The Resolution of Newcomb's Paradox

by Chris Langan

We begin by defining Newcomb's problem. Enter Newcomb's Demon, a rather close relative of Maxwell's Demon (the similarity goes well beyond a shared affinity for black boxes). Newcomb's, Demon - call him ND for short - is a paragon of intelligence, a genius's genius whose acuity transcends the temporal boundaries of merely ephemeral human beings. Like many geniuses, he displays a certain playfulness: his "hobby" is testing his own infallibility in the prediction of human behaviour by making a generous offer to random human playthings. This offer is couched as a "choice"; though ND's human subjects are in fact forbidden to predicate their choices on external events, they are tacitly allowed to use deterministic or nondeterministic internal strategies computed by corresponding classes of neural events (you may already see the superficiality of the distinction between inward and outward events; I include it only as a customary ingredient of the formulation).

Suppose that ND has chosen you as an experimental subject, and that you know that he has never failed to achieve the result he predicted for any subject throughout the extensive history of his experimental run. Nothing appears rigged; in particular, the data are plausibly distributed with respect to the critical behavior of experimental subjects (if Nozick's data are valid, at a ratio of approximately 2.5/1). The experimental set-up is minimal. You are led to a table on which are two boxes. One is transparent, and in it you can clearly see $1000. The other is dead black and totally opaque. ND tells you that he has placed $1,000,000 in the black box if and only if he has predicted that you will take only it and not the other (transparent) box. Otherwise - if he has predicted that you will try to "outsmart" him and take both boxes - the black box has been left empty. ND earnestly assures you that this is no joke; he does not intend to fail. Just in case there are any doubts about this, he offers to take you through a few trial runs using monopoly money and a couple of spare boxes (naturally, you will have to leave the room between these "rehearsals"). You take him up on it. He wins no matter what you think or do. Finally, he tells you that play time is over; next time, the deal is for real. He adds that, lest you suppose he has an infinite amount of time to waste on you, he is giving you a reasonable but finite amount of time to make your move.

Having been around the block, you have already perceived that this is all a kind of game in which ND is the "house", and you his opponent. Of course, since ND is playing for the pure satisfaction of being right while you are playing for big money, a minor conceptual adjustment is in order. But, all things considered, you reason that you should nonetheless apply the standard theory of games in order to maximize your gain. Your reasoning then becomes a little more involved.

You know of certain principles of rational decision that seem to apply to this game. One is known as "the maximization of sub­jective expected utility", and states that you should try to maxi­mize the sum of the products of the utility (monetary value) of each distinct possible outcome by the probability of that outcome, given your "move". Since ND's past performance and serious intent seem to bring the probability that he has correctly predicted your choice close to unity, this principle is telling you in no uncer­tain terms to take only the black box.

But there is a similar rule, called the "dominance principle", which also seems to apply to your predicament. It states that if, for every possible move (or prediction) of your opponent, there is a single countermove as advantageous to you as any other, and more so in at least one case, then you should make that move. You know that ND has already made his move: there is either $l,000,000 in the black box, or there is nothing. In either case, your eyes tell you unequivocally that there is $1000 in the clear box. You reason that you have nothing to lose, and precisely $1000 to gain, by taking both boxes. The dominance principle merely drapes a veil of reason over what your eyes and instincts have already told you: ($1,001 million or $0.001 million) beats ($l million or $0). Conclusion: take both boxes.

But this seems to constitute a dilemma. These two principles of rationality, which are supposed to be mutually consistent, are in this case telling you to do opposite things. Feverishly - there is a lot of money at stake for you - you rack your memory for other rules which might help you reconcile or at least decide between them. What about the gambler's fallacy? No matter how many times ND has won in the past, isn't his string of victories really just an extremely unlikely random sequence? Time only runs in one direction, so how can he possibly claim to be more than a lucky (and very rich) guesser? If this is the case, then you certainly don't want to embrace the fallacy of assuming that the outcome of your imminent trial is in any way dependent on chose preceding it. If nothing else comes to mind, and quick, you're going to have to go the dominance route and take both boxes.

