With the notion of a probabilistic causal model in place, as discussed in the previous section, we are now in a position to modify Halpern and Pearl’s definition so that it can handle probabilistic preemption. Specifically, suppose

that Mis a probabilistic causal model and that ~u is the actual context, that is,

it is the set of values that the exogenous variables in M _{have in the actual}

world (or, more generally, the world of evaluation). The analysis that I wish to propose as the natural extension of Halpern and Pearl’s definition to the case

of probabilistic actual causation is PC46,47,48:

46 _{Like Halpern and Pearl’s AC, PC relativizes the notion of actual causation to a model (in this}

case, a probabilistic model). If one takes model-relativity to be an objectionable feature, then one could avoid it by saying that ~X ¼ ~x is an actual cause of ’ simpliciter, provided that there exists at least one appropriate probabilistic causal model relative to which PC is satisfied (cf. Section 4, above). Most of the criteria for an appropriate deterministic SEM that have been advanced in the literature (see Hitchcock [2001a], p. 287; Halpern and Hitchcock [2010], pp. 394–9;Blanchard and Schaffer [forthcoming], Section 1) apply just as well to probabilistic causal models.

47 _{PC could be stated more simply than it is in the main text. This is because condition PC2(a) is, in}

fact, redundant given PC2(b). The inequality appealed to in PC2(b) is required to hold for all
subsets ~Z0_{of ~}_{Z . Where ~}_{Z}0_{¼ ;, this inequality is identical to the one that is appealed to in}

PC2(a). Despite the possibility of simplification, the version of PC stated in the main text is in a way more perspicuous because it makes clear the formal analogy between AC and PC, with PC2(a) isolating the contingent probabilistic dependence requirement made by PC just as AC2(a) constitutes the contingent counterfactual dependence requirement made by AC. Isolating the contingent probabilistic dependence requirement in condition PC2(a) also helps lend clarity to the discussion below of how PC handles the probabilistic preemption scenario, as well as other interesting probabilistic causal scenarios. Thanks to an anonymous referee for pressing me to say more about why PC is stated in its present form.

48 _{If—as was suggested in the discussion of Section 6—counterfactual dependence is taken to be a}

limiting case of probabilistic dependence (when probabilistic dependence is understood, as it is here, in terms of counterfactuals about probabilities), it is plausible that PC and AC yield equivalent verdicts in deterministic causal scenarios, where all probabilities are ones or zeros.

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PC: ~X ¼ ~x is an actual cause of ’ in ðM

; ~uÞ (that is, in model M _{given}

context ~u) if and only if the following three conditions hold:

PC1. Both ~X ¼ ~x and ’ are true in the actual world (or, more generally,

the world of evaluation).

PC2. There exists a partition, ð ~Z; ~W Þ, of Y (that is, the set of endogen-

ous variables in the model M_{), with ~}_{X ~}_{Z and some setting}

ð~x0_{; ~}_{w}0_{Þ}_{of the variables in ð ~}_{X ; ~}_{W Þ, such that where in the actual}

world Zi¼zi for all Zi2 ~Z, the following holds:

(a) Pð’jdoð ~X ¼ ~x & ~W ¼ ~w0_{ÞÞ}_{>}_{Pð’jdoð ~}_{X ¼ ~}_{x}0_{& ~}_{W ¼ ~}_{w}0_{ÞÞ. In}

words, if the variables in ~W had taken the values ~W ¼ ~w0_{,}

then the probability of ’ would be higher if the variables in ~X

took the values ~X ¼ ~x than if the variables in ~X took the

values ~X ¼ ~x0_{.}

(b) Pð’jdoð~X ¼ ~x & ~W ¼ ~w0_{& ~}_{Z}0

¼~zÞÞ>Pð’jdoð~X ¼ ~x0_{& ~}_{W ¼ ~}_{w}0

ÞÞ

for all subsets ~Z0_{of ~}_{Z. In words, if the variables in ~}_{W had taken}

the values ~W ¼ ~w0_{, and the variables in ~}_{X had taken the values}

~

X ¼ ~x, and all of the variables in an arbitrary subset of ~Z had

taken their actual values, then the probability of ’ would still

have been higher than if the variables in ~W had taken the values

~

W ¼ ~w0_{and the variables in ~}_{X had taken the values ~}_{X ¼ ~}_{x}0_{.}

PC3. ~X is minimal; no strict subset ~X0_{of ~}_{X is such that if ~}_{X is replaced}

by ~X0_{in PC2, then no change to the values of the counterfactual}

probabilities that are appealed to in PC2 results. Minimality en-

sures that only those elements of the conjunction ~X ¼ ~x that are

relevant to the probabilities of ’ appealed to in PC2 are con- sidered part of a cause; inessential elements are pruned.

