Combining the Causal Judgments of Experts with Possibly Different Focus Areas.

Combining the Causal Judgments of Experts with Possibly Different Focus Areas
Meir Friedenberg
Department of Computer Science
Cornell University
meir@cs.cornell.eduJoseph Y. Halpern
Department of Computer Science
Cornell University
halpern@cs.cornell.edu
Abstract
In many real-world settings, a decision-maker must combine
information provided by different experts in order to decide
on an effective policy. Alrajeh, Chockler, and Halpern (2018)
showed how to combine causal models that are compatible in
the sense that, for variables that appear in both models, the
experts agree on the causal structure. In this work we show
how causal models can be combined in cases where the ex-
perts might disagree on the causal structure for variables that
appear in both models due to having different focus areas.
We provide a new formal deﬁnition of compatibility of mod-
els in this setting and show how compatible models can be
combined. We also consider the complexity of determining
whether models are compatible. We believe that the notions
deﬁned in this work are of direct relevance to many practical
decision making scenarios that come up in natural, social, and
medical science settings.
1 Introduction
In many real-world settings, a decision-maker must com-
bine information provided by different experts in order to
decide on an effective policy. For example, when deciding
policing and criminal justice policy, it may be necessary to
consult different experts specializing in areas such as crimi-
nology, psychology, sociology, and economics. Intelligently
combining the information provided by the various experts
is necessary if the decision-maker hopes to select the best
course of action.
Much work has been done on combining simple proba-
bilistic judgments of different experts. However, we are in-
terested in settings where a decision-maker wants to choose
an action in order to induce a particular outcome, so we are
interested in the setting where experts provide models of the
causal relationships between different factors. Despite the
clear importance of combining causal models in real-world
situations, there has been very little work on how to combine
models with this extra structure.
Much work has been done on the related problem of
learning causal models: given data and possibly some prior
structured knowledge, extract the causal model that best
ﬁts the given information (see, e.g., (Claassen and Hes-
kes 2010; 2012; Hyttinen, Eberhardt, and Jarvisalo 2014;
Copyright c
2018, Association for the Advancement of Artiﬁcial
Intelligence (www.aaai.org). All rights reserved.Tillman and Spirtes 2011; Triantaﬁllou and Tsamardinos
2015); Triantaﬁllou and Tsamardinos (2015) provide a good
overview of work in the area). In a certain sense, if all of the
experts we consult have learned their models in this manner
and can provide us with all of their data, then the best thing
to do is simply to learn a new causal model from the union
of all the data. However, in many real world settings, this is
completely impractical. Experts develop intuitions based on
years worth of data, training, and discussions; providing all
of this background information to the decision maker may
be infeasible.
On the topic of combining causal models without data,
Bradley, Dietrich, and List (2014) proved an impossibility
result. Given a set of desiderata for combining causal mod-
els, they show that no algorithm satisﬁes them all. They fur-
ther examine ways of circumventing the impossibility result
by weakening some of those conditions.
It is perhaps not too surprising in retrospect that it will
sometimes be impossible to combine causal models, as two
models can explicitly disagree on every causal relationship.
In the work most related to this, Alrajeh, Chockler, and
Halpern (2018) (ACH from now on) provide conditions for
thecompatibility of models and show how to combine mod-
els that meet their compatibility conditions. They deﬁne a
dominance relation according to which a model M1domi-
nates model M2with respect to a variable Cif the two mod-
els agree on the causal dependence of Con the other vari-
ables shared by the two models, but model M1has perhaps
a more detailed picture of the exact way that the effects are
mediated. Two models are compatible if, for every variable,
one of the models dominates the other; the combined model
takes the causal information from the dominant model for
each variable C. ACH also provide a way of assigning prob-
abilities to causal models in settings where not all models
under consideration are compatible.
The present work can be seen as providing an approach
complementary to that of ACH. Philosophically, the ap-
proach presented by ACH is intended to allow for combina-
tion of models where the modelers fundamentally agree on
the causal relationship between the variables they both dis-
cuss, but go into different levels of detail as to how some of
those relationships are mediated. But consider, for instance,
the following scenario: a medical scientist is interested in
the conditions under which a particular reaction occurs, and
consults with two experts. The ﬁrst specializes in the exact
mechanism by which this reaction occurs; the second spe-
cializes in how one of the reactants gets produced. Because
of their different focus areas, they in fact do not agree on
everything; each of them is more aware of the details of the
reaction that she studies, and thus has more understanding
of what factors can cause that reaction to occur differently.
Our intuition tells us that there should be a way of combin-
ing these models to get the true expertise of both modelers,
but with the ACH approach, these models would in fact have
to be deemed incompatible.
In this work, we allow for combining models where the
modelers disagree due to their different focus areas. Intu-
itively, if the ﬁrst modeler considered more possibilities than
the second, and her conclusion can explain the observations
of the second, then we accept the conclusion of the ﬁrst mod-
eler. To this end, we deﬁne a new notion of a “can explain”
relation and provide new formalizations of compatibility and
model combination relative to this notion.
The rest of this paper is organized as follows. In Section 2,
we review the basic framework of causal models, and extend
them so as to accommodate focus areas. In Section 3, we de-
ﬁne our approach to combining these models. Section 4 con-
tains an approach to weighting models in settings where the
models under consideration are not all compatible. We char-
acterize the computational complexity of the can-explain re-
lation that we deﬁne in Section 5. Section 6 concludes.
2 Causal Models with Focus
In this section, we review the framework of causal models.
We largely follow Halpern and Pearl ( ?), but extend their
basic framework so as to allow the models to express focus
areas.
We assume that a situation is characterized by the val-
ues of a number of variables. There are structural equa-
tions describing the effect that the variables have on each
other. Among the variables, we distinguish between exoge-
nous variables (whose values are determined by factors out-
side of the model) and endogenous variables (whose values
are determined by other variables in the model).
Acausal model with focus is a tupleM= (S;F;G),
whereSis asignature ,Fis a set of structural equations ,
andGis a focus function . The signatureSis itself a tu-
ple(U;V;R). HereUis a (ﬁnite but non-empty) set of ex-
ogenous variables and Vis a (ﬁnite but non-empty) set of
endogenous variables. Ris arange function mapping ele-
ments ofU[V to the (ﬁnite but non-empty) set of values
they can take on. We assume without loss of generality that
jR(C)j>1for all variables C. (If a variable can take on
only one value, then it can neither be a cause nor have its
value be caused by another variable, so we can remove it and
get a semantically equivalent model.) Fassociates with each
endogenous variable X2V a function denoted FXsuch
thatFX: (U2UR(U))(Y2VfXgR(Y))!R (X);
that is,FXdetermines the value of X, given the values of
all the other variables in U[V . For example, we might have
FX(u;y;z ) =u+y, which is usually written as X=U+Y.
Thus, ifY= 3 andU= 2, thenX= 5, regardless of how
Zis set.Up to now, we have essentially described the Halpern-
Pearl ( ?) model. In our setting, though, we add an addi-
tional focus function that intuitively tells us what variables
the modeler considered when trying to determine the struc-
tural equation for each variable. In practice, we might ex-
tract such information from the modeler herself or from
the published experiments of the modeler. Formally, we let
G: (U[V )!2(U[V )be a function that, given a variable
C, gives us the set of variables that the modeler considered
as possibly having an effect on C. We require for all Cthat
C =2G(C), as a variable cannot have a causal effect on it-
self. It may seem surprising at ﬁrst that we deﬁne this func-
tion even for exogenous variables, which are not affected by
other variables in the model. We think that it is more natu-
ral to do so, as it is possible that the modeler considered the
possibility of the variable being endogenous before deciding
that it wasn’t affected by any other variables in the model.
We deﬁneBto be a parent ofCif there exists some set-
ting of all the variables in (V[U )fB;Cgsuch thatCtakes
on some value c1for a valueb1ofBand takes on a differ-
ent valuec2for some other values b2ofB. LetPar(C)be
the set of parents of C. Thus, the parents of Care exactly
those variables that might have a direct effect on C. We re-
quire thatPar(C)G(C)for every variable C. A modeler
cannot have an equation for Cshowing that Bhas a direct
inﬂuence on Cunless the modeler considered Bas a possi-
bly having an effect on C.
