FORMAL METHODS IN POLITICAL PHILOSOPHY FIRST POST

This is it! The launch of my new blog. Please let me know your reaction. Am I going to fast? Too slow? Is everything clear? This is really an on-line course of lectures, not a snarky blog, so I need your guidance. Here we go.

The Use and Abuse of Formal Models in Political Philosophy

Introductory Remarks

The purpose of these remarks is to introduce you to the technical foundations of a number of formal methods of analysis that have come to play a large role in the writings of philosophers, economists, political theorists, legal theorists, and others, and then to show you how these formal methods are misused by many of those theorists, with results that are conceptually confusing and quite often covertly ideologically tendentious. I am going to expound these materials carefully and with sufficient detail to allow you to master them and make your own judgments about the appropriateness of their use.

There are three distinct bodies of material with which we shall be dealing. Each has grown out of a different intellectual tradition, uses different methods of formal analysis, and finds different application by philosophers, political, theorists, and so forth. Quite often they are confused with one another, and my impression is that the people who use them frequently do not understand the distinctions among them, but we shall treat them separately.

The first body of material is Rational Choice Theory. When people talk about maximizing utility or calculating the expected value of an alternative or discounting an outcome by its risk, they are drawing on Rational Choice Theory.
The second body of material is Collective Choice Theory. When people talk about the paradox of majority rule or Arrow's Theorem or Pareto Optimality they are drawing on Collective Choice Theory.

The third body of material is Game Theory. When people talk about strategies or zero sum or prisoner's dilemma, they are drawing on Game Theory.

I am going to ask you to be patient, because this is going to take a while. By the time we are done, I may have written a short book. As we proceed, I will make some reading suggestions for those of you who wish to pursue the subject in greater depth, but everything you will need to know to follow my exposition will be contained in these pages.

The order of exposition is going to be as follows:

1. Some preliminary technical matters, principally concerning different kinds of orderings.

2. The elements of Rational Choice Theory.

3. The elements of Collective Choice Theory, including a formal proof of Kenneth Arrow's General Possibility Theorem, which is the central formal result in the field.

4. The elements of Game Theory, maybe [if you have the stomach for it] including a formal proof of von Neumann's Fundamental Theorem concerning two person zero-sum games with mixed strategies.

5. A formal analysis of John Rawls' claims concerning choice in the Original Position in A Theory of Justice.

6. A formal analysis of Robert Nozick's claims in Anarchy, State, and Utopia.

7. The misuse of Game Theory in nuclear deterrence theory.

8. A general discussion of misuses of the formal materials in treatments of so-called Free Rider problems and other matters. Depending on your endurance and interest, I may at this point discuss the use of formal models and materials in legal theorizing and other areas.

A REALLY, REALLY IMPORTANT REQUEST: I cannot see your eyes, so I cannot tell when they glaze over, either from boredom because I am going too slowly, or from confusion because I am going too fast. So I need to hear from you if either of those things is happening.

Part I. Preliminary Technical Matters

A. Scales of Measurement [For more detail, see S. S. Stevens, Handbook of Experimental Psychology, originally published in 1951 but re-issued and updated.]

Let us suppose that we have a finite set of discrete elements of any sort, which we will call S =(a, b, c, ..., n). The elements might be different amounts of money, different flavors of ice cream, different bowls of ice cream [not the same thing, of course], different candidates in an election, and so forth.

We may wish to impose an ordering on the set. The very simplest ordering we can impose is a nominal ordering, or a labeling. To each element, we assign a label or name [hence "nominal"]. Two or more elements may receive the same name, but each element receives only one name.

Such an ordering is said to be complete if every element in S is labeled. The ordering creates what are called equivalence classes, which is to say, subsets of elements all of which bear the same label or name. This labeling exhaustively and mutually exclusively divides S into subsets. Obviously, two elements are in the same equivalence class if and only if they have the same name. Every element is in one, and only one, equivalence class. With a nominal ordering, nothing more can be deduced from the labeling than the simple fact that two elements are in the same equivalence class if and only if they bear the same label. The essential fact about this very simple measure is that it is complete. Every element bears a label. For any two elements, either they are in the same equivalence class or they are not. Trivially, each element is in the same equivalence class with itself. Thus, every element is in some equivalence class.

The next step is to introduce a binary relation, R, over the set of elements. xRy is construed variously as meaning "x is equal to or greater than y," or "(someone is) indifferent between x and y or prefers x to y," or even "x is hotter than or is the same temperature as y," and so forth. All of these have the same formal structure.

Let us suppose the following two propositions are true for R and for S:

(i) for all x and y in S, xRy or yRx. This says that R is complete. Notice that from this, it follows that for all x, xRx. [Just as a trivial exercise, here is how we prove that xRx. Since for any x and y, xRy or yRx, take the case in which x=y. Then substituting, we have xRx or xRx, which is logically equivalent to xRx. That is the sort of baby logic steps we will be taking many of in what follows]. This property of an element bearing a relation to itself is called reflexivity, and a relation of which it holds is said to be reflexive.

