Wednesday, May 5, 2010


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.


  1. Go slow on the mathematical logic! Also, could I ask that the paragraphs be spaced away from each other? It's just mildly easier to read, but helps when it's heavy stuff.

    Loving it so far!

  2. For some reason, Google has altered the procedures for posting a segment, making it harder to configure, but I will certainly try to separate paragraphs, etc.

  3. Thank you! It is very helpful to have important basic concepts explained from scratch, and the formal theory presented in orderly progression from there (almost always introductory guides gloss over important details, and advanced guides assume knowledge of basic concepts).

    As for the example intended to show that transitivity is questionable, I'm not sure the optometrist case shows this. One could grant that it shows "PERCEIVED to be clearer than or equally clear as" is non-transitive, while insisting that "IS clearer than or equally clear as" is transitive. Perhaps one can respond similarly for preferences, in cases where it is hard to discern whether one likes one thing more than another.

  4. I don't agree about the optometrist example, although maybe I did not make it clear enough. :) The doctor is asking, "Is it clearer to you." That really is not transitive. Obviously, heis not asking, which lens has greater magnification, because you are not in a position to answer that.

  5. Yes, that's clearer. :D What I find striking about that counterexample is that it seems generalizable to all vague predicates of a given form, impugning the transitivity of "is balder than or equally bald as" for instance. (So I was trying out a sort of epistemicist response to vagueness earlier.)

  6. Teminological question: shouldn't "minimally discriminable difference" have the acronym MDD rather than MMD? I've run into the term before, but can't find evidence of an acronym either way through a quick search of the web.

  7. Sigh. I am such a klutz. Of course it should be MDD! I lay in bed last night thinking just that, wondering whether I had got it right. Obviously not. Oh well, I think it was clear, anyway.

  8. I wonder if the size of S is related to the completeness of a preference structure on S. People formulate preferences on the basis of finite experiences. Perhaps they can still be said to have preferences about countably infinite sets -- but preferences about uncountable sets strike me as inaccessible.

    Still, the following seems plausible: that a complete preference structure on a (possibly uncountable) set can always be "approximated as well as you like," by a countable but incomplete preference structure. If so, then the size of the set wouldn't matter so much after all. However, to make this work, we still need to introduce a norm that gives meaning to "approximation as well as you like"...

  9. One difference between neoclassical economics and the Austrian economics of von Mises is that Austrian economics doesn't assume transitivity of preferences.

  10. For most of the literature, S is finite, so none of this arises. But when we get to Luce and Raiffa's formal development of cardinal utility functions, we will be talking about preferenes between pairs of probability distributions over the eleemnts of a finite set, S, and those are at the very least countably infinite, and maybe uncountably infinite [I can't remember]. The danger is that one may reverse the logical order and say "I prefer this lottery over the elements of the set S over that lottery, BECAUSE its expected value is larger," rather than saying, "Because I can express consistent preferences over those lotteries [and other things as well], I can construct a cardianl utility function which makes it possible to carry out expected utility calculations."

  11. Thanks. For me, the pace is just right and the examples are extremely helpful.

  12. Thanks for doing this! I've just recommended your site to a bright young student of mine. His name is 'Greg Keenan'. I trust he will introduce himself when he gets here. (Hint, hint, Greg!)

    -Dustin Locke

  13. This is great, but the font is massive. Is there any way to make it smaller?

  14. The font just works for us older guys, thank you.

  15. For those of us who have been away from (or never experienced) formal logic, I have to echo Jian thoughts of going a little slower or having more of an introduction / explanation for the logic section(s)

  16. As I am sure you have noticed, most of the comments above are rather technical and minor points from people who demonstrably already have a very strong grasp of the subject already, who are doubtfully the audience getting the most out of this!

  17. This is a great project. Thanks for taking the time. I would love it if you'd discuss the use of formal models and materials in legal theorizing (as you mentioned you might).

  18. First of all, welcome all! This is tremendously eciting. I feel as though I threw a party and a lot of folks turned up.

    Leo, I apologize for the font size, but it is really helpful for me. Be patient with us old guys.

    Lord Bacon [really?], I am really torn. I will do my best to slow down and clarify, but this is hard stuff, and takes some getting used to. Stay with me for another couple of posts and see what you think.

    Hrafninn [that can't be a name, right?], I will eventually get to the law, but I need to go through the foundational stuff before I start applying it. By the time I get there, you should be able to do it yourself. At least, that is my hope.

