The Coordinated Attack Problem by Sidney Felder

Transcription

1 The Coordinated Attack Problem by Sidney Felder This problem, which has become a classic in the theory of Distributed Processing, arises in connection with a hypothetical situation in which a force at Headquarters and a force at Outpost are both being prepared for an assault on a strong enemy position located between them. The enemy position must be attacked in the next 72 hours, and the prospects for victory are reasonably good if and only if both the forces at Headquarters and the forces at Outpost attack simultaneously. After the Commander of Allied Forces at Headquarters chooses a time of attack more or less at random, he will relay this decision to Outpost. Unfortunately, the only means for transmitting this directive is through the dispatch of a runner who must first relay H-Hour (the hour of the attack) to Outpost, and then return to Headquarters to confirm that the message was received byoutpost. We assume that it is known by all concerned 1) that the messenger must traverse a section of contested territory in which, in both directions of journey, the likelihood that he will be intercepted or killed is quite substantial, and 2) that the risk of interception on any given crossing does not decrease with the number of successfully completed traversals. Although it is probably obvious that the messenger must complete at least one round trip before any action can be taken, reasonable concern that the reader may come to doubt the necessity for a round trip (after seeing what follows) justifies the following elaboration. Now, it is certainly the case that once the messenger reaches Outpost, Outpost knows (the planned) H-Hour. However, until (or unless) the messenger returns to Headquarters with the information that Outpost now knows the time of attack, Headquarters must worry that its directive was intercepted en route to Outpost, and hence that Outpost does not know the time of attack. Because of this, Headquarters will not be able to launch the assault with confidence of support until the messenger s return verifies that the messenger reached Outpost. Suppose now that the runner has completed his round trip, and is now relaxing at Headquarters. Outpost knows the time of attack, and Headquarters now knows that Outpost knows the time of attack. Does anything further need to be known or done? Unfortunately, although Headquarters now knows that its directive got through to Outpost, Outpost does not yet know that the messenger was not intercepted on his way back to Headquarters, and hence (Outpost) does not yet have confirmation that Headquarters is aware that Outpost received the initial directive. Thus just as Headquarters could not know that its message to Outpost was received by Outpost until the messenger returned to Headquarters, Outpost cannot verify that its own confirmation that it received Headquarter s directive ever made it back to Headquarters until Headquarter s acknowledgment of the receipt of confirmation arrives back at Outpost. Because Outpost knows that Headquarters will not launch its assault until the messenger returns to Headquarters with the confirmation that its directive was received by Outpost, Outpost cannot yet attack. Since this entire train of reasoning is accessible to Headquarters, Headquarters knows that Outpost will not attack until a messenger arrives at Outpost confirming that Headquarters knows that its initial directive was received by outpost and hence Headquarters cannot yet attack. In other words, Outpost cannot yet order an attack because it has not yet confirmed that Headquarters knows that the directive got through to Outpost, and Headquarters cannot yet order an attack because it is aware of Outpost s uncertainty. The only conceivably adequate remedy for this situation is Headquarter s dispatch of a messenger to Outpost to inform Outpost that the return trip from Outpost to Headquarters was successfully concluded. Once Outpost again receives a messenger, Outpost will know that Headquarters knows that Outpost knows the Course Notes Page 1

2 tentative time of attack. This is all fine, but unless the messenger returns to Headquarters, Headquarters cannot infer that the messenger was not intercepted en route to Outpost, and hence cannot conclude that Outpost knows that Headquarters knows that Outpost knows the tentative time of attack. (After all, from Headquarter s point of view, a state of affairs in which the messenger never arrived atoutpost is no better than a state of affairs in which the messenger was never sent: The situation as stipulated is such as to imply that if it is necessary to send a message, it is also necessary to verify that the message reached its destination. Expressed in yet another way, the same uncertainty that stays Headquarter s hand and compels it to send a messenger a second time will stay Headquarter s hand at least until the time that the messenger returns a second time). Clearly, the messenger (or the messengers, collectively) must complete two round trips before Headquarters can confirm that Outpost knows that the messenger has