SlowGuess/NeurIPS_2023_OCR huggingface dataset:
AC1 is fulfilled because  $A_{1} = 1$  and  $V = 1$  actually happened. AC2 is established by verifying that the following two claims hold:  $(M, \vec{u}) \models [A_{1} \leftarrow 1, A_{2} \leftarrow 1]V = 1$  and  $(M, \vec{u}) \models [A_{1} \leftarrow 1, A_{2} \leftarrow 0]V = 1$ . Since  $\vec{W} = \varnothing$ , verifying AC3 is easy: we need to find a single intervention on the variables other than  $V$  such that they result in  $V = 0$ . The intervention  $[A_{1} \leftarrow 0, A_{2} \leftarrow 0]$  does the job.
To evaluate the Epistemic Condition requires making some assumptions about the assassins's probability attributions. It sounds reasonable to assume that, without evidence to the contrary, each assassin attributed a higher probability to them shooting causing the outcome than them not shooting causing the outcome. Therefore the Epistemic Condition is also fulfilled for each assassin, and thus the BvH definition arrives at the right verdict for this example.
We continue with the approach pursued by Halpern Kleiman-Weiner (HK) [17], which uses the modified Halpern & Pearl definition of causation [16]:
Definition 6 (HP).  $\vec{X} = \vec{x}$  HP-causes  $Y = y$  w.r.t.  $(M,\vec{u})$  if there exists a  $\vec{W} = \vec{w}$  so that the following conditions hold:
AC1.  $(M,\vec{u})\vDash \vec{X} = \vec{x}\wedge \vec{W} = \vec{w}\wedge Y = y.$
AC2. There is a setting  $\vec{x}^{\prime}$  such that  $(M,\vec{u})\models [\vec{X}\leftarrow \vec{x}^{\prime},\vec{W}\leftarrow \vec{w}]Y\neq y$
AC3.  $\vec{X}$  is minimal; there is no strict subset  $\vec{X}'$  of  $\vec{X}$  such that  $\vec{X}' = \vec{x}''$  satisfies AC2, where  $\vec{x}''$  is the restriction of  $\vec{x}$  to the variables in  $\vec{X}'$ .
Note that, contrary to the direct NESS definition, the HP definition allows for conjunctive causes  $\vec{X} = \vec{x}$ , instead of merely atomic causes  $X = x$ . The minimality condition (AC3) is there to prevent irrelevant events to be added to such conjuncts. We can retrieve a definition of causation for atomic events by simply considering any conjunct  $X = x$  that appears in an HP-cause  $\vec{X} = \vec{x}$  to be a cause as well, which is indeed what Halpern suggests himself repeatedly [16].
The heart of the HP definition is AC2: it states that the outcome  $Y = y$  counterfactually depends on the cause  $\vec{X} = \vec{x}$  given that we intervene to hold fixed a suitably chosen set of variables  $\vec{W}$  at their actual values  $\vec{w}$ . To see how this definition works, let us apply it to Example 1.
First we try substituting  $\vec{X} = \vec{x}$  with  $A_{1} = 1$ . Alas, this will not allow us to get  $A_{1} = 1$  as a cause of  $V = 1$ . We start with choosing  $\vec{W} = \emptyset$ , and we get that  $(M, \vec{u}) \models [A_{1} \leftarrow 0]V = 1$ . The reason is that  $\vec{u}$  encodes the actual context, in which  $A_{2} = 1$ , and thus also  $V = 1$ . Yet what is required for AC2 would be  $(M, \vec{u}) \models [A_{1} \leftarrow 0]V = 0$ . The only other choice for  $\vec{W} = \vec{w}$  would be  $A_{2} = 1$ , and that does not work either:  $(M, \vec{u}) \models [A_{1} \leftarrow 0, A_{2} \leftarrow 1]V = 1$ .
Second we try  $A_{1} = 1 \wedge A_{2} = 1$ . If this works, then AC3 is satisfied due to the fact that neither of the conjuncts themselves satisfies AC2.  $\vec{W}$  has to be  $\varnothing$ , since there are no other variables. Thus what remains is to find counterfactual values for  $A_{1}$  and  $A_{2}$ . As they are binary, the only option is to consider  $A_{1} = 0 \wedge A_{2} = 0$ . Clearly, for this choice AC2 is satisfied, as  $(M, \bar{u}) \models [A_{1} \leftarrow 0, A_{2} \leftarrow 0] V = 0$ . Therefore  $A_{1} = 1$  is an HP-cause of  $V = 1$ .
