The second journal to which I submitted my paper is well known for extremely rapid rejections, and my paper was no exception. However, unlike the editor of the previous journal, the editor of this journal gave me a reason, not enough value added for a general economics audience, and suggested a different journal to which I might submit it.

Now, if by value added he means interest, then he may well be correct. And certainly the normal presumption in mainstream economics is that agents are the best judges of what is good for them, so it would probably be bad form for me to insist that the general economics audience ought to care more about the foundations of microëconomics.

The editor in question is a macroëconomist, part of a minority in economics who like to think about economic aggregates. But he's one of that noble sort of macroëconomist who seek solid microfoundatons for their macroëconomics, so I'm less able to make a case that he has a bias against microëconomic theory than if he were one of those Keynesians who insist that aggregates can or must be explained immediately one in terms of others.

Anyway, although I'm unhappy with another rejection, I'm pleased that it is explained, and in terms that indicate that it is not being waved-away as foolish nonsense.

I incorporated some changes to my paper that I wished (almost immediately after I'd submitted it) that I had made sooner. Then I chose the next journal to which to submit it, read their submission guidelines, made some changes in the form of the citations, and submitted it to that next journal.

I submitted a version of Indifference, Indecision, and Coin-Flipping to a journal this morning. In that context, I'm going make the working version more generally available.

For those who want to wade through the mathematics but find mysterious some of the notation or the formal notion of a Cartesian product or of a relation, there is a quick-and-dirty explanation of some of that infrastructure.

For those who, instead, want to get the basic ideas without carefully following the math or confronting the proofs, there is a Gentler Guide. It would be helpful, though perhaps not necessary, to have the original paper at hand when reading the Gentler Guide.

I might someday write a Gentler-Still Guide, with even less of the mathematical formalities, but such a treatment would either replace those formalities with verbal constructions that were themselves difficult to follow, or would necessarily discard an even greater amount of the content from the original paper.

If one has a question or questions (concerning this work) not answered by one of the three papers, a comment to this 'blog entry is about as good a way of asking as any.

By far, most submissions to the better reputed economics journals are simply rejected, and I have submitted to what may be the most prestigious economics journal. I am hoping to receive either a simple acceptance or a directive to revise and resubmit. (With economics papers in general, the latter is more common that the former, and is typically seen as good news.) If the paper is simply rejected, then (assuming that no fatal and irreparable flaws were found in the work) the next thing to do is to modify it, as much as seems reasonable, in light of any comments from the referees (reviewers) and from the editor, and submit it to one of the best remaining journals that I think might accept it.

It can take months for referees to actually read a given paper, and between the time that a paper is first submitted to its first journal and it is published in some journal (assuming that the paper is indeed ever published) can be a matter of years. Unsurprisingly, I hope for a faster resolution than that.

Working on that less formal explanation of my paper has paid another dividend.

The model in the paper has involved six propositions, which function like axiomata, specifically about choices concerning lotteries. I have known for some time that these weren't orthogonal — that there was overlap amongst what they said. It is at best unfortunate to have that sort of redundancy in foundational propositions. But I'd not seen how to reduce it.

This morning, after describing those propositions less formally, and then considering what to say about the theoremata that follow, I started to see how to turn one of those propositions into a theorem. After a bit of musing and fretting, I've effected the change.

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.

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})]

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.

As noted earlier, I've been reading Subjective Probability: The Real Thing by Richard C. Jeffrey. It's a short book, but I've been distracted by other things, and I've also been slowed by the condition of the book; it's full of errors. For example,

It seems evident that black ravens confirm (H) All ravens are black and that nonblack nonravens do not. Yet H is equivalent to All nonravens are nonblack.

Uhm, no: (X ⇒ Y) ≡ (¬X ∨ Y) = (Y ∨ ¬X) = (¬¬Y ∨ ¬X) = [¬(¬Y) ∨ ¬X] ≡ (¬Y ⇒ ¬X) In words, that all ravens are black is equivalent to that all non-black things are non-ravens.[1]

The bobbled expressions and at least one expositional omission sometimes had me wondering if he and his felllows were barking mad. Some of the notational errors have really thrown me, as my first reäction was to wonder if I'd missed something.

Authors make mistakes. That's principally why there are editors. But it appears that Cambridge University Press did little or no real editting of this book. (A link to a PDF file of the manuscript may be found at Jeffrey's website, and used for comparison.) Granted that the book is posthumous, and that Jeffrey was dead more than a year before publication, so they couldn't ask him about various things. But someone should have read this thing carefully enough to spot all these errors. In most of the cases that I've seen, I can identify the appropriate correction. Perhaps in some cases the best that could be done would be to alert the reader that there was a problem. In any case, it seems that Cambridge University Press wouldn't be bothered.

[1]The question, then, is of why, say, a red flower (a non-black non-raven) isn't taken as confirmation that all ravens are black. The answer, of course, lies principally in the difference between reasoning from plausibility versus reasoning from certainty.

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.