Arrow's Impossibility Theorem

[Catmatic Leftist]

So the Arrow's theorem was brought up in a recent discussion I've been having to "prove communism wrong."

His post:

In the fields of economics and political theory, there is something called Arrow's Theorem. The theorem basically states that there is no such thing as a democratic process that produces social preferences such that they respect a minimal level of democracy, do not violate universally held views, and cannot be manipulated by politicians who control the political agenda. This flies straight into the face of the popularly held belief that the "will of the people" is expressed through democracy, by denying the existence of any kind of "will of the people".

To make things more precise, imagine a so-called "ideal" democracy (Note, I said ideal and not direct). In an "ideal" democracy, we would expect the following axioms to hold:

1. The "Universal Domain" (UD) Axiom: This axiom states that we cannot dictate to people what their preferences will be. In other words, no one can tell you what social policy to like and what social policy to not like. It ensures that all possible individual preferences are admissible, which is exactly what we would expect from a democracy.

2. The "Pareto Unanimity" (PU) Axiom: The second axiom is that unanimously held views are respected when social decisions are made. For example, if everyone in the society agrees that citizens have the right to free speech, then the democratic process cannot take away that right. If everyone agrees that eating babies should not be legal, then the democratic process cannot make eating babies legal. (I hope )

3. The "No Dictatorship" (ND) Axiom: The third axiom is quite obvious in its meaning. But we can put it into more precise terms. The axiom disallows a certain type of dictator, called an Arrow Dictator. An Arrow Dictator is an individual such that if everyone else in the society agrees that x is better than y, and he thinks that y is better than x, then the "democratic" process always picks y as the social choice (in other words, the dictator always gets his way). Obviously, this kind of dictator is a very powerful kind of dictator.

4. The "Rationality" (R) Axiom: This axiom states that all preferences in the society must be rational. What this means is that if an individual prefers x to y and y to z, then he must prefer x to z. Otherwise, he will be stuck in a loop of preferences, and will never be able to choose among the alternatives. Note that this does not have to have any relation to quantity. For example, if individual A prefers a 20% tax rate to a 10% tax rate, and if he prefers a 10% tax rate to a 30% tax rate, and if that implies that he prefers a 20% tax rate to a 30% tax rate, then his preferences are said to be rational.

5. The "Independence of Irrelevant Alternatives" (IIA) Axiom: This one may be a bit difficult to understand. Suppose that, for a particular preference profile in the population, the social choice process results in social preferences that pick x over y. Then it must be the case that the same social choice process results in social preferences that will still pick x over y for all other individual preference profiles that maintain individual rankings of x and y as they were in the original preference profile of the population.

(The last two axioms are protections against an "agenda setter" who manipulates the democratic process and whose policy decisions approach that of the dictator)

Now, Arrow's Theorem says that no such democracy can exist. If a democracy meets all five of the above axioms, then it must contradict axiom 3, i.e. No Dictatorship. This gives us a set of three plus one alternatives when it comes to what kind of democracy we want:

(this is where you make your choice in the poll above)

1. Mind Control Democracy: This is a democracy that rejects the first axiom. It tells you what you will and will not like. Let's say that they control your mind through advertisements, or chemicals in the water supply, or just the simple every-day mind control ray.

2. Mob Rule Democracy: This type of democracy can result in policy decisions that disagree with universally held views. For example, if everyone believes that citizens should have the right to free speech, it has no problem with taking that away. Property rights? Not immune. Your limbs, freedom, family, life, etc.? Nothing is sacred.

3. Illusory Democracy: This is a democracy that rejects axiom 4 or 5. Although all political decisions appear to be democratic, they are in fact manipulated by an "agenda setter" politician whose policies approach those of the dictator.

4. Dictatorship: Screw democracy! All hail the dictator! This is a democracy that rejects axiom 3, and is, therefore, not a democracy at all. (I should say, however, that a dictatorship can meet all of the other 4 axioms.)

To me, this seems like circular logic and makes haphazard assumptions and has faulty premises, but I'm at a loss on how to word my argument. I'd really appreciate everyone's help. :)

[Kotze]

Note that probability isn't mentioned anywhere in such essays.

Arrow's theorem is usually mispresented somewhat like that by neoliberals. This text is actually a bit worse than the usual dreck, so I believe whoever wrote that is a victim of this indirect pseudo-knowledge. (You know that game kids play, Telephone?) Arrow himself describes his findings as similar to the concept of friction. You can't get rid of the problem entirely, but of course that doesn't mean that all approaches at curbing it are worseless.

I remember a particularly disingenious text by Paul Samuelson who had the gall to use that theorem in a text to argue against direct democracy and in favour of indirect voting, because when you vote on a package of decisions your preference strength plays a role, never mind that politicians aren't legally bound by their campaign promises.

I also encountered that theorem in a neonazi text (which was less stupid than what you copied here), arguing that a homogenous society would be less likely to encounter the problematic situation pointed out by Arrow — which is a very plausible argument, and, whatever their intent was, a very good argument for socialism.

Independence of Irrelevant Alternatives demands this in plain language: The counting rule should work in a way that if you erase any of the losers from the ballots and count them again, the winner shouldn't change. It's about avoiding that losers spoil the election.