In the probabilistic preemption case described in Section 6 above, PC correctly counts C ¼ 1 as an actual cause of D ¼ 1. To see this, note that the actual context (that is, the set of actual values of the exogenous variables) is simply ~

u ¼ fCI ¼ 1; BI ¼ 1g. Let ~X ¼ fCg, with ~x ¼ fC ¼ 1g and ~x0_{¼ fC ¼ 0g. Let}

’be D ¼ 1. In the actual world, C ¼ 1 and D ¼ 1, so condition PC1 is satisfied.

If PC2 is satisfied, then PC3 will also be satisfied because ~X ¼ fCg has no

(non-empty) subsets; and if PC2(a) is satisfied, then this implies that, in the

circumstances ~W ¼ ~w0_{, the values of the variables in ~}_{X ¼ fCg make a differ-}

ence to the probability of ’. So everything hinges on whether PC2 is satisfied.

To see that PC2 is satisfied, let ~Z ¼ hC; S; Di, let ~W ¼ hB; T i, and let

~

w0_{¼ fB ¼ 1; T ¼ 0g. First note that PC2(a) is satisfied because:}

PðD ¼ 1jdoðC ¼ 1 & B ¼ 1 & T ¼ 0ÞÞ > PðD ¼ 1jdoðC ¼ 0 & B ¼ 1 & T ¼ 0ÞÞ: ð8Þ

In words, the probability that McCluskey would have died if Corleone had issued his order, Barzini had issued his order, but Turk hadn’t shot is greater

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than the probability that McCluskey would have died if Corleone had not issued his order, Barzini had issued his order, but Turk hadn’t shot. In fact, given the stipulations of the example, the former probability is approximately 0.45, while the latter is approximately 0. It is important to bear in mind here the non-backtracking nature of the counterfactuals. In particular, the prob- abilities are those that would obtain if Turk’s not shooting were brought about by an intervention, small miracle, or local surgery that does not affect whether or not Sonny shoots. This is what is indicated by the doðÞ operator.

To see that PC2(b) is satisfied, note that if it had been the case that C ¼ 1, B ¼ 1, and T ¼ 0, then the probability of D ¼ 1 would have been higher, even if S had taken its actual value S ¼ 1, than it would have been if C ¼ 0, B ¼ 1, and T ¼ 0. That is,

PðD ¼ 1jdoðC ¼ 1 & B ¼ 1 & T ¼ 0 & S ¼ 1ÞÞ > PðD ¼ 1jdoðC ¼ 0 & B ¼ 1 & T ¼ 0ÞÞ: ð9Þ In words, if Barzini had issued his order but Turk hadn’t shot, then the prob- ability of McCluskey’s death would have been higher if Corleone issued his order even if Sonny had shot, than it would have been if Corleone hadn’t issued his order. Indeed, given the stipulations of the example, the former

probability is approximately 0.5, while the latter is approximately 0.49

So PC2(b) is satisfied. We have already seen that PC1 and PC2(a) are satisfied, and that PC3 is satisfied if PC2 is. Consequently, PC yields the cor- rect verdict that C ¼ 1 is an actual (probabilistic) cause of D ¼ 1.

