During the years when I was wrestling with my paper on indecision, there was at least one paper published that attempts to operationalize incomplete preferences, Indifference or indecisiveness? Choice-theoretic foundations of incomplete preferences by Kfir Eliaz and Efe A. Ok in Games and Economic Behavior vol 56 (2006) 61–86. I didn't come across this paper until mine was essentially ready for submission, but I dropped-in a note distinguishing how they sought to operationalize indecision from how I did. What they did was interesting, but I'm distinctly uncomfortable with it.

They propose a scenario in which a Mrs. Watson is selecting one item to be shared by her two children. The children's preferences are thus:

which is to say that A prefers y to z and z to x, while T prefers x to y and y to z. The authors tell us that if y is available then the parent will not select z, as y is preferred to it by both children. Thus, if the choice were only between y and z, then y would be selected. However, should y not be available, and the parent forced to choose between x and z, then there is no clear choice between x and z. Yet if z is not available, there is also no clear choice between x and y. Thus, we have the oddness that y is preferred to z, neither x nor z is preferred to the other, and neither x nor y is preferred to the other. The authors associate this odd confluence with indecision amongst some of the choices. The parent resolves the choice over x and y or over x and z by flipping a coin. (The authors do not recognize coin-flipping as a third option, and what they say elsewhere implies that indifference would also be resolved thus.)

The authors write

It seems quite difficult to argue that there is something clearly wrong in the way Mrs. Watson deals with her two choice problems here.

What the parent has done here is implicitly embraced a sort of democratic process; she knows how the children would vote (if not voting strategically), and casts those votes on the behalf of each, settling ties with the flip of a coin. Well, it's well established that, in the absence of everyone having the same preferences, even if each participant is him- or herself rational, democratic processes don't produce rational choice structures. (For example, if some votes are decided before others, then the order in which things are decided determines the ultimate outcome.) In fact, it was famously shown (with Kenneth Arrow's Impossibility Theorem) that this isn't just a problem for ordinary democracy, but for any collective decision-making process. (I'm not here counting dictatorship as a collective decision-making process.)

So the parent's attempting to meet the desires of her two children is, in this case, formally equivalent to her attempting to meet the desires of one child who is just crazy. (Parents of multiple children will likely have little difficulty in grasping that point intuïtively.)

I'll make more plain the insanity specifically here. Under a decision-making process in which one would simply trade each unit of every x_n for a unit of x_n+1, by m-stage trades, one can be got to trade all of one's x₀ for x_m, even if one prefers x₀ to x_m. If whether each trade is effected depends upon the flip of a coin, then it might take more offers to get one to trade-away all of one's x₀ for x_m, but that's still where one can be led. (A patient exploiter could get one to surrender all of one's x₀ for x_m, and then monetize that by selling the x₀, with credit for trade-in of the x_m.)

The authors acknowledge that their scenario involves social choice theory, but assert that the problem can be generalized to other problems of choice with multiple criteria. However, the example that they give is one of an agent attempting to follow given rules (for awarding a fellowship); what we should see is that whatever rules might produce the same sort of structure are, again, loopy (in at least two senses of that word).

Now, in some case, perhaps in a great many cases, the sane chose to humor the insane; they even bend to the will of the lunatic; but the choice made in yielding to regulation by another is not, properly, the same choice as that of the regulator. Nor would I otherwise be comfortable with a model that could not find indecision except where one can find this sort of irrationality lurking in the background if not in the foreground.

In his Philosophical Theories of Probability, Donald Gillies proposes what he calls an intersubjective theory of probability. A better name for it would be group-strategy model of probability.

Subjectivists such as Bruno di Finetti ask the reader to consider the following sort of game:

• Some potential event is identified.
• Our hero must choose a real number (negative or positive) q, a betting quotient.
• The nemesis, who is rational, must choose a stake S, which is a positive or negative sum of money or zero.
• Our hero must, under any circumstance, pay the nemesis q·S. (If the product q·S is negative, then this amounts to the nemesis paying money to our hero.)
• If the identified event occurs, then the nemesis must pay our hero S (which, if S is negative, then amounts to taking money from our hero). If it does not occur, then our hero gets nothing.

Di Finetti argues that a rational betting quotient will capture a rational degree of personal belief, and that a probability is exactly and only a degree of personal belief.

Gillies asks us to consider games of the very same sort, except that the betting quotients must be chosen jointly amongst a team of players. Such betting quotients would be at least examples of what Gillies calls intersubjective probabilities. Gillies tells us that these are the probabilities of rational consensus. For example, these are ostensibly the probabilities of scientific consensus.

