Posts Tagged ‘randomness’

Decision-Time for the Donkey

Monday, 6 May 2013

Yester-day, I finished reading the 1969 version of Choice without Preference: A Study of the History and of the Logic of the Problem of Buridan's Ass by Nicholas Rescher, which version appears in his Essays in Philosophical Analysis. An earlier version appeared in Kantstudien volume 51 (1959/60), and some version has or versions have appeared in later collections. I have only read the 1969 version, and some of the objections that I raise here may have been addressed by a revision.

The problem of Buridan's ass may not be familiar by name to all of my readers, but I imagine that all of them have encountered some form of it. A creature is given a choice between two options neither of which seems more desirable than the other. The question then is of how, if at all, the creature can make a choice. In the classical presentation, the creature is a donkey or some other member of the sub-genus Asinus of Equus, the choice is between food sources, and a failure to make a choice will result in death by starvation. The problem was not first presented by the Fourteenth-Century cleric and philosopher Jean Buridan, but it has come to be associated with his name. (Unsurprisingly, my paper on indifference and indecision makes mention of Buridan's ass.)

Rescher explores the history of the problem, in terms of the forms that it took, the ultimate purposes for which a principle were sought from its consideration, and the principles that were claimed to be found. Then he presents his own ostensible resolution, and examines how that might be applied to those ultimate purposes.

One of the immediate problems that I have with the essay is that nowhere does Rescher actually define what he means by preference. I feel this absence most keenly when Rescher objects that there is no preference where some author and I think there to be a preference.

As it happens, in my paper on indifference and indecision, I actually gave a definition of strict preference: (X1 pref X2) = ~[{X2} subset C({X1,X2})] which is to say that X1 is strictly preferred to X2 if X2 is not in the choice made from the two of them.[1] So, in that paper, strict preference really just refers to a pattern of choice. I didn't in fact define choice, and I'll return to that issue later.

The Merriam-Webster Dictionary essentially identifies preference as a gerund of prefer, and offers two potentially relevant definitions of prefer:

  1. to promote or advance to a rank or position
  2. to like better or best
The first seems to be a description of selection as such. The second might be taken to mean something more. But when I look at the definition of like, I'm still wondering what sense I might make of it other than an inclination to choose.

I'm not claiming that Rescher is necessarily caught-up in an illusion. Rather, I'm claiming, first, that he hasn't explained something that is both essential to his position and far from evident; and, second, that his criticism of some authors is based upon confusing their definitions with his own.

When I used the notion of a choice function C( ) in my paper, my conception of choice was no more than one of selection, and that's what I was unconsciously taking Rescher to mean until, towards the end of his essay, speaking of decisions made by flips of coins (and the like), he writes

In either event, we can be said to have "made a choice" purely by courtesty. It would be more rigorously correct to say that we have effected a selection.

Well, no. This isn't a matter of rigor, whatever it might be. The word choice can rigorously refer to selection of any sort. It can also refer to selection with some sort of care, which seems to be what he had in mind.

Some of the authors whom Rescher cites, and Rescher himself, assert that when a choice is to be made in the face of indifference, it may be done by random means. Indeed, Rescher argues that it must be done by such means. But he waits rather a long time before he provides any explicit definition of what he means by random, and he involves two notions without explaining why one must invoke the other, and indeed seemingly without seeing that he would involve two distinct notions. When he finally gives an explicit notion, it to characterize a choice to be made as random when there is equal weight of evidence in favor of each option. However, when earlier writing of the device by which the selection is to be made, he insists

The randomness of any selection process is a matter which in cases of importance, shall be checked by empirical means.

Now, one does not test the previously mentioned equal weight of the evidence by empirical means. An empirical test, instead, adds to the fund of evidence. We can judge the weight of the present evidence about the selection device by examining just that present evidence. The options are characterized by equal plausibility, yet Rescher has insisted that the selection device must instead be characterized by equal propensity. It isn't clear why the device can't simply also be characterized by equal plausibility.[2]

Rescher makes a somewhat naïve claim just before that insistence on empirical testing. For less critical choices, he declares

This randomizing instrument may, however, be the human mind, since men are capable of making arbitrary selections, with respect to which they can be adequately certain in their own mind that the choice was made haphazardly, and without any reasons whatsoever. This process is, it is true, open to possible intrusions of unrecognized biases, but then so are physical randomizers such as coins.