PC also yields the intuitive verdict that B ¼ 1 (Barzini’s order) is not an actual cause of D ¼ 1. In order to get the sort of contingent probabilistic dependence of D ¼ 1 upon B ¼ 1 required by condition PC2(a), it will be ne- cessary to include in the antecedents of the relevant counterfactuals the fact that at least one variable on the Corleone process—that is, either C or S— takes (the non-actual value) 0. The trouble is that, in such circumstances, if B

49 _{I mentioned in Footnote 18 that the version of Halpern and Pearl’s condition AC2(b), given in}

(Halpern and Pearl [2001]) and stated in Section 4, above, has a probabilistic analogue—namely, my PC2(b)—that is superior in its handling of probabilistic preemption to the obvious prob- abilistic analogue of the somewhat different version of AC2(b) given in (Halpern and Pearl [2005]). Though I won’t go into the details (readers familiar with Halpern and Pearl’s ([2005]) account should be able to surmise them for themselves), the obvious probabilistic analogue of the later version of AC2(b) would allow that Corleone’s action was a cause only if PðD ¼ 1jdoðC ¼ 1 & B ¼ 1 & S ¼ 1ÞÞ > PðD ¼ 1jdoðC ¼ 0 & B ¼ 1ÞÞ. But this inequality does not hold in the present case, since PðD ¼ 1jdoðC ¼ 1 & B ¼ 1 & S ¼ 1ÞÞ & 0:5, while PðD ¼ 1jdoðC ¼ 0 & B ¼ 1ÞÞ & 0:81. Since Corleone’s action is a cause, PC2(b) yields the cor- rect result, while the obvious probabilistic analogue of the later version of AC2(b) does not. So PC2(b) is the correct condition. As I also noted in Footnote 18, insofar as we are interested in developing a unified account of deterministic and probabilistic causation, this would appear to give us an additional reason for preferring the original version of AC2(b) in the deterministic case.

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and T took their actual values, B ¼ 1 and T ¼ 0, then the probability of D ¼ 1 would be no higher than if B took the value B ¼ 0. This is contrary to the requirement of condition PC2(b).

For example, consider the obvious partition ~Z ¼ hB; T ; Di and ~W ¼ hC; Si,

and consider the assignment ~w0_{¼ hC ¼ 1; S ¼ 0i. Condition PC2(a) is satisfied}

for this partition and this assignment. In particular, it is true that

PðD ¼ 1jdoðB ¼ 1 & C ¼ 1 & S ¼ 0ÞÞ > PðD ¼ 1jdoðB ¼ 0 & C ¼ 1 & S ¼ 0ÞÞ: ð10Þ That is to say, in circumstances in which Corleone issues his order but Sonny doesn’t shoot, the probability of McCluskey’s dying would be higher if Barzini issued his order than if Barzini didn’t issue his order. Given the stipulations of our example, the former probability is approximately 0.81, while the latter is approximately 0.

But notice that PC2(b) is not satisfied for this partition and assignment of

values to ~W . For take ~Z0_{¼ fT g ~}_{Z, and observe that}

PðD ¼ 1jdoðB ¼ 1 & C ¼ 1 & S ¼ 0 & T ¼ 0ÞÞ

PðD ¼ 1jdoðB ¼ 0 & C ¼ 1 & S ¼ 0ÞÞ: ð11Þ

That is to say, in circumstances in which Corleone issued his order but Sonny didn’t shoot, if (as was actually the case) Barzini issued his order, but Turk didn’t shoot, the probability of McCluskey’s death would have been no higher than it would have been if Barzini hadn’t issued his order in the first place. Intuitively, this is because, in circumstances where Corleone issues his order but Sonny doesn’t shoot, Barzini’s order only raises the prob- ability of McCluskey’s death because it raises the probability of Turk’s shoot- ing. So (in circumstances in which Corleone issues his order but Sonny doesn’t shoot), the probability of McCluskey’s death if Barzini had issued his order but Turk had not shot would have been no higher than if (in the same cir- cumstances) Barzini simply hadn’t issued his order.

