On the Paradoxical Connection between Money and Brains(or, Zen and the Art of the Deal)

by Christopher Michael Langan

The last issue of Noesis, by Chris Cole, presents the "Allais Paradox" inthe context of "Bayesian regression". It is surmised that the idiosyncrasiesof human mental "circuitry", particularly those involving hidden or falseassumptions, may obstruct rational decision-making, and that thesequirks may somehow be responsible for variations in the "subjective valueof money". Yet, what if the implied inconsistency exists not only in theminds of those to whom the paradox poses its alternatives, but within theminds 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 tothe 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 ahypothetical human subject:

A) An 89% chance of an unknown amount of money x,a 10% chance of $1 million, anda 1% chance of $1 million;

OR

B) an 89% chance of x,a 10% chance of $2.5 million, anda 1% chance of nothing.

The subject is to choose between A and B, deterministically and from pureself-interest. The paradox involves a supposedly irrational tendency forsubjects 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 theoutcomes of repeated trials involving different subjects, each of whom isqueried for both test-values of X; call this statistic S. In addition, there is asupposed tendency for S to become constant and rational as all themonetary amounts are reduced by several powers of ten. Let this tendencybe reflected in the metastatistic M(S).

First, any paradox calls for an extension of the frame within which it isdefined. Can such an extension be given in the present case? It can,provided the formulation of the paradox includes or implies terms whoseown definitions generate the extension. Since the paradox involves adecision, it is decision-theoretic. It thus involves expectation and risk,certain aspects of which go beyond any so-called "rationality" whichfalsely absolutizes the decisive criteria.

Consider the "independence axiom" on which the stock version ofrationality is defined: "The rational choice between two alternativesdepends only on how those alternatives differ." Now, difference is anelementary kind of relation, and relations among variables cannot alwaysbe autonomously defined. There seems to be an assumption that thedifference between A and B is context-free, or independent, even thoughit involves at least one context-sensitive criterion (risk). That, and not theputative irregularities in subjective decision-functions, is the strangestaspect of this paradox; how could a rational analyst ignore so basic aconcept? It is almost as though the topic of money were acting as a neuralnoisemaker, a cue for hunger to exile logic. Concisely, the alternativescontain a joint variable (the unknown amount x) which concealscontextual information on which a critical decisive predicate (risk) isdefined. Depending on the value given this variable, risk may or may notdiffer between the alternatives. The axiom cited above, whose formulationis consistent with this possibility, must obviously be reinterpreted in lightof it.

Nature abhors absolute inconsistency; paradox, an inherentlycomputational concept, is always a matter of computational deficiency("always" requires some qualiflcation, applying to the inferior realm ofnatural temporalities). Paradoxes are useful whenever they allow us tolocate and remedy erroneous or incomplete thought-patterns. But it is

sometimes a little too easy for the analysts of paradox to locate thesedeficiencies not within themselves, but with those around whom theparadox is explicitly formulated.

Whether the paradox resides with the analysts or their subjects, it iscomputational. The question remains, do the implied mental deficienciesreflect human limitations as opposed to the broader limitations ofcomputative devices, or can reasonably smart humans also learn toresolve this paradox? If so, we can eliminate one more doubt concerningthe ultimate generic identity of human and mechanical computation, anidentity first explored by Alan Turing in his celebrated paper on thecriteria for mechanical simulation of human behavior (i.e., those of the"Turing Test").

If it can be shown that whether a universal deterministic machine canresolve a paradox (solve a problem by logical induction) in time < ndepends partly on intrinsic factors (e.g., size) rather than only on itsproper or improper programming by fallible humans - or that the mindsof human programmers can themselves create and run the right programand thereby achieve understanding on the level of correctly programmeduniversal machines comparable in general parameters like size and speed- the hunan-mechanical equivalency thesis will be strengthened, at leastwith respect to our (humanistically-flawed?) axioms of computation. Theissue is not whether certain kinds of neural nets - e.g., those with thecapacity 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 principle immune toflaws in the axioms thought to underlie deterministic procedures, cannotwith certainty be said to share human intellectual limitations, and thatsome nondeterministic machines actually emulate whatever "oracular"abilities humans may ultimately come to possess in the course of sensoryand mental evolution. Moreover, we must consider that human brains areto some measure self-programming, and must widen our analysis todefine and account for all self-programming machines. But this is easy,since the class of self-programming machines is virtually identical withthat of freely-programmable (universal) machines, given a means for theinput and program—conversion of environmental data.