As if on cue, your neurons begin again to flicker. It occurs to you that you could make no decisions whatever, and would know nothing at all, without relying on various scientific axioms. But axioms are mutually independent, or virtually random with respect to each other. None of them can be deduced from other axioms. Doesn't accepting these axioms - including the dominance principle or whatever more primitive concepts underlie it - thus amount to something very much like the gambler's fallacy? That is, isn't one accepting as a premise that certain future events will be like similar events from the past, but without being able to prove it? You are now in the Minotaur's recreatorium without a thread.

You are also roughly as well-informed on what you should do as any professional scientist or logician who has ever published an analysis of Newcomb's paradox. Fortunately, I am no professional, and need grind the favorite axe of no publisher, professor, or em­ployer. The trail that unwinds below is freshly blazed; though a bit steep in spots, it can be negotiated purely on the strength of an open and reasonably sharp mind.

The first step towards resolving the paradox is to provide a logical scaffolding from which Co construct the mathematical model necessary for sound inference. Past arguments involving the problem have used either the standard decision-theoretic model of people playing a game, or the linear "arrow of time" fundamental to classical physics. These two models have been taken to imply opposite solutions, and this suggests that they be somehow unified in an extended "meta-model" which adequately relates the concepts to each other. It has been less obvious just what this higher model should be.

The solution of problems, and the resolution of paradoxes, are inherently computational activities. What, then, could be a better setting for this resolution than a computative one? And what could possibly be a more fitting preface than a brief introduction to the abstract theory of computation?

Consider an acceptor F = (Q, S,d,qo,A). Q is a finite nonempty set of internal states, S is an alphabet whose symbols are concat­enated as strings, qo c Q is the initial state, and A c Q is the set of accepting states triggered only by input recognizable to F. The transition mapping d, which governs the way in which F changes states, is deterministic if d:Q x S —> Q, but nondeterministic if d:Q x S —> 2Q (where 2Q is the set of all subsets of Q). In the nondeterministic case, d will be written dn, for clarity. In terms of human psychology, we might regard the 5-tuple of F as its "categorical imperative", or accepting syntax, and say that F projects this syntax onto its universe. Nothing in the universe of F is recognizable to it but the particular input strings (sense data, facts) which cause it to pass through some q e A; they are its phenomenal "reality", a subset of the noumenal metareality of the wider universe in which strings are representationally gener­ated and entered by programmers. The restriction to finite Q is pragmatic and amenable to conditional relaxation.

If F is deterministic, it accepts (recognizes) a string s e S* if and only if d(qo, s) e A. Since we have defined d only for the individual symbols s e S, we must define an extended transition function d': where l is the null string, d'(q,l) = q; and for all q e Q, s e S, and s e S* (where S* is the set of varirepetitional permutations of the d e S) , d'(q,ss) = d' (d' (q, s) ,s). Thus, the accepting behavior of F is defined inductively for s-quantized string extensions; the way in which we recognize and assimilate new bits of information within our reality is specified in d'. Were we to widen the discussion to imagination, conceptualization, theorization, or other intrinsic computations, we would need to consider "ideas"; we would have to generalize from recognition to abstraction by means of a nonlocal or self-iterating, input-free extension of d. If the reference to "strings" seems to imply a dimensional limitation on input, this too can be generalized.

Where F is nondeterministic, it accepts a string s e S* if and only if dn'(qo,s) n A = ~Ø. The nondeterministic extension dn' of dn is defined by induction: dn'(qo, l) = {q} , and dn'(q,ss) = Uq'cdn'(q,s) dn(q',s). I.e., q' is one of the possible successors of q under dn' given s; the unextended mapping dn on singletons of S* then determines the image under dn' of s plus an adjoint symbol s, given q'. This is a classical recursive definition. It describes a stepwise probabilistic ramification of computative potential whose complexity depends on dn.

Nondeterminism is not always restricted to Q x S; under certain conditions, either Q or S can be extended, or A shifted within Q. This. of course, entails a modification of F, unless F is defined to allow for parametric extension and adjustment. To this effect, let F' be such an open extension. To be meaningful in mechanistic contexts such as those in which acceptors are usually considered, F' must exist within an appropriate mechanistic extension of the computative environment of F. Organisms, being mechanical in the deterministic sense, need not be distinguished in this extension. Nondeterminism can be used to subtly manipulate recognition, thus cryptically modifying an acceptor's reality. Nondeterministic recognition can help to explain the ability of an acceptor to rapidly sieze certain kinds of higher-order phenomena, or even interact with higher-order agencies ordinarily insensible to it.