Nor is there any other partition, ð ~Z; ~W Þ, of the endogenous variables

fC; B; S; T ; Dg such that PC2 is satisfied. In particular, none of the remaining

variables on the Barzini process, {T, D}, can be assigned to ~W instead of ~Z if

PC2(a) is to be satisfied, for the values of each of these variables screens off B from D, so the result would be that PC2(a) wouldn’t hold for any assign-

ment, ~w0_{, of values to variables in ~}_{W . On the other hand, reassigning all or}

some of the variables on the initial Corleone process, {C, S}, to ~Z will not

affect the fact that PC2(b) fails to obtain. This is because no matter what

subset of {C, S} we take ~W to comprise, and no matter what values ~w0_{are}

assigned to that subset by interventions, the probabilistic relevance of B to D remains entirely by way of its relevance to T. So it will remain true that where

~

W ¼ ~w0_{, if B ¼ 1 and T ¼ 0, then the probability of D ¼ 1 would be no higher}

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than if B ¼ 0, in violation of PC2(b). (Again, it is important to remember that

the relevant worlds where ~W ¼ ~w0_{and B ¼ 1 and T ¼ 0 hold are those in which}

T has the value T ¼ 0 as the result of an intervention or similar, rather than T’s value being influenced in the usual way by the value of S.)

So PC gives the correct diagnosis of probabilistic preemption. It does so on intuitively the correct grounds. Specifically, the reason that Corleone’s order is counted as a cause is that (i) given Turk’s non-shooting, Corleone’s order raised the probability of McCluskey’s death; and (ii) there is a complete causal process running from Corleone’s order to McCluskey’s death. This is indicated by the fact that for arbitrary subsets of events on the Corleone process, it is true that (in circumstances in which Turk doesn’t shoot), if Corleone had issued his order and the variables representing those events had taken their actual values, then the probability of McCluskey’s death would have remained higher than if Corleone had never issued his order in the first place.

By contrast, Barzini’s order isn’t counted as a cause because, although (i) given Sonny’s non-shooting, Barzini’s order would raise the probability of McCluskey’s death; nevertheless, (ii) there is no complete causal process from Barzini’s order to McCluskey’s death as indicated by the fact that if Barzini had issued his order and Sonny hadn’t shot but (as was actually the case) Turk didn’t shoot, then the probability of McCluskey’s death would have been no higher than it would have been if (Sonny hadn’t shot and) Barzini hadn’t issued his order in the first place.

It was noted above that Halpern and Pearl ([2005], p. 859) suggest that their

definition AC might reasonably be adjusted in light of the contrastive nature of many causal claims. Indeed, as noted above, several philosophers have argued rather convincingly that actual causation is contrastive in nature

(for example,Hitchcock [1996a],[1996b];Schaffer [2005],[2013]), and specif-

ically that causation is a quaternary relation, with the cause, the effect, a set of alternatives to the cause, and a set of alternatives to the effect as its relata. In the present context, this would mean that the primary analysandum is not

‘ ~X ¼ ~x is an actual cause of ’’, but rather ‘ ~X ¼ ~x rather than ~X ¼ ~x0 _{is an}

actual cause of ’ rather than u0_{’, where ~}_{X ¼ ~}_{x}0_{denotes a set of formulas of the}

form ~X ¼ ~x0_{, such that for each such formula ~}_{x 6¼ ~}_{x}0_{, and where u}0_{represents}

a set of formulas of the form ’0_{, such that for each such formula, ’ is incom-}

patible with ’0_{.}

The case for turning PC into an analysis of a four-place relation is just as compelling as the case for the corresponding modification of AC. As it stands, where the cause and/or effect variables are multi-valued, PC (just like the unmodified AC) is liable to run into difficulties. Consider a case where Doctor can administer no dose, one dose, or two doses of medicine to Patient. Let M be a variable that takes value M ¼ 0 if no dose is administered, M ¼ 1 if one dose is administered, and M ¼ 2 if two doses are administered.

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Suppose that Patient will recover with chance 0.1 if no dose is administered, with chance 0.9 if one dose is administered, and with chance 0.5 if two doses are administered (two doses is an ‘overdose’, which would adversely affect Patient’s natural immune response). Let R be a variable that takes value R ¼ 1 if Patient recovers and R ¼ 0 if she does not. Suppose that the context is such that Doctor is equally disposed to each of the three courses of action. We can represent the (exogenous) intentions of Doctor that give rise to this disposition using a (exogenous) variable, D, that takes value D ¼ 1 if Doctor has these intentions and D ¼ 0 if she does not. Suppose that Doctor in fact administers two doses of medicine, and Patient recovers.

