Fifty years ago, Nozick introduced the philosophical world to a puzzle which he learnt from the physicist William Newcomb. With some superficial alterations, the puzzle goes like this: An evil genius provides you with an opaque envelope containing a cheque. This cheque is yours to keep. Now: the decision you face is whether to give her £1,000 or not. Remember, she’s an evil genius, so you want to deprive her of as much money as possible. So why would anyone think about giving her £1,000? Because: unbeknowst to you, several years ago, she took a scan of your brain and made a prediction about what you would do in this very situation. If she predicted that you would hand over the £1,000, then she placed a £1,000,000 cheque in your envelope. If she predicted that you would keep your money, then she only wrote you a cheque for £1,000. These predictions of hers are quite reliable. Given that you pay her £1,000, you’re 60% sure that this is what she predicted you’d do. And, given that you don’t pay her anything, you’re 60% sure that this is what she predicted you’d do.

Presented with this puzzle, some advise you to pay the evil genius £1,000 – doing so makes it more likely that she wrote you the cheque for £1,000,000! Others advise you to keep your money – you’ve already got the cheque, so paying her now accomplishes nothing! The puzzle has attracted an incredible amount of attention, and for good reason. It sits at the intersection of a cluster of rich philosophical questions about the nature of instrumental rationality and its relationship to free will, predictability, probability, causation, success, restraint, planning, and much more besides. Each of these questions is well covered in this new collection of essays. It deserves to be read by anybody interested in any of these topics.

Almost everything about Newcomb’s problem is debated and controverted, and most of these debates and controversies are well represented in Newcomb’s Problem. Some deny that cases with this structure ever actually arise. This position is taken up in José Luis Bermúdez’s contribution. If you are at all like this reviewer, you will find this position initially surprising. You may think that consequentialist voters face Newcomb’s problem each election. If you find yourself going to the polls, this gives you evidence that others like you are also going to the polls, and therefore, it gives you evidence that your candidate will win, though you’re astronomically unlikely to make any difference to the outcome of the election. Calvinists face a kind of Newcomb problem with each new temptation – for resisting temptation makes it more likely that they are predestined for heaven, though it was settled long ago whether their name is written, and nothing they do now can change that. Bermúdez doesn’t deny that people face these choices, but he does deny that these choices are Newcomb problems. The trick is that Bermúdez only treats a decision as a Newcomb problem if it is a case in which two prominent decision theories, evidential decision theory (EDT) and causal decision theory (CDT), make different predictions.

EDT, by the way, tells you to take whichever action gives you the best news about the world. It says to choose whichever action has the highest conditional expected value. Conditional on what? Conditional on the action being chosen. So, if Pr is your probability function, and V is your value function, specifying how desirable a proposition is, then EDT says to perform whichever act, A, maximizes the expectation ∑_iPr(S _i|A) · V(S _i & A), where {S _i}_i is a set of mutually exclusive and jointly exhaustive states of nature. CDT, on the other hand, tells you to take whichever action you expect to do the most to improve the world. It says to choose whichever option has the highest unconditional expected value. That is, it says to perform the act, A, which maximizes ∑_iPr(S _i) · V(S _i & A). (Causalists call this quantity ‘the utility of A’.)

The standard understanding is that EDT advises you to pay the evil genius the £1,000, while CDT tells you to keep your money. And this is what Bermúdez denies. He thinks that, in realistic cases like this, EDT will advise you to keep your money, since, in realistic cases like this, you will be in a position to obtain all readily available information about what was predicted before making your decision. Then, the act you ultimately choose won’t provide you with any new information about the prediction, so that EDT and CDT will agree.

As I say, nearly everything about Newcomb’s problem is controversial – even what Newcomb’s problem is. This particular controversy hides beneath the surface of this collection of essays, as different contributors use the name ‘Newcomb’s problem’ in inconsistent ways. (This will no doubt perplex newcomers to Newcomb.) Bermúdez stipulates that you face Newcomb’s problem only if EDT and CDT give different recommendations about how you should choose. In his contribution, James M. Joyce provides 5 conditions which your decision problem must meet in order for you to be facing Newcomb’s problem. These criteria do imply that EDT and CDT will diverge in their recommendations, but Joyce’s characterization is nonetheless logically independent of Bermúdez’s. Reuben Stern suggests in his contribution that, if you think you shouldn’t pay the evil genius, then you should think that, maybe, the evil genius can’t predict your choice. Given Joyce’s definition, if you think that your actions are or may be unpredictable in this way, then you are not facing Newcomb’s problem. In their contribution, Huw Price and Yang Liu argue that, in Newcomb’s problem, you should regard a decision to pay as causing the evil genius to predict that you would pay. In that case, EDT and CDT would agree. So the decision which Price and Liu call ‘Newcomb’s problem’ is definitionally not a Newcomb problem, as that term is used by Joyce and Bermúdez.

In part, these different uses of ‘Newcomb’s problem’ are symptomatic of the wide variety of philosophical issues and interests raised by the puzzle. Different authors aim to address different philosophical questions when they discuss Newcomb’s problem, and this leads them to understand the decision differently.