A causal model with focus with exogenous variables U
and endogenous variables Vcan be represented by a pair of
graphs onU[V . In the ﬁrst graph, called the parent graph ,
the edge setEconsists of edges from the vertices in Par(C)
toC, for each endogenous variable C. In the second graph,
called the focus graph , the edge set E0consists of edges to
each vertex Cfrom the members of G(C). Pictorially, we
can depict this representation with directed edges for the el-
ements ofEand crossed-out directed edges for the elements
ofE0E. We call a model recursive oracyclic if the par-
ent graph does not contain any cycles. In cases where the
model is acyclic, given a context ~ u(i.e., a setting of the ex-
ogenous variables), the values of all the endogenous vari-
ables are uniquely determined by the structural equations.
As is standard in the literature, we restrict our discussion to
acyclic models in this work.
Given a model M, an endogenous variable X2V, and a
valuex2R(X), we deﬁneMX xto be the model that is
the same asMexcept that the equation for Xis replaced by
X=x. We can think of the model MX xas describing the
result of intervening to set Xtoxin modelM.
Take a causal formula to be one of the form [Y1
y1;:::;Yk yk]', whereYj2U[V and'is a Boolean
combination of primitive formulas of the form X=x,
whereXis an endogenous variable and x2R(X).1In the
special case where k= 0, we identify [ ]'with the formula
'.
We now deﬁne what it means for a causal formula 'to
be true in a causal setting (M;~ u)consisting of a causal
1In previous work, each Yiis taken to be an endogenous vari-
able. For our purposes, it is useful to also allow Yto be exogenous.
modelMand a context ~ u, written (M;~ u)j=', by induc-
tion on the structure of '. For a primitive event X=x,
(M;~ u)j=X=xifX=xin the unique solution to the
equations in Mgiven context ~ u(the solution is unique since
we are dealing with acyclic models, so the setting of the ex-
ogenous variables determines all other variables). The truth
of a Boolean combination of primitive events is deﬁned in
the obvious way. If k1andYkis an endogenous variable,
then
(M;~ u)j= [Y1 y1;:::;Yk yk]'iff
(MYk yk;~ u)j= [Y1 y1;:::;Yk1 yk1]':
IfYkis an exogenous variable, then
(M;~ u)j= [Y1 y1;:::;Yk yk]'iff
(M;~ u[Yk=yk])j= [Y1 y1;:::;Yk1 yk1]';
where~ u[Yk=yk]is the result of replacing the value of Ykin
~ ubyyk.
We now show how to model the example from the intro-
duction in this framework.
Example 2.1. Recall the basic scenario: a medical scien-
tist is trying to understand under what conditions a par-
ticular reaction occurs, and consults with two experts. The
ﬁrst specializes in the exact mechanism by which this re-
action occurs; the second in how one of the reactants gets
produced. The scientist then wants to combine the informa-
tion provided by the two experts. The models provided by
the experts are depicted in Figure 1, where expert ipro-
vides model Mi. The main difference between these mod-
els is that expert 1takes into account the effect that tem-
peratureTcan have on reaction C, while modeler 2, who
does not, takes into account the effect temperature can have
on the production of reactant B. Formally, the parame-
ters of these two models are deﬁned as follows: for the
ranges, we haveR1(T) =R2(T) =fFreezing;Cool;Hotg,
R1(A0) =R1(B0) =R2(A0) =R2(B0) =f1;2;3g,
R1(A) =R1(B) =R2(A) =R2(B) =f1;:::; 5g, and
R1(C) =R2(C) =ftrue;falseg. We haveG1(A) =fA0g,
G1(B) =fB0g, andG1(C) =fA;B;Tg, whileG1(A) =
fA0g,G1(B) =fB0;Tg, andG1(C) =fA;Bg. This is
how we model the fact that expert 1 does not take into ac-
count the effect that temperature ( T) can have on B, while
expert 2 does not take into account the effect that temper-
ature can have on C. The structural equations in M1are
deﬁned by taking A=A0,B=B0, andC= ((T=
Freezing )^(A+B9))_((T=Cool)^(A+B
5))_((T=Hot)^(A+B4)). InM2, the structural equa-
tions areA=A0;B=B0+ 2ifT=Freezing andB=B0
otherwise; and C=true ifA+B5andC=false
otherwise.u t
3 Combining Causal Models with Focus
In this section we turn to the question of combining causal
models. We deﬁne a new relation and show how it can be
used to deﬁne compatibility and combination.
3.1 The “can-explain” relation
We want to capture the intuition that if modeler iconsidered
the causes of some variable Cmore carefully than modeler
𝑀𝑀1
BA
C𝑀𝑀2 𝑀𝑀1⊕′𝑀𝑀2
B’ A’ T
BA
CB’ A’ T
BA
CB’ A’ TFigure 1:M1andM2are the models of the two scientists
trying to understand reaction C.
j, theni’s analysis is preferable. Roughly speaking, we pre-
fer modeleri’s structural equation for Coverj’s ifi’s model
can explain (in some appropriate sense) j’s observations.
Before going on, we introduce some notation conven-
tions that will simplify the exposition. When we write
Mi, we assume that the model Mihas components
((Ui;Vi;Ri);Fi;Gi), andPari(C)refers to the parents of
variableCin modelMi. Also, for a model M, we writeFM
to denote theFfunction in model M, and similarly for the
other components of the model.
Deﬁnition 3.1. M1can explainM2with respect to C, writ-
tenM1CM2, if
(a)R1(C) =R2(C),
(b)G2(C)G 1(C), and
(c) for all exogenous settings ~ u2forM2and all interven-
tionsG2(C) =~ xthere is a context ~ u1inM1such that
if(M2;~ u2)[G2(C) ~ x](C=c)then (M1;~ u1)
[G2(C) ~ x](C=c).
This relation ought really be called can explain and has
considered everything considered by , but for the sake of
brevity we use simply can explain .
The intuition here is that expert 2 (whose knowledge
is characterized by M2) has considered carefully the ef-
fect onCof all the variables in G2(C)and has observed
that those in Par 2(C)have an effect on C, while those in
G2(C)Par 2(C)do not. She has not bothered considering
the effect of the variables not in G2(C)onC, because she
is reasonably sure that they have no effect (but could turn
out to be wrong about this). Expert 1 (whose knowledge is
characterized by M1) can explain expert 2’s observations (at
least, with regard to C) if she has also considered at least all
of the interventions that expert 2 has considered, and can ex-
plain all of expert 2’s observations in the sense of condition
(c) of Deﬁnition 3.1.
We conclude this subsection with two technical results
that highlight useful properties of the Crelation. Say that
M1andM2areC-compatible if eitherM1CM2or
M2M1. We now show that Cis transitive when re-
stricted toC-compatible models.
Proposition 3.2. IfM1CM2,M2CM3, andM1and
M3areC-compatible, then M1CM3.
Proof. Assume that M1CM2,M2CM3, and by way
of contradiction, that M16CM3. BecauseM1andM3
areC-compatible, it must then be the case that M3C
M1. SinceR1(C) =R2(C)andR2(C) =R3(C), we
have thatR1(C) =R3(C). And sinceG2(C)G 1(C),
G3(C) G 2(C), andG1(C) G 3(C), we get that
G1(C) =G2(C) =G3(C). Now consider any interven-
tionG3(C) =~ x. BecauseM2CM3, we know that any
value ofCthat can be achieved in M3under intervention
~X=~ xcan also be achieved in M2under the same inter-
vention; that is, for all contexts ~ uand valuesc2R(C), if
(M3;~ u)j= [G3(C) ~ x](C=c), then there exists a con-
text~ u0such that (M2;~ u0)j= [G3(C) ~ x](C=c). But
becauseG2(C) =G3(C)andM1CM2, it follows that
there exists a context ~ u00such that (M1;~ u00)j= [G3(C)
~ x](C=c). Thus, condition (c) of Deﬁnition 3.1 holds, so
M1CM3.
