Sunday, March 19, 2006

A Case for Maximin and A Case for Utilitarianism

Each without veils.

Maximin says that when choosing between options for society we should only look at the lowest level of command over primary goods (think roughly money) that people have. It is like chosing based on what I would choose if I didn't know whose shoes I would fill (veil of ignorance) and I were infinitely risk averse.

One argument for maximin is that one chould choose as if etc.

Case for maximin roughly due to John Roemer.

1. Hobbes, Locke Rousseau etc argue that just institutions are institutions which could be (and maybe were) willingly created by free people who formed a social contract. The idea is that we are born free but choose to live according to societies rules because for each of us the advantages outweigh the costs so we are all free under laws that we freely accept.

2. David Hume says this is like a contract between a man who has been Shanghaied and is now on a boat out at see and the captain of the ship who gives the man his freedom back and asks if he prefers to obey or to try to swim to shore. Dave's got a point there.

3. So we do have to offer people passage home back to the state of nature. If a group of people say we want to opt out and start over, they have to be given that chance. A society is just only if there is no such group.

4. This means that just societies are not societies which would be chosen over all other possible societies by everyone under the veil of ignorance requiring us to understand choice under uncertainty. It means that they must be in the core, that is they can contain no blocking coalition with the passage home principle for blocking.
The jargon term "corps" is not useful for this presentation and is defined in use above.

5. The idea is that a society is just if there is no disgruntled subgroup of any size who would rather start over (just them not the contented ones).

6. So what does "all over again" mean ? Well at the least it means with the ability to make institutions constitutions etc if the group which wants out agrees. It also requires plenty of empty land as humanity had a short million years ago. I also think it requires the group to divide up time as well as land. That is they don't all have to be alive at the same time and they can say "look I don't like nature all that much I want to live in the 22nd century of the new and better society" so long as they can find people who are willing to live in the other 21st century and the new 1 million BCE. Finally and very importantly, I do not think we have a natural right to the fruits of our innate abilities if such abilities differ across people. That means the disgruntled can block a planned society if they want a new IQ so long as they can agree on a new IQ for each of them so that the distribution of IQ is the same as in the original society.

7. For a society with an infinite number of citizens this implies that a just society follows maximin. For a finite number of citizens, teeny tiny groups don't block. Oddly this is Rawls' view which is usually not mentioned in the simple maximin rule.

Case for utilitarianism (almost identical to an argument made by Peter Hammond).

This is horribly boring and definitely incomprehensible. I am trying to do math with plain text. It is definitely not worth the effort to read what follows. Still having typed it I will post it.

Don't read on. It isn't worth it. Don't say in comments that I didn't warn you or you are a double plus ungood spammer.

Here utilitarianims means we should act in order to maximize total happiness. It also means we should make choices as we would if we chose the way Von Neuman and Morgenstern consider rational and we chose under the veil of ignorance, that is we didn't know who we were going to be.

In fact the veil of ignorance is the argument John Harsanyi used when he claimed he had proven that utilitarinism was the be all and end all of ethics.

Peter Hammond made an argument (which does not seem to be downloadable although it seems to be cited here) based on linear algebra.

This is a similar argument.

I'm going to assume that we are choosing under uncertainty. There is a probability distribution of different outcomes. There are N people in the world (N is finite and they don't all have to be alive at the same time N is at least 30 billion and I hope a whole lot more). One aspect of outcomes is a level of happiness for each person.
Each person would make choices in order to maximize his or her expected happiness.
Alternatively, I can assume that if no one else were affected by the choice each person Should make choices in order to maximize his or her expected happiness. a unit of happiness is called a util.

It is just and right for society and every person in it to choose according to a rule such that

1. if everyone is indifferent between two choices they are equally good.
2. If everyone prefers one choice to another it is better.
3. everything is symmetric over people so the rule can be described without using any names. That is the social choice rule is fair, anonymous, a law before which everyone is equal.

Then the right thing to do is to act so as to maximize a positive constant times the sum of utils.

The proof goes like this. Assumptions 1 and 2 implies that the social choice rule is maximize the expected value of a linear function of the N vector of happiness of each of the N people.

Clearly given this result 2 and 3 imply that the function is the sum of utils.

How does it work ?

well all the Von Neuman and Morganstern assumptions apply to the social choice rule by an obvious application of assumptions 1 and 2, because they are all about indifferent or prefer and stuff so the social choice rule consists of maximizing the sum of the expected value of some function of the outcome. Call that outcome social welfare (it is a number).

Call the N vectors of how happy each person is in an outcome a happiness vector. Call the function of outcomes which gives how happy individual i is as a function of the outcome the individual i vector.

Consider all the possible N happiness vectors. Here possible does not mean conceivable. The probability density of reaching each possible N vector must be positive. This is what "possible" means in this post.

Choose one possible outcome. Call it the zero outcome. in the zero outcome person i has happiness happy0_i. subtract happy0_i from the happiness in each possible outcome. Subtract social welfare in the zero outcome from the social welfare function. Without loss of generality we have a society choosing what to do when one possible outcome is that everyone has happiness zero and social welfare is zero.(happiness and social welfare can both be positive or negative so the zero outcome is not necessarily the worst outcome).

There is a subset of M happiness vectors with M less than or equal to N so that every possible happiness vector is a linear combination of those M happiness vectors and the M vectors are linearly independent. That such a subset exists is a well known theorem.

Now consider social choice over outcomes in which only those M outcomes and the zero outcome are possible. The social welfare function is a linear combination of the individual i vectors. This is a well known result and can be proven by induction on M. Recall M is finite.

That is any function of those M outcomes is a linera combination of the individual i vectors because there must be M linearly independent i vectors since there are M linearly independent happiness vectors where only those M outcomes are possible.

Now every other happiness vector is a linear combination of the M happiness vectors. Consider any outcome call it outcome_1. the happiness vector of outcome 1 is a linear combination of the M happiness vectors. If social welfare of outcome 1 is not the same linear combination of social welfare in the M chosen outcomes, I can construct a violation of instruction 1.

Let's say social welfare_1 is greater. Consider probability distribution A over outcomes where each of the M outcomes occurs with probability greater than a positive number epsilon and both the zero outcome and outcome 1 occur with probability greater than epsilon times the absolute value of the sum of the coefficients in the linear combination and less than one minus the absolute value of that sum of coefficients . In probability distribution B each of the M outcomes is epsilon less likely and the probability of outcome 1 is increased by epsilon times the sum of the coefficients in the linear combination and the probability of the zero outcome is changed so that the sum of all probabilities is greater than 1. Given the choice of outcome A, outcome B is an ok probability distributions since there are no negative probabilities or probabilities greater than 1.

Given the definitions of probability distributions A and B, B is better than A even though everyone is indifferent between A and B. This violates assumption 1.

Similarly social welfare in state 1 can not be less than the linear combination of social welfare in the M states.

This is true for any state chosen to be state 1. This means that social welfare is a linar combination of individual happiness. Assumptions 2 and 3 imply that social welfare is a postive constant times the sum of utils.

Assumptions 1, 2 and 3 imply that crude vulgar add it up utilitarianism is the be all and end all of right and wrong.

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