We can now formulate a definition of responsibility that is closely inspired by HK.
Definition 7 (HK Responsibility). An agent who performs  $A = a$  is responsible for outcome  $O = o$  w.r.t. a responsibility setting  $(M, \bar{u}, \mathcal{E})$  if:
- (Causal Condition)  $A = a$  HP-causes  $O = o$  w.r.t.  $(M, \vec{u})$ .
- (Epistemic Condition) There exists  $a' \in \mathcal{R}(A)$  so that  $\operatorname*{Pr}(O = o|[A \leftarrow a]) > \operatorname*{Pr}(O = o|[A \leftarrow a'])$ .
In addition to disagreeing about the definition of causation, the HK definition also disagrees with the BvH definition about the epistemic condition: rather than requiring that the agent failed to minimize the probability of causing the outcome, the HK definition focuses on the agent failing to minimize the probability of the outcome simpliciter.
Note that both HK and BvH's epistemic condition satisfy our Responsibility Schema: an agent who believes that they failed to minimize a probability that they could have minimized, thereby also believes that they could have avoided satisfying the respective epistemic condition. Given that the epistemic condition is a necessary condition for being responsible, they also believe that they could have avoided being responsible for the actual outcome.
Let us apply the HK definition to Example 1. We already established that each  $A_{i} = 1$  is an HP-cause of  $V = 1$ , so the Causal Condition is met. Further, as long as each assassin attributes a strictly positive probability that the other assassin may fail to shoot, we get that  $\operatorname*{Pr}(V = 1|[A_i\leftarrow 1]) > \operatorname*{Pr}(V = 1|[A_i\leftarrow 0])$ , so that the Epistemic Condition is satisfied as well. (What if the assassins are certain the other assassin will shoot? We come back to this in Section 5.) Therefore the HK definition also arrives at the correct verdict for this example.
4 The Causal Condition
Before discussing the problems with NESS- and HP-causation, I present CNESS-causation [2]. As a first step, we define NESS-causation as the transitive closure of direct NESS-causation, which is how it was conceived by Wright [33]. In addition, we pay explicit attention to the path along which the causal influence is transmitted.
Definition 8 (NESS).  $X = x$  NESS-causes  $Y = y$  along a path  $p$  w.r.t.  $(M, \vec{u})$  if the values of the variables in  $p$  form a path of direct-NESS causes from  $X = x$  to  $Y = y$ .
The Counterfactual NESS definition (CNESS) takes the NESS definition and adds a subtle counterfactual difference-making condition: there should be a counterfactual value so that it would not NESS-cause the outcome along the same path as the actual value, nor along any subpath.
Definition 9 (CNESS).  $X = x$  CNESS-causes  $Y = y$  w.r.t.  $(M, \vec{u})$  if  $X = x$  NESS-causes  $Y = y$  along some path  $p$  w.r.t.  $(M, \vec{u})$  and there exists a  $x' \in \mathcal{R}(X)$  such that  $X = x'$  does not NESS-cause  $Y = y$  along any subpath  $p' \subseteq p$  w.r.t.  $(M_{X \leftarrow x'}, \vec{u})$ .
With all the definitions of causation at hand, I now motivate my choice for the CNESS definition by going over some well-chosen examples. We start with a case of Late Preemption.
Example 2 (Late Preemption). We return to our two assassins, but this time  $\text{Assassin}_1$  is slightly faster, so that their bullet kills Victim, who collapses and thereby dodges  $\text{Assassin}_2$ 's bullet.