(ii) for all x, y, and z in S, if xRy and yRz then xRz. This property is called transitivity, and it will turn out to be the single most important property of relations like R.

Just to be absolutely clear what we are talking about here, suppose we interpret the relation xRy to mean (someone) prefers x to y or is indifferent between x and y. Then (i) says that for any two members of the set S, the person in question either prefers x to y or is indifferent between them, or else prefers y to x or is indifferent between them. If this is still a bit puzzling, think of x and y as real numbers and R as meaning "is equal to or greater than." (ii) says that if the person in question prefers x to y or is indifferent between them, and also prefers y to z or is indifferent between them, then that person also prefers x to z or is indifferent between them.

A binary relations like R is said to establish a weak ordering on the set S. It is weak because it allows for indifference. Starting with the relation R, we can also define a relation P on S, like this: xPy means xRy and not yRx. P here stands for "prefers," and a relation like P is said to establish a strong ordering on the set S. To get an intuitive handle on these very important little symbols, think of it this way. xRy says that x is at least as good as y, and maybe better. xPy says that x really is better than y [whatever "better" means here.] So R is weak and P is strong. Later on, when we come to Collective Choice Theory, we will be saying a lot about weak and strong orderings.

A relation, R, over a set, S, for which i) and (ii) hold is said to be an ordinal ordering. In discussions of these matters in philosophy, economics, and political theory, it is often taken as a fundamental test of a person's rationality that his or her preferences exhibit at least an ordinal ordering over all available alternatives.

Some economists, using what is called a theory of "revealed preference," even argue that everyone must have a preference structure that at least satisfies the first condition, and thus is complete, because confronted with any two alternatives, x and y, a person will either choose one, thus showing that she prefers it to the other, or else will be indifferent between the two. But that, I will argue much later, is a covertly tendentious thesis made more plausible by the formalism. Think Sophie's Choice. [I.e., first you force a woman to choose which of her two children you are going to kill, and then you say, "So, that shows she prefers the one to the other." I am going to have a good deal to say about this sort of thing down the line.]

By the way, "ordinal" because the ordering merely establishes which of the elements is first, second, third, fourth, etc. according to the relation R, and these are what are called "ordinal numbers."

It may not be obvious at first glance, but preference structures do not always exhibit transitivity, and hence are not even ordinal. Indeed, the casual assumption of transitivity is actually an enormously powerful and simplifying assumption.

Let me give an elementary and non-controversial example here, and save the controversial examples for later. All of us, I assume, have had our eyesight checked at the optometrist's office. You shut one eye, the room is darkened, and you look through a complicated gadget at a chart of rows of letters, each line smaller than the one above. The doctor flips lenses in front of your open eye, and asks "Which is clearer, one, or two?" Sometimes you can see a difference, and sometimes you just say, "They are the same." The two lenses may actually have different degrees of magnification, but the difference is simply too small for you to notice. Experimental psychologists say that the difference between the two is then below your "minimal discriminable difference," or MMD. Now, it is obvious that with a little work, the optometrist could line up a series of lenses, each successive pair of which falls below your MMD, but the first and last of which are clearly discriminable. If we interpret R in this case to mean "is clearer than or is equally as clear as," it would be true that for any adjacent pair, m and n, mRn and nRm, but for the first, a, and the last, q, it would not be the case that aRq and qRa. In other words, the relation "is clearer than or equally as clear as" would not be transitive.

The same thing might manifestly be true of someone's preferences. What all this means is that it is very powerful and quite probably false to assume that someone has a transitive preference ordering over a set of available alternatives. But people who use this sort of formalism almost never realize that fact. Indeed, it is quite often the case that people introduce this formalism without even feeling any need to say that they are assuming transitivity. This is a simple example of what I mean when I say that the formalism can conceal powerful and dubious assumptions.

Ordinal preference orders encode the order of someone's preferences, but not the intensity of that preference. Compare voter A with voter B in the 1992 presidential election. Voter A is a fanatic George H. W. Bush supporter. She doesn't really like either Clinton or Perot, but despite Perot's kookiness, prefers him by a hair to Clinton. Voter B is torn between Bush and Perot, neither of whom he loves, but he finally decides to go with Bush. He hates Clinton and wouldn't vote for him even if Mao Tse-Tung were the alternative. These voters have identical ordinal preference structures: Bush first, Perot second, Clinton third. That is all you need to know to figure out how they will vote, but obviously for all sorts of other purposes this ordinal preference ordering fails to embody a great deal of important information. In particular, this ordering will not tell you how either voter might behave in other political contexts besides voting, such as donating money, working for a campaign, lobbying, and so forth.