completed one round trip. And, if the completion of a first round trip is an indispensable precondition for anyone s attack, the completion of a second round trip is likewise. Where are we after the second complete circuit is traversed? Certainly, Headquarters, the endpoint of the two circuits, knows 1) that the second round trip was completed and 2) that Outpost knows that the first round trip was completed. However, it is clear that Outpost cannot yet know whether two whole circuits were completed, and it is equally clear that Headquarters understands that Outpost cannot yet know whether two round trips were completed. Is Outpost s knowledge that two circuits were completed really a necessary precondition for coordinated action? If so, is Headquarter s confidence that Outpost possesses this knowledge also a prerequisite for action? That we must answer both questions in the affirmative should be evident once one considers the following: If, as was demonstrated in the preceding paragraph, Headquarters cannot safely launch its assault unless two complete round trips are completed, Outpost, which certainly can deduce that this is the case, cannot launch its own attack with any expectation of support unless it knows that the second round trip was completed. Because Headquarters is itself capable of making this deduction, Headquarters cannot begin its attack with any confidence that its forces will be joined by those of Outpost. This implies that the messenger must be dispatched yet again, in order to convey (to Outpost) the information that the second round trip was completed. However, if, as was just argued, Headquarters must have confirmation that Outpost knows that the second round trip was completed before the attack can proceed, the mere dispatch of a messenger for the third time cannot in itself provide the requisite confirmation. The messenger must return from this journey before any attack is possible: Until the messenger reports back to Headquarters after completing the third two-way circuit, Headquarters will have no way of knowing whether the messenger ever made it to Outpost the third time. Again, however, because Outpost (and Headquarters) realizes that Headquarters cannot attack unless a third round trip is completed, it directly follows that Outpost s forces will not move unless it receives confirmation, by messenger, that a third circuit was completed. Headquarters of course realizes this, and is thus compelled to send out a messenger for the fourth time, to inform Outpost that the third complete circuit was completed. But wait: Because we have shown both that Outpost cannot attack unless it receives this information, and that Headquarters knows this, it is clear that Headquarters cannot move its forces before it has confirmation that the messenger was not intercepted before reaching Outpost the fourth time. Thus the messenger must complete four round trips before anyone can move. Now, Outpost know this, and of course Headquarters knows that Outpost knows this, and thus Outpost must receive confirmation that the fourth complete circuit was traversed before it dares move its forces. Thus Headquarters must dispatch the messenger a fifth time... Obviously, the same reasoning applies at every finite stage, and hence no matter how many circuits Course Notes Page 2

3 a messenger has completed, neither Headquarters nor Outpost will be able to infer that any attack it might launch will receive the assistance necessary to secure success. The mode of inference generating this regress may be summarized in the following formulae: 1) If X can validly infer that it does not yet possess sufficient information to justify launching an attack, it can deduce that the fact that it has come to this conclusion is deducible by Y; and 2) If, at any stage, it is necessary to dispatch a messenger from X to Y, it is necessary that X receive confirmation that the messenger arrived at Y,and necessary that Y receive confirmation that the messenger arrived back at X. (Needless to say, the fact that these formulae are valid in this context depends upon the peculiar character of the communications environment we have posited). Course Notes Page 3

4 Prisoner s Dilemma In this game, two men, A and B, are arrested for a crime (of which, let us assume, they are guilty), and are being interrogated in separate rooms. After each interrogator opens up by saying, Your confederate is being told the same thing I am telling you now, each prisoner is acquainted with his bleak situation, which, in essence, reduces to the following: 1) If both prisoners confess to their crime, each will receive a prison sentence of 5 years; 2) If both prisoners remain firm and refuse to confess, they will both be locked up for 1 year for some lesser charge that can be proven against them; 3) If A confesses and B does not, A will be set free, and B will be put away for 15 years; and 4) If B confesses and A does not, B will be set free, and A will be condemned to 15 years imprisonment. Tw o parenthetical points are essential here. The first is that despite the symmetry of this game, this is not a zero sum game a