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This is as good a place as any to call into question the easy assumption that the possession of a complete ordinal preference structure is the most elementary test of one's rationality . A great deal is at stake here, much more than you might think. Let us start slow. The theory of rational choice has its roots in analyses of gambling behavior, of economic behavior, and -- to some degree -- of political behavior. Now, when we are talking about the way in which professional gamblers decide how to play their cards or place their bets, it makes sense to assume that they can define a complete preference order over the available alternatives. That is to say, the various possible outcomes offered by a gambling game are plausibly described as commensurable with one another. The outcomes are, after all, simply different wins or losses of amounts of money. The same is true of people engaged in economic activities. But these are relatively limited and specialized arenas of human activity. There are many other arenas in which it is not so obvious that rational individuals have complete preference orders over available alternatives.

Consider, as an example, the terrible choice presented to the central character in William Styron's novel Sophie's Choice. [I know the story from the movie of the same name, starring Meryl Streep.] A Gestapo gauleiter overseeing the loading of Jews onto trains taking them to the death camps offers Sophie a choice. She may save one of her two children from certain death, but she must choose which one will survive. His posing of this choice is clearly an act of satanic sadism. There are two ways of thinking about this situation. The natural, and I suggest, rational way to think about it is as a tragedy in which a woman is presented with a terrible situation that will destroy her life no matter what she does. To choose either child is impossible. To fail to choose one is to condemn them both to death. Religion may have something useful to say about this situation. Literature may. Perhaps nothing can. But for sure Rational Choice Theory is no help. But Rational Choice Theory says that she must have some preference order or other over the three outcomes, and her choice, whatever it is, reveals that preference.

Let me put this in a summary fashion, and ask you to think about what I say. Perhaps later on we can discuss it. The assumption of a complete ordinal preference order is presented in the literature as an innocuous premise that gets the more complex and interesting arguments going. But in fact it is, covertly, a highly questionable proposal to extend a form of economic rationality into areas in which it arguably does not belong. By accepting the formalism, someone unwittingly buys into this powerful encroachment of the economic into arenas of human experience in which it has no place. Imagine coercing a man into acting dishonorably, and then saying that his agreement reveals exactly what price he places on his honor. It would be more true to the human reality to say that by this act of coercion, you have besmirched his honor, which henceforth is worth nothing to him. The outcome of the choice you forced on him is not a rational choice but shame.

The defining characteristic of capitalism is the reduction of all human activity to market relations. Too often, Rational Choice Theory functions as a covert and seductive rationalization of the capitalist ethos, which then seems, because of the apparent neutrality of the formalism, to be equivalent to rationality tout court.

It is not necessary to limit ourselves to complete orderings -- orderings which establish the individual's or society's preference for any two alternatives whatever. We can also define partial orderings, and these have in fact played an important role in Economics and other disciplines. I will only say a few words here, and return to this subject down the line. The Sophie's Choice example has shown us that sometimes individuals cannot say, for two alternatives, which one they prefer. It is not that they are indifferent between the two. The two are simply, in their minds and hearts, not comparable. How many lives is it worth to save the only score of Bach's B Minor Mass? The question makes no sense to us, no matter what phony scenarios we cook up in a philosophy essay.

A similar problem arises when we are trying to compare different social distributions of wealth. If Situation B offers everyone more wealth than Situation A, then we can be pretty sure there will be unanimous agreement that B is better than A. Indeed, if people are willing not to be envious of what others get, then we might be able to secure unanimity for the proposition that B is better than A if B offers everyone at least as much as A does, and offers at least one person more. [Why begrudge her the extra if it isn't coming out of your share?] But what about the case in which B makes some people better off and others less well off than they were in A? There may just be no answer in that case.

Thanks to Vildredo Pareto [1848 - 1923], when B makes everyone better off than they were in A, we say that B is Pareto Preferred to A. Obviously, if B is Pareto Preferred to A and C is Pareto Preferred to B, then we should expect that C will be Pareto Preferred to A. So this Pareto or Unanimity ordering is transitive but not complete. If some way of distributing things is such that there is no alternative distribution that is Pareto Preferred to it, then we say that it is Pareto Optimal. Don't be misled by the enticing sound of the word "optimal." If we assume that everyone has positive marginal utility for money, so that taking even a little bit away from someone makes her less well off, then a social distribution that gives everything to one man and nothing at all to anyone else is Pareto Optimal, because any re-distribution will involve making at least one person worse off, namely the person who had everything and now has slightly less. In case you are wondering why this matters, I will just point out that when economists describe a market as efficient, they mean that it produces a Pareto Optimal outcome. Not too heart warming.