  19. Long story with the web name (I am apparently genealogically related to Lord Byron, but wouldn't dare be so pompous in name. I am also the Sorrell from email). I agree the balance of slow and fast is always in the horns of a dilemma with a diverse audience but was just pointing it out for those who come across this without experience and for whom this would truly benefit.

  20. Here's a tip for anyone who is discontent with the size of the font, whether you want it larger or smaller, you can usually change that on your own end.

    Most web browsers use the keyboard shortcuts Ctrl+- and Ctrl++ to decrease and increase the font size being displayed. That is, hold down the Ctrl key and then press - to lower the size, or + to increase it.

  21. Thanks for the clarity and examples!

    One question: how important is it that more than one element can have the same name....that is, there is not a unique mapping one to one. Will you explore this more later?

    Lovely. :-)

  22. Lucid and elegant. This is a wonderful read. Please, keep them coming!

  23. Roberto, That is awesome. The things you learn.

    Now, I have a serious question. I want to scan some diagrams onto my computer [I can do that, in jpeg or pdf format, among others]. Then I want crop them so that I get rid of all the empty space around them, and then I want to insert them into my narrative.


  24. Okay, maybe i'll retract my comment on trying to go slow. If there are parts which need more pointing out, if there are sufficient comments asking, there can always be an adjunct blog post explaining the mathematical logic, right? :)

  25. Getting rid of empty space is called cropping. There are many image manipulator programmes that will allow you to do that, and usually the software that allows you to scan the pictures comes with basic cropping features.

  26. thanks for doing this. I would definitely be interested in seeing as many of the formal proofs as possible.

  27. There is also a quicker, related way to increase and decrease font size: hold down the control key while scrolling up and down with the scroll wheel on your mouse et voila!

  28. Perhaps premature: in your optometry example, I believe P, as you define it, is transitive (but not complete) even though R is not. Why not just use P instead of R so that vagueness doesn't undermine transitivity?

  29. Walking John, the reason is that for certain purposes, such as Arrow's Theorem, the proof only goes through using R, not P. Also, when we get to Amartya Sen's brilliant extension of Arrow's Theorem, we will need R.

  30. Thanks for the tutorial. I like the speed and look forward to more.

    I am interested in transitivity, so let me ask a question about this assumption. As you point out, the assumption that preference orderings are transitive is usually in the context of rational choice theory--where the transitivity of her preferences is one test of an individual's rationality in holding these preferences. So the assumption is related to the idea that it would be irrational to simultaneously hold a set of judgments like the following: aPb, bPc, cPa.

    I agree that in your optometrist case the ordering would not exhibit transitivity. Suppose the ordering is as follows (to simplify a bit): aRb, bRa, bRc, cRb, aRc, not-cRa. And I grant that the case does not show me the patient is irrational. But I am suspicious that this does not bear on cases where the assumption of transitivity is used as a measure of rationality in holding preferences.

    Consider whether the patient should be given pause if he were to be shown a sheet documenting his responses, like the list above. Suppose further that the doctor points out the intransitivity of these judgments by noting relevant entailments. In the case as originally presented, there is no need to assume that the patient holds all of these judgments in mind all at once. But in the revision of the case, the sheet is supposed to do just this, induce the patient to hold all judgments before his mind's eye at once.

    Now I think that the patient should be given pause in the revised case because he will feel rational pressure to revise one of his responses. And I think this pressure can be explained by the fact that he now has all judgments before his mind's eye at once and wants to avoid affirming contradictions. This set of judgments entails the judgment cRa, which contradicts not-cRa. This is revealed when the patient considers all judgments at once. One notices this with the set before one all at once, but not from looking at the individual judgments as they are formed. So it is understandable why the original case did not involve a pressure to revise his judgments. This suggests to me that where an individual is supposed to have a set of judgments in mind all at once, the assumption of transitivity is not doing heavy lifting that cannot be explained by other features of the case (e.g., that the agent is able to recognize the relevant entailments, hold all of the judgments in mind all at once, and is motivated to revise contradictory sets of beliefs). The assumption simplifies things by standing in for the sorts of features noted in the parentheses. But it does not add anything beyond what these other features already build into the case.

    I wonder what you think about all of this.