game in which one person s gain is necessarily equal to the other player s loss under all eventualities. It is true that the payoffs in this game are such that the outcome that is most satisfactory for one of the prisoners is the least satisfactory for the other. However, it isnevertheless not a zero sum game because 1) the sum of the payoffs to both players in one eventuality (say, the case in which both confess) is not equal to the sum of the payoffs in every one of the other eventualities and 2) an outcome that is satisfactory for one of the players is not necessarily unsatisfactory, let alone equally unsatisfactory, for the other. Most interpersonal situations in real life are not zero sum games. Indeed, most situations of conflict are not zero sum games. This holds true even inthe extreme case of war. Both sides fight a war to win, but they wish to win in order to achieve certain objectives, objectives that often have nothing to do with the fate of the other belligerent. (Perhaps the most important observation ever made about war was Karl von Clausewitz s celebrated proposition War is a continuation of politics by other means ). A nation far prefers to win a war suffering small casualties rather than large casualties: The fact that the enemy has suffered even greater casualties is rarely much consolation, and indeed it is frequently the case that a nation prefers to win a war inflicting small casualties rather than large casualties. Furthermore, I think it is fair to say that it will normally be the case that a nation prefers to lose a war with small casualties than to win a war with large casualties, and even fair to say that it is sometimes the case that a nation will prefer to lose a war inflicting small casualties than to win a war inflicting large casualties. The second general remark is a warning against a certain over-generalization. We make the common sense assumption that both prisoners find each of their options, and each of the possible outcomes, quite unpalatable. We suppose also, for the sake of argument, that each prisoner is sufficiently unconcerned with the fate of his accomplice that he is able to rank the desirability of the possible outcomes in rough accordance with their material implications for himself. However, it cannot be emphasized too strongly that this assumption about each prisoner s primary or exclusive dedication to his own material welfare is introduced solely as a concession to narrative convention and imaginative simplicity, and is completely inessential to the general logical structure of this situation. Exactly the same game-theoretic situation would exist, and exactly the same arguments would hold, if we were to substitute for this assumption the supposition that each prisoner is concerned primarily or exclusively with the other s welfare. Any account of human interaction, whether it is a normative theory that defines how people should behave if they are to be considered rational or ethical, or a purely descriptive theory, that purports to tell us how people actually do behave under familiar conditions, must make room for the possibility of conflict even if selfishness and malevolence are completely absent. Let me take care of it. No, let me take care of it. I insist. Course Notes Page 4

5 No, you re too generous... Or, two scientists may both be trying to save the world, but have different theories about how this is most likely accomplished. Each may scheme, and even fight, to have his own scheme accepted and the other scheme rejected, though each is consciously and subconsciously sincerely concerned only about the welfare of humanity. In the case defined above, A s ranking of outcomes (from most preferred to least preferred) is: 1) A confesses and B does not confess; 2) both A and B do not confess; 3) both A and B confess; 4) A does not confess and B does confess. B s order of preference, of course, reverses the rankings of (1) and (4). If we were operating at a more abstract level, each outcome would be represented by an ordered pair of numerical values (one for A, and one for B) rather than by a description of an ordered pair of concretely described states of affairs whose degree of desirability for A and B is a matter for our interpretation. The employment of pure numbers as payoffs has the advantage of fixing the subjective desirabilities of the outcomes for each of the agents once and for all, and of making clear, in particular, that the numerical values of each of the possible outcomes for an agent is to be conceived as the resultant of the values of all discernible desirable and undesirable aspects of the state of affairs associated with the outcome. This implies that whatever concern one agent has for another is already taken into account in the number defining the desirability that the agent assigns to each outcome. The main interest of this situation for those who think about strategic rationality in any ofits varied manifestations and applications lies in the manner in which the course of action that would be best for each prisoner if it could be assumed that both would follow the same course of action fails to