Science in Christian Perspective
Newcomb's Problem and
EDWARD B. DAVIS
Department of History and Philosophy of Science
Bloomington, Indiana 47405
From: JASA 36 (March 1984): 9-12.
Newcomb's box problem, a well known puzzle in decision theory, may be useful as a model of God's prior knowledge of freely made human choices. This paper, which is intended to stimulate discussion on the nature of divine omniscience, contains three sections. (1) Newcomb's problem is presented and partially analyzed. (2) It has been argued elsewhere, on the basis of Newcomb's problem, that free choices must be inherently unpredictable for all beings, including God; the very possibility of divine omniscience as traditionally understood has been called into question. I refute this argument here. (3) Two types of infallible beings are proposed: a superscientist, who must calculate in order to predict; and the biblical God, who already knew the outcome of all free choices before the world was created, and who has already acted upon that knowledge. Only the latter type of being is compatible with real human freedom. Predestination is best understood as divine action predicated upon prior knowledge of human decisions.
Suppose that on a table in front of you are two boxes, B, and B2. B, is transparent and contains a $1000 bill, but B, is opaque. Across the table sits a being who has correctly predicted your choices in the past and has correctly predicted the choices of many of your friends as well. To the best of your knowledge he has never been wrong. The being gives you a choice between two actions:
You are not permitted to take only the transparent box B1 and you are not permitted to look inside the opaque box B2 until you have made your choice. The being tells you that the contents of B, depend upon what he has predicted you will choose: if he has predicted (P,) that you will choose only one box (C,), then he has already put one million dollars ($M) inside B2 (perhaps to reward your moderation or your faith in his abilities); if he has predicted (P2) that you will choose both boxes (C2), then be has not put anything inside B2, leaving it empty (perhaps to penalize your greed or your lack of faith in his abilities). Knowing all this, what would you choose to do, C, or C2? Before reading any further, take some time to decide.
A strong argument can be made for either choice. Robert Nozick, the philosopher who first published this puzzle known as "Newcomb's Problem," gives this version (A,) of the argument for taking only one box (C1:1
A, If I take what is in both boxes, the being, almost certainly, will have predicted this and will not have put the $M in the second box, and so I will, almost certainly, get only $1000. If I take only what is in the second box, the being, almost certainly, will have predicted this and will have put the $M in the second box, and so I will, almost certainly, get $M. Thus if I take what is in both boxes, 1, almost certainly, will get $1000. If I take only what is in the second box, 1, almost certainly, will get $M. Therefore I should take only what is in the second box.
But it is possible to argue (A,) for choice C2 just as convincingly (see Fig. 1):
A2 The being has already made his prediction, and has already either put the $M in the second box, or has not. The $M is either already sitting in the second box, or it is not, and which situation obtains is already fixed and determined. If the being has already put the $M in the second box, and I take what is in both boxes I get $M + $1000, whereas if I take only what is in the second box, I get only $M. If the being has not put the $M in the second box, and I take what is in both boxes I get $1000, whereas if I take only what is in the second box, I get no money. Therefore, whether the money is there or not, and which it is already fixed and determined, I get $1000 more by taking what is in both boxes rather than taking only what is in the second box. So I should take what is in both boxes.
The arguments above are incompatible, that is, they cannot both be correct at the same time. Your preference will depend on how you interpret the information given in the problem regarding the nature of the being. There seem to be at least three possible interpretations consistent with the amazing record of successful guesses:2
I that there is some connection between the Predictor's predictions and the choosers' choices which accounts for his past success, and which will ensure his success on other occasions, such as this one;
11 that there is some connection between the Predictor's predictions and the choosers' choices which accounts for his past success, without being such as to ensure his success on other occasions, such as this one;
III that there is no connection between the Predictor's predictions and the choosers' choices, so that his past success is sheer chance, mere coincidence, and hence does not ensure, or even suggest, his success on other occasions, such as this one.