So much for ordinal preference orders, at least for the moment. Now things get somewhat more complicated, but also a good deal more important. The next step up, after nominal and ordinal orderings, is cardinal orderings. Since this is going to require a little technical work, let me first explain what is at stake. Both Rational Choice Theory and Game Theory [but not Collective Choice Theory] involve talking about people doing something called "maximizing expected utility," or "discounting the value of an outcome by its risk" and so forth. These calculations require that we be able to assign cardinal numbers, or magnitudes, to different outcomes or alternatives, and that we be able then to do things like adding them, subtracting them, multiplying them by other numbers, etc. Now, you cannot add or subtract or multiply or divide ordinal numbers. It makes no sense to ask, "Is Second the average of First and Third?" in the way that you might ask "Is 2 the average of 1 and 3?" If you have ever been involved in trying to work out a system to decide which team in a track meet has won over all, or which country has won over all in the Olympics, you will understand this. Does a whole raft of silver and bronze medals count for more or for less than a small pile of gold medals? Are a gold and a bronze equal to two silvers? The questions are meaningless. To carry out any of these calculations, you need an interval scale, also called a cardinal ordering.

An interval scale is an assignment of numbers to the elements of an ordinal ordering in such a way that the intervals are equal [hence "interval scale." This is actually a gross simplification of the correct definition, but I don't want to scare people away, and this will suffice.] A good example is the Fahrenheit temperature scale. The elements here are, let us suppose, readings provided by a thermometer of the temperature of different bodies of water. We can first impose a nominal ordering by grouping together the bodies of water that are [or maybe feel] the same temperature. We then impose an ordinal ordering by arranging the equivalence groups in a hierarchy from hottest to coolest. Thus far, all we have is the information that this body of water is hotter than that one, or maybe that this body of water is at least as hot as that one [i.e., weak rather than strong ordering]. Now, suppose we can actually answer the following question for any four bodies of water, a, b, c, and d: Is the difference between the temperature of a and the temperature of b at least as large as the difference between the temperature of c and the temperature of d? Notice I said any four bodies of water. In other words, I am asking about intervals of temperature, not just temperatures. If I have enough information to answer that question for any tetrad of bodies of water, then I can define a cardinal measure of temperature,. I can say, for example, using the Fahrenheit scale, that the difference or interval between fifty degrees and sixty degrees is the same as the difference or interval between twenty degrees and thirty degrees. So it makes sense to say, "It is ten degrees cooler today," regardless of what the temperature was yesterday.

We are here performing an arithmetic operation on the labels assigned to the elements of the set [namely subtraction]. But you cannot perform arithmetic operations on ordinal numbers. For that you need cardinal numbers [i.e., real numbers] like 1, 2, 3, and 4. So, this sort of scale is called a cardinal scale.

This right here is one of the most important things I am going to explain, so if it is not clear, I expect to hear from you.

The last step, which is only important for a few purposes, is to define what is called a ratio ordering or a ratio scale. A ratio scale is just like a cardinal scale but with one thing added: with a ratio scale, we have enough information to say, for any four elements of our set, a, b, c, and d, whether the ratio of a to b is equal to or greater than the ratio of c to d. Or, going back to the symbolism we used above, whether a/bRc/d. Now a little experimentation will show you that a ratio scale requires that you be able to identify some point as the zero point, or origin. A Fahrenheit temperature scale is not a ratio scale. The zero point in the Fahrenheit scale is chosen arbitrarily. Therefore, it makes no sense at all to say that in a Fahrenheit scale, the ratio of twenty degrees to ten degrees is the same as the ratio of eighty degrees to forty degrees. [For those of you who know some Physics, that sort of statement does make sense in a Kelvin scale of temperature, where the zero point is what is called absolute zero.]

Why am I going on about this? Well, for one reason, because it will turn out, way down the line, that without knowing this stuff you cannot understand what a zero-sum game is.

B. Transformations

Now we are going to talk about transformations. Technically, a transformation is a one-one mapping of a set onto another set, but we can just think about a transformation as a rule for assigning new labels or numbers to a set of elements.

1. A permutation is a re-assignment of the labels attached to the elements that preserves their grouping into equivalence classes. Initially, you will recall, we attached labels to the elements of our set, S. Two elements that got the same label were then in the same equivalence class. So if we label people by their last names, all the Millers go together, all the Tailors go together, and so forth. We could now relabel everyone, say by translating their names into another language [so that all the Millers become Muellers, and all the Tailors become Schneiders.] That would change everyone's name, but it would not change the groupings into equivalence classes. All the Millers were together before, and they are still together now that they are all Muellers. What is more, no two people who were in different groups before are in the same group now. The official jargon for this state of affairs is that the labeling is invariant under a permutation.