correspond to the course of action that would be best for either one of the prisoners under the assumption that the actions of the two prisoners are mutually independent. Thus assume that, prior to their arrest, the two prisoners had agreed not to confess. Prisoner A either trusts his confederate or he does not. In the former case which corresponds to the case in which A assumes that B will not confess A s choice of action in effect becomes a choice between confessing and going free on the one hand, and not confessing and spending a year in prison on the other. Under this assumption about B s course of action, A s most advantageous course of action is clearly to confess. Assume, contrarily, that A does not trust his confederate, and hence thinks it probable that B, despite his promise, will confess. In this case, A s choice of action boils down to a choice between confessing and spending 5 years in prison on the one hand, and not confessing and spending 15 years in prison on the other. Again, confessing is more advantageous for A than not confessing. Thus confession yields a better outcome for A than does non-confession no matter which course of action B adopts. In the language of game theory, confession is the dominant strategy, meaning that confessing is more advantageous for A than not confessing whatever assumption is made about B s actions. With notable exceptions, it is generally agreed that confession is the rational course of action in Prisoner s Dilemma and all structurally identical situations. (From a slightly more abstract point of view, any situation with the above structure constitutes a Prisoner s Dilemma ). Again, this is under the assumption that the agents desires are as stipulated: We are not claiming that these desires are themselves rational despite what many people think and advocate, there is no good argument that rational behavior is behavior motivated solely by self-interest. Not surprisingly, many find it troubling that the prisoners are doomed to a joint outcome that is significantly less desirable to each of them than is the joint outcome that would be produced if the prisoners could somehow coordinate their actions. Although no amount of prior consultation or agreement can change this situation, Thomas Hobbes celebrated solution to an important class of variants of this problem is suggestive. (Hobbes describes this solution in his great work of political philosophy, The Leviathan Course Notes Page 5

6 (1651) ). Suppose, contrary to previous assumption, that before their separation and interrogation, the prisoners are given the option of signing a contract that commits each to the strategy of not confessing. This agreement is witnessed by an official who has the power (and manifestly, as evidenced by the gleam in his eye, has the intention) to place in prison for 25 years any prisoner who, after signing the contract, confesses to the crime in question. (We assume of course that the terms of the contract will be enforced only under the condition that both prisoners sign it, and that the prison term for breach of said contract runs consecutively with whatever prison terms are drawn in consequence of the outcome of the game proper). Clearly, it is in the interest of any prisoner who signs the contract to maintain silence: After one signs the binding contract, it is better to not confess than to confess whatever one assumes about the other prisoner s course of action for the simple reason that both a prison term of 5 years and a prison term of 15 years are less bad than a prison term of 25 years. More interestingly, if both prisoners can verify that both of their signatures are on the contract, it is preferable that the agreement be offered than that it not be offered. And most interestingly, it is most advantageous if the contract is involuntarily imposed upon the two prisoners meaning that is best that it simply be stipulated that the contract is in force, and that breach of contract will result in the 25 years imprisonment mentioned above. Again, the issue of whether or not the prisoners are given a chance to communicate outside any context in which any agreement between them can be enforced is irrelevant to the structure of the game or its resolution. This should be clear once it is understood that even in the variant of Prisoner s Dilemma in which it is supposed that both A and B are informed that A will receive information about what B has done before A has acted, the best course of action for A will be to confess whatever course of action B has taken, and, consequently, the best course of action for B, in anticipation, is also to confess. The outcome in which both prisoners confess occupies a point of stable equilibrium with respect to all possible variations in either prisoner s assumptions or knowledge concerning the identity of the other s action. (I have my own, quite different, approach to Prisoner s Dilemma, but an exposition of it would be far too lengthy to present in this introductory context). Course Notes Page 6