More briefly: is the Predictor absolutely infallible (Explanation 1), extraordinarily reliable (Explanation II), or incredibly lucky (Explanation III)?
It is important to realize that you will decide what to choose on the basis of which type of being you believe you are dealing with, regardless of whether or not you are correct to believe it. Suppose, then, that you perceive the being to be absolutely infallible (type 1), a conclusion which you feel is justified by his past record. Given your belief, it would be perfectly rational for you to pick just the second box, as argued above (A,). But suppose that you believe that the past record of the being is all a fake-something is fishy; the players have all been confederates of the being or you have been otherwise duped. Or perhaps his record is pure luck; while highly improbable, it is not impossible. Thus you believe that the being is of type III-he is utterly unable to predict your choice in any sense of the word. Since you believe that there is no connection whatsoever between what you will choose and what he has already predicted, then you should choose both boxes by argument A,. It may be, however, that you are an agnostic about the being's abilities: you may think it likely that there is some reason other than chance or deception that accounts for his past successes, yet you do not think that this reason makes it certain that he will predict your choice correctly this time. Although you would estimate that the being's chance of a correct prediction is much better than 50%, it is not quite 100%. Nozick, for one, opts for this case, recommending the choice C, (both boxes) with considerable hesitation. Isaac Levi has shown that Nozick's argument is faulty, but he has also recommended C, on the basis of the maximin principle.3 Without further information, as Levi has shown, the problem permits more than one correct strategy; one can defend either C1 or C2 if one believes the being is of type II.
The Possibility of Perfection
The true nature of the being is in fact unknowable. For example, one would be unable to know for certain that the being is infallible even if an infinite number of trials were possible. And though a limited number of trials could readily show that the being is fallible, one could still not distinguish with absolute certainty between an extremely lucky being and a truly good predictor.' Nevertheless if one were to continue to find that the being has correctly predicted choices in repeated trials, one would seem justified in concluding that the being is either of type I or of type II5.
George Schlesinger has argued, however, that one could never be correct to believe that the being is the least bit reliable, let alone infallible, even if he has a long and unbroken string of successes, since the assumption of a perfect predictor, he argues, leads to a contradiction.6 Briefly, his argument is this. Suppose that the Newcomb game is to be played by three parties: a being who has never been wrong, a chooser, and an observer ("Smith") who always has in view the contents of the opaque box B, (perhaps there is a window on one side of the box). If we assume that Smith is a perfectly intelligent person who always advises the chooser to do what is in the chooser's best interest, then what would he advise? if Smith finds that B, is empty, then he will surely advise that both boxes be taken, so that the chooser will not go away empty handed. If on the other hand Smith finds that B, contains $M, then he will still advise that both boxes be chosen, since this will result in an extra $1000 for the chooser. Now under the rules of the game Smith is not permitted to communicate his advice, but this is irrelevant since the chooser knows what that advice would be: under all circumstances, choose both boxes. Thus it is in the chooser's best interest to choose C,. However if the being with the perfect record of predictions is assumed to be infallible, then by argument A, (above) the best strategy is to choose C1. This is a contradiction. To avoid a contradiction, Schlesinger asserts that no being-not even one with a perfect record-should ever be assumed to be a perfect predictor of free choices; indeed he goes beyond this, denying any competence whatsoever to the being. The only way for a being to insure his infallibility, argues Schlesinger, would be for him to prevent the chooser by force from contravening his prediction, but then the chooser would not be free.
Is Schlesinger correct? If so, then free choices are in principle unpredictable for all beings, including God, a conclusion which Schlesinger takes as fundamental to his theiSM.7 His argument is flawed, however. The contradiction does not arise by assuming that a being who has not erred in the past will not err in the future; the contradiction arises because Smith's (uncommunicated) advice is assumed to be in the chooser's best interest when in fact it is not. Smith advises the chooser to take both boxes only because that is what Smith would do if the choice were his own. It is not what Smith would do if he were in the chooser's place-and the fact that he is not is crucial, for it creates an asymmetry-it is what Smith would do if he were given the opportunity to choose knowing what he does about the contents of B2 and, therefore, about the nature of the prediction. That knowledge makes all the difference because a revealed prediction is no longer in force. In essence, then, one must be well informed (as Smith is) in order to make out better by choosing C2 than by choosing C1. Since the chooser is not well informed (he does not know the contents of B2), he is better off to choose C,. And since Smith knows that the chooser is not well informed and therefore can not expect to fool the predictor, his advice should be to choose Q, regardless of what he sees. Hence the contradiction vanishes.
