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Causal decision theory’s predetermination problem


It has often been noted that there is some tension between engaging in decision-making and believing that one’s choices might be predetermined. The possibility that our choices are predetermined forces us to consider, in our decisions, act-state pairs which are inconsistent, and hence to which we cannot assign sensible utilities. But the reasoning which justifies two-boxing in Newcomb’s problem also justifies associating a non-zero causal probability with these inconsistent act-state pairs. Put together these undefined utilities and non-zero probabilities entail that expected utilities are undefined whenever it is a possibility that our choices are predetermined. There are three ways to solve the problem, but all of them suffer serious costs: always assume that, contrary to our evidence, the outcome of our present decision-making is not predetermined; give up the reasoning that justifies unconditional two-boxing in Newcomb’s problem; or allow epistemically impossible outcomes to contribute to expected utility, leading to the wrong results in a series of cases introduced by Ahmed (Br J Philos Sci 65(4):665–685, 2014a, Evidence, decision and causality, Cambridge University Press, Cambridge, 2014b). However they choose to respond, causal decision theorists cannot remain silent: the intuitive tension between decision-making and the possibility of predetermination can be made precise, and resolving it will require giving up something. Causal decision theorists have a predetermination problem.

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  1. While evidential decision theory tells us that what matters is the evidence that our acts give us about which state obtains, CDT tells us that what matters is what our acts will causally promote. An act can be evidence for a state which it does not causally promote. For example, our choice is evidence about, but does not causally influence, the prediction in Newcomb’s problem. EDT tells us that in such situations we must use the conditional probability of the state obtaining if the act is performed. CDT, on the other hand, tells us that in such situations we should use the unconditional probability of that state obtaining—or, equivalently, that we should use causal probabilities defined so that they coincide with unconditional probabilities when there is no causal influence from our act to the state—since our act cannot make the state any more or less likely in the relevant causal sense.

  2. In fact this may only be a predetermination problem—I do not wish to rule out the possibility that predetermination might cause other problems for decision-making, or CDT specifically (perhaps to do with ought-implies-can principles or similar)—but it is the predetermination problem this paper is about.

  3. For some discussions of this tension see, for example, Nozick (1969, p. 141), or Fernandes (2016) and references therein. There is also a debate on the relationship between belief in determinism and decision-making (usually called deliberation) in the free will literature, see McKenna and Pereboom (2016, § 12.3) for a short introduction to this debate and further references. Finally, there is the debate about whether it is possible to have credences about one’s own acts while engaged in decision-making. The negative answer to this question seems to be at least partly motivated by the worry that believing that one’s choices are predetermined might conflict with decision-making. See Hájek (2016) for a critical introduction to that debate. Note that most of these discussions are explicitly phrased in terms of global determinism, rather than predetermination of particular choices. This is, however, a mistake. It is no help to my decision-making that there might be indeterministic events at other times and places in the universe; the relevant question is always whether this particular choice is predetermined.

  4. I do not intend the Predetermination Problem, or the trilemma it gives rise to, as an objection to CDT, but as a challenge to be solved by further argument and refinement of our intuitions. Those already skeptical of CDT might, perhaps, turn it into an objection to CDT. But, even if not, I hope that they can learn something from understanding the Predetermination Problem and the challenge it raises for CDT.

  5. For example, the most obvious understanding of Lewis (1981) is committed to using subjunctives to define utilities as in the SubU solution; interventionist decision theories like Pearl (2000) are plausibly understood as being committed to the FFCD solution; as, more explicitly, is Joyce (2016, p. 226); and at least two authors, Cantwell (2010, 2013) and Edgington (2011), have suggested versions of CDT which endorse the PPII solution.

  6. A decision problem is sometimes just taken to be a set of acts and states. I prefer to think of a decision problem as a concrete situation facing a decision maker, and the corresponding set of acts and states as (one among) the correct description(s) of that situation.

  7. This might be different if the argument concerned an intuitive and pre-theoretic notion of ‘act’. But in the present context we are clearly using ‘act’ as a theoretical term, and we should not try to settle debates by defining our technical terms to make our preferred position trivially or analytically true.

  8. Hedden (2012), amongst others, suggests that we can solve a variety of problems to do with the epistemic availability of options by taking the options in decision making to be the decisions themselves, instead of the actions which issue from them. I am sympathetic to this position in so far as it solves a number of problems involved with trying and failing to perform an act—since one generally cannot try and fail to make a decision—and have no objection to its being substituted here. However, it will not help to solve the Predetermination Problem. The Predetermination Problem arises because the world might be such that we are predetermined to choose or perform some option; making the options in decision making the decisions themselves will not help, because we might be predetermined to make a particular decision. Hedden (2012, p. 354) suggests that if an agent is predetermined to make a particular decision then subjective oughts of the kind that CDT is interested in do not apply to them. He does not suggest what an agent should do when they are unsure if this is their situation; this is just what is at stake in the Predetermination Problem. We can, charitably, assume that he is suggesting something like the FFCD solution. But that is an additional commitment over and above the claim that the options in decision making are decisions themselves; a solution to the Predetermination Problem is still needed on such an account.

