the Paradoxical Connection between Money and Brains
(or, Zen and the Art of the Deal)
by Eric Hart
The last issue of Noesis, by Chris Cole, presents the "Allais Paradox" in the context of "Bayesian regression". It is surmised that the idiosyncrasies of human mental "circuitry", particularly those involving hidden or false assumptions, may obstruct rational decision-making, and that these quirks may somehow be responsible for variations in the "subjective value of money". Yet, what if the implied inconsistency exists not only in the minds of those to whom the paradox poses its alternatives, but within the minds of those who perceive that it does?
I have not had the time to acquire an extensive analysis of this paradox, and must rely on Chris's account of it. Accordingly, readers are referred to the last edition of the newsletter as preparatory reading for what follows (see editorial).
The paradox involves two alternatives, one of which may be claimed by a hypothetical human subject:
A) An 89% chance of an unknown amount of money x,
a 10% chance of $1 million, and
a 1% chance of $1 million;
B) an 89% chance of x,
a 10% chance of $2.5 million, and
a 1% chance of nothing.
The subject is to choose between A and B, deterministically and from pure self-interest. The paradox involves a supposedly irrational tendency for subjects to switch from A to B as the value of x changes from $1 million to $0. This tendency would be evident in a statistic representing the outcomes of repeated trials involving different subjects, each of whom is queried for both test-values of X; call this statistic S. In addition, there is a supposed tendency for S to become constant and rational as all the monetary amounts are reduced by several powers of ten. Let this tendency be reflected in the metastatistic M(S).
First, any paradox calls for an extension of the frame within which it is defined. Can such an extension be given in the present case? It can, provided the formulation of the paradox includes or implies terms whose own definitions generate the extension. Since the paradox involves a decision, it is decision-theoretic. It thus involves expectation and risk, certain aspects of which go beyond any so-called "rationality" which falsely absolutizes the decisive criteria.
Consider the "independence axiom" on which the stock version of rationality is defined: "The rational choice between two alternatives depends only on how those alternatives differ." Now, difference is an elementary kind of relation, and relations among variables cannot always be autonomously defined. There seems to be an assumption that the difference between A and B is context-free, or independent, even though it involves at least one context-sensitive criterion (risk). That, and not the putative irregularities in subjective decision-functions, is the strangest aspect of this paradox; how could a rational analyst ignore so basic a concept? It is almost as though the topic of money were acting as a neural noisemaker, a cue for hunger to exile logic. Concisely, the alternatives contain a joint variable (the unknown amount x) which conceals contextual information on which a critical decisive predicate (risk) is defined. Depending on the value given this variable, risk may or may not differ between the alternatives. The axiom cited above, whose formulation is consistent with this possibility, must obviously be reinterpreted in light of it.
Nature abhors absolute inconsistency; paradox, an inherently computational concept, is always a matter of computational deficiency ("always" requires some qualiflcation, applying to the inferior realm of natural temporalities). Paradoxes are useful whenever they allow us to locate and remedy erroneous or incomplete thought-patterns. But it is sometimes a little too easy for the analysts of paradox to locate these deficiencies not within themselves, but with those around whom the paradox is explicitly formulated.
Whether the paradox resides with the analysts or their subjects, it is computational. The question remains, do the implied mental deficiencies reflect human limitations as opposed to the broader limitations of computative devices, or can reasonably smart humans also learn to resolve this paradox? If so, we can eliminate one more doubt concerning the ultimate generic identity of human and mechanical computation, an identity first explored by Alan Turing in his celebrated paper on the criteria for mechanical simulation of human behavior (i.e., those of the "Turing Test"). If it can be shown that whether a universal deterministic machine can resolve a paradox (solve a problem by logical induction) in time < n depends partly on intrinsic factors (e.g., size) rather than only on its proper or improper programming by fallible humans - or that the minds of human programmers can themselves create and run the right program and thereby achieve understanding on the level of correctly programmed universal machines comparable in general parameters like size and speed - the hunan-mechanical equivalency thesis will be strengthened, at least with respect to our (humanistically-flawed?) axioms of computation. The issue is not whether certain kinds of neural nets - e.g., those with the capacity for emotion - are inoptimal for solving some kinds of problems, but whether they can solve them at all.