2. A monotone transformation is a re-labeling that preserves an ordinal ordering. Suppose we take the items labeled first, second, third, and fourth, and now label them fifth, eleventh, nineteenth, and fortieth. No information in the original ordering has been lost, and none has been gained. In the formalism of the relation R, for any two elements a and b in S that have been relabeled a' and b' respectively, aRb if and only if a'Rb'. So the ranking has not been changed by the transformation. Again, the official way to say this is that the ordinal ordering, R, is invariant under a monotone transformation.

3. A linear transformation is a transformation that preserves a cardinal ordering. A linear transformation of a relation R on a set of elements S = (a,b,c,....n) is a relabeling of each element a in S such that the new label, a' equals the old label a times some constant plus another constant. Or: a' = aq + r. This is called a linear transformation because the expression (a' = aq + r ) is the formula for a straight line drawn on the a and a' axes. A little elementary algebra will show that this transformation preserves an interval scale or cardinal ordering on the elements of S. Remember: This means that it preserves equality of intervals between pairs of elements. Here is how we prove this:

Take four elements, a, b, c, and d, such that (a-b) = (c-d), Now impose a linear transformation on S. That means:

a' = aq + r

b' = bq + r

c' = cq + r

d' = dq + r.

Notice that we have imposed the same linear transformation on each element. In other words, the constants q and r are the same in each case.

By hypothesis (a - b) = (c - d)

Substituting the transformed labels, we get (aq+r - bq-r) = (cq+r - dq-r)

or: (aq-bq) = (cq-dq)

Dividing both sides by q, we get (a-b) = (c-d) Ta Da!

It is just too bloody hard for me to import scanned diagrams into this blog, but if I could, I would draw you a graph with a line on it, showing you that the line a'=qa+r cuts the a axis at r and cuts the a' axis at -r/q. So long as you re-label the elements of S so that the labels satisfy the equation a' = qa+r, for any a in S, it makes no difference which set of labels you use, because they all encode the same information. This is what it means to say that a cardinal ordering is invariant under a linear transformation.

A linear transformation does two things. It changes the size of the intervals [but not the equality of different pairs of intervals], and it changes the zero point. The classic example of a linear transformation is the formula for converting temperature from Fahrenheit to Centigrade. The formula, as everyone knows who travels to Europe and wonders whether to wear a sweater or not, is F degrees = 9/5 Centigrade degrees + 32. So, if Le Monde says it is going to be 20 degrees today in Paris, that means 9/5(20) + 32 or 68, so no sweater. Zero degrees in centigrade is the temperature at which water freezes, but in Fahrenheit, that is 32 degrees. And so forth. Each degree Centigrade is equal to 9/5 Fahrenheit degrees. Why does this matter? Once again, it will turn out to be crucial when we come to give a correct definition of a zero sum game, and for many other purposes besides.

Finally, and uninterestingly, a ratio transformation is a transformation of the form a' = qa. This the formula for a line that goes through the origin of the graph. The transformation just changes the size of the intervals but does not change the zero point. And obviously, a'/b' = qa/qb = a/b. This one doesn't matter much, but I put it in for completeness' sake.

O.K. We have nominal, ordinal, interval, and ratio scales, and we have permutations, monotone transformations, linear transformations, and ratio transformations.

Now, as Portnoy's analyst says in the last line of the novel, let us begin.

## Friday, May 7, 2010

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If I might ask, what's the non-simplified way to introduce interval scales? Is it through mentioning the mathematics of (and I had to look this up) affine space? I ask, because I've always glossed interval scales to myself in the way you describe, blissfully unaware that there might be more to the matter.

ReplyDeleteNothing so fancy. I would just need to do a little algebraic manipulation to derive the formula for the transformation, but it is too much trouble, and I think might drive people away.

ReplyDeleteOh, ok, that's simple enough. I'm familiar with that transformation from my contact with basic mathematical logic, as I imagine a number of your readers might be.

ReplyDeleteRe: your first paragraph, please allow a "principled objection" to treating economic activities just like a casino...when economic activities treat humans as objects and subject them to market coercion...as I'm sure you are aware...! Even so, it is very difficult to choose NOT to play the economics game. :-)

ReplyDeleteThanks very much for doing this! After moving away from the Marxism of my high school and college days, I've gotten back into it a bit as I've started to do serious work in political philosophy and think hard about economics.