7 Prisoner s Dilemma Repeated Consider the composite situation that, in effect, corresponds to 100 repetitions of Prisoner s Dilemma. Suppose that two individuals, A and B, both know that they will be confronting a sequence of exactly 100 situations each of which is structurally identical to Prisoner s Dilemma. The payoffs obviously cannot be the prison sentences described in Prisoner s Dilemma proper (unless each of the prison terms is scaled down, or each of the prisoner s lifespans are scaled up, by a factor of something like 100). Consequently, the numbers attached to the prison terms in Prisoner s Dilemma proper are best regarded, in this multi-stage context, as direct representations of the comparative undesirabilities, for each player, of each of the possible outcomes; and, correspondingly, the total payoff ofthis whole 100 term series for each player X is defined as the sum obtained by accumulating the individual payoffs (to X) yielded by the outcomes of each distinct play of the composite game. From now on, we substitute the term defection for the action previously described as confession, and substitute the term cooperation for the action previously described as non-confession. Thus if A defects and B cooperates on the first play of this composite game, A has accumulated zero units of unpleasantness and B has accumulated fifteen units of unpleasantness; and if in addition A and B both defect in the second play of the game, A has accumulated a total of 5 units of unpleasantness and B has accumulated a total of 20 units of unpleasantness; and if both A and B in addition choose to defect at each of the 98 remaining plays of the game, A will end up with a total of 495 units of unpleasantness and B will wind up with a total of 510 units of unpleasantness. (For the sake of narrative flow, we will henceforth employ monetary terminology in our descriptions of the players payoffs, and set a unit of unpleasantness equal to the loss of one dollar). Because both players are able to keep track of how each player acted in every prior play of the hundred term series, it is at least at first glance plausible that the players collectively possess the capacity to enforce their own agreements. Thus suppose that A makes the following appeal to B before the first play: If both of us were to cooperate on every single one of the plays of this game, we would both emerge 100 dollars poorer than we began. On the other hand, if we both adopt the course of action that is appropriate for a one-time play of this game (i.e., for Prisoner s Dilemma itself) on every one of the 100 plays of this game, we will both be 500 dollars poorer than we began. Clearly, the first eventuality is the far more favorable one for each of us. I suggest, therefore, the following: On the first play of the game, I will take the cooperative course of action, in anticipation that you will do the same. If you take advantage of my amiable and generous confidence in your good sense, and adopt the course of defection rather than that of cooperation in order to save yourself a dollar on that single occasion, I will take itasasign that you have not understood the logic of the situation confronting us, and hence I will be forced to conclude that you will choose the non-cooperative option on each subsequent play. At that point, I will have no alternative but to protect myself against terrible losses by taking the path of defection on the remaining 99 plays of the game. Thus instead of loosing only 100 dollars through the entire course of play by adopting the strategy I have suggested, you will in consequence of your short-sighted refusal to cooperate on the first play condemn yourself to the loss of the much larger sum of 495 dollars. Because I realize it is possible that the force of this reasoning may make a greater impression after you discover that your failure to cooperate with me has lost you more money after just a fourth of the series of plays have been run through than you would have lost through the entire series of plays if you had instead cooperated from the beginning, I will interpret a cooperative move in any two successive plays as an indication that you have come to your senses, at which point I will myself re-enter upon the path of cooperation whose logic I have just sketched out. (The reason that Course Notes Page 7

8 I have insisted that you must make cooperative moves twice in succession before I will reciprocate is that I wish to ensure that you will be strongly deterred from non-cooperation on the first play of the hundred term sequence). A is suggesting, therefore, that because each player is in a position to respond to the other s action, the two players together form an internal enforcement mechanism that obviates the necessity for the intervention of a Hobbesian deus ex machina. There is, unfortunately, a disturbing line of reasoning that seems to completely undermine this very plausible argument for mutual cooperation. Suppose that both players follow A s suggestion for the first 99 plays of the game, and that both now face the final play of the series. From a strategic point of view, this terminal play is distinguished from all of its predecessors by the fact that neither player can respond to