Actually, empirical testing of attempts by people to generate random numbers internally show very marked biases, such that it's fairly easy to find much less predictable physical selectors.

Rescher's confusion of notions of randomness is entangled with a confounding taxonomy of choice which is perhaps the biggest problem with Rescher's analysis. The options that he allows are

  1. decision paralysis
  2. selection favoring the first option
  3. selection favoring the second option
  4. random selection, in which random entails a lack of bias
And, proceeding thence, he seems to confuse utterly the notion that choice without some preference somewhere is impossible with the notion that choice without some preference somewhere is unreasonable. In any case, Rescher insists that only the last of these modes of selection is reasonable, and this insistence would tell Buridan's ass that it must starve unless it can find a perfectly unbiased coin![3] Reason would be a harsher mistress than I take her to be!

Another term that Rescher uses without definition is fair and its coördinates, as when he writes

Random selection, it is clear, constitutes the sole wholly satisfactory manner of resolving exclusive choice between equivalent claims in a wholly fair and unobjectionable manner.

I certainly don't see that random selection should be seen as wholly satisfactory (though I believe it to often be the least unsatisfying manner), and I don't know what Rescher imagines by fair. My experience is that when the word fair is used, it is typically for something more appealing than justice to those inclined to envy. In the case of allotments by coin-flip, there may be no motivation for envy ex ante, but things will be different ex post. People do a great deal of railing against the ostensible unfairness of their luck or of that of another.

I recall one final objection, which moves us quite out of the realm of economics, but which I have none-the-less. One of the applications of these questions of choice without preference (or, at least, without preference except stemming from meta-preference) has been to choices made by G_d. In looking at these problems, Rescher insists that G_d's knowledge must be timeless; I think that he ought to allow for the possibility that it were not.

[1] That might seem an awkward way of saying that X1 is strictly preferred to X2 if only X1 is in the choice from the two of them, but it actually made the proofs less awkward to define strict preference in this odd manner.

[2] Even if one insists that the selection device must be characterized by equal propensity, there is in fact little need for empirical testing, if one accepts the presumptions that a coin may be considered to have unchanging bias and that flips of a coin may be independent one from another. Implicitly making these assumptions, my father proposes a method for the construction of a coin where the chances of heads and of tails would be exactly equal. One starts with an ordinary coin; it comes-up heads sometimes, and tails others. Its bias is unknown; at best approximated. But, whatever the bias may be, says my father, in any pair of flips, the chances of heads-followed-by-tails are exactly equal to the chances of tails-followed-by-heads. So a pair of flips of the ordinary coin that comes-up heads-tails is heads for the constructed coin; a pair of flips of the ordinary coin that comes-up tails-heads is tails for the constructed coin; any other pair for the ordinary coin (heads-heads, tails-tails, or one or both flips on edge) is discarded.

[3] I don't know that my father could explain his solution to a donkey. I've had trouble explaining it to human beings.

Randomness and Time

Sunday, 20 February 2011

When someone uses the word random, part of me immediately wants a definition.[1]

One notion of randomness is essentially that of lawlessness. For example, I was recently slogging through a book that rejects the proposition that quantum-level events are determined by hidden variables, and insists that the universe is instead irreducibly random. The problem that I have with such a claim is that it seems incoherent.

There is no being without being something; the idea of existence is no more or less than that of properties in the extreme abstract. And a property is no more or less than a law of behavior.

Our ordinary discourse does not distinguish between claims about a thing and claims about the idea of a thing. Thus, we can seem to talk about unicorns when we are really talking about the idea of unicorns. When we say that unicorns do not exist, we are really talking about the idea of unicorns, which is how unicorns can be this-or-that without unicorns really being anything.

When it is claimed that a behavior is random in the sense of being without law, it seems to me that the behavior and the idea of the behavior have been confused; that, supposedly, there's no property in some dimension, yet it's going to express itself in that dimension.

