpret Arrow's theorem to mean that democracy is a futile
exercise. For example, the anarchist Robert Paul Wolff uses an argument
similar to Arrow's theorem to show that all democracies must be tyrannical to
at least some of its members.

But Arrow's theorem is first and foremost an exercise in logic. It is grossly
oversimplified, and therefore should not be treated as realistic simulation of
real-world voting systems. We should be very careful when drawing political
conclusions from logical proofs.

There are several reasons why most contemporary political theorists don't give
a damn about Arrow's theorem, despite its logical plausibility.

1) Arrow's theorem assumes everyone's preferences to be fixed points, and only
cares about finding a curve that fits all of those points. But people's
preferences are not fixed. People are always changing their minds, often in
response to the shifting preferences of others. Many political theorists in
the "deliberative democracy" camp (the dominant model since the early 90s)
argue that the whole point of a democratic discussion is to get people to
reconsider their pre-existing preferences and find some sort of middle ground.

2) It's not even clear why an ideal procedure would need to satisfy all of the
preferences, or even most of them. If making everybody happy were as simple as
designing an election procedure, we would have gotten rid of politics a long
time ago! You don't even need 3 or more preferences to arrive at a conflict.
Two people with one preference each, that directly contradict each other,
would be enough to produce a situation where no procedure can satisfy them
all. In other words, there's nothing new here. Time to move on.

3) Arrow's theorem is somewhat effective in explaining how the actual share of
seats in a lawmaking body can end up being very different from the number of
votes that each party received in a first-past-the-post voting system with 3
or more major parties, such as UK and Canada. But there are much simpler, more
intuitive ways to explain that.

All in all, Arrow's theorem was a neat response to the political theory of the
mid-20th century, when people assumed democracy to be simply a matter of
efficient curve-fitting. But political theory has come a long way since then,
partly in response to problems like Arrow's theorem. In the new academic
milieu, Arrow's theorem isn't as relevant as it used to be.

On the other hand, I can sort of imagine how Arrow's theorem might find a new
use in designing distributed computer systems. Since computers aren't as
fickle as human politicians, the logical conclusions of Arrow's theorem might
be more relevant there. It's good to see that the HN thread so far focuses
more on technical details than on grand, mostly irrelevant political
narratives.

~~~
logicchop
(1) Not quite. The "deliberative democracy" camp is not interested in the
measurement of group preference, and are instead interested in consensus
building, political "rationality" (in hopefully some eventually-stabilizing
sense), and so on. That is not a response to Arrow and his associates, it is
just a different topic.

(2) It is abundantly clear why a "procedure" should satisfy all of the
requirements of the related theorems: they are trivial, intuitive, and
absolutely spot-on. There is a reason why these results are surprising, and
not just some arbitrary theorems concerning uninteresting axioms.

(3) Arrow's theorem has nothing to do with explaining anything. It is an
impossibility result in mathematics.

The rest of your post seems just dismissive of the problem, rather than
directly critical of it. ("All in all.." \-- as if these results were just
passing fads and now we've got our sense back??)

~~~
kijin
If we read early works (from the late 80s) in what is now called "deliberative
democracy", we can see that it began as a response to certain models of
democracy that emphasize preferences and procedures -- up to and including the
participatory models of the 70s and early 80s. Although deliberative democracy
is not a direct response to Arrow's impossibility theorem in particular, it
was intended to sidestep its troubling implications as well as other problems
with the older models.

There are two major factions within the deliberative democracy camp. One is
indeed interested in consensus building and eventually-consistent rationality.
This is the "Rawlsian" faction led by Gutmann and Thompson. The other faction,
however, focuses more on actual practices of negotiation through which pre-
existing preferences and power structures are transformed. This is the
"critical theory" faction led by John Dryzek and the late Iris Marion Young.
Personally, I think the latter remains closer to the original aims of
deliberative democracy and presents a better contrast to the older models it
was intended to transplant. The Rawlsians just took the opportunity to cram
their own agenda into democratic theory, as they always do with everything
they touch.

Your claims (2) and (3) seem to contradict each other. If it is so abundantly
clear that a procedure that satisfies Arrow's conditions is desirable, why do
you say that Arrow's theorem is just a mathematical result that doesn't
explain anything IRL?

Arrow's theorem is surprising and troubling only if you believe in some sort
of sacred relationship between the trinity of democracy, voting, and
satisfaction of all pre-existing preference. To the contrary, I find it both
trivial and intuitive that it is impossible to satisfy all of the preferences
of all human beings, and I would be very surprised and troubled if someone
claimed to be able to do so.

~~~
logicchop
I think you are a bit confused. What would Arrow's theorem explain? There is
no empirical phenomena that we are puzzled about that Arrow's theorem solves
(unless you are wondering: Why is it so hard to come up with a voting system
that doesn't have the potential for goof-ball results? - Answer, because it is
impossible..)

I don't know what you mean by the "trinity" in which "satisfaction of all pre-
existing preference" is a part.. That's obviously not relevant; we aren't
interested in a system that satisfies preferences. What we are interested in
is a system that (1) isn't a dictatorship, (2) is fair [i.e., everyone counts
equally], (3) allows us to decide any kind of potential matter _as a group_.

If you look at the conditions this way it should be glaringly obvious what
this has to do with the (possibility) of democracy..

~~~
kijin
If we're not interested in a system that satisfies preferences, then Arrow's
impossibility theorem is irrelevant.

The conditions, of course, sound obvious and intuitive. Of course we don't
want a dictatorship, and of course we want everyone to count as equally as
possible. But it takes a very specific interpretation of your third condition
to bootstrap the rest of Arrow's theorem. You have to interpret it in a way
that emphasizes _translating fixed individual preferences into group
preferences as straightforwardly as possible_. If you care about that, then
yes, you should be worried about Arrow's theorem. Otherwise, Arrow's theorem
is just a cool thought experiment that helps explain why said interpretation
is wrong.

Deliberative democracy currently happens to be the most popular model of
democracy among political theorists, and they don't care about Arrow's theorem
because whatever preferences people have before they enter the democratic
"procedure" isn't worth jack shit to them. As a result, Arrow's theorem is
much less relevant to the possibility and fate of democracy as currently
understood than it was 60 years ago.

------
fithisux
Do you know an exposition for mathematicians? Preferably in front of a
paywall. What is the mathematical content?