Current research implies that generic equivalency obtains within thenatural 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 absolutelimitations) of human thinkers. To demonstrate this, it will suffice toresolve the paradox within our own neural parallel-processing networks,or "brains". The framework has been erected above; all that remains is tofill it in.

Let us begin by relating the economic and computational aspects of theparadox. Because the choice to be computed depends on collectiveeconomic criteria, which in turn depend in part on the computativecharacteristics of the individuals locally situated in the economy, neitheraspect is independent of the other; they are tautologically united. Thealgorithm to be employed amounts to a binary deterministic game-theoretic function F with the pair (A,B) as its argument and with certainplayer-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 essentiallygame-theoretical basis, F(A,B) incorporates a risk criterion. Risk, ofcourse, is defined as the probability of inverse gain, or of loss (theevaluation of risk, being probabilistic, is one part of this matter to whichBayesian inference obviously applies, though economic reasoning ingeneral is shot through with probabilistic regressions). Observe that sincethe subject is not allowed to play repeatedly, F cannot reduce risk tonegative expectation, but must somehow incorporate the "worst case": amaximum loss of $1 million.

Assessments of risk can themselves be risky. We live in a country - and ina world - where observation demonstrates time and again that exposedmoney is a target for numerous grasping hands. The average person isencouraged by this to distinguish sharply between monetary exposure andnonexposure. A scheme's risk, one observes, is consistently understated topotential investors by those who want their money, and one can trust anevaluation of risk only as far as one trusts the evaluator. Money is not athing which breeds trust among men.

It follows that we will observe a repolarization of F, as reflected in thestatistic S(F), along with any change concerning the presence of (nonzero)risk. Allais paradox displays such a disjunction: when the unknownamount 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 theposer, who may or may not have a fair coin in his hand). This is becausethe subject is effectively given the megabuck (via alternative A) and

offered what amounts to a double-or-nothing bet on it (alternative B).Where the variable amount is 0, however, nothing has been given withany certainty, and risk cannot be materially defined. The concept of"material definition" is pivotal: in the same way that a bird in the hand isworth two in the bush, money possessed with certainty is qualitativelydistinct from - and subjectively more valuable than - money for which oneis merely hoping. Since the rational evaluation of risk must take accountof distinctions of value for the subject, to whom money in the control ofanother is usually valueless, the aphorism takes on mathematical force.

Notice that while the distinction between zero and nonzero risk can betaken to involve psychological factors, these are strictly behavioral:conditioning (compare "programming") by negative reinforcement(compare "past input") within the local environment. Even digitalcomputers are "psychological" to this extent. So the distinction may besafely understood as computational in the general sense, and no humanpeculiarities need be blamed.

On the other hand, what if we were to let the subject use his own "faircoin"? While trust ceases to matter, other factors remain. The actualmaterial loss of $1 million equates to financial ruin for most people. Butwhat is the exact point at which an unacceptable loss becomesacceptable? 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 localrelativizations affecting F, as well as to effects associated with thesimultaneous reduction of all the amounts (a precise description of whichexceeds the level of this presentation). To avoid taxing the reader'sattention and/or facility with mathematics, this phase of the discussionwill be kept as informal as possible.

It has been conjectured that arithmetic was originally a mercantilecreation, born of trade and profit. If this is so, there may still exist asimplistic tendency to equate money with inventory against an infinitemarket (if twenty bolts of cloth are worth a talent of silver by decree ofAssurbanipal, then how many talents will buy enough cloth to outfit twolegions?). However, we now understand that economies seldom functionso simply, and that money does not admit of linear valuation. Sincehuman beings exist in economic settings which vary according to localcriteria, they cannot rationally ignore those criteria in their valuation 0fmoney. That is, they must "subjectivize" the value of money in accordance

with their goals and current positions in the economy, by rules abstractedfrom their past experience and what they know of financial theory. Theefficiency with which they do this, of course, again admits df analysisapart from considerations of detective psychology or the basic inferiorityof neural mentation.

Similarly, the rate at which investment generates capital is not constant asthe level of investment rises within a given financial setting. Expansion isseldom continuous; there are certain discrete monetary thresholds whichmust be reached before the successive phases of expansion can occur, andwindows of opportunity are often labeled with particular antes. Theeconomic and game-theoretic complexities which come into play caneasily boggle any reasonable set of computative parameters...a fact well-known to those who attempt to construct computative models capable ofany but the most general economic predictions. If the subject resorts tooversimplification - or even nondeterminism - in making his decision, weneed not count it any great wonder. But one fact is obvious: one cannotalways 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 ofthe regressive uncertainties that characterize open economies.