Having thus formalized the logical abstraction of recognition - i.e., the passive phase of organo-mechanical cognition - we now proceed to the output behavior of computative automata, or to the active phase of cognition. Consider a transducer M = (S,Q,T,d,m), where S is a finite nonempty input alphabet, Q a finite nonempty state set, T a finite nonempty output alphabet, d:Q x S —> Q the state-transition function, and m: Q x S —> T the output function. A computation of M has internal and external phases; through m, the output that M delivers back to its outward universe depends on strings of d-iterated transitional internal states. Thus, m is a functional of the function d of input. Together, m and d totally determine the behavior of M. They can be extended from S and T to S* and T* as for the acceptor F: d'(q,l) = q, m'(q,l) = l; and d'(q, ss) = d( d'(q,s),s), m'(q,ss) = m'(q,s) m(d'(q,s), s). Where appropriate, we can add to M an initial state ("reset control") qo: M = (S,Q,T,d',m',qo), to be regarded as a separator of locally independent computations, and put it at the disposal of a function r c M which determines computational relevancy.

Considered as a robotic brain, M T-behaves according to m, but Q-reasons towards its decisions along paths generated by d. Where d is deterministic, m may be related to it as a "timing function" according to which any computation can be arrested (input aborted, regression terminated) and converted to output on passing certain tests. Where the duration of the computation is determined with d, input becomes the only variable. Where input as well is fixed in content and scheduling, the entire system is tightly determined. As Laplace might have observed, predicting the behavior of deter­minate mechanisms requires only data, the means to acquire it, and a valid scientific methodology to organize and interpret it. While the situation is actually more complex, the fact remains that were one to play a deterministic game with a deterministic transducer like M, one would need only a detailed knowledge of its input and programming to predict the outcome, given analytic tools adequate for that purpose and consistent with one's own constraints (e.g., the amount of time available for analysis and strategy). If one's object in the game were merely the validation of one's prediction, so much the easier to win.

Suppose instead that M has nondetenninistic output capping mn, where state-transition may or may not be deterministic. Then the prediction of output entails control of mn by the predictor. To win a game of prediction, one must now control mn as well dn Sn to the extent chat it is output-critical; one must take over where the probabilistic mn leaves off. Since whichever control the transducer has over itself resides in mn and dn, one must in effect deprive it of self-control. The relevance to "free will" is obvious.

Computation is purposive. The purpose of an acceptor is pure recognition; no action is explicitly predicated on its internal transitions. The purpose of a transducer is conversion of input to output; yet, such a conversion is aimless unless algorithmic. Like yin and yang, acceptors and transducers are complementary; only together can they begin to resemble functional systems of organic complexity. In order to model organic systems, transducers must be endowed with goals and algorithms comparable to the ends and means of living beings. Algorithms are themselves purposive procedures which model both acceptance and transduction. The problems which comprise their input are scanned by preliminary steps for certain kinds of information, which must in turn be accepted as parameters by subsequent steps, and so on to the output stage (at which point the algorithm delivers its answer). The mechanistic representation of an algorithm must allow for the innate structure of a device, considered apart from the algorithm itself; this structure may have variant and invariant aspects. The algorithm simply conforms variables to purpose given the invariants. As the definitions of F and M might lead one to expect, this generally involves importing to M the set A c Q defined for F.

Human beings, it is said, are self-programming. Their thought is polyalgorithmic; useful algorithms are either meta-algorithmically constructed, or selected from a learned store, to deal with input. If learning, construction, and selection are deterministic, then they characterize a deterministic meta-algorithm not fundamentally different from any other deterministic algorithm we might study. If they are nondeterministic, then they are characteristic of a nondeterministic meta-algorithm, and likewise. It follows that the formal transductive model of human nature withstands any objection from the relative complexity of human mentality or behavior.