  9. See Nozick (1969) for the canonical statement of Newcomb’s problem. My version differs only in inconsequential details.

  10. Of course, opponents of CDT will not agree, but they can still gain something from the rest of this paper: understanding the Predetermination Problem can help us understand CDT even if we disagree with it by helping us to understand costs that CDT must take on which do not arise directly from Newcomb’s problem. I will also assume throughout that we can use monetary values as a directly proportional substitute for utilities. This is obviously unrealistic, but it will not harm the case to be made below.

  11. It would be more standard to use the phrase “is causally independent of” here instead of “are not causally influenced by”. However, in the present context it is very important to bear in mind the asymmetrical nature of the relationship we are interested in: it is possible for which state obtains to causally influence one’s choice while at the same time one’s choice has no causal influence on which state obtains. The phrase “causally independent” tends to suggest a symmetry we should avoid suggesting here. I thank an anonymous reviewer for pointing out this possible confusion.

  12. Specific agents in specific situations might be sure their choice is not predetermined. But CDT is supposed to apply to all rational agents. So unless it is irrational to believe your choice might be predetermined the Predetermination Problem will arise.

  13. It is sometimes suggested that something is wrong with this partition of states. However, since, for the moment, we are assuming partition invariance it is necessarily the case that if CDT gives an undefined expected utility with this partition it will do so with any other.

  14. If you are paraconsistently minded you might object at this point that we can make sense of the truth of contradictions in the actual world. Unfortunately I do not have the space here to discuss using a paraconsistent logic to solve the Predetermination Problem. I take it, however, that this possibility will be attractive to very few—giving up classical logic (or even just the law of non-contradiction) would be a very high price to pay to solve the Predetermination Problem.

  15. You might be suspicious that the Predetermination Problem is really about simply considering what we will do in the future; that it would arise with any state which entails that you will perform some particular act, not just those in which you are predetermined to perform some particular act. For example, in Newcomb’s problem, you might think that the state “I will two-box and the prediction is two-boxing” would be a problem—because it is inconsistent with one-boxing—even though it has nothing specifically to do with predetermination. However, when the fact that I will take two boxes is a fact about the future not entailed by the past and the laws of nature, it is sensible to say that if I take only one box it will cause me not to be in a world where I take both boxes; the causal probability of taking two boxes if I take one box is zero when my taking two boxes is not entailed by anything, including the past and the laws of nature, outside my causal control. Only when the fact that I will two-box is entailed by the past and the laws of nature, or something else outside my causal control, is the causal probability that I will two-box if I one-box non-zero. The Predetermination Problem is specifically to do with predetermination, not the mere existence of facts about what we will do.

  16. All three solutions can be understood in either a revisionary mood, suggesting a change to CDT that will solve the problem, or in a descriptive mood, as pointing to something that CDT is already, more or less implicitly, committed to; I will remain neutral on this distinction here—though the reader should bear in mind that this might make a difference to the plausibility of the solutions.

  17. We could, equivalently, exclude such \(S_i\) from the probability space over which causal probabilities are defined. This might be preferable on the grounds that it allows us to distinguish between propositions with zero probability because they pick out sets of measure zero and propositions which the decision maker takes to be impossible.

  18. Perhaps God can have evidence that is incompatible with predetermination, but no mere mortal could.

  19. I thank an anonymous reviewer for pointing out this possibility.

  20. For example, we might suggest that causal probabilities be defined so that the following equalities hold in Newcomb’s problem, using the states from Table 1: \(P(S_1 || \text {One-Box}) =P(S_1) + 0.5\times P(S_3) \), \(P(S_2 || \text {One-Box}) = P(S_2) + 0.5\times P(S_4)\), \(P(S_3 || \text {One-Box}) = 0\), \(P(S_4 || \text {One-Box}) = 0\), \(P(S_5 || \text {One-Box}) = P(S_5) + 0.5 \times P(S_3)\), and \(P(S_6 || \text {One-Box}) = P(S_6) + 0.5 \times P(S_4) \). That is, we start with the assumption that the states are not causally influenced by our choice—so that the causal probabilities are just the unconditional probabilities of the states. Then we set the causal probabilities of the inconsistent act-state pairs to zero. Finally we ensure normalization by splitting the unconditional probability of the inconsistent act-state pairs equally among the consistent act-state pairs which agree about the prediction (\(S_1\), \(S_3\), and \(S_5\) all agree that the prediction is one-boxing; \(S_2\), \(S_4\), and \(S_6\), that it is two-boxing). And mutatis mutandis for two-boxing. Using the causal probabilities so defined will always lead to two-boxing in Newcomb’s problem, no matter how high your credence that your choice is predetermined.