Thus, the subjective value of money reflects certain actualities within andwithout the immediate economic environment of the individual. A setamount is necessary to even become a "player" in certain social andfinancial circles, including those to which a given subject aspires. Becauseaspirations, which vary greatly among people, act like gauge modulatorson the scale of monetary value, it is meaningless to speak of howalternatives differ "intrinsically" and without respect to outside influenceson the scale of differences. In reality, money - and much else as well - hasno measurable intrinsic value at all. This is a lesson dear to the hearts ofeconomists, some of whom (e.g., the U. S. Federal Reservists) are actuallyin the practice of adjusting its value according to global criteria withinwhich local variables differ (thus, regulation of the total supply of moneycan be used to control its average value over the space of regulation, butnot the small-scale "subjective" variations which occur therein). If youmust convince yourself of this lesson, try eating a hundred-dollar bill thenext time you get hungry, or thinking of any direct use for it in which it isirreplacable by something which costs far less than its face value. As theonly such applications involve showing it to other people, its valuedepends on their subjectivizations of it.

Because the value of money is defined on economic criteria, and becauseeconomic criteria devolve locally to the subjective values of people, moneyon the local scale has subjective value only. It is defined on the psychologyof need and happiness, and it comes as no surprise when subjectivefactors enter into judgments involving it. Were we to replace "money"with anything whose value obeys a constant function of quantity, thechoices would remain rational (in the naive sense) up to the point ofsubjectivization. But as subjectivism is unavoidable in economicreasoning, no such monetary replacement can exist (a fact I demonstrateelsewhere along with more detailed findings on various intriguing logicaland quantitative phenomena in economy-like systems).

Now we reach a seeming distinction between brains and universalmachines. Any set of goals can be programmed into the latter, but thewants 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 theirfunctional generality; a random or nondeterministic machine producesoutput as a random function of input. This forces us to limit ourcomparison to functional machines. The distinction then

comes to resemble that between ROM and programmed RAM, and thusbecomes amenable to standard computational analysis. Concisely, humannature is ultimately a compression of nurture, or environmentalconditioning, primed by the pure will to survive on the individual andcollective levels. Since, even if we attempt to abstract all "nature" out ofour machine, we are forced to unite it with a programmatic agency havingas much nature as we do, the distinction is ultimately irrelevant toequivalency. Simply note that the physiomorphic distinctions amonghuman brains, inasmuch as they are the products of natural selection,reflect past local variations within the system of selective principles andmay be regarded as programmatic distinctions. Because we areconsidering only deterministic rational processes, we can for now ignorethe effects of nondeterministic lacunae - which may be an essential part ofwhat makes us human - in the selective process (and in any subjectivesubstitute for F thereby generated).

So gain and risk are subjectively evaluated as functions of the punctuatedand fluctuating monetary thresholds associated with particular systems ofgoals and computational parameters, both of the latter being determined

as functions of location and conditioning on genetic and environmentallevels. This admits of mathematical formalization, and we are temporarilyspared a pronouncement of incorrigible eccentricity by our ability tofollow this reasoning and thus override our respective mental quirks (andwhatever limitations they seem to entail).

Money has stable value only as an averaging function on human scales ofwants yet, because humans are diverse and variously-situated, individualscales need not conform to this average, out are instead part of whatdetermines it. The problem resides in our tendency to try to predicateeconomic decisions on the value of money, rather than vice versa (whichis how it actually is on the scale of this formulation). So - apart from theconsideration of risk - it is not the "unusualness of the amounts" whichconfuses, but the subjective and polymorphic nature of money.Rationality must account for this, and reasoning which treats money asthough it had absolute intrinsic value is flawed and therefore irrational. Itthus falls before a higher reality in which Allais paragox straightens andvanishes like a well-shot arrow.

Note that the resolution possesses a certain undeniable beauty.Economies depend in part on the neural parameters of brains whichevolved from a primal Darwinian "economy" of survival, which in turnevolved from the system of invariants by which the universe isconstructed. By simple extension, we can think of this system of laws asthe "anthropic" configuration of the "Mind of God". The implied model isa hellix linking generality with specificity, and closing ultimately uponitself in a teleological loop.

And thus can obscure paradoxes lead to profound insights. PreciselyBecause he is capable of such insights can man learn now to optimizeeconomies in the promotion of global happiness, and apply reason andcompassion where abuse now prevails. By defining the connectionbetween human nature and free economics - the laws at once governingbrains, economies, and other systems - we extend the narrow conceptualpath leading towards human fulfillment. But more on this below.

(All contents copyrighted – 1989, C. M. Langan)

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