The requirement in Proposition 3.2 that M1andM3are
C-compatible is necessary, as we show below (see Exam-
ple 3.8).
Deﬁnition 3.3. M1CM2iff either (a) C2U1\U2,
R1(C) =R2(C), andG1(C) =G2(C)or (b)C2V1\V2,
R1(C) =R2(C),G1(C) =G2(C), andF1(C) =F2(C).2
The next result shows that, in a sense, Cis anti-
symmetric.
Proposition 3.4. M1CM2andM2CM1iffM1C
M2.
Proof. The fact that M1CM2impliesM1CM2and
M2CM1follows easily from the deﬁnitions, using the
fact thatPari(C)Gi(C).
To prove the opposite implication, suppose that M1C
M2andM2CM1. We ﬁrst show that Ccannot be in
eitherU1\V 2orU2\V 1. Suppose, by way of contradiction,
thatC2U 1\V 2. Consider an intervention G2(C) =~ x.
BecauseCis exogenous in M2, there must exist contexts
~ u26=~u02such that (M2;~ u2)[G2(C) ~ x](C=c2)and
(M2;~ u0
2)[G2(C) ~ x](C=c0
2)for somec2andc0
2such
thatc26=c0
2. Now consider this same intervention in M1.
SinceM1CM2andM2M1, we have thatG1(C) =
G2(C). By deﬁnition, Par 1(C)G 1(C). Thus there must
exist a unique c1such that, for all exogenous settings ~ u1in
M1,(M1;~ u1)[G2(C) ~ x](C=c1). But because c26=
c0
2, there cannot be contexts ~ u1and~ u0
1such that (M1;~ u1)
[G2(C) ~ x](C=c2)and(M1;~ u0
1)[G2(C) ~ x](C=
c0
2). This contradicts the assumption that M1CM2. A
similar argument shows that Ccannot be inU2\V 1.
It is almost immediate from the deﬁnition of Cthat if
M1CM2,M2CM1, andC2(U1\U 2)[(V1\V 2),
thenR1(C) =R2(C)andG1(C) =G2(C). It follows that
2TechnicallyF1(C) =F2(C)is not deﬁned ifU1[V16=U2[
V2; all we mean is that Par 1(C) =Par 2(C),R1(D) =R2(D)
for all D2Par 1(C), and (M1; ~ u1)[Par 1(C) ~ p](C=c)
iff(M2; ~ u2)[Par 2(C) ~ p](C=c)for all ~ u1; ~ u2;and~ p.ifC2U 1\U 2, thenM1CM2. It remains to show that if
C2V 1\V 2, thenF1(C) =F2(C).
So suppose that C2V1\V2. IfPar 1(C) =Par 2(C),
then sinceG1(C) =G2(C)andPari(C) Gi(C)for
i= 1;2, it follows that (M1;~ u1)[G2(C) ~ x](C=c)
iff(M2;~ u2)[G2(C) ~ x](C=c)for all contexts
~ u1and~ u2, soF1(C) =F2(C). On the other hand, if
Par 1(C)6=Par 2(C), then without loss of generality there
is some variable D2Par 1(C)Par 2(C). There must thus
exist two interventions G2(C) =~ xandG2(C) =~ ythat dif-
fer only on the value of Dsuch that for some c2R(C),
we have (M1;u0
1)[G2(C) ~ x](C=c)and(M1;u0
1)
[G2(C) ~ y]:(C=c)for all exogenous settings u0
1in
M1. BecauseD =2Par 2(C), we know that interventions
G2(C) =~ xandG2(C) =~ ywill give the same value of Cin
M2for all settings of exogenous variables u2. Thus, it is not
the case that M1can explainM2with respect to C, giving a
contradiction.
So we have in all cases that M1CM2, as desired.
3.2 Combining compatible models
We now turn to compatibility and combination of causal
models. We start by deﬁning a simpliﬁed notion of compat-
ibility and an operator 0that gets us most of the way there.
Unfortunately, as we show, 0has a small shortcoming, so
we then modify it to get a more reasonable operator .
Deﬁnition 3.5. M1andM2arecompatible if, for allC2
(U1[V 1)\(U2[V 2), eitherM1CM2orM2CM1.
IfM1andM2are compatible then, for each variable C,
we intuitively want the combined model to take all of the
information for Cfrom the model that best explains C. So
ifM1can explain M2with respect to C, then we want the
combined model to use M1’s focus function and structural
equation (if Cis endogenous in M1) forC. Formally, the
combined model M10M2= ((U;V;R);F;G)is deﬁned
as follows:
 U[V = (U1[V 1)[(U2[V 2)(so the exogenous and
endogenous variables in the combined model comprise all
the endogenous and exogenous variables in M1andM2).
A variableUis exogenous in M10M2if it is exogenous
in one ofM1orM2, sayMi, and either does not appear
inM3i(i.e., the other model) or it appears in M3ibut
MiUM3i; the remaining variables are endogenous.
Formally,U= (U1(U2[V 2))[(U2(U1[V 1))[fC:
9i2f1;2g(C2UiandMiCM3igandV= (V1
(U2[V 2)))[(V2(U1[V 1))[fC:9i2f1;2g(C2
ViandMiCM3i)g.
ForC2(U1[V 1)(U2[V 2), setR(C) =R1(C),
F(C) =F1(C), andG(C) =G1(C).
Similarly, for C2(U2[V 2)(U1[V 1), setR(C) =
R2(C),FC=F2(C), andG(C) =G2(C).
ForC2(U1[V 1)\(U2[V 2), we must have either
M1CM2orM2CM1. IfMiM3i, then set
R(C) =Ri(C),F(C) =Fi(C), andG(C) =Gi(C).
(By Proposition 3.4, this is well deﬁned: if M1CM2
andM2CM1, thenR1(C) =R2(C),F1(C) =
F2(C), andG1(C) =G2(C).)
Returning to Example 2.1, it is easy to check that the
modelsM1andM2are compatible. Specﬁcally, we have
M1CM2andM2BM1. (For all other variables D,
we haveM1DM2andM2DM1.) Thus, we can com-
bineM1andM2to get the model M10M2depicted in
Figure 2
𝑀𝑀1
BA
C𝑀𝑀2 𝑀𝑀1⊕′𝑀𝑀2
B’ A’ T
BA
CB’ A’ T
BA
CB’ A’ T
Figure 2: Taking into account what each scientist focused on
allows us to combine M1andM2to get the model shown on
the right.
In the ACH approach, M1andM2would be declared
incompatible. Since no information about who considered
what possibilities is available, expert 2is assumed to have
come to the conclusion that Tdoes not directly affect C,
and therefore be in fundamental disagreement with expert 1.
In our setting, though, we can take advantage of the focus
information to determine whether there is truly a fundamen-
tal disagreement. The disagreement may just be a result of
the fact that one of the experts was not focusing on certain
variables. In situations where this information is available, it
can allow us to take more complete advantage of the differ-
ent areas of expertise that different experts may have.
Slightly more generally, the ACH deﬁnition is designed
to take into account situations where one expert’s model is
more detailed in terms of the topology of the causal graph,
that is, where the causal relationship is considered to be me-
diated by variables the other modeler was simply not aware
existed. In our setting there is more information available, al-
lowing us to consider another sense in which one modeler’s
understanding might be locally more detailed than another’s,
namely, situations where one expert can explain the other’s
observed results by taking into account the fact that the other
was not focusing on certain variables.
This notion of combination is commutative and, when de-
ﬁned, associative:
Proposition 3.6. Given three pairwise compatible models
M1,M2, andM3,
(a)M10M2=M20M1;
(b) ifM1CM2orC2(U1[V 1)(U2[V 2), thenM10
M2CM1;
(c) ifM3is compatible with M10M2andM1is
compatible with M20M3thenM10(M20M3) =
(M10M2)0M3.