Opponents of subjectivists such as di Finetti have long argued that the sort of game that he proposes fails in one way or another to be formally identical to the general problem for the application of personal degrees of belief. Gillies doesn't even try to show how the game, if played by a team, is formally identical to the general problem of group commitment to propositions. He instead belabors a different point, which should already be obvious to all of his readers, that teamwork is sometimes in the interest of the individual.

Amongst other things, scientific method is about best approximation of the truth. There are some genuine, difficult questions about just what makes one approximation better than another, but an approximation isn't relevantly better for promoting such things as the social standing as such or material wealth as such of a particular clique. It isn't at all clear who or what, in the formation of genuinely scientific consensus, would play a rôle that corresponds to that of the nemesis in the betting game.

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Karl Popper, who proposed to explain probabilities in terms of objective propensities (rather than in terms of judgmental orderings or in terms of frequencies), asserted that

Causation is just a special case of propensity: the case of propensity equal to 1, a determining demand, or force, for realization.

Gillies joins others in taking him to task for the simple reason that probabilities can be inverted — one can talk both about the probability of A given B and that of B given A, whereäs presumably if A caused B then B cannot have caused A.

Later, for his own propensity theory, Gillies proposes to define probability to apply only to events that display a sort of independence. Thus, flips of coins might be described by probabilities, but the value of a random-walk process (where changes are independent but present value is a sum of past changes) would not itself have a probability. None-the-less, while the value of a random walk and similar processes would not themselves have probabilities, they'd still be subject to compositions of probabilities which we would previously have called probabilities.

In other words, Gillies has basically taken the liberty of employing a foundational notion of probability, and permitting its extension; he chooses not to call the extension probability, but that's just notation. Well, Popper had a foundational notion of propensity, which is a generalization of causality. He identified this notion with probability, and implicitly extended the notion to include inversions.

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Later, Gillies offers dreadful criticism of Keynes. Keynes's judgmental theory of probability implies that every rational person with sufficient intellect and the same information set would ascribe exactly the same probability to a proposition. Gillies asserts

[…] different individuals may come to quite different conclusions even though they have the same background knowledge and expertise in the relevant area, and even though they are all quite rational. A single rational degree of belief on which all rational being should agree seems to be a myth.

So much for the logical interpretation of probability, […].

No two human beings have or could have the same information set. (I am reminded of infuriating claims that monozygotic children raised by the same parents have both the same heredity and the same environment.) Gillies writes of the relevant area, but in the formation of judgments about uncertain matters, we may and as I believe should be informed by a very extensive body of knowledge. Awareness that others might dismiss as irrelevant can provide support for general relationships. And I don't recall Keynes ever suggesting that there would be real-world cases of two people having the same information set and hence not disagreeing unless one of them were of inferior intellect.

After objecting that the traditional subjective theory doesn't satisfactorily cover all manner of judgmental probability, and claiming that his intersubjective notion can describe probabilities imputed by groups, Gillies takes another shot at Keynes:

When Keynes propounded his logical theory of probability, he was a member of an elite group of logically minded Cambridge intellectuals (the Apostles). In these circumstances, what he regarded as a single rational degree of belief valid for the whole of humanity may have been no more than the consensus belief of the Apostles. However admirable the Apostles, their consensus beliefs were very far from being shared by the rest of humanity. This became obvious in the 1930s when the Apostles developed a consensus belief in Soviet communism, a belief which was certainly not shared by everyone else.

Note the insinuation that Keynes thought that there were a single rational degree of belief valid for the whole of humanity, whereäs there is no indication that Keynes felt that everyone did, should, or could have the same information set. Rather than becoming obvious to him in the 1930s, it would have been evident to Keynes much earlier that many of his own beliefs and those of the other Apostles were at odds with those of most of mankind. Gillies' reference to embrace of Marxism in the '30s by most of the Apostles simply looks like irrelevant, Red-baiting ad hominem to me. One doesn't have to like Keynes (as I don't), Marxism (as I don't) or the Apostles (as I don't) to be appalled by this passage (as I am).

Probability is one elephant, not two or more formally identical or formally similar elephants.

Towards my next paper, I've been thinking a lot about decision-making where one has uncertainty but not quantified probabilities or even necessarily a total ordering of possible outcomes by plausibility. Most recently, I've tried to formalize the notion of when, without quantified probabilities, one lottery may be said to be fairer than another, and of a simple rule for selecting the fairer of two coins (as in my previous paper I have made considerable use of orderings of coins by entropy).

Yester-day, in the context of such ponderings, I arrived at some interesting, simple complementarity rules. Consider two actions, each paired with one considered outcome. Between these, various plausibility relations may obtain — the first pair may be more plausible than the second, the second may be more plausible than the first, they may be equally plausible, their relative plausibility may be unknown, or the relation may be a union of two or three of the aforementioned (eg one pair may be more-or-equally plausible). In any case, whatever that relation, the same relation will hold if we reverse the order of the pairs and take the logical complement of the outcomes. Here's an example of the formal expression of one of these rules where I'm using the same notation that I did in in my entry of 19 August, and represents the relationship of the left side being more plausible than the right side.