  31. Ben, This raises complicated questions that I do not want to try to answer here. I will try to weave them into my narratuve. By the way, I wouid caution against the use of the phrase "gesture at." It is a cliche that obscures clear thought :)

  32. I look forward to how you separate rational choice from "collective choice" and "game theory." Seems to me these three categories are not really categories (disjoint sets?) but theories along a continuum. Game theory, as much as rational choice, requires an ordering of preferences -- though it does not require these preference orderings to be permanent.

  33. Collective choice assumes ordinal preferences. Game Theory assumes cardinal preferences. If you want to see the sort of trouble you can get into by confusing the two, look at my article on Steven Strasnick's supposed proof og Rawls' Difference Principle in ehe Journal of Philosophy for December 1976.

  34. In terms of nuclear determent, I thought you might find this interesting:

  35. This onetime mathematician is having a high old time here, Bob.

    Ben's comment brings to mind what seems an essential element of how Rat Choice works (and possibly other Theories as well), namely that the answers you get (as in the optometrist example) depend in significant part on the questions put to the patient. That is, the Theory is not a Theory of How Things Are but of How Adepts May Be Taught to Think Things Are.

  36. On speed, I think you're striking a great balance so far; as long as future posts have brief recaps I think people will find it fairly easy to pick it up as they go along.

    A formatting suggestion: you might consider bolding section headings just to make them stand out more clearly and to make it easier to find things in the post. (Depending on whether you think it worth the extra time, it might also be helpful to some if you bold words that you are defining or explaining and that will turn out to be important.)

  37. Brandon, I bolded a whole lot of stuff and then the bolding disappeared when I transferred it to the blog. This is a learning experience.

  38. I have a, probably inane, question that doesn't even really bear on my understanding of the part of the post that doesn't concern nominal ordering. (Is there a nifty name for the ordering that results when R is introduced over a set, but we don't know whether it is complete or transitive?) Why is it that every element is in one, and only one, equivalence class? For example, why can't you have a name "chocolate or vanilla" and a name "chocolate and vanilla" and then the element vanilla-chocolate swirl fits in two equivalence classes? Is it just stipulated that this can't be?

  39. I saw an answer to how to create an image file above, but not to the question of how to get it onto the blog page, so here goes a partial reply.

    Probably your blog host has a help page about how to drop an image file onto a page. In HTML it is pretty easy to include images in a page, and I imagine that the text editor for blogger includes an easy way to do this as well. It likely involves uploading the target image file (as a jpeg or gif) to a directory somewhere and then plugging the URL for that file into the blog editor. Alternatively, some editors even allow you to take something from your hard drive and upload and link it at once. Usually these have a browse button which when you click it lets you browse your own computer for the file you want to upload and link to.

    Sorry I can't be more specific as I haven't used blogger to create a blog.

  40. This is tremendous. I am looking forward to the next installment. I find the pacing just right.

    Coming from a liberal arts background, I find it surprising (and slightly baffling) that such highly formal principles constitute the foundation of modern political philosophy. My lay intuition is that applying such formalism to human affairs won't work. It seems wildly presumptuous to schematize so rigidly something as vague and little understood as preferences and desires. But I am prepared to abandon my skepticism if future installments demonstrate that such models are capable of producing nontrivial results.

    Speaking for myself, I would love to read a short historical exposition of the rise of these models in contemporary philosophy. It needn't be longer than a couple of paragraphs. Perhaps others would find it useful, too.

    Finally, a note regarding graphics. There are some very good web-based image editors. These can't compete with the likes of Photoshop or GIMP in terms of functionality, but they are more than adequate for simple operations like cropping and resizing. You could try Pixlr Express (at or Picnik ( Hope this helps!

  41. I am a little unclear on the transitivity Dr. example perhaps because it is not presented as the typical a>b>c therefore a>c. It seems you are saying m=n but a> or <q. How does this show the relation is not transitive?

  42. The point is that for EVERY adjacent pair, m and n, both mRn and nRm, so by transitivity, it would follow that for the first and last elements, a and q, aRq and qRa. But for those elements, aPq, which is to say, aRq and not qRa
    Is that clearer?.

  43. Some of us in my program have been dabbling in beginning game theory for the last few months on and off (though more "off" than we'd like, honestly). The fact that you are blogging with a forum, and in doing that finding a middle ground between a book and an interactive lecture, makes this really helpful. Thanks for doing it.

  44. Hello, I'm very new to the field, but it interests me a lot. I have a got a mathematics/economics background, but I'm studying philosophy now. I look forward to reading your next posts. In the meantime, would you be able to advice some books on this subject? Thanks!