Proof. Commutativity is immediate from the deﬁnition of
0.For part (b), suppose that M1CM2. ThenCis ex-
ogenous in M10M2iffCis exogenous in M1. More-
over,RM10M2(C) =R1(C),FM10M2(C) =F1(C)if
C2V 1, andGM10M2(C) =G1(C), so it immediately fol-
lows thatM10M2CM1. A similar argument applies if
C2(U1[V 1)(U2[V 2).
For part (c), observe that to show that M10(M20
M3) = (M10M2)0M3, it sufﬁces to show that
M10(M20M3)C(M10M2)0M3for allC2
(U1[V 1[U 2[V 2[U 3[V 3). We do this by considering,
for each variable C, how many models it appears in.
First consider the case where Cis in only one of the
three models (i.e., C2Ui[Vifor exactly one i2
f1;2;3g). Assume without loss of generality that Cis in
M1. Then it follows almost immediately from our deﬁni-
tions thatM10(M20M3)CM1C(M10M2)0M3,
soM10(M20M3)C(M10M2)0M3. Similarly, in
the case where Cis only in two models, assume without
loss of generality that Cis inM1andM2. Then it fol-
lows immediately that (M10M2)0M3CM10M2,
so ifM1CM2then(M10M2)0M3CM1and if
M2CM1then(M10M2)0M3CM2. It is also im-
mediate that M20M3CM2, so ifM1CM2then
M10(M20M3)CM1. Now consider the case where
M2CM1. It must be the case that either M20M3C
M1orM1CM20M3because they are compatible. If
M20M3CM1thenM10(M20M3)CM2and
we are done. On the other hand, if M1CM20M3then,
becauseM20M3CM2, we know that M1CM2.
But then because we assumed M2CM1we get by
Proposition 3.4 that M2CM1and soM10(M20
M3)CM1CM2. Thus, in al cases, we have that
(M10M2)0M3CM10(M20M3).
Finally, ifCis in all three models, by Propositions 3.2 and
3.4, for some choice of i,j,kwe haveMiCMjMk.
Suppose that M1CM2CM3(the argument is al-
most identical in all other cases). It follows from part (b)
thatM10M2CM1. BecauseM3is compatible with
(M10M2), we know that either (M10M2)CM3or
M3C(M10M2). In the ﬁrst case, it follows immediately
from part (b) that (M10M2)0M3CM1. In the second
case, since (M10M2)CM1by part (b) and M3C
(M10M2)by assumption, it follows that M3CM1.
And sinceM1CM2CM3, we have that M1CM3
by transitivity (Proposition 3.2), so it follows from Proposi-
tion 3.4 that M3CM1. But then from part (b) we have
that(M10M2)0M3CM3cM1. It is easy to show
by similar reasoning that M10(M20M3)CM1. So
we get thatM10(M20M3)C(M10M2)0M3,
completing the argument.
One natural question to ask is whether this deﬁnition of
combination is guaranteed to preserve acyclicity. Unfortu-
nately, this is not the case, as the following example shows.
Example 3.7. Consider the models M1andM2in Figure 3,
where
 U 1=fA;Cg,V1=fB;Dg,U2=fB;Dg, andV2=
fA;Cg;
all variables are binary (i.e. have range f0;1g);
 G 1(C) =G1(A) =G2(B) =G2(D) =;,G1(B) =fAg,
G1(D) =fCg,G2(A) =fDg, andG2(C) =fBg;
inM1,Ais the parent of BandCis the parent of D,
and inM2,Bis the parent of CandDis the parent of A.
The details of the equations do not matter; for simplicity,
suppose that in M1we haveB=AandD=C, while in
M2we haveA=DandC=B.
Thus,AandCare exogenous in M1, whileBandDare
exogenous in M2. It is easy to see that, despite the fact that
both models are acyclic, when we combine them we get a
cyclic model.u t
𝑀𝑀1 𝑀𝑀2 𝑀𝑀1⊕′𝑀𝑀2
A
B D
CA
B D
CA
B D
C
Figure 3: Although the models M1andM2are acyclic, the
combined model M1M2contains a cycle.
We can, however, provide a simple and efﬁcient test to
guarantee that the combined model will be acyclic. Let
G1= (U1[V 1;E1)be the parent graph for model M1and
letG2= (U2[V 2;E2)be the parent graph for model M2.
LetG0= ((U1[V1)[(U2[V2);E1[E2). In linear time, we
can compute whether G0contains any cycles. If it does not,
thenM10M2is guaranteed to be acyclic. This is a sufﬁ-
cient but not necessary condition for acyclicity, as edges can
be deleted via our combination process. In practice, though,
we suspect this condition will hold in most cases of interest
where the combined model is indeed acyclic.
3.3 Combination as least upper bound
When we combine two models, we would like the combined
model to be the simplest model that can explain both. Un-
fortunately, this may not be the case for M10M2. Indeed,
even ifM1andM2are compatible, M10M2may not be
able to explain both Mifor all variables Cthat appear in
Mi. It follows from Proposition 3.6 that if MiCM3ior
C2(Ui[Vi)(U3i[V 3i), thenM10M2CMi, so
(by Proposition 3.4) M10M2can explainMiwith respect
toC. But, as the following example shows, if M1CM2
andCappears inM2,M10M2may not be able to explain
M2with respect to C.
Example 3.8. Consider the models M1andM2depicted in
Figure 4, where the range of all variables is ftrue;falseg;
the focus set of each variable consists of just its parents, as
deﬁned in the parent graph; in M1, the structural equations
are such that C=AXORB, while inM2,A=Dand
B=D. Then inM10M2, all three of these equations
hold.
It is easy to see that M10M26CM2. The problem is,
to explain the value C= 0,AandBneed to have different
𝑀𝑀1 𝑀𝑀2 𝑀𝑀1⊕′𝑀𝑀2
D
B A
CD
B A
CB A
CFigure 4: Models M1andM2whereM10M26CM2.
values, and there is no context in M10M2that gives them
different values. Intuitively, although M2can explain M1
wth respect to each of AandBindividually, it cannot ex-
plain them both together. In particular, the setting A=false
andB=true cannot be explained in M2. We have not de-
ﬁned what it would mean to explain a setting involving more
than one variable; this is because our intuition for “can ex-
plain” is based on the assumption that experts are testing one
variable at a time.
This example also shows that Cis not necessarily tran-
sitive: we have M10M2CM1andM1CM2, we do
not haveM10M2CM2. This does not contradict Propo-
sition 3.2, since M10M2andM2are not compatible. u t
The fact that M10M2may not be able to explain both
M1andM2is somewhat disconcerting. However, the situa-
tion is not quite as bad as it appears.
Deﬁnition 3.9. M1dominatesM2, writtenM1M2, if
M1CM2for allC2U 2[V 2.
Note that if M1dominatesM2, then we must have that
U1[V 1U 2[V 2.
Theorem 3.10. IfM1andM2are compatible, then M10
M2dominates both M1andM2iffM10M2is the unique
least upper bound of fM1;M 2g.
Proof. Suppose that M10M2M1andM10M2
M2. Then, by deﬁnition, M10M2is an upper bound of
fM1;M 2g, so now we must show that, for any other up-
per boundM0offM1;M 2g, we haveM0M10M2.
We ﬁrst note that the variables in M10M2are precisely
(U1[V1)[(U2[V2). For each variable CinM10M2, there
exists somei2f1;2gsuch thatGM10M2(C) =Gi(C)and
eitherFM10M2(C) =Fi(C)orCis exogenous in both
M10M2andMi. Moreover,ParM10M2(C) =Pari(C).
Thus, given an intervention GM10M2(C) =~ xinM10M2
and context~ usuch that (M10M2;~ u)j= [GM10M2(C) =
~ x](C=c), there exists a context ~ u0inMisuch that
(Mi;~ u0)j= [GM10M2(C) =~ x](C=c). SinceM0Mi,
it follows that there exists a context ~ u00inM0such that
(M0;~ u00)j= [GM10M2(C) =~ x](C=c). It follows that
M0CM10M2. SinceCwas arbitrary, it follows that
M0M10M2.
