Yester-day, as I was in the course of writing-up a somewhat less formal discussion of the ideas in my decision-theory paper, I reälized that I had fudged something important in that paper.

More specifically, I had erroneously treated an important property of one relation as ex definitione. That property does follow from the combination of the definition with some propositions in the paper, but it's not a fully observable property, whereäs I had tried to make each of the relations as observable as was possible, even at the cost of using somewhat cluttered definitions.

I was, unsurprisingly, very unhappy with the reälization. The confusion had been an act of incompetence on my part, in an area where competence is quite important to me. It meant that I had increased the potential burden upon those who have been or will be kind enough to check-over that work. And I didn't immediately know how much work I would have to do to repair the model.

As to the last, when I buckled-down and started the revision, it proved to be fairly easy. I removed the mistaken assertion in the discussion of the definition, and inserted a quick proof of the property amongst the theoremata.

Had the error not been caught before the paper were submitted to a journal, it might well have caused the paper to be rejected. A referee might have been bright enough to pick-up on the mistake, but (for various possible reasons) not seen that it weren't truly fundamental to the work.

One lesson that is reïnforced by this experience is the value, for one's own understanding, of explaining ideas to others.

The current version of my paper operationalizing and formalizing preferences that are not totally ordered can probably be pushed out the door now. I'm running or going to run it past two more academic economists whom I know before I submit it to a journal, but that's just pronounced caution, and more concerned with the quality of the exposition than with that of the ideas themselves.

Unfortunately, while I'm reasonably sure that the work is correct, I no longer have a gut sense that it's important. Intellectually, I see this lack of such a sense as stemming from a confluence of three things.

The first two are my extended efforts to understand and to communicate matters clearly; these efforts result in those matters now seeming very clear and thus seeming rather obvious to me. I have to remind myself that it was all very murky when I began.

But, additionally, through my life, I've repeatedly had the experience that I just don't feel the significance of completed work. I'm not sure why. Perhaps it's just constitutional anhedonia. I do know that things such as ceremonies and celebrations don't help.

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I've cobbled-together a sort of cheat-sheet for some of my readers who might not be familiar with some of the notation, mathematical notions, and economics jargon that I use in the paper; I'll make it openly available when I make the paper itself more openly available. (In the mean time, anyone with access to the paper who wants a copy of the cheat-sheet should let me know.) At some point, I hope to write-up a sort-of informal translation of the ideas of the paper into far less technical language.

In ordinary decision theory, the name weak preference is used for a relation that could be defined as the complement of the inverse of strict preference, or as the union of strict preference with indifference. In ordinary decision theory, these two are equivalent. Typical symbols used for this relationship are ≽, ≿ and ⪰.

In my paper, I noted the conventional conception of weak preference as the aforementioned union; later, I defined it for my purposes such that

(X₁ ≽ X₂) ≡ [{X₁} ⊆ C({X₁, X₂})]

which is the complement of the inverse of strict preference.

Well, I've decided that I was just asking for trouble using that name and that symbol, because people would have trouble not thinking of it as the union of strict preference with indifference, and thence think that a distinct relation of indecision is precluded a priori. So I've switched to the name non-rejection and the symbol ⊀.

There will probably still be people who ask how this relation differs from that of weak preference, but that will be more an expression of their cleverness than of their confusion, and it will be easier to offer an explanation how the relation could be seen as weak preference, how it should not be seen as weak preference. (I may try to squeeze some preëmptive discussion into the paper, though I am bumping-up against size limits.)

Addendum (2009:04/18): I added a preëmptive discussion, so that even fewer people will get confused.

After he read my paper, Anthony told me something that I already knew — that the Discussion section was brief, and the Conclusion sudden. I had been both sick of working on the paper, and having trouble thinking about it in natural language. So those parts were… lacking. Anthony's gentle remarks increased my sense that they were inadequate.

I have expanded and reörganized the Discussion section (cannibalizing part of the Conclusion), and added to the Future Work section between it and the Conclusion. In that context, the Conclusion should seem less sudden, and I have added some thoughts to it (as well as having taken one from it).

Anthony also suggested that the paper could be made more accessible by discussion of the historical background of the problem, and of real-world examples. But, as I told him, I fear to alienate experts by such discussion; and I have since learned that I am already bumping-up against the size-limits for a submission to the journal of my choice.

Anyway, the latest version of my paper is at

The mathematical expressions that appear (tiled) in the background of this 'blog; are from a paper on which I have been working (off-and-on) over a long time. I'm far from perfectly happy with the present state of the paper, but I've finally put-together something like a complete draft of it, in PDF, at

There are some temporary issues with presentation of this draft:

• The layout needs to be fixed, by the insertion of page breaks, so that things such as section headings are pushed to a next page, rather than orphaned.
• The OpenOffice formula editor does not support use of the relational symbols ≻, ≽, ≿, or ⪰. (For now, I am representing strict preference with pref and weak preference with wpref; later, I will output the paper as LAΤΕΧ and then tweak the formulæ to give me ≻ for strict preference and one of the other three for weak preference.)

At this point, I welcome comments, from experts and from non-experts, both on the underlying content and on how I have expressed myself.

Up-Date (2009:04/08): Professor Gamst of UCSD caught an error in what had been formula (10). It was easily patched. I have up-dated the on-line version of the paper.

In conversation to-day with the Woman of Interest, I said something that I have often said jocularly

I'm not so much a fool as you think.

It's actually a line in translation from a play, Policja, by Sławomir Mrożek. The line is spoken by the character of the General, after his paranoiac nature has kept him from being killed in an explosion.

I only saw that play once, back in late '71 when it was broadcast on WNET, with John McGiver in the rôle of the General. But that one line has very much stuck with me, and repeatedly been used by me, ever since.

Much of my time of late has been going into my paper on operationalizing a model of preference in which strict preference and indifference don't provide a total ordering.

Quite a while ago, I reälized very precisely what sort of system the assumptions would have to imply; I mistakenly presumed that I would relatively quickly identify sufficient assumptions (beyond those already recognized). But, at this point, I have a sufficient assemblage, each member of which is, taken by itself, at least passably acceptable. Jointly, however, there's an issue of factoring.

The paper derives its results from three sets of propositions. The first and second sets seem perfectly fine to me, and I don't expect them to provoke much dispute. The third set are more ad hoc. For the purposes of the paper they function as axiomata, but some or all of them would more ideally be derived from deeper principles (the pursuit of which, however, would be mostly a distraction from my goals).

It's amongst this last set of propositions that the factoring problem exists. One of them used to play an important rôle; right now it's doing nothing but occupying space. I'd remove it, except that I suspect that, in conjunction with the very principle that seemed to make it superfluous, it renders redundant another principle which feels even more ad hoc.

At the same time, I am now wrestling with what sort of discussion to provide after presenting the theoremata. I just don't seem to be in much of a frame-of-mind to ruminate.