For Stern, the primary question he aims to explore with Newcomb’s problem is about the nature of choice – whether, when we form an intention to act in a certain way, our actions will be unpredictable. In the jargon, his question is whether our intentions to act may constitute interventions. He believes that this question can help settle the debate over whether or not to pay the evil genius.

For Price and Liu, the primary question is what they term ‘the Euthyphro question for causalists’: ‘Is it the causal connection between C and E that makes it rational to do C to achieve E? Or does the rationality of the latter somehow constitute or ground the fact that C causes E?’ (160, with minor notational changes.) Price and Liu argue for the second answer: facts about causation are grounded in or constituted by facts about rational choice. Their interest in Newcomb’s problem comes from the agency theory of causation, according to which the relation of causation is a kind of secondary quality which rational agents project onto the world. Agency theorists say that C causes E when it would be rational to choose to bring it about that C, were E the only thing you desired. Newcomb’s problem, then, is of interest because it may provide a case in which we project a backwards causal relation onto the world. This is why Price and Liu understand Newcomb’s problem differently from some other contributors. If Newcomb’s problem is to present a case of backwards causation, then we had better not define Newcomb’s problem to be a case in which there are no backwards causal relations, as Joyce does.

For Joyce, the primary question in play in discussions of Newcomb’s problem is which theory of rational choice is correct. His central goal is to defend his preferred version of CDT against some putative counterexamples to the theory. To illustrate the kind of counterexamples Joyce considers: suppose that you must drink one of two vials of liquid which the evil genius has placed in front of you. One contains an odourless, flavourless lethal poison, while the other contains harmless water (though you know not which). If she predicted that you’d choose the left vial, then she put the lethal poison on the left. If she predicted that you’d choose the right vial, then she put the lethal poison on the right. Once again, she is 60% reliable. If you choose to drink the vial on the left, L, you’ll be 60% sure that it contains the poison. And if you choose to drink the vial on the right, R, you’ll be 60% sure that it contains the poison. But you are given a third alternative: you may, if you wish, pay a pittance to use a coin flip to determine which vial you drink. While the evil genius is 60% reliable at predicting how you’ll choose, she’s unable to predict the outcome of the coin flip. If she predicted that you’d follow the coin flip, F, then she simply flipped her own coin to decide where to put the poison. Following the coin, you’ll think that the vial you drink from is as likely to contain the poison as not, 50%.

Faced with this decision, it’s natural to reason as follows: the coin will give me a 50% chance of living; whereas, if I choose either left or right on my own, I’m 60% likely to choose the poison. A 10% chance of life is worth a pittance to me, so I should pay. This reasoning has some pull, but CDT says that it is impermissible to pay the pittance. Joyce defends CDT by objecting to the reasoning above and attempting to disarm the intuition. The defense is compelling. As a lapsed causal decision theorist who still longs for the hymns and incense of causalist high mass, I wanted to be convinced. Unfortunately, I have some reservations—let me briefly explain them here.

On Joyce’s treatment, you should deliberate until you are equally likely to select the left and the right vial (and certain to not pay the pittance to follow the coin). At that point, you will be indifferent between the two options L and R, and uncertain which you will ultimately choose. This state of indifference is one from which it is difficult to rationally extricate yourself. If you attempt to choose the left vial then you’ll give yourself evidence that the poison is on the left, and so the utility of L will drop below the utility of R. And if you attempt to choose the right vial, then you’ll give yourself evidence that the poison is on the right, and the utility of R will drop below the utility of L. So, even though you’re indifferent between L and R, pursuing either option would give you new information which would change your rational evaluation of the options. So it appears that CDT does not allow you to rationally choose either L or R. Joyce deals with this as follows: from the deliberative perspective in which you think you’re just as likely to choose L as R, you should (irrevocably) pick one of the two. Picking, for Joyce, is a special way of choosing. He explains: ‘“Pick” is a term of art for a choice process which selects one from a set of equally good acts in a way that is not sensitive to differences in utility. (Think Buridan’s ass!) Picking is inherently arational’ (p. 150). Thus: when deciding between L, R and F, rational deliberation ends in a draw between L and R. At that point, rational deliberation hands over causal responsibility for decision-making to an arational picking process. Because rational deliberation has ended, you are no longer deciding on the basis of options’ utilities, so when you find yourself inclining towards picking left, this may lead you to evaluate right as the more choiceworthy option, but it won’t lead you to reconsider picking left (as it would if you were choosing rationally).