The advice which Schlesinger would have Smith offer-to take both boxes-is useless because the chooser does not know why it is being given: he does not know if Smith sees $0 or $M in B2.8 He can find out in one way only, by making his choice, but the very act of choosing will logically determine the state that obtains (if the being is infallible). It will not physically determine the state; the predictor's action has already done so. Neither will it change the state; Smith will note that the
contents of B2 remain the same after the chooser makes up his mind. By choosing C1, for example, one does not make it true that there is $M in B2-this has been true for some time, as Smith can testify-but one is offering conclusive evidence that $M will be found in B2 when it is opened.
Is Schlesinger correct? If so, then free choices are
in principle unpredictable
for all beings, including God
Since the being knows for certain what the chooser will do, isn't the choice in some sense "inevitable"? Isn't it "bound to occur?" Yes, it is. But this need not imply (although it may) that the chooser is programmed to choose in a certain way. That depends upon the nature of the being's infallibility. Apart from the trivial case of a being who forces the chooser to "decide" in harmony with a previous prediction, I can
conceive of at least two radically different ways in which a being could be an infallible predictor of choices: in one case, physical determinism holds; in the other, logical determinism holds.9 It could be that the being is a superscientist who predicts choices on the basis of his present knowledge of the chooser's state of mind, deriving the future choice by some nomic statement(s).10 All that would be required for infallibility would be a set of valid universal generalizations, full
An infallible and omniscient being has sure and certain knowledge of a choice before it is made. From his perspective, it is not really a choice; it is an event that will occur.
But there could be another kind of infallible predictor. Rather than a Laplacian Divine Mathematician who needs to calculate in order to predict, the being might simply know the future in the same way that he knows the past. He would have no need of deterministic laws (although such laws might still exist). For him, it would simply be a matter of knowing; the future would be as certain as the past. Events would not be inevitable in the mathematical or physical sense, but in the logical sense: if he knows that they will happen, then they will happen. The biblical God is just such a being. As Calvin put it,11 that which God has determined, though it must come to pass, is not, however, precisely, or in its own nature, necessary. "12 In this case one's choice would not be determined by inexorable laws and could not be predicted by extrapolating a current state of mind into the future. It would be significantly free: if one were given the same choice under the same conditions in some other identical world, the outcome could be different. No matter what one decides, one would do so freely, only one could not escape being "seen" by God as he makes his free choice. 12 Since God would therefore have completely reliable foreknowledge of one's choice, He could have already prepared one's reward before the foundation of the world. He would not make a calculation; He would not make a prediction and then control the choice to insure its agreement; He would simply foresee free choices as if they had been made prior to His preparations.
But God does not play Newcomb games with His creatures. He does not offer cash incentives for people to take only one box. He offers salvation to all who will humbly trust Him and freely choose Christ; the same He did foreknow and did also predestinate to be conformed to the image of His Son, that He might be the firstborn among many brethren.13REFERENCES
1. This quotation and the following one are from p. 115 of Nozick's article, "Newcomb's Problem and Two Principles of Choice," in Essays in Honor of Carl G. Hempel, edited by Nicholas Rescher et al. (Reidel, 1970), pp. 114-146. Nozick attributes the problem to Dr. William Newcomb of the Livermore Laboratories in California. Like many others, I first read of the Newcomb problem in Martin Gardner, "Free Will Revisited, with a Mind-bending Paradox by William Newcomb," Scientific American 229(l) (July 1973), 104-108. Reader responses to the problem (including one by Isaac Asimov) are analyzed by Nozick in another Gardner column, "Reflections on Newcomb's Problem: a Prediction and Free-Will Dilemma," Scientific American 230(3) (March 1974),102-106.