whatever action is taken by the other in this play of the game, and hence neither player can punish the other for failing to cooperate on this final play, and neither player need fear retaliation for any move he might then make. In other words, the possible payoffs of the single remaining play are now all there is to consider, and hence both players are in the same situation as that defined in the original Prisoner s Dilemma. Consequently, both players know that they will adopt the non-cooperative strategy when they reach the terminal play, and each can predict that the other player will defect in the terminal play. Now, assume that the players are about to make their moves in the ninety-ninth play of the series. Since each player knows that the other player will defect on the final play whatever combination of moves were made by the players prior to the hundredth play, each can infer, in particular, that the other player will not cooperate on the final play whether or not this co-player defects or cooperates on the ninety-ninth play. This removes any rational incentive for cooperating on the ninety-ninth play that might be derivable from the fact that the players can respond to each other s moves on upcoming plays. Both players are thus free to treat the ninety-ninth play of the game as the final play. Thus each player can deduce before reaching the ninety-ninth play that no player whose only incentive for cooperation is the logic of deterrence will cooperate on the ninety-ninth play. Since the fact that neither player will cooperate on the ninety-ninth play is a foregone conclusion by the time at which the players face a final decision concerning the proper course of action for the ninety-eighth play, each knows that the other s action both on the ninety-ninth and on the hundredth play is independent of whatever happens on the ninety-eighth play, and hence each player is forced to conclude that it is rational to defect on the ninety-eighth play. This conclusion, obviously, can be deduced before the ninety-seventh play, and hence each player will have to take the other s defection on the final three plays as a given at this prior stage. This implies, in particular, that each player is rationally bound to make the same move on the ninety-seventh play that he would have made if the later plays had not been supposed to exist. In other words, because both players are logically committed to non-cooperative moves on the ninety-eighth play and beyond, the possibility that one player can retaliate for the other s non-cooperation on the ninety-seventh play (by choosing non-cooperation rather than cooperation on the ninety-eighth play) can likewise be written-off by both players before they make their choices concerning the ninety-seventh play. Each cycle of reasoning inexorably moves the boundary between the regime of cooperation and the regime of defection one step backwards, until we eventually reach the point at which all hundred stages of the game are in the future, yet to be played. Because this reasoning is accessible to both players in advance of their first play of this composite game, each player knows before even the first play of the game that the other is rationally compelled to defect every time, and hence is himself rationally bound to defensively anticipate these defections by his own course of exceptionless defection. Although the theoretical interest and suggestiveness of these results are great, the fact that these results stand in apparent contradiction to our ordinary reliance upon the efficacy of collective self- Course Notes Page 8

9 enforcing deterrent mechanisms in all areas of life is not difficult to understand. All particular games are ultimately embedded in the Great Game of Life, whose boundaries, in relation to most ordinary situations, are typically not determinate enough to exclude the possibility of retaliation. And indeed, this point is obliquely confirmed by the fact that the practically distinctive character of the special situations to which the above reasoning applies is well understood. Think, for example, of the three time loser ; Mr. Bigelow in the film noir classic D.O.A. ; the man whose acts have already incurred the supreme penalty; the despot who believes that he can eliminate the last remaining obstacle to world domination by a surprise attack; the murderer who encounters a man carrying a valuable object alone in the woods; etc. Course Notes Page 9

