Liberal Paradox

Every solution that we can think of is bettered by some other solution,
given the Pareto principle and the principle of liberalism, and we seem to
have an inconsistency of choice.
Amartya Sen, The Impossibility of the Paretian Liberal

In 1970, Amartya Sen’s paper demonstrated that given a certain set of circumstances, there could exist no social choice method (more or less a voting system) that would both respect individual rights, and make sure that the option it chose was not pareto dominated by an unchosen option.

Vilfredo Pareto has a couple of ideas named after him, but the one I’m using here is one that is often called ‘efficiency’. Imagine two people are arguing over what breakfast cereal to buy. From Sammys point of view, Frosties is better than Crunchy Nut is better than Cornflakes which is better than All-Bran. From Mothers point of view, Cornflakes is better than Crunchy Nut which is better than All-Bran which is better than Frosties.

Sammy: Frosties > Crunchy Nut > Cornflakes > All-Bran
Mother: Cornflakes > Crunchy Nut > All-Bran > Frosties

What should they choose? Well, there’s an argument to be made for most options, but one thing that they shouldn’t choose is All-Bran. If they took All-Bran home, they’d realise before they got there that both Sammy and Mother would have been much happier with Cornflakes or Crunchy Nut. Nobody wanted All-Bran. I’m going to call ‘Pareto dominated’ any option where there are other possibilites that all parties prefer. It seems obvious that whatever system we want making our choices for us shouldn’t be chosing options that are Pareto dominated.

Anyway, back to the Liberal Paradox. The problem is that there can be no social choice system that respects rights and can be guaranteed not to end up picking a pareto dominated option. Sen proved this with two book readers, Prude and Rude and a single copy of Lady Chatterlys Lover.

Prude is a bit of a censor and prefers that nobody reads it, but if someone must read it, Prude considers that he himself is more likely to withstand it’s influence than Rude. Rude wants to read it a lot, but to be honest thinks it’d do Prude a world of good and so would prefer Prude to read it. So the preferences look like this:

Prude: nobody reads it > Prude reads it > Rude reads it
Rude: Prude reads it > Rude reads it > nobody reads it

We’ve decided to combine these preferences in some way to come up with who should get the book. But really, we want to give Prude the right to not read it if the choice is between nobody reading it and Prude reading it – it’d be a cruel government that forced Prude to read in that case. We also want to give Rude the right to read it if otherwise the choice would be nobody reads it. These are rights we’re giving the individuals, because we’re liberals.

So, we can’t choose Prude reads it because we’d be infringing his rights, we can’t choose nobody reads it, because we’d be infringing Rudes rights, we’re left with only one choice – Rude reads it. That seems like a good solution.

But that’s no good either. Both Prude and Rude would prefer that Prude read it over Rude, so it’s a Pareto dominated alternative.

And that’s a problem that can happen any time you try to come to a community decision on a topic in which individuals have rights that must be respected. There are a few ways around this problem. You can ensure that you only vote on topics that don’t infringe rights (if we’re giving Rude the right to read the book, then why should Prude be allowed to say that he shouldn’t?), you could accept the possibility of choosing an outcome that is suboptimal for everyone, or you could accept that sometimes, you’ll have to infringe someones rights to choose the majority decision.

Navin Kartik, in Liberalism and Pareto Efficiency: Sen’s Paradox argues that the preferences are not independant of other peoples actions, e.g. in a real life version of the above, one person will probably be in the queue at the book shop first, and whomever gets there first gets the book without infringing on the others rights. This is quite true, but it’s not particularly relevant to the paradox, since in that example there is no attempt to use a social choice method. Instead we’re using the “early bird” gets the chance to determine the state of the world method. A dictatorship method based on opportunity. In situations where social choice methods are used (the paradox I think is really trying to be applied to democracy and governance), the decision is made in an instant taking into account the preference profiles fed into it. You can only make an escape via noninterdependence by splitting up the decision into parts that are made at different times.

The paradox isn’t that surprising – I see it as just stating that a community that’s trying to find the best solution for the whole community might choose things that are at odds with the rights of its members. What may be surprising is that following the rights of the individuals can result in choices that are bad for everyone, even those individuals.

What has happened is that giving individuals rights (to be decisive) has meant that in those choices that they are acting like individuals in competition to see their vision of reality come true.

There is a classic example of how following individuals choices can result in a situation worse for everyone.

Al Capone and Dick Turpin are both in the holding cells. They’re about to be taken for questioning. If both of them stick to the story they’ve agreed, there is little evidence to hold them, and they’ll both go to jail for four months for tax evasion. If either of them want to though, he can turn Kings evidence in which case, the other will serve 10 years, and he’ll go free. If they both voluteer to turn against the other, then they’ll both go to prision for 4 years.

It’s the prisioners dilemma. If both of them operate from a free choice to chose their own best interests, then they will choose the Nash equilibrium, and they’ll both go down for 4 years, paying the price for choosing a pareto dominated solution. If they give up their right to choose, and submit their preferences to a social choice method, it’ll look like this:

Al Capone: Al betrays > both hold > both betray > Dick betrays
Dick Turpin: Dick betrays > both hold > both betray > Al betrays

Both betraying – the Nash equilibrium – is pareto dominated by both sticking to the story, and a pareto social choice method would take that into acount. The most likely choice here for any sensible social choice method is that both stick to their story.