Did Doctor’s administering two doses of medicine cause Patient to recover? I think the natural reaction is one of ambivalence. After all, while it is true that Patient’s recovery was more likely given that Doctor administered two doses than it would have been if she had administered zero doses, it was less likely than if Doctor had administered one dose. If we focus on the fact that Doctor could have administered just one dose, we might be inclined to say that Patient recovered despite Doctor’s action. If we focus on the fact that Doctor could have administered zero doses, we might be inclined to say that Patient recovered because of Doctor’s action. One plausible interpretation of our ambivalent at- titude is that actual causation is contrastive in nature, and ‘Doctor’s adminis- tering two doses of Medicine caused Patient to recover’ is ambiguous between ‘Doctor’s administering two doses of Medicine rather than no doses caused Patient to recover’ (to which most people would presumably assent) and ‘Doctor’s administering two doses of Medicine rather than one dose caused Patient to recover’ (to which most people would presumably not assent).

Yet, as it stands, PC delivers the unequivocal result that Doctor’s action (M ¼ 2) was an actual cause of Patient’s recovery (R ¼ 1), where the variable

set for our model is {D, M, R}. To see this, let ~X ¼ fMg, let ~x ¼ fM ¼ 2g, and

let ’ be R ¼ 1. Consider the partition ð ~Z; ~W Þ of the endogenous variables such

that ~Z ¼ hM; Ri and ~W ¼ ;. Condition PC1 is satisfied because M ¼ 2 and

R ¼ 1 are the actual values of M and R (or rather the values that obtain in the world in which our causal scenario plays out). If condition PC2 is satisfied, then condition PC3 is satisfied because if PC2(a) is satisfied, then this implies that (in the relevant circumstances) the value of M makes a probabilistic dif- ference to that of R, and there are no (non-empty) subsets of {M}. Condition PC2(a) is satisfied because it requires only that there be one alternative value

of M such that if M took that value (and the variables in ~W took some

possible assignment ~W ¼ ~w—something that trivially holds because there

are no variables in ~W in this case),50then the probability of R ¼ 1 would be

50

In what follows, I shall leave this parenthetical qualification implicit in all cases where ~W is empty.

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lower than if M had taken M ¼ 2. In this case, M ¼ 0 is such a value. So PC2(a) is satisfied. Condition PC2(b) is rather trivially satisfied: since there are no

variables in ~Z nM; R, PC2(b) just reduces to the requirement that if M had

taken the value M ¼ 2, then the probability of R ¼ 1 would have been higher than it would have been if M had taken the value M ¼ 0, which clearly holds in the example given. So PC2 is satisfied. We have already seen that PC1 is satisfied, and that PC3 is satisfied if PC2 is satisfied. Consequently, as it stands, PC implies that Doctor’s action (M ¼ 2) was an actual cause of Patient’s recovery (R ¼ 1).

The unequivocal nature of PC’s verdict contrasts with the verdict of intuition, which is equivocal. Thus, as was the case with AC, it would seem desirable to modify PC so that it can capture the nuances of our contrastive causal judge-

ments. This is easily achieved. To turn PC into an analysis of ~X ¼ ~x rather than

~

X ¼ ~x0_{being an actual cause of ’, we simply need to require that PC2 hold not}

just for some non-actual setting of ~X , but for precisely the setting ~X ¼ ~x0_{.}

This revised version of PC yields the intuitively correct verdict that M ¼ 2 rather than M ¼ 0 was an actual cause of R ¼ 1. Specifically, taking the rele- vant contrast to M ¼ 2 to be M ¼ 0, the revised version of PC is satisfied for

precisely the same reason that taking ~X ¼ ~x0_{to be M ¼ 0 showed the original}

version of PC to be satisfied. The revised version of PC also yields the verdict that M ¼ 2 rather than M ¼ 1 is not a cause of R ¼ 1. This is because the revised version of PC2(a) is violated when we take M ¼ 1 to be the contrast to M ¼ 2. This is because it’s not the case that if M had taken the value M ¼ 2, then the probability that R would have taken R ¼ 1 would have been higher than it would have been if M had taken the value M ¼ 1 (in fact it would have been lower in the example given). So the revised PC yields the desired verdicts