10 Newcomb s Problem This problem or paradox, proposed by the physicist William Newcomb in the early 1960 s, was first discussed in print (by Robert Nozick) in 1969 in the Festschrift for Carl Hempel, and became widely known after appearing (sometime in the summer of 1973) in Martin Gardner s wonderful puzzle column in Scientific American. The following formulation is more or less standard. Tw o boxes are placed before an individual: Box A, which is transparent, and can be seen to contain $1,000; and box B, which, though opaque, is demonstrably tamper-proof and is known to contain either $1,000,000 or some materially equivalent quantity of worthless paper. The agent must choose between 1) the option of taking the contents of both boxes (option A), and 2) the option of taking the contents of box B alone (option B). It is assumed that the agent is not permitted to peer into box B before making his final selection. However, he knows 1) that its contents are determined by the prior irrevocable action of a perhaps superhuman but not supernatural being (the Predictor) that has placed the sum of $1,000,000 in box B if and only if it had predicted that the agent would choose to open only box B; and 2) that the Predictor has had a very high rate of success in forecasting the selections of agents confronted with the same situation. It should be noted that even among those who see nothing absurd or ill-defined in the scenario presented above, there is no consensus about the correct solution Tw o points in the above require some amplification. The assumption that the contents of the box B is determined by the prior action of the Predictor is to be interpreted as meaning that the contents of this box is assumed not to change during the entire interval from the moment twenty minutes ago, twenty hours ago, or a thousand years ago at which either a million dollars or a stack of blank paper was placed in the box by the Predictor through the moment right after one has already made one s choice at which one examines box B s contents. In order to make this point as vivid as possible, one may imagine that the back of the box is transparent, and that a trusted friend of yours is observing its contents during the entire period from the instant at which you are presented with the problem to the moment at which you examine its contents. Although your friend is assumed to be incapable of communicating with you during the whole of the period of presentation and decision, he will be able to assure you afterwards that the contents of box B whatever it was remained invariant during the whole of the relevant interval. (One may suppose that you are awarded a million dollars if your friend observes any change). The second point one must be clear about is that the hypothesized accuracy of the Predictor is not the result of a mere statistical trick. For example, it may happen to be the case 1) that ninety percent of the people who have been presented with this problem have decided to select both of the boxes, and 2) that the Predictor in fact never puts a million dollars in box B. Thus although the Predictor has a quite impressive track record of ninety percent accuracy in the narrow sense that the Predictor s choice and the agent s choice correspond (or match ) ninety percent of the time, the degree of prescience that must be attributed to the Predictor under this scenario is not quite what the above brief description of this problem might at first glance suggest. We are assuming, on the contrary, that the Predictor s methods are very powerful, on an individual-to-individual basis. In statistical terms, we are supposing 1) that a great mass of highly intelligent and sophisticated people (of which you are a good representative) have confronted this problem; 2) that a great number of these people have chosen both boxes and agreat number of people have chosen the single box B (the precise break-down is irrelevant); and 3) that the Predictor has a ninety percent accuracy rate both in correctly predicting the choices of those who have gone on to select both boxes, and in correctly predicting the choices of those who have gone on to select the single box option B. Inother words, Course Notes Page 10

11 ninety percent of those who have selected both boxes end up with only a thousand dollars (the remaining ten percent of this class wind up with a million and one thousand dollars), and ninety percent of those who have selected only box B end up with a million dollars (the remaining ten percent of this class come away with nothing. The argument for choosing just one box is completely straightforward: Look, it s simple 1) Persons who select both boxes tend to gain a thousand dollars, and persons who select only box B tend to gain a million dollars; 2) By hypothesis, you are a good representative of the class of persons for which these statistics are accurate; and hence 3) You have a far higher probability of becoming a millionaire if you select only box B than if you select both boxes. The probability that the million dollars will be found in box B if you select only box B is.9, and the probability that the million dollars will be found in box B if you select both boxes is.1. If a level ofninety percent accuracy does not seem quite high enough to merit foregoing the chances of acquiring both the contents of box A and the contents of box B, you may set the statistical reliability of the Predictor to any degree of accuracy you please. Surely, if the error-rate of the Predictor is only one in a billion, it would be crazy not to choose the single box. The argument for choosing both boxes is a little less straightforward. Look, the million dollars is either already in the box, or not in the box, before you make your selection. In other words, if the money was in box B before you selected