world. A world with a rank of 0 is reasonably normal, one with a rank of 1 is somewhat normal, one with a rank of 2 is quite abnormal, and so on. Given a ranking on worlds, the statement "if p then typically q**" is true if in all the worlds**
of least rank where p is true, q **is also true. Thus, in one model where people do not**
typically put either poison or antidote in coffee, the worlds where neither poison nor antidote is put in the coffee have rank 0, worlds where either poison or antidote is put in the coffee have rank 1, and worlds where both poison and antidote are put in the coffee have rank 2.

Take an *extended causal model* to be a tuple M = (S, F, κ), where (S, F**) is a**
causal model, and κ **is a** *ranking function* **that associates with each world a rank.**
In an acyclic extended causal model, a context ~u **determines a world, denoted** s~u.

X~ = ~x **is a** cause of φ in an extended model M *and context* ~u if X~ = ~x **is a cause of**
φ according to Definition 1, except that in AC2(a), there must be a world s **such**
that κ(s) ≤ κ(s~u) and X~ = ~x′ ∧ W~ = ~w is true at s**. This can be viewed as a**
formalization of Kahneman and Miller's [1986] observation that "an event is more likely to be undone by altering exceptional than routine aspects of the causal chain that led to it".

This definition deals well with all the problematic examples in the literature.

Consider Example 6. Using the ranking described above, Bodyguard **is not a cause** of Victim's survival because the world that would need to be considered in AC2(a),
where Assassin poisons the coffee, is less normal than the actual world, where he does not. We consider just one other example here (see [Halpern 2008] for further discussion). EXAMPLE 7. Consider the following story, taken from (an early version of) [Hall 2004]: Suppose that Billy is hospitalized with a mild illness on Monday; he is treated and recovers. In the obvious causal model, the doctor's **treatment is a cause**
of Billy's recovery. Moreover, if the doctor does not **treat Billy on Monday, then** the doctor's omission to treat Billy is a cause of Billy's being sick on Tuesday. But now suppose that there are 100 doctors in the hospital. Although only doctor 1 is assigned to Billy (and he forgot to give medication), in principle, any of **the other** 99 doctors could have given Billy his medication. Is the nontreatment **by doctors** 2–100 also a cause of Billy's being sick on Tuesday?

Suppose that in fact the hospital has 100 doctors and there are variables A1, . . . , A100 and T1, . . . , T100 in the causal model, where Ai **= 1 if doctor** i is assigned to treat Billy and Ai = 0 if he is not, and Ti = 1 if doctor i **actually treats**
Billy on Monday, and Ti **= 0 if he does not. Doctor 1 is assigned to treat Billy;**
the others are not. However, in fact, no doctor treats Billy. Further assume that, typically, no doctor is assigned to a given patient; if doctor i **is not assigned to**
treat Billy, then typically doctor i does not treat Billy; and if doctor i is **assigned**
to Billy, then typically doctor i **treats Billy. We can capture this in an extended**
causal model where the world where no doctor is assigned to Billy and no doctor