Newcomb's object-transducer MN naturally includes an acceptor: MN (S,Q,qo,d,A,m,T). Recognition is phasic; a string must often be "pre-accepted" for M to tell whether to accept or reject it. Ordinarily, this tentative phase of recognition is easily computed by the physical entities whose behavior is predicted by ND. To be "real", an input-quantum s must simply possess a certain first-order predicate, "reality", which - this being a self-validating tautology - induces a type-theoretic predicate stratification like that involving the old Cretan, Epimenides. To this sine qua non of recognition corresponds a primary element q1 of A; no input-quantum failing the q1-test is reified, whereas all those passing are relayed to Q - q1. Higher-order recognition of "passed" quanta then proceeds at a rate determined by the respective computational demands of the stratified-algorithmic phases of dM. Corresponding to the structure of A are various ordered states analogous to q1 within their respective levels of acceptance.

Let us narrow the definition of MN in a way consistent with Newcomb's problem. Suppose that associated with mM is a threshhold value b > 0 below which output is nil, but above which a decision will be finalized and implemented. With each qi c Q we associate a pair of strength coefficients ai, and ai2, to be incremented and decremented according to a strategic dM, appropriate to the Newcomb decision-theoretic context; these represent the current tendency, given the present amount of input, for MN to output either possible behavior (taking one or both boxes, respectively). The ai divide Q into three classes XQ, YQ, and ZQ, with membership conditions a, > a2, a, < a2, and a, = a2 respectively. To each qi is attached a total weight ai = |ai, - ai2|. As soon as the output condition [a = b] is met, a decision and behavior result which correspond to the Q-class of the current state (note that dM, being strategic, precludes q c ZQ, or "indecision", at the output deadline). Thus, the states of Q c MN are preferential and impetal, the graded fore-images of the outputs they favor. This is just a convenient way to view the internal configurations to which they correspond, and does not violate the general definition of transducers. Nor, for that matter, does it violate the way human beings perceive their own decision-making processes.

The question posed by Newcomb's problem involves the computative analysis, by a predictive agency with computative characteristics, of the computative analysis undertaken by a transducer on a given input. That input is the problem itself, presented in the manner prescribed by the formulation. This situation, which defines a computative regression, is recursive and inductively extensible. The regression in turn defines the only soluble context for the higher-level "paradox" generated by the problem. This context translates as mechanism. The mechanism is a stratified automaton G containing both the predictor and its object-transducer as sub-automata. Whether "free will" is defined determlnistically as mere outside non-interference in m and d, or nondeterministically as the ability of MN to override any exogenous restriction of mn or dn, its mechanism is contained in that of G.

Logical diagonalization of the formal computational language generated by the accepting syntax of MN directly implies that certain structural aspects of G may be unrecognizable to MN. In particular, those aspects involving MN-relativized nondeterminacy, as well as those involving certain higher-order predicates of the nondistributive, nonlocal organizations involving mM and dM, are formally undecidable to it and need not be recognized directly by it with any degree of specificity. To understand why, consider the extent to which a common computer "recognizes" the extended system including its cpu, its other components, its programmers, and the environment it inhabits. In fact, it can recognize nothing that does not conform to its input-to-output transformational grammar. Even if it were self-analytic, such analysis could be limited to a schematic electronic syntax which overlooks the material elements of which it is constructed. In any case, it can make sense of nothing but strings of input translated and rearranged according to the internal stratification of its hard and soft programming.

You, your purposes, and your dependencies are undecidable to it, and so are the mechanisms by which you can predict and control its behavior. It matters not who formulates this undecidability; if the machine's internal logic is inadequate to do so, yours surely is not (currently, most mechanical acceptors are nongeneralistic, treating complementation as negation and negation as rejection; this bars the tools of diagonalization from their computations). Should it ignore your higher prerogatives, you could "diagonalize" it - if nothing extrinsic to the machine were to stop you - with a sledgehammer whose effects on it do not depend on its acceptance. By analogy, Newcomb's object-transducer MN cannot preclude G on grounds of "insensibility". Nor, for chat matter, can we.

There are many self-styled experts on undecidability who have expressed the opinion that all attempts to reify Godel's theorem along paranormal lines reflect a misunderstanding of its "real nature". Such experts are quite correct in that a misunderstanding exists, but the misunderstanding is all theirs. What the theorem forces by juxtaposing truth and derivability (or consistency and completeness) is a hierarchical stratification of classes of truth functions and the inferential syntaxes which parametrize them. This stratification follows that of G, fractionating computative reality along with the "truth" to which it corresponds.