Joyce notes that CDT gives different advice if, instead of choosing between L, R and F, you are asked to choose between following the coin, F, and not following the coin – which is just a choice between L and R, {L, R}. Presented with this second decision, {F, {L, R}}, CDT will tell you to choose F. He suggests that, when we feel pulled to pay the pittance, we are confusing the decision {F, L, R} with the decision {F, {L, R}}. I’m not sure about this – but rather than disputing the source of the intuition, I’d like to focus on CDT’s disparate treatment of the two cases. I don’t want to criticize CDT for treating a sequential decision like {F, {L, R}} differently from an all-at-once decision like {F, L, R}. Instead, I’m interested in the reason why it treats the two cases differently. When making the choice between F and {L, R}, CDT evaluates the arational pick between L and R in prospect – in the decision {F, {L, R}}, picking between L and R is not something that your current self may do, but rather, something that your future self may do. When CDT evaluates acts in prospect, it evaluates them the same way as EDT does. That is, when CDT evaluates a future option, it takes into account correlations between that option and states beyond your control. And your pick, unlike the coin flip, is positively correlated with poison. By choosing {L, R}, you may save yourself a pittance, but you also cause yourself to be more likely to end up with poison. On the other hand, by choosing F, you lose yourself a pittance, but you also cause yourself to be less likely to end up with poison. Choosing {L, R} makes you the kind of person who’s 60% likely to drink poison. Choosing F makes you the kind of person who’s only 50% likely to drink poison. And making yourself a person like that is worth a pittance.

How do things change when you face the decision {F, L, R}? The causal consequences of F remain unchanged. The coin flip is still causally downstream of your choice, and it is still uncorrelated with poison. So choosing F still makes you the kind of person who’s only 50% likely to drink poison. And the arational process of picking between L and R still has a 60% probability of culminating with you drinking poison. Your picks are still biased towards poison. Still, when making the choice {F, L, R}, even though you know that picking between L and R is 60% likely to lead you to death, picking between L and R is not one of your options. Your options are L, R and F. And while one of L and R – the one you will actually choose – has a 60% chance of leading you to death, one of L and R – the one you won’t actually choose – has only a 40% chance of leading you to death. Before picking, you won’t know which is which, since you’ll think that you’re equally likely to pick either; so you’ll think both L and R are 50% likely to lead you to death. As neither costs a pittance, both are preferable to F.

On Joyce’s view, in the decision {F, L, R}, rational deliberation concludes in a draw between L and R. Before abdicating responsibility, rational deliberation’s final act is to entrust the arational picking process with the responsibility of deciding between L and R. In the decision {F, {L, R}}, on the other hand, rational deliberation never steps down; it concludes by choosing to not allow your future self to pick between L and R. What I have a hard time understanding is why rational deliberation should conclude differently in these two decisions. I can see that the resolution of rational deliberation in the second decision is a choice; whereas, in the first, rational deliberation concludes before a choice is made. But I can’t see why this difference should make a difference. In both decisions, rational deliberation may or may not enable an arational picking process which is biased towards poison. In both cases, whenever rationality reigns, this picking would take place in the future. If rational deliberation should evaluate picking in prospect in the decision {F, {L, R}}, then it seems to me that it should similarly evaluate picking in prospect in the decision {F, L, R}. True, in the second decision, picking commences immediately after rational deliberation concludes, during what we may wish to call the same decision; whereas, in the first, picking commences only after another round of rational deliberation, during a separate and later decision. But in both cases, whenever rationality holds the reins, picking lies in the future. And in both cases, picking is correlated with death. So I have a hard time seeing the rational grounds for treating picking differently in the two cases.

To dramatize my confusion, suppose you’ve reached the deliberational perspective in which you think you’re just as likely to pick L as R. You ask Joycean CDT: ‘What do I do now?’. It says: ‘Pick between L and R.’ You reply: ‘But I thought that picking between L and R wasn’t one of my options. Moreover, even if picking between L and R were one of my options, I definitely shouldn’t choose it, since it has a 60% probability of leading me to death.’ CDT: ‘Of course you shouldn’t choose to have your future self pick between L and R. If that were an option, you should avoid it like the plague! But it’s not one of your options. In contrast, both L and R are options, and both of them are permissible. So picking between L and R will lead you to a permissible option. So you may permissibly pick between L and R – though, just to reiterate, by no means should you choose to put yourself in a position to pick between L and R.’ At the end of the day, I just have a hard time making sense of this advice. And that is why I still have reservations about the view, despite Joyce’s compelling defence.

Unfortunately, there is so much more of interest in Newcomb’s Problem than I am able to discuss in any depth in this brief review. Chrisoula Andreou relates Newcomb’s problem to Kavka’s toxin puzzle and Quinn’s puzzle of the self-torturer in a discussion of whether rationality could require you to bring about an outcome which you disprefer to another outcome you could have had instead. Ahmed defends paying the evil genius the £1,000 by noting that those who do so tend to end up with more money than those who don’t. Melissa Fusco has an ingenious discussion of time bias in causal decision theory. She notes that causalists will gladly keep their money, but will be distressed to learn that they kept their money. Fusco uses some delightful examples to operationalize these preferences before defending this form of time-bias. Preston Greene defends paying the evil genius even when the envelope is see-through, and you can see that she has written you a cheque for only £1,000. Robert Stalnaker has a fascinating discussion of the relationship between causal decision theory and game theory. And there’s still more. The collection is a must-read for anyone interested in any of these topics.