Abstract
The paper argues that on three out of eight possible hypotheses about the EPR experiment we can construct novel and realistic decision problems on which (a) Causal Decision Theory and Evidential Decision Theory conflict (b) Causal Decision Theory and the EPR statistics conflict. We infer that anyone who fully accepts any of these three hypotheses has strong reasons to reject Causal Decision Theory. Finally, we extend the original construction to show that anyone who gives any of the three hypotheses any non-zero credence has strong reasons to reject Causal Decision Theory. However, we concede that no version of the Many Worlds Interpretation (Vaidman, in Zalta, E.N. (ed.), Stanford Encyclopaedia of Philosophy 2014) gives rise to the conflicts that we point out.
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Notes
Jeffrey (2004, p. 99).
Mermin (1981, pp. 407–408).
Price (1996).
Bell (1977, p. 100).
Reichenbach (1984, p. 157).
See Lewis (2006) for a survey of these responses.
But not however to superluminal signaling: as is widely recognized (e.g. Redhead 1987, pp. 113–115), there is no way in which somebody operating the first receiver could exploit these correlations to send any sort of signal to somebody operating the second. This fact opens the door to a kind of ‘peaceful co-existence’ between nonlocality and relativity, if we take the latter to be claiming only that there is no superluminal signaling, not that there is no superluminal causality of any sort. On the other hand there is something unsatisfactory about taking the relativistic restriction against superluminal causality to be a principle only about signaling, for as Bell himself wrote: ‘the “no signaling ...” notion rests on concepts that are desperately vague, or vaguely applicable. The assertion that “we cannot signal faster than light” immediately provokes the question: Who do we think we are? We who make “measurements,” we who can manipulate “external fields”, we who can “signal” at all, even if not faster than light? Do we include chemists, or only physicists, plants, or only animals, pocket calculators, or only mainframe computers?’ (Bell 1990, p. 245)
Of course, in the de Broglie–Bohm theory, this parameter dependence is made experimentally inaccessible by the epistemic uncertainty in the real locations of the corpuscles. This relies on the probabilities for the corpuscles’ locations taking their equilibrium values, as given by the Born rule (see Holland 1993, chap. 11). However, there is the theoretical possibility of non-equilibrium states; in these, an experimenter can in principle take advantage of parameter dependence to signal non-locally (see Valentini 2002).
Holland (1993, pp. 471–476).
Hume (1949, p. I.iii).
In (22) and (35) we are adopting the notational conventions that \((\hbox {a, b}) < (\hbox {c, d})\) iff \(\hbox {a} < \hbox {c}\) and \(\hbox {b} < \hbox {d}\), and (a, b) = (c, d) iff a = c and b = d. Also, if P and Q are options then \(\hbox {P} \succ _{\mathrm{EDT}} \hbox {Q}\) means that EDT strictly prefers P to Q. Similarly, \(\hbox {P} \succeq _{\mathrm{EDT}}\) Q means that EDT reckons P at least as choiceworthy as Q. Similarly with ‘CDT’ replacing ‘EDT’.
For EDT this is clear from (19), (21) and the fact that every other option in Table 2 gets V-score 0. For CDT it follows from the fact that 12hom and 12het both weakly dominate all of the other four options.
Of course, she must also believe that the prior state either causes or shares a common cause, or is acausally correlated, with one’s present choice to set the receivers this or that way. But this is not implausible: given that one already opts for (A) and so has swallowed hidden variables themselves, baulking at the idea that the experimenter lacks retrocausal powers (that being the only alternative to (A2)) is arguably straining at a gnat.
Remember, (B2) is not denying that switching either receiver to this or that setting has any causal influence on its own reading (and neither are (A2) or (C2)). Rather what it denies is any superluminal, retroactive or common causality between anything going on in the region of one receiver, including its setting, and anything going on in the region of the other.
For the continuous case, read \(\hbox {Cr }(\hbox {Ch }(\hbox {yy}\vert \hbox {O}_{1}) = \hbox {x}))\) as F\(^{\prime }\) (x): the first derivative of the cumulative distribution function \(\hbox {F (x) = Cr (Ch} (\hbox {yy}\vert \hbox {O}_{1}) \le \hbox {x}\)).