In this case  $\text{Assassin}_2$  obviously did not cause Victim's death, and is thus not responsible for the outcome (despite the fact that their act itself is of course still blameworthy). BvH only allows variables for strategies and are thus unable to capture this result, since the asymmetry between both assassins is not a matter of strategy. As illustrated at length by Halpern [16], using causal models this poses no problem. The equation  $V = BH_1 \lor BH_2$  expresses the fact that either bullet hitting Victim would be fatal;  $BH_1 = A_1$  and  $BH_2 = A_2 \land \neg BH_1$  captures the asymmetry between both assassins:  $\text{Assassin}_2$ 's bullet only hits Victim if  $\text{Assassin}_1$ 's bullet does not. In the context at hand, we have that  $A_1 = A_2 = BH_1 = V = 1$ , and  $BH_2 = 0$ . We now go through the various definitions to verify that  $A_1 = 1$  NESS-causes, CNESS-causes, and HP-causes  $V = 1$ , whereas  $A_1 = 1$  does not directly NESS-cause  $V = 1$ , thereby showing that the direct NESS definition is too simplistic.
We start by verifying that  $A_{1} = 1$  does not directly NESS-cause  $V = 1$ . By itself  $A_{1} = 1$  does not form a sufficient set for  $V = 1$ , for setting both of the  $BH$  variables to 0 guarantees that the Victim survives:  $(M, \vec{u}) \models [A_{1} \leftarrow 1, BH_{1} \leftarrow 0, BH_{2} \leftarrow 0]V = 0$ . In fact, in this context, any sufficient set for  $V = 1$  has to contain  $BH_{1} = 1$ , yet  $BH_{1}$  is sufficient for  $V = 1$  all by itself. Thus  $A_{1} = 1$  is not a necessary member of any sufficient set for  $V = 1$ . Still,  $A_{1} = 1$  NESS-causes  $V = 1$  along  $p = \{A_{1}, BH_{1}, V\}$ , because  $A_{1} = 1$  directly NESS-causes  $BH_{1} = 1$  and  $BH_{1} = 1$  directly NESS-causes  $V = 1$ .
To establish CNESS-causation requires having a look at the counterfactual setting  $(M_{A_1\leftarrow 0},\vec{u})$ . In this setting we get that  $A_{1} = 0$ ,  $A_{2} = 1$ ,  $BH_{1} = 0$ , and thus  $BH_{2} = 1$  (as well as  $V = 1$ ). (Informally: if  $\text{Assassin}_1$  had not shot, then  $\text{Assassin}_2$ 's bullet would have hit and killed Victim.) Here  $A_{1} = 0$  directly NESS-causes  $BH_{1} = 0$ ,  $BH_{1} = 0$  directly NESS-causes  $BH_{2} = 1$  (since it forms a sufficient set together with  $A_{2} = 1$  and  $A_{2} = 1$  does not suffice on its own), and  $BH_{2} = 1$  directly NESS-causes  $V = 1$ . Therefore  $A_{1} = 0$  NESS-causes  $V_{1} = 1$  along  $p^{*} = \{A_{1}, BH_{1}, BH_{2}, V\}$ . (Take note of this surprising finding. We come back to it in Example 3.) Since  $p^{*} \notin p$ , we get that  $A_{1} = 1$  CNESS-causes  $V = 1$  (whereas  $A_{1} = 0$  does not CNESS-cause  $V = 1$  in the counterfactual setting, since  $p \subseteq p^{*}$ ).
To see that  $A_{1} = 1$  HP-causes  $V = 1$ , it suffices to note that  $(M, \vec{u}) \models BH_{2} = 0$  and  $(M, \vec{u}) \models [A_{1} \leftarrow 0, BH_{2} \leftarrow 0]V = 0$ . Lastly, I leave it to the reader to verify that  $A_{2} = 1$  is not an HP-cause of  $V = 1$ , and nor is it a direct NESS-cause of anything. Because of the latter,  $A_{2} = 1$  is not a NESS-cause or a CNESS-cause of anything either.
Modifying the BvH definition so that it uses NESS-causation instead of direct NESS-causation is not a solution, for the NESS definition itself is problematic, as the following example shows. (In the appendix I discuss one more example, a so-called "Frankfurt-case", to show that BvH's reliance on strategies as opposed to events forms another source of problems.)
Example 3. We revisit the counterfactual setting of Example 2 in which  $\text{Assassin}_1$  does not shoot, so that Victim is killed by  $\text{Assassin}_2$ 's shot.