We have thus shown that M10M2is a least upper bound
offM1;M 2gifM10M2M1andM10M2M2.
Uniqueness is straightforward: if M0is another least upper
bound offM1;M 2gthen, by Proposition 3.4, it follows that
M0CM10M2for allC2U 1[V 1[U 2[V 2, soM0=
M10M2. The converse is also immediate: if M1M2is
not an upper bound of both M1andM2, it certainly cannot
be a least upper bound of fM1;M 2g.
So where does this leave us? Our goal is to combine
the information of experts. If a decision-maker believes that
modelsM1andM2both provide useful information, then
she would want to work with a model that somehow com-
bines this information. As Example 3.8 shows, the problem
withM10M2is that it does not necessarily combine all
the information in M1andM2. To deal with this problem,
we simply deﬁneby takingM1M2=M10M2if
M10M2Mifori= 1;2, and otherwise say that M1
andM2are incompatible and M1M2is undeﬁned. In-
tuitively, in the latter case, there is no clear way to explain
both models, so more experiments are necessary. It is easy
to check that Proposition 3.6 holds for , with no change
in proof. Moreover, by Proposition 3.10, when it is deﬁned,
M1M2is the least upper bound of fM1;M 2g.
We conjecture that if M1M2is not deﬁned, then
fM1;M 2gin fact has no least upper bound. This is the case
in the models of Example 3.8. Consider the models M0
1and
M0
2, whereM0
1is identical to M1except that it includes the
variableD, andGM0
1(A) =GM0
1(B) =fDg, andM0
2is
just likeM2except thatGM0
2(C) =fA;Bg. It is easy to
check thatM1andM2are both upper bounds on fM1;M 2g,
and there is no upper bound M0offM1;M 2gsuch that
M0
1M0andM0
2M0.
If this conjecture is correct (and we have shown that it
is in a number of special cases), then it shows that if we
think ofas an information ordering, then M1M2, when
it is deﬁned, is the model that combines the information in
M1andM2and has no additional information; if it is not
deﬁned, then there is no such model.3
3.4 Explanation complexity and combination
complexity
Recall thatM1CM2if, for every intervention G2(C) =
~ x, valuec2R(C), and context ~ u2, there exists a context
~ u1such that if (M2;~ u2)[G2(C) ~ x](C=c)then
(M1;~ u1)[G2(C) ~ x](C=c). However, in principle,
we could use a different context ~ u1to explain each possi-
ble intervention on G2(C). We might be reluctant to accept
explanations that are complicated, in the sense of requiring
too many different contexts; if an overly complicated expla-
nation is needed to reconcile two models, we may instead
prefer to simply declare them incompatible. The following
deﬁnitions of explanation complexity andcombination com-
plexity capture these intuitions.
Deﬁnition 3.11. M1can explainM2with respect to Cusing
a setU0of contexts ifM1can explainM2with respect to C
using only contexts u0
1drawn fromU1; that is, we just modify
3We remark that we can deﬁne an analogue of for the notion
of combination considered by ACH, and show that M1M2as
ACH deﬁne it is the least upper bound M1andM2with respect
to the ACH notion. Thus, thinking in terms of least upper bound
seems like a useful way to think of combining models.Deﬁnition 3.1 so that all the contexts u1in condition (c) are
drawn fromU0. The complexity of M1’s ability to explain
M2with respect to CisminfjU0j:M1can explainM2with
respect toCusingU0g.
Example 3.12. Consider the models in Figure 5. In all of
these models,R(A) =R(B) =R(A1) =R(A2) =
R(A3) =f0;:::; 10g,R(D) =f0;:::; 30g, andR(C) =
f0;:::; 60g. In modelM1on the left, we have the structural
equationsC=A+BifD1andC= 2(A+B)if
D= 0; in modelM2, we haveC=A+B; in modelM3
on the right, we have C=D; and in model M4, we have
C=A1+A2+A3. In the low-complexity models on the
left, the complexity of M1’s ability to explain M2with re-
spect toCis1, as every intervention can be explained by
the value of Dsimply having been 1the entire time. For the
high-complexity models on the right, though, the complex-
ity ofM3’s ability to explain M4with respect to Cis30; for
each intervention, Dmust take on precisely the right value in
M3for each particular outcome of Cto be observed. Thus,
we would be more hesitant to combine the high-complexity
modelsM3andM4. Combining them implicitly assumes
thatM3andM4are compatible, and, in particular, that M3
can explainM4with respect to C.u t
𝑀𝑀1 𝑀𝑀2Low Complexity
𝑀𝑀3 𝑀𝑀4High Complexity
𝐴𝐴2𝐴𝐴3 𝐴𝐴1
CB A
CD
B A
CD
𝐴𝐴2𝐴𝐴3 𝐴𝐴1
C
Figure 5: The two models on the left have low explanation
complexity with respect to Cwhereas the two on the right
have high explanation complexity.
We can extend the notion of explanation complexity to the
combination complexity of two models.
Deﬁnition 3.13. Thecombination complexity of two com-
patible models M1andM2is the minimum cardinality jU0j
taken over all setsU0such that, for all C2(U1[V1)\(U2[
V2), eitherM1can explain M2with respect to CusingU0
orM2can explainM1with respect to CusingU0.
A decision-maker may want to consider only explanations
that have complexity less than or equal to some threshold or
model combinations that have complexity less than a thresh-
old. In the next section, we show how combination complex-
ity can be used to weight models.
4 Weighting and Combining Expert
Opinions
Given a collection of models, it may be impossible to com-
bine all of them, but possible to combine a variety of dif-
ferent subsets of them. ACH proposed a way to assign con-
ﬁdence to different possible combined models based on the
decision-maker’s conﬁdence in the original models. Here we
provide a way to extend this to our setting.
We start with a collection of pairs (M1;p1);:::; (Mn;pn)
whereMiis a causal model with focus and piis a value
in(0;1]. Here the intuition for each pair should be that Mi
was the model proposed by expert iandpiis the decision-
maker’s degree of conﬁdence that expert i’s model is cor-
rect. More precisely, piis not the decision-maker’s degree
of conﬁdence that the assumptions built into Miare correct,
but her conﬁdence that, for each variable Cand intervention
Gi(C) =~ x, if(Mi;~ u)[Gi(C) =~ x](C=c)then ex-
pertiindeed observed a world where the variables in Gi(C)
were~ xandCdid have value c. Following ACH, we deﬁne
Compat =fIf1;:::;ng:the models infMi:i2
Igare mutually compatible gand deﬁneMI=i2IMifor
allI2Compat . The mutual compatibility of a setMof
models is deﬁned inductively on the cardinality of M. If
jMj= 1thenMis automatically mutually compatible, and
ifjMj= 2thenMis mutually compatible if the two mod-
els inMare compatible. If jMj =nthenMis mutually
compatible if every subset of cardinality n1is mutually
compatible and, for each M2M ,Mis compatible with
M02M :M06=MM0.
One simple way to weight the combined models, pro-
posed by ACH, is to assign model MIprobability
pI=Y
i2IpiY
j=2I(1pj)=N; (1)
whereNis simply a normalization term to get the proba-
bilities to sum to 1. Thus,pIcaptures the intuition that the
agents inIperformed their experiments correctly while the
agents not in Imay have made a mistake in one or more of
their experiments, where the probabilities of agents having
made a mistake are treated as being mutually independent.
LetMI=fMi:i2Ig. In our setting, we may also
want to take into account how complex it is to combine the
models inMIwhen assigning MIa probability; if com-
bining the models in MIrequires a large set of contexts to
make all of the necessarily explanations, then we may have
less conﬁdence that the combined model captures the true
state of the world. To formalize this idea, we ﬁrst generalize
Deﬁnition 3.13 in the obvious way: the combination com-
plexity of a setMis the minimum cardinality jU0jof a set
U0such that all explanations made during the combination
process can be made using U0. The combination complexity
of a singleton set is deﬁned to be 1.