The stratification of G induces stratum-relativizations of computative time and space. Thus, the timetype in which MN computes recognition and output is a mere subtype of that in which it is programmed. Dynamical "arrows of determinacy" which are inviolable to MN, being programmed into its accepting syntax, have no force whatsoever to the programmatic agencies themselves. This applies just as well to "metrical" restrictions embodied in the MN-syntax; these may allow MN to recognize nothing but an artificial submetric of the metric in which these agencies define their own existence. MN and its reality might consist of quanta with higher-dimensional interpretations as the termini of channels for the transmission of information between strata. Metatemporal predicates may exist with respect to which those of MN, are definite only in a mutual sense; predicates which MN accepts as "before" and "after" could be the programmatic projections of "in front of" and "in back of", or any other G-consistent higher-prepositional relationships.

There can thus exist a mechanism x c G through which a predictor like ND could measure and/or control the mappings d, m c MN in ways directly insensible to MN. Where in G relative to MN would such a predictor have to be located? Precisely where access is available. Sirnplistically, we might characterize the predictor-M relationship as one of proper inclusion, where it is understood that prediction is direct rather than second-hand, and programmatic in the passive and active senses. That is, a programmer mentally internalizes the structure of that which he programs, and this internalization amounts to computative inclusion. The fine structure of G, while to a degree analytic, is a natter of some complexity. For now, it will suffice to have demonstrated the possibility of x and its utility to well-situated G-subautomata. Because G is structured to allow for relativized deteririnacy and nondeterminacy, the solution is invariant with respect to argumentation involving mind-brain dichotomy. That is, such dichotomies reduce to distinctions of determinacy and nondeterminacy, and may be treated in kind.

Restricted dominance, which relies on probabilistic independence derived from the lower-order, local istic dynamical timetype of MN's artificially restricted "reality", is revealed under G-extension to be 'itself dominated by utility. That is, the subjective utility of MN forces the assimilation by dM of this entire demonstration, which disables restricted dominance and thus frees the strategic component to recognize higher patterns among observed data. The principle of restricted dominance, though valid as long as the reality of MN remains unbreached, loses all force in the presence of exodynamic influence.

Let's sum it up. You can be modeled as a deterministic or nondeterministic transducer with an accepting syntax that can be diagonalized, or complemented by logical self-negation. ND can be modeled as a metalogical, metamechanistic programmatic agency, some insensible part of whom surrounds you in a computational space Including physical reality, but not limited thereto. This space is the mechanistic equivalent of the computative regression around which Newcomb's problem is essentially formulated. The existence of this space cannot be precluded by you on the grounds that you cannot directly observe it, nor can it be said by you to deny ND a mechanism of control and prediction of your thought and behavior. Additionally, you have an open-ended run of data which lowers to 1/ 8 the probability that NO is "just lucky". This implies that mechanism does indeed exist, and warrants the adjunction to the axioms of physics an independent, empirical physical axiom affirming that mechanism. This then implies that ND can predict or control human thought and behavior (a somewhat weaker implication, you will notice, than "omniscience"). ND possesses means, motive, opportunity...and you. You are "possessed" by Newcomb's Demon, and whatever self-interest remains to you will make you take the black box only. (Q.E.D.)

Do not let incredulity cheat you of understanding. It may be hard to accept the idea of a mathematical proof of the possibility of demons; if so, you may take solace from the fact that this does not of itself imply their actual existence. Remember that Newcomb's problem includes a hatful of hypothetical empirical data that may have no correspondents in your sector of reality. The proof just given merely prevents you from generally precluding the paranormal experiences reported by others on logical grounds. That is, it restrains you from any blanket distribution of too narrow a brand of "rationalism" over reality. Because I have long been exploring the ramifications of the logic this proof employs, I already know how easy it would be to deal with whomever might dispute it on "scientific" or other grounds, regardless of their prestige or reputed intelligence. There is a certainly a strong correlation between rationalistic dogma and the persecution of its foes. Fortunately, it is not too late for "rationalism" to catch up with the advances of twentieth-century logic.