ReplyDeleteThe definition of linear transformation that you use doesn't match up with the definition mathematicians use in, for example, linear algebra. I would call your linear transformations `affine transformations' (and your ratio transformations are what I would call `linear transformations'; they're not so interesting in one-dimensional spaces, but very interesting once you have two or more dimensions to work with). Affine transformations do preserve the ratio of intervals, but your proof is a bit garbled. Presumably you want to show that, if (a-b)=(c-d) then (a'-b')=(c'-d'). You've either proved the converse or begged the question. Here's a correct proof:

Assume: (a-b)=(c-d)

Multiply by q and distribute: (aq-bq)=(cq-dq)

Add 2r, distribute, and re-associate: (aq+r) - (bq+r) = (cq+r) - (dq-r)

Apply def'n of transformation: a' - b' = c' - d'

QED

Also, I'm not sure what you mean when you say that a linear transformation (in your sense) preserves a cardinal ordering. You haven't given a formal definition of a cardinal ordering -- which is fine, you're not writing this mathematicians -- but it looked like you take a cardinal ordering to be defined in terms of addition and multiplication. So I'd assume that you mean a linear transformation commutes with addition and multiplication, ie, if a+b=c then a'+b'=c' and if a*b=c then a' * b' = c'. But neither of these holds in general.

Noumena, it looks to me as though you are right about this. Sorry. As for "cardinal," I mean simply that this is an ordering that contains enough information for us to perform meaningful arithmetic operations on the utility indices, such as calculating the mathematical expectation of a gamble. If, as between two game strategies, the M. E. of the first is at least as great as the M. E. of the second, that relationship will be preserved under an affine [or as I have been saying linear] transformation. No?

ReplyDeleteIf the Pareto-better distribution where only one person is made slightly better must exclude envy, then the Pareto-worse distribution where only one person is made slightly worse off and everyone else better off must exclude empathy?

ReplyDeleteYou say that the Pareto ordering is transitive but not complete. I'm guessing this means that there are some social distributions on which it has nothing to say about whether they are better than or the same as others. Or is it just the unanimity ordering that's not complete? (Sorry, I am jumping ahead without clearly understanding the concepts, so I'm willing to wait as you present them.)

The Pareto ordering is a unanimity ordering, and it is not complete. If some are better off and others are worse off, it just has nothing to say about their relative merits. I think you have it right.

ReplyDeleteAbout the Sophie's choice issue: are you going to talk about the different interpretations of rational choice theory at some point?

ReplyDeleteIt is my understanding that your objection only works if you interpret revealed preference theory as being committed to saying that choices reveal real, underlying states. The objection is irrelevant if we interpret talk of such states behaviouristically, as Samuelson, etc. do.

It is my experience/suspicion that economists tend to be realists about such states until challenged, but then, annoyingly, claim they have no such commitments at all and actually believe in the behaviourist interpretation.

Personally I think neither strategy works. Will you discuss this?

I am going to try to when I get to Nozick. As I see it [and this is true for other things as well], these really smart economists stake out very strong claims for what they are doing. When they are challenged, they retreat to impregnable but totally uninteresting reinterpretaions of their theories. As soon as you back off, they are out there making the strong claims again. For example, all the nonsense about marginal productivity and the counterproductivity of minimum wage laws, which depend for their plausibilty on Euler's theorem about linear homogeneous functions, even though they know perfectly well that the theorem never applies to an actual capitalist system. It is a very high powered, sophisticated shell game.

ReplyDeleteOK, thanks. I do think this issue really matters. Revealed preference was explicitly formulated in behaviourist terms and is still seen as such by the economists who care about such things. (This makes the name of the theory almost impossibly annoying.) It is my understanding that Becker and those who are realists are a small minority.

ReplyDeleteThis means that economists who read this will shrug off this sort of challenge with a few remarks about behaviour and money-pump arguments. Given how often this happens I think a discussion of this dialectical move is vital. I would also quite like to know your views on this topic.

JP, I think I do not know enough about this debate to speak usefully about it, but I would welcome a guest post if you want to take a shot. I hate to admit it, but there is a limit to the number of things I actually something about! :)

ReplyDeleteThanks, that is very kind of you. I know a fair bit about the topic, but it lies well outside my area of expertise. I will have a look at the recent literature. If I think I can say something informative I will write it up and pass it along.

ReplyDeleteHey, I have lived my life outside my area of expertise!!

ReplyDelete