Now here's the thing, suppose the ballots allow voters to rate the candidates and the one with the highest average rating wins. If you erase losers from the ballots, this doesn't change the fact that the one with the highest average still has the highest average. This satisfies IIA. Does this mean that this method is immune to a current losing by running too many candidates?
—It would be true if the ratings people give you were completely independent of who else runs, but of course, that's not a realistic assumption. That's not to say that taking the average is the worst voting method ever. If we give each ballot a weight of 1 and count the ratings of a ballot fractionally, then even under the unrealistic model assumption, this can't guarantee IIA. The extreme model tells us in a roundabout way something about reality, that fractional voting is worse at dealing with multiple similar candidates than average ratings.

Another criterion: If there are only 2 candidates that run (or 2 competing proposals), the one preferred by the bigger group should win.

When we look at ballots with ratings, and we erase everybody except the winner and some other person, we can guarantee that the winner has the higher average rating, but we cannot guarantee that the one with the higher average rating is preferred to the other by a bigger group than vice versa. (Example: 60% give candidate X a rating of 3 and candidate Y a rating of 2, whereas 40% give Y a 9 and X a 0. Majority opinion and average rating give different results in that instance.) That's not to say that this is very likely. If people vote in an informed strategic manner, this method is very likely to give majoritarian results. When people are more honest, it returns results that respect preference strength more, and that doesn't sound like a bad thing either. That average ratings has a big risk of returning crappy results from both a majoritarian and utilitarian point of view requires not just a mix of informed strategic voters and honest voters, it requires that this mix is very different among different currents.

Anyway, given that economic mechanisms usually tend towards a wealth distribution that has a few very well-off people, as a rule of thumb we can expect not-so altruistic majoritarian opinions about redistribution to also be okay from a utilitarian point of view. So let's take a look at majoritarian methods.

The thing with majorities is that they can be cyclic, a majority prefers A to B and a majority prefres B to C and a majority prefers C to A, all at the same time, eg.

46 A>B>C
30 B>C>A
24 C>A>B

76 B>C
70 A>B
54 C>A

So whatever the method chooses here as a winner, if the method is as majoritarian as possible, we can construct a situation with only the winner and 1 other candidate, where the winner turns into the loser. Or we go the other way around, we have a majoritarian method in a case with 2 candidates and we add a non-winning candidate, who spoils the election.

That doesn't mean that cyclic majorities are very common.
4. The "Rationality" (R) Axiom: This axiom states that all preferences in the society must be rational. What this means is that if an individual prefers x to y and y to z, then he must prefer x to z. Otherwise, he will be stuck in a loop of preferences, and will never be able to choose among the alternatives. Note that this does not have to have any relation to quantity. For example, if individual A prefers a 20% tax rate to a 10% tax rate, and if he prefers a 10% tax rate to a 30% tax rate, and if that implies that he prefers a 20% tax rate to a 30% tax rate, then his preferences are said to be rational.I prefer the term transitivity here. You know what's funny? When we assume instead of that "rationality" that people see the competing proposals as points on the same line — not meaning that everybody has the same opinion what's best, but just where on the spectrum something is — then the median-opinion proposal stays at the median if we delete non-median proposals.

For one-dimensional decisions, methods that select the median are majoritarian and don't have an IIA problem here. These methods are called Condorcet Methods.

[syndicat]

Arrow's argument presupposed an individualist methodology. this is clear from Arrow's name "Rationality" for his particular rule. This rule is a reasonable rule of consistency for individual preferences.

But what Arrow fails to consider is that there can be a collective rationality. In other words, when people are in situations where they are making decisions for a group, such as at a union meeting or a meeting of the members of their condo association, a family dicussion, or whatever, people will often adopt explicitly a "we" mentality, that is, they will talk and argue about what is best for this collective subject.

There is no reason this collective rationality should be "reducible" to the individual preferences of the individuals taken separately from the collective decision-making context. of course, in their participation in the meeting and voting and so on, they will have preferences for the decisions that are made. but their preferences in regard to "what we should do" do not have to be reducible to their self-regarding preferences, that is, their preferences for what they want for themselves.

[Catmatic Leftist]

Thank you for the replies, everyone. I am still looking for more responses and I greatly appreciate them. :)

[Kotze]

But what Arrow fails to consider is that there can be a collective rationality.
(...)
There is no reason this collective rationality should be "reducible" to the individual preferences of the individuals taken separately from the collective decision-making context.If it wasn't for cyclic majorities, there would be a perfect answer to the demands in Arrow's theorem. Cyclic majorities can happen, whether people are acting according to what they perceive as best for themselves individually or best for a group. Arrow isn't the one failing here :P

[jake williams]

The whole argument sounds incoherent and arbitrary.

Axiom 2 doesn't make any sense. Is anyone advocating a society where EVERYONE believes X, but X isn't implemented?

The whole shtick about "mob rule democracy" is that "everyone", except a democratic majority of people, support the protection of capitalist property rights, and so it's undemocratic to allow that democratic majority to protect their own interests. Honestly, that's basically the whole point of the argument.