  21. This might fail if the probabilities \(P(S_1 || \text {One-Box})\) and \(P(S_2 || \text {One-Box})\) decrease with increasing utility of the associated act-state pairs for every rational agent. This is not a requirement that causal decision theorists should rush to place on causal probabilities. First, it would require that all rational agents consider increasing utility always less likely, which is an implausible rational constraint (it must be all rational agents since causal decision theorists are committed to all rational agents two-boxing, not just those with particular beliefs about the relationship between utilities and probabilities). Second, allowing such dependence between these probabilities and utilities causes all sorts of problems elsewhere, particularly for the Principal Principle and normalization of credences, because it allows the probability of a state to depend on something—its utility—which is not a feature of that state (the utility a state has is a relationship between that state and features of the agent in the actual world).

  22. Note that while simplifying the numerator of the inequality above to its maximum value $1,001,000 is useful in understanding what is going on, the inequality which must be satisfied for one-boxing to be endorsed is the more complex one. The relevant value of a in the more complex inequality can be equal to, or less than, zero—that is, there are many definitions of causal probability meeting the PPII solution conditions that will directly endorse one-boxing in Newcomb’s problem when you are sure enough that your choice is predetermined.

  23. Unless, perhaps, supporters of the PPII solution can explain why, despite appearances, our intuitions in the standard Newcomb’s problem are based on an assumption that our choice is not predetermined. Note that supporters of the FFCD solution will also claim that our intuitions in Newcomb’s problem are based on the assumption that our choice is not predetermined. But since they take this assumption to be required for rational decision-making it seems less difficult for them to explain why our intuitions are based on it than for supporters of the PPII solution who argue that we can reason perfectly well while considering the possibility of predetermination, and that when we do so we will sometimes come to endorse one-boxing. Supporters of the FFCD solution also maintain that one should two-box no matter how high one’s degree of belief that one’s choices are predetermined—it is merely that no matter what this degree of belief one’s decision-making probability should be zero in one’s choice being predetermined.

  24. It is natural to think of outcomes as giant conjunctions, in which case this will be the case when the act and state are each conjuncts.

  25. I am here using the terms ‘subjunctive’ and ‘indicative’ rather loosely. All I take the former to imply is that the relevant answers to this question need not hold fixed what the decision maker takes to be the facts. In particular, it may be that what would happen if I did A on the supposition that \(S_i\) requires a violation of that very supposition. Whereas answers to indicative questions must hold any supposition fixed.

  26. Indeed, this suggestion is mathematically equivalent to a version of the PPII solution. Namely, the one where we assign the inconsistent act-sate pairs zero causal probability and then simply normalize the other causal probabilities in proportion to the unconditional probabilities of the states involved.

  27. I am somewhat modifying Ahmed’s actual argument here. In particular I am including the possibility that our choice is not yet determined, which Ahmed ignores. The modification does no harm and makes the argument more general.

  28. I thank an anonymous reviewer for this journal for encouraging me to deal with both of these worries.

  29. Note that these are miracles relative to \(S_i\), not relative to the worlds at which the subjunctive conditionals are assessed. That is to say, the account of subjunctives will have to allow that the worlds at which we assess the consequences of an act contain violations of the laws of nature of the worlds in the state we are considering, but not violations of their own laws—which are logically impossible.

  30. Does one of these solutions deserve the name causal decision theory better than others? I am not one to put much stock in names. But, more importantly, I think this question, and the implication that whichever does deserve the name therefore has a dialectic advantage, gets things the wrong way around. Causal decision theory was introduced to systematize the intuitions behind two-boxing in Newcomb’s problem—we don’t (or at least shouldn’t) endorse two-boxing in Newcomb’s problem because we have an antecedent commitment to the relevance of causal influence in decision-making. And we certainly shouldn’t endorse two-boxing, or any particular solution to the Predetermination Problem, because of an antecedent commitment to some particular analysis of causation, say in terms of subjunctive conditionals. If it turns out that we have to give up the (exclusive) relevance of causation to best systematize all the relevant intuition then so be it.


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This research was supported by an Australian Government Research Training Scheme Scholarship. Thanks to Al Hájek, Tim L. Williamson, Jeremy Strasser, Wolfgang Schwarz, Nevin Climenhaga, Alex Sandgren, and Edward Elliot for useful comments and discussion.

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Correspondence to Toby Charles Penhallurick Solomon.

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Solomon, T.C.P. Causal decision theory’s predetermination problem. Synthese 198, 5623–5654 (2021).

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  • Causal decision theory
  • Decision theory
  • Determinism
  • Predetermination
  • Free will