(Common-sense examples are easy to generate. For example: If it is more likely that the Beet Weasel will bite than that the Woman of Interest will stay home, then it is more likely that she will depart than that he will refrain from biting. Or: If we don't know whether a given nickle is more likely to come-up heads than is a given quarter, then we don't know whether the quarter is more likely to come-up tails than is the nickle.)

In the context of an irreflexive, antisymmetric, transitive relation, one can identify closeness without measurement. For example, if A is more plausible than B and B is more plausible than C, then B is closer both to A and to C than they are each to the other.

This abstraction of closeness, along with the principles of complementarity, allow one to identify when one coin is more fair than another, without having any quantification of fairness, so long as one can order the plausibilities of outcomes across coins. One simple rule is to pick the coin whose most likely outcome is less likely than the most likely outcome of other coins; an equivalent rule would be to pick the coin whose least likely outcome is more likely than the least likely outcomes of other coins.

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BTW, the aforementioned previous paper is still in the hands of the editors of the journal to which I submitted it a bit less than four months ago. I've not had any word from them. But, while this journal did not provide a time-frame, other journals give frames such as six months. (A friend recently had one of her submissions rejected at just before the three-month mark.) It is at least somewhat plausible that, by the time that said previous paper is published somewhere, I will have this next paper ready to submit to a journal.

Last night, I finished the clean-up of a LAΤΕΧ version of my paper on incomplete preferences. From remarks by a person more knowledgeable about ΤΕΧ than I, it seemed that my best option in dealing with the under-sized angle brackets was to just fall back to using only parentheses, square brackets, and braces for taller delimiters. And most width problems were resolved by expressing formulæ over more lines. Unfortunately, these changes leave the formulæ harder to read than in the original.

This after-noon, I completed the submission process to one of the two specialized journals recommended by the advising editor who rejected it at the previous journal to which I submitted it. The submission process for this latest journal required that I name the other journals to which I'd submitted the paper. As simultaneous submissions are disallowed, basically they were asking for a list of which journals has rejected the paper. I gave it. (I didn't tell them that the third had been suggested by the second, nor that theirs had been suggested by the fourth.)

Anyway, I'm back to waiting for a response.

My latest submission of my paper, to a yet more specialized journal, has met with a fate similar to that of my previous submissions:

The advisory editor suggests that the paper does not fit the general readership of [this journal] (see his short report below).

That advisory editor writes

I suggest to the author to submit his paper, which certainly deserves an outlet, to more specialistic journals

and then recommends two in particular. So I will review the guidelines for each, and try to decide to which of them I will make my next submission. I take some solace in the fact that, while my paper is indeed being rejected, editors are suggesting that it truly ought to be published in a respected academic journal.

Recently, when working on my next paper, I was struck by the formal similarity between two expressions.

Using to represent outcome X of action c, and to represent a relationship of equal plausibility, implies that X_i is certain given c, and implies that X_i is impossible given c. (The two expressions differ in the subscript of the outcome on the right-hand side of the relation.)

The ideas here are simple: If no other outcome contributes plausibility, then X_i is certain; if X_i contributes no plausibility, then X_i is impossible.

After some vacillation over the question of to which of two journals next to submit my paper, I have submitted it to a game theory journal which has published at least one other article attempting to operationalize incomplete preferences. (I think that attempt rather less satisfactory than mine.)

I have, alternately, been considering submitting to an older journal, based in Europe, which focusses primarily on mathematical microëconomic theory, but I decided both that they would be more likely to reject the paper as too specialized, and that my paper would be less widely read if published there.

Good L_rd! In response to my submission, the editor of the third journal responded

While I find the paper interesting, I feel it is too specialized a topic for a general audience journal such as [ours].

The thing is that, unlike the previous two journals, which cover economics in general, this is a journal of microëconomics. Yet, like the editor of the previous journal, the editor still feels that the paper is too specialized for the audience. (Though, as noted, this third journal was recommended by that editor of the second.)

I need to figure-out just who won't think it too specialized.

I have submitted my paper to a third journal, that recommended by the editor who rejected it at the previous journal.

This third journal is one from an association which, like many, charges a lower submission fee to its members. Even with the annual dues and on the assumption that I only made one submission in a year, I would still save money, so I joined. However, after I registered and paid, I learned that it could take up to four weeks for my membership information to be recorded and provided to me. Hence, this delay between submissions. I'm not sure that the money saved was worth that delay.