Notice that nondeterministic machines, which are in
Current research implies that generic equivalency obtains within the natural realm. But if so, this is a truth of little use to us here, since Allais' paradox boils down to the mere bad habits (but not the absolute limitations) of human thinkers. To demonstrate this, it will suffice to resolve the paradox within our own neural parallel-processing networks, or "brains". The framework has been erected above; all that remains is to fill it in.
Let us begin by relating the economic and computational aspects of the paradox. Because the choice to be computed depends on collective economic criteria, which in turn depend in part on the computative characteristics of the individuals locally situated in the economy, neither aspect is independent of the other; they are tautologically united. The algorithm to be employed amounts to a binary deterministic game-theoretic function F with the pair (A,B) as its argument and with certain player-specific valuational (as opposed to expressly quantitative) parameters. In other words, the game varies to some extent with the "subjective" situation of the player. Being formulated on an essentially game-theoretical basis, F(A,B) incorporates a risk criterion. Risk, of course, is defined as the probability of inverse gain, or of loss (the evaluation of risk, being probabilistic, is one part of this matter to which Bayesian inference obviously applies, though economic reasoning in general is shot through with probabilistic regressions). Observe that since the subject is not allowed to play repeatedly, F cannot reduce risk to negative expectation, but must somehow incorporate the "worst case": a maximum loss of $1 million.
Assessments of risk can themselves be risky. We live in a country - and in a world - where observation demonstrates time and again that exposed money is a target for numerous grasping hands. The average person is encouraged by this to distinguish sharply between monetary exposure and nonexposure. A scheme's risk, one observes, is consistently understated to potential investors by those who want their money, and one can trust an evaluation of risk only as far as one trusts the evaluator. Money is not a thing which breeds trust among men.
It follows that we will observe a repolarization of F, as reflected in the
statistic S(F), along with any change concerning the presence of (nonzero)
risk. Allais paradox displays such a disjunction: when the unknown amount is 0,
there is no risk. Yet when this amount is raised to $1 million, the probability
of significant material loss becomes ".01" (or so says the poser, who
may or may not have a fair coin in his hand). This is because the subject is
Notice that while the distinction between zero and nonzero risk can be taken to involve psychological factors, these are strictly behavioral: conditioning (compare "programming") by negative reinforcement (compare "past input") within the local environment. Even digital computers are "psychological" to this extent. So the distinction may be safely understood as computational in the general sense, and no human peculiarities need be blamed.
On the other hand, what if we were to let the subject use his own "fair
coin"? While trust ceases to matter, other factors remain. The actual
material loss of $1 million equates to financial ruin for most people. But what
is the exact point at which an unacceptable loss becomes acceptable? This
distinction represents another potential irregularity, which we may for present
purposes associate with the metastatistic M. That is, the test-values of x may
be subject to various local relativizations
It has been conjectured that arithmetic was originally a mercantile creation, born of trade and profit. If this is so, there may still exist a simplistic tendency to equate money with inventory against an infinite market (if twenty bolts of cloth are worth a talent of silver by decree of Assurbanipal, then how many talents will buy enough cloth to outfit two legions?). However, we now understand that economies seldom function so simply, and that money does not admit of linear valuation. Since human beings exist in economic settings which vary according to local criteria, they cannot rationally ignore those criteria in their valuation 0f money. That is, they must "subjectivize" the value of money in accordance with their goals and current positions in the economy, by rules abstracted from their past experience and what they know of financial theory. The efficiency with which they do this, of course, again admits df analysis apart from considerations of detective psychology or the basic inferiority of neural mentation.
Similarly, the rate at which investment generates capital is not constant as the level of investment rises within a given financial setting. Expansion is seldom continuous; there are certain discrete monetary thresholds which must be reached before the successive phases of expansion can occur, and windows of opportunity are often labeled with particular antes. The economic and game-theoretic complexities which come into play can easily boggle any reasonable set of computative parameters...a fact well-known to those who attempt to construct computative models capable of any but the most general economic predictions. If the subject resorts to oversimplification - or even nondeterminism - in making his decision, we need not count it any great wonder. But one fact is obvious: one cannot always translate less money than one needs into enough to meet his goals, even if the "objective" difference in amounts appears relatively small, without information that is often difficult to acquire and fallible by dint of the regressive uncertainties that characterize open economies.