Since \(\hbox {Cr }(\hbox {O}_{1} \rightarrow \hbox {yy}) + \hbox {Cr }(\hbox {O}_{1} \rightarrow \hbox {nn}) + \hbox {Cr }(\hbox {O}_{1} \rightarrow \hbox {yn}) + \hbox {Cr (O}_{1} \rightarrow \hbox {ny}) =1\), it is possible to write U (iihom) as \((1 - \hbox {z)x} + \hbox {z}(1-\hbox {x}) = \hbox {x} - \hbox {z}\), where \(1 - \hbox {x} = \hbox {Cr (iihom} \rightarrow \hbox {yn}) + \hbox {Cr }(\hbox {iihom} \rightarrow \hbox {ny})\). So if \(1 - \hbox {x} > 0\) then \(\hbox {x} < 1\), hence there is some \(\hbox {z}^{*} < 1\) s.t. \(\hbox {z}^{*} > \hbox {x} \ge 0\). So if \(\hbox {z} \ge \hbox {z}^{*}\) then \(\hbox {U (iihom)} < 0 = \hbox {U (Q)}\).
Note that points (i)–(iii) suffice to derive a Bell inequality for the Ch function. This does not contradict the predictions of quantum mechanics so long as the chances given by Ch do not reflect long-run relative frequencies. It is permitted if, for example, Ch represents single-case chances that vary from case to case. And that is what Ch should represent if conditional chance matters to Causal Decision Theory: for CDT is supposed to be sensitive to the tendency of a setting to causally promote this or that outcome in the particular decision situation to which you are applying it. The situation here is similar to that in Newcomb’s problem, where, even though there is a long-run correlation between one’s choosing one box and this having been predicted, the latter is, on any occasion, conditionally independent of the former with respect to the appropriately causal chance function. This is consistent with the claim that chances control long run frequencies if, as in the Newcomb case and as here, these one-off conditional chances vary from one occasion to the next.
To see this consider that (57) takes the form xy + (1 \(-\) x)(1 \(-\) y) = 1. So for (0, 0) \(\le \) (x, y) \(\le \) (1, 1) the only solutions are x = y = 0 and x = y = 1.
It’s also worth contrasting the construction in this paper with two other attempts (the only ones known to us) to exploit violations of the Bell inequalities in order to make EDT and CDT disagree. Berkovitz’s example (1995) assumes that the agent rejects all of the (A)- and (B)-hypotheses and instead believes in a prior instruction set that is uncorrelated with her setting of the receiver. It therefore depends on a theoretical assumption that is demonstrably false and so is no more realistic than the supernaturalistic Newcomb cases on which we had been seeking an improvement.
Cavalcanti’s argument (2010), which invokes the CHSH arrangement (Clauser et al. 1969), appears to mischaracterize the causal theory. His case depends crucially on there being two agents, one at each wing of the experiment. But his calculation of the U-score of any option available to one of these agents treats both agents’ choices as actions i.e. ignores their evidential bearing on anything other than their effects. (A formal symptom of this is the symmetric treatment of the terms ‘\(\hbox {A}_{\mathrm{R}}\)’ and ‘\(\hbox {B}_{\mathrm{G}}\)’ in his Eq. (16).) But this is a mistake: from the point of view of either experimenter the other agent’s choice—which is not up to her—itself partly characterizes the ‘state of nature’, and her credence should reflect this. Cavalcanti’s reasoning that the causalist must bet against quantum mechanics in these scenarios (2010, pp. 585–586) is therefore invalid.
In any case Cavalcanti’s argument concerns only the case in which the agent believes in a prior instruction set (i.e. the analogues of what we called (A)-type interpretations of the Stern–Gerlach experiment). He does mention (2010, p. 589) his own belief that CDT’s advice in these cases would carry over to the case where the agent rejects any hidden variables (in particular to the case that I called (B2)); but he gives no good argument that this is so. (There is a one-sentence argument to this effect at 2010, p. 589, which however the already-mentioned mischaracterization of CDT entirely vitiates.) It turns out that his suspicion is correct. But it has taken some work to show this, including the invention of a totally new family of problems D (i, z).
Everett (1957).
At any rate this is so if the agent’s credences satisfy (46). If they do not satisfy (46) then there is some other situation D (i, z) for \(0 < \hbox {z} < 1\) in which EDT and CDT give conflicting advice to anyone that accepts (A2), (B2) or (C2)—see Table 4. And it is in this scenario that we can then expect CDT to underperform relative to EDT, and the forthcoming remarks in the main text go through mutatis mutandis for it.
See e.g. Gibbard and Harper (1978, p. 369).
For an example of this explicit stipulation see Joyce (1999, p. 149). Of course there are some who deny that the stipulation is coherent on the grounds that my present act can only be symptomatic of its effects (Price 2012, p. 510). On that view it is hard to see that EDT and CDT ever diverge; but then it is unclear what is attractive about a distinctively causal formulation of their common content, at least on a non-Humean conception of causality.