Giving people rights, and allowing them to exercise them in a non naive way, will lead to a situation that seeks the nash equilbrium, just like any non cooperative game. Sometimes giving up those rights to find a solution best for the community will also result in a better result for every individual in that community.

If you start from the position that individuals start with all rights, then analysing the situations where your rights are keeping you from pareto optimality can be a hint that it might be better to form a community and give up those specific rights to it. This is the rule that causes humans to band together into communities.

If you start from a position where the community bestows the rights upon individuals, then you need to be aware that the dark side of this is that sometimes the preferences of the majority are to hurt, torture and kill one or two of the communities members.

One way of modelling this would be to consider it as a iterated game. Make the rights something that a social choice function decides, but give the members of the society the knowledge that this is an iterated game, and sometimes they will be in the minority. To do this, we’ll need to attach preference values to each of the choices. These are called utility values. We can model the situation to discover if an individual will, when asked, vote to grant a specific right to all citizens or to withhold that right.

d = Disutility to me of having my right infringed by the others
u = Average utility to me of the social choice function being able to select scenarios where citizens have this right infringed.
p = Probability of being the one who has his rights infringed for any given iteration.
i = Iterations.

d(1 – (1-p)^i) – (u * i) > 0 : I will prefer to bestow this right to the citizen
d(1 – (1-p)^i) – (u * i) < 0 : I will prefer not to bestow this right to the citizen Societies are most likely to bestow rights under this model when the disutility is very high (such as death / torture), or when the members consider that they have a high chance of being in the minority. Perhaps they have fled from persecution in another place, or they are very diverse. There must also be a reasonably large number of iterations (the life expectancy in the society must be reasonably high). Seen this way, although rights may lead to some iterations having pareto dominated results, after all the iterations, we should expect that the final set of decision states is not pareto dominated by any other set, since the society would only choose to bestow the right if enough individuals thought it would be, in the long run better for them.

Dog Incompleteness

This is the story of how I discovered that not every dog has his day.

I got a fantastic pair of dogs the other day, they’re always having pups, but the most amazing thing is that I’ve trained them to give birth to whatever kind of dog I specify. So I go to them one day, and I say “I want a dog born on the 22nd of February with 14 teeth and black hair and blue eyes, and sure enough, the dog is born.

Well I got to thinking about all the things I could do with this amazing dog breeding trick, and I thought about collecting the full set. It took me a bit, but eventually I realised that there was a never ending list of properties and abilities of dogs. I mean, there’s “propensity to chase cats” for example, but there’s also “psychic dogs”, and then theres “propensity to chase dogs that chase cats” and “propensity to chase dogs that chase dogs that chase cats” and so on. Or theres “dogs that can do only the 2 times table in their heads” or “dogs that can do only the 3 times table in their heads” and so on. So I began to realise that if you gave me any list of properties of dogs, I’d always be able to think of ones you’d missed.

Well, that put paid to the idea of collecting the full set, which was a disappointment, as you can imagine.

One day a man from the Pedigree office came round, and he said to me “You’ve got a large number of dogs that haven’t been named, if you want them registered, you’ve got to have them named”, so I went to work, and named all the dogs that I’d had born, quite a big task by then. I mean, you start ok with Rex, Bowser, Gnasher etc, but after the hundredth or so you start having to tax your creativity.

A couple of days later, I was thinking, and I realised that it’d be handy if I just worked out all the names of all the potential dogs that could be born, so I didn’t have to think up new names. Even if I used a meaningless string of letters or numbers, at least they’d have a name, and it’d keep the Pedigree office happy. Although there’s a never ending list of possible dogs, theres also a never ending list of possible names, so there should be no problem right?

So, I got a piece of paper, and down one side, I listed the names that I’d come up with in alphabetic order, and down the other side I listed the properties of the dogs. I was only able to make a start on each of those lists, but I figured that once I had a system, it’d be obvious how to continue it.

It went kinda like this


Name | Psychic | Eats Cats | does 2xTable | does 3xTable…
Rex1 No No No No
Rex2 Yes No No No
Rex3 No Yes No No
Rex4 Yes Yes No No

Wed Aug 29 2001 at 22:40:48
Well everything was fine until one day the Pedigree office turned up and complained about a dog that I’d had born with no name. I said “Look, it can’t have, cus I listed all the names out, and it’s all done”. The officer showed me the list and he said “Look, sir, this dog you’ve got, well it’s not called the first name, cos it’s different from Rex1 in that it’s Psychic, it’s different from Rex2 in that it eats cats, it’s different from Rex3 in that it can do the 2x table….”. And he went on. I saw what had happened in an instant. The dog born had the opposite characteristic to the first named dog in the first property, and to the second named dog in the second property, and to the third named dog in the third property, etc. Even if I listed the names forever, There’d still be at least one dog with no name.

I slowly came to realise something big, wonderful, and profound. We can never fully describe all the dogs that might exist.

First published on everything2.com under the name delFick