  45. Well, you could get hold of Luce and Raiffa, Games and Decisions, for a really solid technical treatment.

  46. So in ordinary language, it is equivalent to saying: I am indifferent between A and B, I am also indifferent between B and C, but I am not indifferent between A and C, thereby rejecting transitivity?

  47. Exactly, Darwin 13, except that in the example I gave, you have a string of indifferences, the end points of which atre not indifferent, and that violates transitivity.

  48. This seems extraordinarilly close to the Sorites paradox, has anyone tried accounting for this with, for example, Łukasiewicz's fuzzy logic?

  49. Thank you very much for taking the time to explain such interesting but potentially difficult material so clearly and lucidly. This is fantastic!

  50. In response to the above discussion of why the optometrist example exhibits nontransitivity: Maybe an example of nontransitivity using a strong ordering P would be more striking.

    Say that 5 desserts are ordered a, b, c, d, e, from sweetest (a) to least sweet (e). "xPy" means that if a restaurant offers me a choice between x and y (and nothing else), then I would choose x. Say I have a sweet tooth, and if I have to choose between two adjacent desserts, I will choose the one that is *slightly* sweeter. So: aPb, bPc, cPd, dPe.

    But I have a bit of guilt about my sweet tooth, and when the issue of my sugar consumption is placed front-and-center in my mind -- such as it would be when I have to make a choice between the strikingly different a and e, then I will be health-conscious and choose the less sweet dessert. So, ePa, violating transitivity.

  51. Re: Daniel
    I don't think that your example really works because the agent clearly has other preferences (ie. not wanting to be seen having a sweet tooth) that are influencing his decision yet are not in the formalism.

    You could change the example a little bit and maintain the strict ordering just by making it more like the optometry case: you prefer sweeter things, but each dessert is a tiny bit more sweet than the last, yet it is by so little that you do not notice. Hence, you are indifferent to adjacent desserts. However, perhaps there is a very long line of desserts that are each progressively sweeter. In this case, although you are indifferent to the smaller differences, you will not be indifferent to the larger ones.

  52. Game theory does not need cardinal preferences -- Cardinal preferences are often used as a pedagogical tool but the proofs are always based on utility functions that follow the usual rules about rank ordering-- see e.g. the Fudenberg and Tirole Game Theory text. The issue is a little bit more complicated for mixed strategy or Bayesian equilibria but even there I do not see the necessity of cardinal preferences. Almost everything that can be achieved by using numbers can also be achieved by using ranks in utility functions. So I think you are making a mistake by classifying differences between Game Theory and Rational Choice models according to whether cardinal or ordinal utility is used.

    Oddly enough it is a mistake made by Austrian economists like Von Mises and Rothbard as well.

  53. Re: J. Vlastis--

    Is your point that "I choose dessert x over y" cannot be a formalizable preference ordering, because it somehow is a "complex preference" (composed of two "simple preferences") and only these "simple preferences" are deserving of formalization?

    If so, that seems like a dangerous route to go... it would introduce a kind of deep and quirky psychological assumption.

    More generally: Do you believe that there are any *strong* preference orderings P that are not transitive? (I mean just a single person's preferences, not group preferences.)

  54. Re: Daniel

    I'm pretty sure that our preference orderings are not transitive, or at best only some of the are.

    My point is that your example clearly had two different preference relations that were applicable in this situation, one which prefered based on the standard of your sweet tooth and one that prefered based on you wanting to be healthy. Why is this a dangerous route to go? It seems to be the source of a lot of our practical conflicts.

    I was just trying to say that your example could be rigged up so that the best way to describe it is to say that this one preference relation imposes a strong ordering where transitivity does not hold. This example would be just like vagueness examples where someone who is not bald can have one hair removed and still be not bald yet if this were iterated enough times, the person would become bald.

  55. I prefer Locke to Kant, and I prefer Kant to Macchiaveli. But I prefer Macchiaveli to Locke. I prefer Locke's brevity to Kant's loquaciousness, but I prefer Macchiaveli's amorality to Kant's categorical imperative.

    It would seem that transitivity can be expected only when the members of the category are reduced to a single characteristic.

    What am I missing?

  56. Economicus, it is precisely for mixed strategies that we will need cardinal utility. I am afraid I do not see how you can prove von Neuman's fundamental theorem without it. Can you send me an email and explain [it is probably going to be too compicatated for a blog comment].