Exactly how complexity should be taken into ac-
count when assigning conﬁdence scores may be context-
dependent; it is up to the decision-maker who is combin-
ing the models to decide. We propose several simple rules
here. One simple rule that may be relevant in some situa-
tions is to simply use a threshold, and assign conﬁdence 0
to models where the combination complexity or the expla-
nation complexity with respect to any variable Cis above
some constant . (Here and in the following two rules, the
normalization factor Nmust be updated accordingly.) An-
other natural option may be to add a weighting factor to (1)
that is inversely proportional to the combination complexity.If the combination complexity of MIisI, then we set
p0
I=1
IY
i2IpiY
j=2I(1pj)=N:
A third rule that may be useful in some contexts is to assign
complexity weights that are inverse exponential in the com-
bination complexity. Here the conﬁdence scores assigned
would be
p00
I=eIY
i2IpiY
j=2I(1pj)=N:
Example 4.1. Consider three models M1,M2, andM3,
where
 U 1=U3=fA;B;Dg,V1=V3=fCg,U2=fA;Gg,
andV2=fB;Cg;
 R 1(C) =R2(C) =R3(C) =R1(D) =R3(D) =
f0;1;2gandR1(A) =R1(B) =R2(A) =R2(B) =
R2(G) =R3(A) =R3(B) =f0;1g;
 G 1(C) =fA;B;Dg,G2(C) =fA;Bg,G2(B) =fGg,
andG3(C) =fA;B;Dg;
the structural equations are such that, in M1,C=D; in
M2,C=A+BandB=G; and inM3,C= 2ifD= 0
andC= min(1;A+B)ifD= 1orD= 2.
The models in the set fMI:I2CompatgareM1,M2,
M3,M1M2, andM3M2, with combination complexity
5forM1M2(3forM1to explainM2with respect to Cand
2forM2to explainM1with respect to B) and combination
complexity 4forM3M2(2forM3to explainM2with re-
spect toCand2forM2to explainM3with respect to B). Of
course,M1,M2, andM3(viewed as singleton sets) all have
combination complexity 1, by deﬁnition. Consider the sec-
ond weighting rule above, inversely proportional weighting,
with prior conﬁdences p1= 0:85;p2= 0:8;andp3= 0:9.
The assigned conﬁdence scores would then be
p0
M1= (0:85)(0:2)(0:1)=N0:176
p0
M2= (0:15)(0:8)(0:1)=N0:124
p0
M3= (0:15)(0:2)(0:9)=N0:280
p0
M1M2= (1
5)(0:85)(0:8)(0:1)=N0:141
p0
M3M2= (1
4)(0:15)(0:8)(0:9)=N0:280:
Under the third rule, inverse exponential weighting, with
the same prior conﬁdences, the assigned conﬁdence scores
would be
p00
M1= (0:85)(0:2)(0:1)=N0:291
p00
M2= (0:15)(0:8)(0:1)=N0:205
p00
M3= (0:15)(0:2)(0:9)=N0:462
p00
M1M2= (e5)(0:85)(0:8)(0:1)=N0:008
p00
M3M2= (e4)(0:15)(0:8)(0:9)=N0:034:
As expected, the inverse exponential weighting rule is
more complexity averse, and so assigns a greater proportion
of conﬁdence to the uncombined models. u t
These three rules behave in a qualitatively similar manner,
with the importance of complexity being taken into account
in different ways. More generally, let Ibe the combination
complexity ofMIand letQI=Y
i2IpiY
j=2I(1pj). We
believe that there are many reasonable functions f(QI;I)
that can be used to assign a conﬁdence scores to MI; we
leave it up to the decision-maker to decide what function f
is most suitable for a given context. The two requirements
that seem necessary to us is that fbe non-increasing in I
and non-decreasing in QI; that is,f(QI;I)f(QI;0
I)
for ﬁxedQIif0
II, andf(QI;I)f(Q0
I;I)for
ﬁxedIifQ0
IQI. These two rules capture the intuition
that we should not prefer models that are more complicated,
nor should we prefer models that are composed of models in
which we had less prior conﬁdence.
An additional factor that may sometimes play a role is the
likelihood of different endogenous settings occurring. If one
model can explain the other only by using a context that is
very unlikely to occur, then we may not want to assign much
weight to that combined model. Thus, in certain settings it
may also make sense to have the conﬁdence scores depend
on a distribution over exogenous settings.
5 Computational Complexity
We now consider the computational complexity of determin-
ing whether one model can explain another with respect to
C.
Theorem 5.1. Determining whether M1CM2is inP
2,
and is P
2-hard, even in instances where all variables are
binary.
Proof. It is easy to see that the problem is in P
2: the ﬁrst
two conditions in the can-explain relation can clearly be
checked in polynomial time, while, for a ﬁxed intervention
G2(C) =~ xinM2, context~ u2inM2, and context ~ u1inM1,
checking whether (M2;~ u2)[G2(C) ~ x](C=c)and
(M1;~ u1)[G2(C) ~ x](C=c)can be done in polyno-
mial time.
For the lower bound, consider the canonical P
2-hard lan-
guage P
2(SAT)=f8~X9~Y':8~X9~Y'is a closed quantiﬁed
Boolean formula,8~X9~Y'=trueg. We show a reduction
from P
2(SAT)to our language.
Consider a CQBF (closed quantiﬁed Boolean formula)
8~X9~Y'; we show how to transform this into an instance
of our problem. For ease of exposition, we assume that all
variables in ~X[~Yappear in'. LetM2contain exogenous
variables~Xand an endogenous variable C =2~X[~Y. InM2,
the range of all variables is ftrue;falseg,G2(C) =~X, and
the equation for CisC=true. InM1, we haveU1=~X[~Y,
V1=fCg,G1(C) =~X[~Y, and the equation for Cis
C='.
We now show that M1CM2if and only8~X9~Y'
is true. First, suppose that M1CM2. BecauseCis al-
ways true inM2andG2(C) =~X, by condition (c) in the
deﬁnition of the can-explain relation, for all settings of ~X
there must be a setting of the remaining variables such that
C=true inM1. But because the equation for CinM1is
C=', this means that for all settings of ~X, there existsa setting of ~Ysuch that'is true. For the other direction,
suppose that8~X9~Y'is true. ClearlyG2(C)G 1(C)and
R1(C) =R2(C). To see that condition (c) of the deﬁnition
of can-explain holds, consider an intervention ~X=~ xon
~X. Because8~X9~Y'is true, there must be some setting of
the values in ~Ysuch that if ~Xwere set to~ x, thenCwould
evaluate to true inM1. So in the context where ~Yis set
correspondingly, we get that the original intervention would
makeC=true, as desired.
While this result indicates that this computation is likely
to be intractable in the worst case, models that arise in the
physical and social sciences often contain only a small num-
ber of variables, so we would still expect these computations
to be feasible in practice.
6 Conclusion
We have shown how causal models can be combined in in-
stances where experts disagree due to different focus areas.
We deﬁned what it means for one model to be able to ex-
plain another with respect to a given variable and showed
how this can be used to combine two compatible models.
Furthermore, we showed that the model obtained via this
combination process is in fact the least upper bound of the
combined models relative to the natural relation, in some
sense making it the simplest model that can explain the ob-
servations of both experts.
The can-explain relation embodies one way of explaining
why two experts may have different causal models. ACH can
be viewed as modeling a different reason, where M1is “bet-
ter than”M2with respect to a variable Cin the ACH view
if, roughly speaking, M1has a more detailed picture of the
causal relations among the ancestors of C. While we believe
that the can-explain relation captures quite a natural intuition
(as does the ACH notion of compatibility!), there may well
be other reasonable intuitions that are worth exploring. More
generally, this viewpoint suggests that a decision-maker try-
ing to combine experts’ models must think seriously about
the reasons underlying their disagreement before combining
models, and consider a notion of combination appropriate
for these reasons. Since the need to combine expert opin-
ions arises frequently in practice, having a principled under-
standing of the process seems to us critical. We hope that the
results of this paper help in this process.