Thus, the subjective value of money reflects certain actualities within and without the immediate economic environment of the individual. A set amount is necessary to even become a "player" in certain social and financial circles, including those to which a given subject aspires. Because aspirations, which vary greatly among people, act like gauge modulators on the scale of monetary value, it is meaningless to speak of how alternatives differ "intrinsically" and without respect to outside influences on the scale of differences. In reality, money - and much else as well - has no measurable intrinsic value at all. This is a lesson dear to the hearts of economists, some of whom (e.g., the U. S. Federal Reservists) are actually in the practice of adjusting its value according to global criteria within which local variables differ (thus, regulation of the total supply of money can be used to control its average value over the space of regulation, but not the small-scale "subjective" variations which occur therein). If you must convince yourself of this lesson, try eating a hundred-dollar bill the next time you get hungry, or thinking of any direct use for it in which it is irreplacable by something which costs far less than its face value. As the only such applications involve showing it to other people, its value depends on their subjectivizations of it.
Because the value of money is defined on economic criteria, and because economic criteria devolve locally to the subjective values of people, money on the local scale has subjective value only. It is defined on the psychology of need and happiness, and it comes as no surprise when subjective factors enter into judgments involving it. Were we to replace "money" with anything whose value obeys a constant function of quantity, the choices would remain rational (in the naive sense) up to the point of subjectivization. But as subjectivism is unavoidable in economic reasoning, no such monetary replacement can exist (a fact I demonstrate elsewhere along with more detailed findings on various intriguing logical and quantitative phenomena in economy-like systems).
Now we reach a seeming distinction between brains and universal machines.
Any set of goals can be programmed into the latter, but the wants of men and
women are more often determined by such quasi-congenital factors as talent and
ability. Thus, humans are to some degree "pre-programmed", and to
this extent non-universal. On the other hand, universal machines without any
pre-programming are paralyzed by their functional generality; a random or
nondeterministic machine produces output as a random function of input. This
forces us to limit our comparison to functional machines. The distinction then
comes to resemble that between ROM and programmed RAM, and thus becomes
amenable to standard computational analysis. Concisely, human nature is
ultimately a compression of nurture, or environmental conditioning, primed by
the pure will to survive on the individual and collective levels. Since, even
if we attempt to abstract all "nature" out of our machine, we are
forced to unite it with a programmatic agency having as much nature as we do,
the distinction is ultimately irrelevant to equivalency. Simply note that the
physiomorphic distinctions among human brains, inasmuch as they are the
products of natural selection, reflect past local variations within the system
So gain and risk are subjectively evaluated as functions of the punctuated and fluctuating monetary thresholds associated with particular systems of goals and computational parameters, both of the latter being determined as functions of location and conditioning on genetic and environmental levels. This admits of mathematical formalization, and we are temporarily spared a pronouncement of incorrigible eccentricity by our ability to follow this reasoning and thus override our respective mental quirks (and whatever limitations they seem to entail).
Money has stable value only as an averaging function on human scales of wants yet, because humans are diverse and variously-situated, individual scales need not conform to this average, out are instead part of what determines it. The problem resides in our tendency to try to predicate economic decisions on the value of money, rather than vice versa (which is how it actually is on the scale of this formulation). So - apart from the consideration of risk - it is not the "unusualness of the amounts" which confuses, but the subjective and polymorphic nature of money. Rationality must account for this, and reasoning which treats money as though it had absolute intrinsic value is flawed and therefore irrational. It thus falls before a higher reality in which Allais paragox straightens and vanishes like a well-shot arrow.
Note that the resolution possesses a certain undeniable beauty. Economies depend in part on the neural parameters of brains which evolved from a primal Darwinian "economy" of survival, which in turn evolved from the system of invariants by which the universe is constructed. By simple extension, we can think of this system of laws as the "anthropic" configuration of the "Mind of God". The implied model is a hellix linking generality with specificity, and closing ultimately upon itself in a teleological loop.
And thus can obscure paradoxes lead to profound insights. Precisely Because he is capable of such insights can man learn now to optimize economies in the promotion of global happiness, and apply reason and compassion where abuse now prevails. By defining the connection between human nature and free economics - the laws at once governing brains, economies, and other systems - we extend the narrow conceptual path leading towards human fulfillment. But more on this below.
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