We already established for this scenario that  $A_{1} = 0$  NESS-causes  $V = 1$ . Thus if we use the NESS definition, we get the absurd result that  $\text{Assassin}_1$  failing to shoot causes Victim to die. If we then supplement the example so that also BvH's Epistemic Condition is fulfilled, we get that  $\text{Assassin}_1$  comes out as being responsible for Victim's death. (Imagine, for instance, that they mistakenly believe to be holding a flare gun that could sound a warning shot so that Victim ducks for cover to avoid  $\text{Assassin}_2$ 's bullet.) We already established that  $A_{1} = 0$  does not CNESS-cause  $V = 1$ , the reader may verify that the same holds for the HP-definition.
This leaves CNESS-causation and HP-causation as candidates for the Causal Condition. I use Halpern & Pearl's own example to argue against HP-causation [15].
Example 4 (Loader). "Suppose that a prisoner dies either if  $A$  loads  $B$ 's gun and  $B$  shoots, or if  $C$  loads and shoots his gun. Taking  $D$  to represent the prisoner's death and making the obvious assumptions about the meaning of the variables, we have that  $D = 1$  iff  $(A = 1 \land B = 1) \lor C = 1$ . Suppose that in the actual context  $\vec{u}$ ,  $A$  loads  $B$ 's gun,  $B$  does not shoot, but  $C$  does load and shoot his gun, so that the prisoner dies. Clearly  $C = 1$  is a cause of  $D = 1$ . We would not want to say that  $A = 1$  is a cause of  $D = 1$ , given that  $B$  did not shoot (i.e., given that  $B = 0$ ). [emphasis added]
I agree with Halpern and Pearl. A fortiori,  $A$  is not responsible for the prisoner's death, even if  $A$  only loaded the gun because he was convinced that  $B$  would shoot. Now consider the following variant. In the original example,  $C$ 's shot is determined directly by the context. Imagine we add a little twist, so that  $C$  would only fire his gun if  $B$  did not, i.e., the equation for  $C$  is  $C = \neg B$ . The above reasoning regarding  $A$  still applies, and therefore I believe it is a mistake to all of a sudden consider  $A = 1$  a cause of  $D = 1$ . Yet  $A = 1$  now is an HP-cause of  $D = 1$  (as it appears in the HP-cause  $A = 1 \wedge B = 0$ ), and thus  $A$  would be considered responsible for the prisoner's death. The CNESS definition avoids this result (as does the NESS definition): the only candidate sufficient set for  $D = 1$  of which  $A = 1$  could be a necessary part, is  $\{A = 1, B = 1\}$ . So the mere fact that  $B = 0$  in both versions of the example implies that  $A = 1$  is not a NESS cause of  $D = 1$  in either.
I leave a second counterexample to the HP definition for the appendix and refer the reader to [1] for a detailed critical examination of the HP definition. The alternative definition I there presented is in fact very similar to my CNESS definition, although the precise relation is the subject of further investigation.<sup>5</sup> This leads me to suggest adopting the CNESS definition for the Causal Condition.
5 The Epistemic Condition
Recall that the difference between HK and BvH's Epistemic Conditions lies in whether an action minimizes the probability of the outcome occurring (HK) or of it causing the outcome (BvH). Given that one cannot cause an outcome unless the outcome actually occurs, and that vice versa, in many cases the best way to make sure that an outcome occurs is by causing it, both of these conditions often go hand in hand. However, as the following example illustrates, they do not always do so, and when they do not the appeal of HK's condition is stronger.
Example 5. Bombing A bomb  $(B)$  is connected to three detonators  $(D_{1}, D_{2},$  and  $D_{3})$  by two switches  $(S_{1}$  and  $S_{2})$ .  $D_{1}$  is functional if only  $S_{1}$  is on,  $D_{2}$  is functional if only  $S_{2}$  is on, and  $D_{3}$  is functional whenever  $S_{1}$  is on. The equations are thus as follows:  $B = D_{1} \vee D_{2} \vee D_{3}$ ,  $D_{1} = S_{1} \wedge \neg S_{2}$ ,  $D_{2} = S_{2} \wedge \neg S_{1}$ , and  $D_{3} = S_{1}$ . Assassin2 (reasonably) assigns a probability of 0.6 to Assassin1
I tentatively conjecture that the CNESS definition implies my other definition, and not vice versa.
turning on  $S_{1}$ . He decides to turn on  $S_{2}$ , thereby guaranteeing that the bomb will explode. Assassin<sub>1</sub> decides not to turn on  $S_{1}$ , so that the bomb explodes due to the functioning of  $D_{2}$ .