Acknowledgments: This work was supported in part by
NSF grants IIS-1703846 and IIS-1718108, ARO grant
W911NF-17-1-0592, and a grant from the Open Philan-
thropy project.
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Sufficient Conditions for Causality to be Transitive.

Sufﬁcient Conditions for Causality to be
Transitive
Joseph Y . Halpern
Cornell University
Computer Science Department
Ithaca, NY 14853
halpern@cs.cornell.edu
http://www.cs.cornell.edu/home/halpern
Abstract
Natural conditions are provided that are sufﬁcient to ensure that causality
as deﬁned by approaches that use counterfactual dependence and structural
equations will be transitive.
I thank Chris Hitchcock and the anonymous reviewers of the paper for perceptive comments
that greatly inﬂuenced the structure and story of the paper. Work supported in part by NSF grants
IIS-0812045, IIS-0911036, and CCF-1214844, by AFOSR grants FA9550-08-1-0438, FA9550-
09-1-0266, and FA9550-12-1-0040, and by ARO grant W911NF-09-1-0281.
1
1 Introduction
The question of the transitivity of causality has been the subject of much debate.
As Paul and Hall [2013] say: “Causality seems to be transitive. If CcausesDand
DcausesE, thenCthereby causes E.” The appeal to transitivity is quite standard
in informal scientiﬁc reasoning: we say things like “the billiards expert hit ball A,
causing it to hit ball B, causing it to carom into ball C, which then drops into the
pocket”. It then seems natural to conclude then the pool expert’s shot caused ball
Cto drop into the pocket.
Paul and Hall [2013, p. 215] suggest that “preserving transitivity is a basic
desideratum for an adequate analysis of causation”. Hall [2000] is even more
insistent, saying “That causation is, necessarily, a transitive relation on events
seems to many a bedrock datum, one of the few indisputable a priori insights we
have into the workings of the concept.” Lewis [1986, 2000] imposes transitivity
in his inﬂuential deﬁnition of causality, by taking causality to be the transitive
closure (“ancestral”, in his terminology) of a one-step causal dependence relation.
But numerous examples have been presented that cast doubt on transitivity.
Paul and Hall [2013] give a sequence of such counterexamples; Hall [2000] gives
others. I review two such examples in the next section. This leaves us in a some-
what uncomfortable position. It seems so natural to think of causality as transi-
tive. In light of the examples, should we just give up on these intuitions? Paul
and Hall [2013] suggest that “What’s needed is a more developed story, according
to which the inference from “ CcausesD” and “DcausesE” to “CcausesE” is
safe provided such-and-such conditions obtain—where these conditions can typ-
ically be assumed to obtain, except perhaps in odd cases . . . ”. The goal of this
paper is to provide sufﬁcient conditions for causality to be transitive. I formalize
this using the structural equations framework of Halpern and Pearl [2001, 2005].
The properties that I require suggest that these conditions apply to any deﬁnition
of causality that depends on counterfactual dependence and uses structural equa-
tions (see, for example, [Glymour and Wimberly 2007; Hall 2007; Halpern 2015;
Halpern and Pearl 2005; Hitchcock 2001; Hitchcock 2007; Woodward 2003] for
examples of such approaches).
These conditions may explain why, although causality is not transitive in gen-
eral (and is not guaranteed to be transitive according to any of the counterfactual
accounts mentioned above), we tend to think of causality as transitive, and are
surprised when it is not.
2
2 Deﬁning causation using counterfactuals
In this section, I review some of the machinery of structural equations needed
to deﬁne causality. For deﬁniteness, I use the same formalism as that given by
Halpern and Pearl [2005].
2.1 Causal structures
Approaches based on structural equations assume that the world is described in
terms of random variables and their values. Some random variables may have a
causal inﬂuence on others. This inﬂuence is modeled by a set of structural equa-
tions . It is conceptually useful to split the random variables into two sets: the
exogenous variables, whose values are determined by factors outside the model,
and the endogenous variables, whose values are ultimately determined by the ex-
ogenous variables. For example, in a voting scenario, we could have endogenous
variables that describe what the voters actually do (i.e., which candidate they vote
for), exogenous variables that describe the factors that determine how the voters
vote, and a variable describing the outcome (who wins). The structural equations
describe how the outcome is determined (majority rules; a candidate wins if A
and at least two of B,C,D, andEvote for him; etc.).
Formally, a causal model Mis a pair (S;F), whereSis asignature , which
explicitly lists the endogenous and exogenous variables and characterizes their
possible values, and Fdeﬁnes a set of modiﬁable structural equations , relating
the values of the variables. A signature Sis a tuple (U;V;R), whereUis a
set of exogenous variables, Vis a set of endogenous variables, and Rassociates
with every variable Y2U[V a nonempty setR(Y)of possible values for Y
(that is, the set of values over which Yranges ). For simplicity, I assume here
thatVis ﬁnite, as isR(Y)for every endogenous variable Y2V.Fassociates
with each endogenous variable X2V a function denoted FXsuch thatFX:
(U2UR(U))(Y2Vf XgR(Y))!R (X). This mathematical notation just
makes precise the fact that FXdetermines the value of X, given the values of
all the other variables in U[V . If there is one exogenous variable Uand three
endogenous variables, X,Y, andZ, thenFXdeﬁnes the values of Xin terms of
the values of Y,Z, andU. For example, we might have FX(u;y;z ) =u+y,
which is usually written as X=U+Y.1Thus, ifY= 3andU= 2, thenX= 5,
regardless of how Zis set.
1The fact that Xis assigned U+Y(i.e., the value of Xis the sum of the values of UandY)
does not imply that Yis assigned XU; that is, FY(U;X;Z ) =XUdoes not necessarily
3
The structural equations deﬁne what happens in the presence of external inter-
ventions. Setting the value of some variable Xtoxin a causal model M= (S;F)
results in a new causal model, denoted MX=x, which is identical to M, except that
the equation for XinFis replaced by X=x.
Following [Halpern and Pearl 2005], I restrict attention here to what are called
recursive (oracyclic ) models. This is the special case where there is some total
orderingof the endogenous variables (the ones in V) such that if XY,
thenXis independent of Y, that is,FX(:::;y;::: ) =FX(:::;y0;:::)for all
y;y02R(Y). Intuitively, if a theory is recursive, there is no feedback. If XY,
then the value of Xmay affect the value of Y, but the value of Ycannot affect
the value of X. It should be clear that if Mis an acyclic causal model, then given
acontext , that is, a setting ~ ufor the exogenous variables in U, there is a unique
solution for all the equations. We simply solve for the variables in the order given
by. The value of the variables that come ﬁrst in the order, that is, the variables X
such that there is no variable Ysuch thatYX, depend only on the exogenous
variables, so their value is immediately determined by the values of the exogenous
variables. The values of variables later in the order can be determined once we
have determined the values of all the variables earlier in the order.
It is sometimes helpful to represent a causal model graphically. Each node
in the graph corresponds to one variable in the model. An arrow from one node
to another indicates that the former variable ﬁgures as a nontrivial argument in
the equation for the latter. The graphical representation is useful for visualizing
causal models, and will be used in the next section.
2.2 A language for reasoning about causality
To deﬁne causality carefully, it is useful to have a language to reason about causal-
ity. Given a signature S= (U;V;R), aprimitive event is a formula of the form
X=x, forX2V andx2R(X). Acausal formula (over S)is one of the form
[Y1 y1;:::;Y k yk]', where
'is a Boolean combination of primitive events,
Y1;:::;Y kare distinct variables in V, and
yi2R(Yi).
hold.
4
Such a formula is abbreviated as [~Y ~ y]'. The special case where k= 0 is
abbreviated as '. Intuitively, [Y1 y1;:::;Y k yk]'says that'would hold if
Yiwere set toyi, fori= 1;:::;k .