Regarding free will, consider that demons are generally reputed to offer a material reward for the surrender of one's soul. Where we define "soul" as the autoprogrammatic G-extension M' of one's characteristic transductive representation M, this means only that one is rewarded for voluntarily moving aside and letting the demon take over as captain of one's fate. He does this by adjoining to his subject any functional extensions he requires; e.g., d(dn), or m(mn). Or, if restricted determinism prevails but he is not in an especially predictive mood, he might create a parametric extension FND' (analogous to F') enabling a hyperdeterninistic override m ? m of one's deterministic output function. Actually, volition may be irrelevant; one might be shoved aside rather than nudged. Where freedom is defined negatively on restricted determinacy, it is duly stratified.

It has by now occurred to most readers that "demon" is a darkly connoted term that could just as well be replaced with "guardian angel" or something equally comforting. After all, ND does not ask his subjects to commit evil, but only to let him do them a rather large favor with no apparent strings attached. Insisting on "free will" here is reminiscent of the child who stares defiantly at its mother while touching the hot stove she has just finished telling it to avoid. If the child's act has any meaning at all, it is just that poorly timed independence can lead to more painful dependency down the line (surprisingly, Isaac Asimov advocates just this kind of defiance in the Newcomb context, and on what he seems to feel are ethical grounds; but we will deal with ethics later). One cannot be willingly "enslaved" by ND before being enslaved by the money he offers. Who among us always behaves in willful defiance of economic reality? My heart goes out to him, for he is either "enslaved" by an institution, or he is at the absolute mercy of the spontaneous, unsolicited charity of those whose own demons generally'predispose them to pass him by.

Having quenched the fire, we will now "mop up". Professor Nozick himself has remarked on the possible nonrecursivity of scientific laws involved in the determination of behavior. This, he surmises, prevents the inference of predictability from determinism. But this conclusion entails the confinement of these predicates within a restricted syntax, whereas the extensionality of G renders such a constraint powerless. In other words, restricted recursivity is beside the point for agencies defined to possess sufficient access to the channels through which such laws project as restricted reality, and particularly to their source. All ND has to do is run a G-presimulation of interstratum transmission. Note that the computat ive requirements for tills, which correspond to assumptions on the nature of G, are satisfied given ND's efficacy.

Nozick also remarks on the regression of determinism to self-validating premises. Self-validating languages comprise their own inferential schemata. Validation, being inferential to the extent of quantification, follows the G-stratification of inference. So self-validation is also stratified, and such languages correspond to appropriate levels of computative reality. For sufficiently expressive languages, self-validation becomes diagonalistic self-vitiation. Determinism is thus stratified, and so is its counter­part, freedom. The seemingly infinite regression can be brought to closure at an ultimate form of undecidability, which in turn can be locally resolved with a relativized degree of confirmation.

Regarding various remarks that have been submitted to Nozick by members of the physics community, I should explain that the above resolution has a metaphysical generalization of which certain implications may broadly be termed physical. This generalization, as developed by the author, is called the CTMU, or Computation-Theoretic Model of the Universe (the acronym has a mnemonic pronunciation: 'cat-mew). We will later examine chosen aspects of this model, which transcends characterization as a "theory" by virtue of its categoricality with respect to computative entities such as human beings. That is, the CTMU is as close to "absolute truth" as we will ever be privileged to get along computative or intellectual pathways. If I might be indulged a bit of testimony: beyond the CTMU ultimate syntax, the lattice which gives shape to the multilayered layered veil of maya, there is but the light that shines forever.

Christopher Michael Chappelle-Langan

(copyright 1990 by C. M. Langan. No portion of this article is to be reproduced in any form without the author's written consent.)

Newcomb's Problem and Two Principles of Choice
Robert Nozick in Essays in Honor of Carl G. Hempl Editor: Nicholas Rescher. Humanities Press, 1970
Newcomb's Paradox Revisited
Maya Bar-Hillel and Avishai Margalit in The British Journal For the Philosophy of Science, vol . 23, No. 4, pp. 295-304; .November, 1972
Reflections on Newcomb's problem: a prediction and free will dilemma
Martin Gardner and Robert Nozick in the Mathematical Games Department of Scientific American. pp. 102 -106; March, 1974


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