Here we certainly would want to say that  $\text{Assassin}_2$  is responsible for the explosion, and the reason for this seems to be precisely that he knowingly increased the probability of the bomb going off (from 0.6 if  $S_2 = 0$  to 1 now that  $S_2 = 1$ ). There is also no doubt that  $\text{Assassin}_2$ 's action caused the explosion: if he had turned  $S_2$  off, the bomb would not have exploded.
However,  $\text{Assassin}_2$  did act so as to minimize the probability that his act would cause the explosion, regardless of whether one chooses NESS-, HP-, or CNESS-causation. Concretely, for all three definitions of causation,  $\text{Assassin}_2$ 's probability that  $S_2 = 1$  would cause  $B = 1$  is 0.4, whereas his probability that  $S_2 = 0$  would cause  $B = 1$  is 0.6. (The details are worked out in the appendix.)
Note that in case  $S_{1} = 1$ , then  $S_{2} = 0$  would result in the outcome being overdetermined, and thus although the latter action would also be a cause of the outcome, it does nothing to contribute to the probability of the outcome occurring. This is what explains why the two conditions can come apart, and why I take the general moral of this story to be that increasing the probability of the outcome trumps increasing the probability of causing the outcome.
However, it does not follow that the probability of causation is irrelevant, but only that it should fulfill a secondary role. Consider again Example 2, and assume that  $\text{Assassin}_1$  believes that  $\text{Assassin}_2$  will shoot, and thus believes that Victim is facing certain death. (If that sounds too unrealistic, imagine  $\text{Assassin}_1$  is one of ten members of a highly trained firing squad that is executing Victim.) Thus the action of  $\text{Assassin}_1$  had no effect on the probability of the outcome, and would thus not be responsible for Victim's death according to HK's definition. If  $\text{Assassin}_2$  has a similar belief, then we end up with nobody being responsible. I take this to be an unacceptable result. (Fischer & Ravizza reach the same conclusion when likewise discussing a case (Missile 2) in which an agent knows that the outcome will ensue no matter what they do, and yet the agent is still responsible for the outcome by choosing to cause it [12, p. 102].)
The lesson I draw from this is that if one knowingly has the opportunity to reduce the probability of causation without thereby increasing the probability of the outcome, then an agent is responsible if she fails to do so. Therefore I propose the following definition of moral responsibility.
Definition 10 (Responsibility). An agent who performs  $A = a$  is responsible for  $O = o$  w.r.t. a responsibility setting  $(M, \vec{u}, \mathcal{E})$  if:
(Causal Condition)  $A = a$  CNESS-causes  $O = o$  w.r.t.  $(M,\vec{u})$
- (Epistemic Condition) There exists  $a' \in \mathcal{R}(A)$  so that one of the following holds:
1.  $\operatorname{Pr}(O = o|[A \leftarrow a]) > \operatorname{Pr}(O = o|[A \leftarrow a'])$
2.  $\operatorname{Pr}(O = o|[A \leftarrow a]) = \operatorname{Pr}(O = o|[A \leftarrow a'])$  and
$\operatorname*{Pr}(A = a$  CNESS-causes  $O = o) > \operatorname*{Pr}(A = a^{\prime}$  CNESS-causes  $O = o)$
6 Degree of Responsibility
My binary definition of responsibility can be complemented with a definition of the degree of responsibility in order to capture the widely shared sense that responsibility (as well as blame and praise) is a graded notion. Both BvH's and HK's Epistemic Conditions naturally suggest such a definition, and so does my combined condition.