As Cavalcanti points out (2010, p. 569), CDT was itself developed as a response to the standard Newcomb problem, with which EDT seems unable to deal. If the examples in this paper show that CDT has to go, then we might wonder where this leaves EDT. Our own view, which we cannot defend at length here (but see Ahmed 2014) is that EDT is in fact a perfectly adequate normative decision theory—or at any rate it is more adequate than CDT. This commits us to a ‘one-boxing’ position on Newcomb’s problem. We bite that ‘bullet’: one-boxing, which EDT recommends, is superior in Newcomb’s problem for essentially the same, statistical, reason that the ‘het’ options are superior to the ‘hom’ options in Table 6. For a recent defence of this line on Newcomb’s problem see Price (2012). Thanks to a referee for raising this point.
On the other hand this claim is not the contradiction that Maudlin appears to imply it is when he writes: ‘if a theory predicts a correlation, then that correlation cannot, according to the theory, be accidental. A nomic correlation is indicative of a causal connection—immediate or mediate—between the events, and is accounted for either by a direct causal link between them, or by a common cause of both’ (2002, p. 90).
But this argument involves a loaded understanding of ‘accidental’. If ‘accidental according to the theory’ just means not predicted by the theory then of course the claim that no theory predicts accidental correlations is a mere tautology but hardly entails that that theory has any causal commitments. On the other hand if ‘accidental according to the theory’ means has no causal explanation according to the theory then certainly there are theories that predict ‘accidental’ correlations; but this, according to their advocates, reflects the insight that we should stop looking for causal explanations at this level (Van Fraassen 1991, pp. 372–374). Finally, if we simply define ‘causality’ in such a way as to be somehow involved in any nomic connection, then Laudisa’s remark is apt. ‘What we are doing... is nothing but saying that “connected events are connected”... using causal concepts in this case appears then to be a mere labeling devoid of any real physical and philosophical significance’ (Laudisa 2001, p. 229).
Note that on this definition (94) holds good on hypothesis (A2) as well as on hypotheses (B2) and (C2) because on the former hypothesis 111, 110 etc. are respectively equivalent to YYY, YYN etc.
Dummett (1976, p. 53).
The example is from Evans (1980, pp. 276–277).
A possible such position would be an atheistic version of Berkeleian phenomenalism. We usually think that what makes it true, that if I were in my office now then I’d see a desk in my office, is that there now is a desk in my office. But for Berkeley it is the other way around: it is counterfactuals about what I or somebody else would observe that make true the apparent categorical statements about ‘physical’ objects (1985[1710], p. 90 (Principles Sect. 3)). For Berkeley himself the counterfactuals are themselves made true by God’s categorical will; but for the atheist phenomenalist they would have to be barely true. It is for that phenomenalist simply a brute fact, not obtaining in virtue of anything that is actually already there, or in virtue of anyone’s actual present willing, that if I were now in my office I should see my desk (Berlin 1999, p. 43ff).
For details of maximin see Resnik (1987, pp. 26–27); for minimax regret ibid., pp. 28–32.
That stronger demand would certainly rule out at least some of the cases that are of interest to decision theory. E.g. the standard Newcomb Problem (Nozick 1969, pp. 207–208) only generates divergence between EDT and CDT if we are willing to go along with the stipulation that the case involves no backwards causation, even though the phenomena of the problem admit that interpretation if anything does. So there is nothing new about the idea of presenting an example against a background of specific theoretical assumptions.
This assumption is not logically unquestionable; but it is not really contentious either. If it were not the case that most people’s credences are relatively stable across time in the absence of new information, it would be very hard to know anyone’s beliefs about anything in the intervals between explicit avowals.
The authors thank Jeremy Butterfield, Huw Price, and two referees for this journal. AA wrote most of his contribution to this paper whilst on a Visiting Fellowship at the Research School of Social Sciences, ANU, Canberra, and wishes to thank that institution for its hospitality, and also the Faculty of Philosophy, University of Cambridge, for granting him leave for this period. AC wishes to thank the British Academy for its generous support. Some of the material in Sects. 3–5 and 9(i) appears in chap. 6 of A. Ahmed, Evidence, decision and causality (Cambridge University Press 2014) and appears here with the permission of Cambridge University Press.
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Ahmed, A., Caulton, A. Causal Decision Theory and EPR correlations. Synthese 191, 4315–4352 (2014). https://doi.org/10.1007/s11229-014-0536-9
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DOI: https://doi.org/10.1007/s11229-014-0536-9