both boxes, it does not disintegrate after your selection is finalized; and if only a stack of blank paper was in box B before you decided to take only box B s contents, the stack of paper does not suddenly transmute into valuable currency your loyal friend monitoring box B will later vouch for that. So, if box B does contain the million dollars, you will take away both the million dollars hypothesized to be in box B and the thousand dollars known to be in box A if you select both boxes. If box B does not contain the million dollars, you will at least pick up the thousand dollars if you do not allow yourself to be intimidated into passing up the money in box A. In either of the two possible cases, therefore, you will be a thousand dollars wealthier if you chose both boxes rather than just box B. (Compare this to the Prisoner s Dilemma). If this isn t convincing enough, look at this from the point of view of your friend who sees what is in box B the whole time. Whether he sees the million dollars in box B, or sees that its contents are valueless, he would tell you to take both boxes if he could. Although he can t communicate with you, we all know what he hopes you will do! This is not to say that one who belongs to the class of persons confused enough to select just the single box is not very likely to come away from this situation far wealthier than one who belongs to the class of persons who have sense enough to grab both boxes. This is not the only situation anyone s ever heard of in which folly has been rewarded and wisdom punished. I know this sounds odd, but consider: It has been demonstrated that the activity of smoking, if prolonged over decades of time, is dangerous to one s health. Let s assume, counterfactually, the validity of the great statistician Ronald Fisher s cantankerous hypothesis, formulated (perhaps under monetary suggestion) in the late 1950 s, that the high correlation between smoking and certain cardio-vascular and respiratory diseases is the product of a genetic defect that induces both this cluster of diseases and a great enjoyment of smoking. The fact that these medical findings would be grim news for a man who has had to resist a strong impulse to smoke almost every day since the year he succeeded in his attempt to give up smoking is not incompatible with the recognition of the fact that if the man enjoyed smoking, it would be foolish to continue to resist the impulse after this explanation of the correlation was found. Or (for those for whom the fantastic nature of Fisher s reckless hypothesis makes Course Notes Page 11

12 the requisite suspension of disbelief unattainable) consider the statistical correlation said to exist between one s life expectancy and the number of hours per day that one sleeps. Specifically, some studies suggest that the life expectancy ofthose who sleep far more or far less than eight hours a day is less than that of the remainder of the population. Although it is not difficult to think of many plausible explanations for this correlation, it does not seem likely that one who has always tended to sleep over twelve hours a day in blissful unawareness of these statistics will improve his life expectancy byhenceforth forcing himself out of bed after eight hours sleep with the aid of an insistent alarm clock. Newcomb s problem is entirely analogous: The Predictor has pegged you either as the kind of person who tends to select both boxes in this kind of situation, or as the kind of person who tends to select just the opaque box in this kind of situation and indeed you are, and have been, exactly one of these types of persons even ifyou yourself may not know which until after you have made your choice. You are not fooling anyone (except perhaps yourself) by groggily forcing yourself out of bed every day with (what is for you) plainly insufficient sleep and then (sleepily) saying to yourself that you would have gotten up at this time anyway. Analogously, you are not fooling anyone by choosing only the opaque box. An even more blatant example of this statistical phenomenon is seen in the fact that individuals who enter surgery are more likely to die within a few years of the date of the surgery than are individuals in the general population. Now, we all know that there are incompetent surgeons, and that hospitals can be dangerous places, but we are not particularly tempted to embrace the general hypothesis that the science and art of surgery is responsible for a decrease in life expectancy in the early twenty-first century. No, being seriously ill with a more or less operable condition is a common cause of both the surgery and the premature death. Newcomb s problem, like the medical examples discussed above, exemplifies what is technically called the conjunctive fork, a causal structure in which one event C causes events E and F, but neither E and F are the cause or effect of the other. The decision not to take the two boxes is like the decision not to go to the doctor when you have strong reason to believe that you are in fact sick. You should meditate on this problem before we discuss it in class. Course Notes Page 12