Another idea of randomness is one of complexity, especially of hopeless complexity. In this case, there's no denial of underlying lawfulness; there's just a throwing-up of the hands at the difficulty in finding a law or in applying a law once found.

This complexity notion makes awfully good sense to me, but it's not quite the notion that I want to present here. What unites the notion of lawlessness with that of complexity is that of practical unpredictability. But I think that we can usefully look at things from a different perspective.

After the recognition that space could be usefully conceptualized within a framework of three orthogonal, arithmetic dimensions, there came a recognition that time could be considered as a fourth arithmetic dimension, orthogonal to the other three. But, as an analogy was sensed amongst these four dimensions, a puzzle presented itself. That puzzle is the arrow of time. If time were just like the other dimensions, why cannot we reverse ourselves along that dimension just as along the other three. I don't propose to offer a solution to that puzzle, but I propose to take a critical look at a class of ostensible solutions, reject them, and then pull something from the ashes.

Some authors propose to find the arrow of time in disorder; as they would have it, for a system to move into the future is no more or less than for it to become more disorderly.

One of the implications of this proposition is that time would be macroscopic; in sufficiently small systems, there is no increase nor decrease in order, so time would be said neither to more forward nor backward. And, as some of these authors note, because the propensity of macroscopic systems to become more disorderly is statistical, rather than specifically absolute, it would be possible for time to be reversed, if a macroscopic system happened to become more orderly.

But I immediately want to ask what it would even mean to be reversed here. Reversal is always relative. The universe cannot be pointed in a different direction, unless by universe one means something other than everything. Perhaps we could have a local system become more orderly, and thus be reversed in time relative to some other, except, then, that the local system doesn't seem to be closed. And, since the propensity to disorder is statistical, it's possible for it to be reversed for the universe as a whole, even if the odds are not only against that but astronomically against it. What are we to make of a distinction between a universe flying into reverse and a universe just coming to an end? And what are we to make of a universe in which over-all order increases for some time less than the universe has already existed? Couldn't this be, and yet how could it be if the arrow of time were a consequence of disorder?

But I also have a big problem with notions of disorder. In fact, this heads us back in the direction of notions of randomness.

If I take a deck of cards that has been shuffled, hand it to someone, and ask him or her to put it in order, there are multiple ways that he or she might do so. Numbers could be ascending or descending within suits, suits could be separated or interleaved, &c. There are as many possible orderings as there are possible rules for ordering, and for any sequence, there is some rule to fit it. In a very important sense, the cards are always ordered. To describe anything is to fit a rule to it, to find an order for it. That someone whom I asked to put the cards in order would be perfectly correct to just hand them right back to me, unless I'd specified some order other than that in which they already were.

Time's arrow is not found in real disorder generally, because there is always order. One could focus on specific classes of order, but, for reasons noted earlier, I don't see the explanation of time in, say, thermodynamic entropy.

But, return to decks of cards. I could present two decks of card, with the individual cards still seeming to be in mint state, with one deck ordered familiarly and with the other in unfamiliar order. Most people would classify the deck in familiar order as ordered and the other as random; and most people would think the ordered deck as more likely straight from the pack than the random deck. Unfamiliar orderings of some things are often the same thing as complex orderings, but the familiar orderings of decks of cards are actually conventional. It's only if we use a mapping from a familiar ordering to an unfamiliar ordering that the unfamiliar ordering seems complex. Yet even people who know this are going to think of the deck in less familiar order as likely having gone through something more than the deck with more familiar order. Perhaps it is less fundamentally complexity than experience of the evolution of orderings that causes us to see the unfamiliar orderings as random. (Note that, in fact, many people insist that unfamiliar things are complicated even when they're quite simple, or that familiar things are simple even when they're quite complex.)

Even if we do not explain the arrow of time with disorder, we associate randomness with the effects of physical processes, which processes take time. Perhaps we could invert the explanation. Perhaps we could operationalize our conception of randomness in terms of what we expect from a class of processes (specifically, those not guided by intelligence) over time.

(Someone might now object that I'm begging the question of the arrow of time, but I didn't propose to explain it, and my readers all have the experience of that arrow; it's not a rabbit pulled from a hat.)

[1] Other words that cause the same reäction are probability and capitalism.