The obvious graded counterpart of HK's condition is to simply look at the causal effect [22], which in the context of causal strength is referred to as the Eells measure of causal strength of  $A = a$  relative to  $A = a'$ :  $CS_e(o, a, a') = \operatorname{Pr}(O = o|[A \leftarrow a]) - \operatorname{Pr}(O = o|[A \leftarrow a'])$  [13, 28]. Sprenger [28] argues for accepting the Eells measure as a general measure of causal strength, which is in line with the priority that my Epistemic Condition attributes to it. Moreover, when restricted to positive values, this is in fact HK's definition of the degree of blameworthiness. Likewise, the obvious counterpart of BvH's condition is to look at the increase of probability in causing the outcome. Thus I also define the actual causation measure of causal strength as<sup>6</sup>
$$
C S _ {a c} \left(o, a, a ^ {\prime}\right) = \Pr \left(A = a \text {C N E S S - c a u s e s} O = o | [ A \leftarrow a ]\right) - \Pr \left(A = a ^ {\prime} \text {C N E S S - c a u s e s} O = o | [ A \leftarrow a ^ {\prime} ]\right).
$$
<sup>6</sup>Surprisingly, to my knowledge this rather obvious measure of causal strength has been overlooked so far in the literature. (For any definition of causation of course, not just CNESS.)
Taking into account that our Epistemic Condition is a mixture of those of BvH and HK, I suggest the following definition, where the value of  $\alpha$  expresses the relative importance of both measures.
Definition 11 (Degree of Responsibility). The degree of responsibility  $d$  for  $O = o$  of an agent who performs  $A = a$  w.r.t. a responsibility setting  $(M, \vec{u}, \mathcal{E})$  is 0 in case the agent is not responsible, otherwise let  $S = \operatorname{argmin}_{a^* \in \mathcal{R}(A)} \operatorname{Pr}(O = o|[A \leftarrow a^*])$ , and let  $a'' = \operatorname{argmin}_{a' \in S} \operatorname{Pr}(A = a' CNESS$ -causes  $O = o|[A \leftarrow a']$ ), then  $d = CS_{e}(o, a, a'') + \alpha \cdot \max(0, CS_{ac}(o, a, a''))$ .
Informally, this measure works as follows. Among all actions that minimize the probability of the outcome, we take one that minimizes the probability of causing the outcome, and then take a weighted sum of both causal strength measures for that action (where the second measure is ignored if it is negative). This captures the idea that in order to avoid responsibility, the agent should choose an action that makes the outcome as unlikely as possible, and then further select their action so that it makes causing the outcome as unlikely as possible. The following example illustrates this definition.
Example 6. Imagine again our scenario from Example 1, but with the following change:  $\text{Assassin}_1$  is known to be a reliable assassin, whereas  $\text{Assassin}_2$  is known to have second doubts and almost never shoots. In other words, it is reasonable for  $\text{Assassin}_2$  to expect that  $\text{Assassin}_1$  will shoot, and it is reasonable for  $\text{Assassin}_1$  to expect that  $\text{Assassin}_2$  will not shoot. On this particular occasion, both assassins shoot and kill victim.
Although both assassins are responsible according to my definition, it is easy to see that  $\text{Assassin}_1$  is responsible to a higher degree: the measures of actual causation are identical for both and so are their respective probabilities of the outcome occurring given that they shoot (namely 1), but  $\text{Assassin}_1$ 's probability of the outcome occurring given that they do not shoot is far lower, and thus<sup>7</sup>
$$
C S _ {e} ^ {A s s _ {1}} \big (V = 1, A _ {1} = 1, A _ {1} = 0 \big) > C S _ {e} ^ {A s s _ {2}} \big (V = 1, A _ {2} = 1, A _ {2} = 0 \big).
$$
Interestingly, recent studies offer empirical confirmation that the agent's epistemic state does indeed impact people's judgments in precisely this way: in a disjunctive scenario (like ours), an agent who performs an action that is typical (for them) is considered to be more responsible than an agent who acts atypically [18]. The authors contrast this disjunctive scenario, which they have trouble explaining, with a conjunctive one in which both agents' actions are necessary for the outcome to occur, which their account explains quite well. In a conjunctive scenario (in other words, if the equation were  $V = A_{1} \wedge A_{2}$ ), an agent who performs an action that is atypical is considered to be more responsible than an agent who acts typically, flipping the judgments compared to the disjunctive scenario. That is also the verdict of my degree of blameworthiness: in this case, the atypical agent can reasonably expect the outcome to depend on them performing the action whereas the typical agent can reasonably expect that their action has little impact, which translates into a larger measure of causal strength (both  $CS_{e}$  and  $CS_{ac}$ ) for the former. So in contrast to the account of Kirfel and Lagnado [18], my proposal applies equally to both scenarios and can thus be seen as a formal extension of their work.