2. This is taken from Don Locke, "How to Make a Newcomb Choice," Analysis 38 (1978), 17-23; he argues for taking both boxes even though he expects to receive only $1000.
3 . In his excellent article on "Newcomb's Many Problems," Theory and Decision 6 (1975), 161-175, Levi shows that Nozick's statement of the problem is not precise enough to permit of a single correct answer.
4. The problem of distinguishing between an extremely lucky being and a good predictor appears to be logically equivalent to the problem of determining whether a given number is random, and even an infinite number of trials may be insufficient for this. See Gregory Chaitin, "Randomness and Mathematical Proof," Scientific American 232(5) (May 1975), 47-52. 1 want to thank Mr. Dennis L. Feucht for calling my attention to this article.5. See Locke, p. 20.
6. See "The Unpredictability of Free Choices," British Jour. Phil. Sci. 25 (1974), 209-221. A Jewish rabbi and philosopher of science, Schlesinger has written at greater length on these points in his stimulating volume, Religion and Scientific Method (Reidel, 1977). In this latter work (pp. 93ff) he anticipates my objection but fails adequately to deal with it. Other objections to his argument can be found in Andr6 Gallois, "How not to Make a Newcomb Choice," Analysis 39 (1979), 49-53, and James Cargile, " Newcomb's Paradox," British Jour. Phil. Sci. 26 (1975), Z34-239.7, Religion and Scientific Method, Part 11: "Free Will, Men and Machines."
8, Since Smith can not relate anything useful-indeed he can not communicate at all-his presence is a red herring. Schlesinger's argument is merely a restatement of A2 above, and we are left with the same situation: the nature of the being is a matter of conjecture.
9. Donald M. MacKay has discussed Newcomb's problem and these two types of predictors in Science, Chance, and Providence (Oxford University Press, 1978). 1 have independently reached many of the same conclusions, particularly with regard to the efficacy of prayer; since my handling of this would be the same as his, I have not dealt with it here. I am indebted to MacKay for emphasizing the distinction between physical and logical determinism. Anthony Flew's negative review of MacKay's book can be found in British ]our. Phil. Sci. 30 (1978), 183-186. Richard Swinburne, writing in Isis 70 (1979), 284f, is much more sympathetic, but he differs with MacKay (and with me) on petitionary prayer.
10. Nozick postulates such a being, although in his version the being is fallible. Since a being of this type could be defeated easily by basing one's choice upon the result of a chance event such as the roll of a die, Nozick explicitly rules out the use of such events in making one's decision.
11. Institutes of the Christian Religion, trans. by Henry Beveridge (James Clarke & Co., 1953) 1,181.12. James Cargile calls this the "crystal ball" version of the story.
13. Romans 8:29. The interpretation of election suggested here, in which a sinner has true freedom to reject God's grace, is not one with which I am entirely happy. In his article on "Determinism and Free Will (A) Scientific Description and Human Choice," JASA 33 (1981), 42-45, Richard Bube has rejected a view of this sort; many others will no doubt agree with him. But I am not entirely happy with other formulations of election, either. God's providence and predestination are tough nuts to crack-we are not likely to crack them completely in this world (or the next)-and Newcomb's problem suggests a partial solution which I find attractive at this time. In addition to the references previously cited, see Maya Bar-Hillel and Avishai Margalit, "Newcomb's Paradox Revisited," British Jour. Phil. Sci. 23 (1972), 295-304; Andr6 Gallois, "Locke on Causation, Compatibilism and Newcomb's Problem," Analysis 41 (1981), 42-45; and Doris Olin, "Newcomb's Problem: Further Investigations," Amer. Phil. Soc. Quar. 13 (1976), 129-133.