A causal formula  is true or false in a causal model, given a context. As
usual, I write (M;~ u)j= if the causal formula  is true in causal model M
given context ~ u. Thej=relation is deﬁned inductively. (M;~ u)j=X=xif the
variableXhas valuexin the unique (since we are dealing with acyclic models)
solution to the equations in Min context~ u(that is, the unique vector of values for
the exogenous variables that simultaneously satisﬁes all equations in Mwith the
variables inUset to~ u). The truth of conjunctions and negations is deﬁned in the
standard way. Finally, (M;~ u)j= [~Y ~ y]'if(M~Y=~ y;~ u)j='.
2.3 Deﬁning causality
The basic intuition behind counterfactual deﬁnitions of causality is that Ais a
cause ofBif there is counterfactual dependence between AandB: ifAhadn’t
occurred (although it did), then Bwould not have occurred. It is well known that
the counterfactual dependence does not completely capture causality; there are
many examples in the literature where people say that Ais a cause of Bdespite
the fact that Bdoes not counterfactually depend on A(at least, not in this sim-
ple sense). Nevertheless, all the counterfactual deﬁnitions of causality (as well as
people’s causality ascriptions) agree that this simple type of counterfactual depen-
dence gives a sufﬁcient condition for causality. For the purposes of this paper, I
consider only cases where this counterfactual dependence holds.
More formally, say that X=xis abut-for cause of 'in(M;~ u)(where'is a
Boolean combination of primitive events) if (M;~ u)j=X=x^'(so bothX=x
and'hold in context ~ u) and there exists some x0such that (M;~ u)j= [X x0]:'.
Thus, with a but-for cause, changing the value of Xto something other than x
changes the truth value of '; that is,'counterfactually depends on X.
All the complications in counterfactual approaches to causality arise in how
they deal with cases of causality that are not but-for causality. Roughly speaking,
the idea is that X=xis a cause of Y=yif the outcome Y=ycounterfactually
depends on Xunder the appropriate contingency (i.e., holding some other vari-
ables ﬁxed at certain values). While the various approaches to deﬁning causality
differ in exactly how this is done, they all agree that a but-for cause should count
as a cause. So, for simplicity in this paper, I consider only but-for causality and
do not both to give a general deﬁnition of causality
5
3 Sufﬁcient Conditions for Transitivity
In this section I present two different sets of conditions sufﬁcient for transitivity.
Before doing that, I give two counterexamples to transitivity, since these motivate
the conditions. The ﬁrst example is taken from (an early version of) Hall [2004],
and is also considered by Halpern and Pearl [2005].
Example 1: Consider the following scenario:
Billy contracts a serious but nonfatal disease so is hospitalized. Sup-
pose that Monday’s doctor is reliable, and administers the medicine
ﬁrst thing in the morning, so that Billy is fully recovered by Tuesday
afternoon. Tuesday’s doctor is also reliable, and would have treated
Billy if Monday’s doctor had failed to. Given that Monday’s doctor
treated Billy, it’s a good thing that Tuesday’s doctor did not treat him:
one dose of medication is harmless, but two doses are lethal.
Suppose that we are interested in Billy’s medical condition on Wednesday. We
can represent this using a causal model MBwith three variables:
MT for Monday’s treatment (1 if Billy was treated Monday; 0 otherwise);
TTfor Tuesday’s treatment (1 if Billy was treated Tuesday; 0 otherwise);
and
BMC for Billy’s medical condition (0 if Billy feels ﬁne on Wednesday; 1 if
Billy feels sick on Wednesday; 2 if Billy is dead on Wednesday).
We can then describe Billy’s condition as a function of the four possible combi-
nations of treatment/nontreatment on Monday and Tuesday. I omit the obvious
structural equations corresponding to this discussion; the causal graph is shown in
Figure 1:In the context where Billy is sick and Monday’s doctor treats him, MT= 1is
a but-for cause of TT= 0—because Billy is treated Monday, he is not treated on
Tuesday morning. And TT= 0is a but-for cause of Billy’s being alive ( BMC =
0_BMC = 1). However, MT= 1 isnota cause of Billy’s being alive. It is
clearly not a but-for cause; Billy will still be alive if MT is set to 0. Indeed, it
is not even a cause under the more general deﬁnitions of causality, according to
all the approaches mentioned above; no setting of the other variables will lead
to a counterfactual dependence between MT andBMC6= 2. This shows that
causality is not transitive according to these approaches. Although MT= 1 is a
6
rr
r
?S
S
S
S
SS w

 /BMCTTMT
Figure 1: Billy’s medical condition.
cause of TT= 0andTT= 0is a cause of BMC = 0_BMC = 1,MT= 1is not
a cause of BMC = 0_BMC = 1. (Of course, according to Lewis [1986, 2000],
who takes the transitive closure of the one-step dependence relation, MT= 1is a
cause of BMC = 0_BMC = 1.)
Although this example may seem somewhat forced, there are many quite real-
istic examples of lack of transitivity with exactly the same structure. Consider the
body’s homeostatic system. An increase in external temperature causes a short-
term increase in core body temperature, which in turn causes the homeostatic
system to kick in and return the body to normal core body temperature shortly
thereafter. But if we say that the increase in external temperature happened at
time 0 and the return to normal core body temperature happened at time 1, we
certainly would not want to say that the increase in external temperature at time 0
caused the body temperature to be normal at time 1!2
There is another reason that causality is intransitive, which is illustrated by the
following example, due to McDermott [1995].
Example 2: Suppose that a dog bites Jim’s right hand. Jim was planning to det-
onate a bomb, which he normally would do by pressing the button with his right
foreﬁnger. Because of the dog bite, he presses the button with his left foreﬁnger.
The bomb still goes off.
Consider the causal model MDwith variables DB(the dog bites, with values
0 and 1),P(the press of the button, with values 0, 1, and 2, depending on whether
the button is not pressed at all, pressed with the right hand, or pressed with the
2I thank Richard Scheines [personal communication, 2013] for this example.
7
left hand), and B(the bomb goes off). We have the obvious equations: DBis
determined by the context, P=DB+ 1, andB= 1ifPis either 1 or 2. In the
context where DB= 1, it is clear that DB= 1is a but-for cause of P= 2(if the
dog had not bitten, Pwould have been 1), and P= 2is a but-for cause of B= 1
(ifPwere 0, then Bwould be 0), but DB= 1 is not a but-for cause of B= 1.
And again, DB= 1 is not a cause of B= 1 even under a more general notion
of causation. Whether or not the dog had bitten Jim, the button would have been
pressed and the bomb would have detonated.
As I said, I believe that we feel that causality is transitive because, in typical
settings, it is. My belief is based mainly on introspection here and informal polling
of colleagues. Even when told that causality is not transitive, people seem to ﬁnd it
hard to construct counterexamples. This suggests that when they think about their
everyday experience of causality, they come up with examples where causality is
transitive. If there were many counterexamples available in everyday life, it would
be easier to generate them.
I now give two sets of simple conditions that are sufﬁcient to guarantee tran-
sitivity. Speciﬁcally, I give conditions to guarantee that if X1=x1is a but-for
cause ofX2=x2in(M;~ u)andX2=x2is a but-for cause of X3=x3in(M;~ u),
thenX1=x1is a but-for cause of X3=x3in(M;~ u).
The ﬁrst set of conditions assumes that X1,X2, andX3each has a default
setting. We can think of the default setting as the result of doing nothing. This
makes sense, for example, in the billiards example at the beginning of the paper,
where we can take the default setting for the shot to be the expert doing nothing,
and the default setting for the balls to be that they are not in motion. Let the default
setting be denoted by the value 0.
Proposition 3.1: Suppose that (a) X1=x1is a but-for cause of X2=x2in
(M;~ u), (b)X2=x2is a but-for cause of X3=x3in(M;~ u), (c)x36= 0, (d)
(M;~ u)j= [X1 0](X2= 0) , and (e) (M;~ u)j= [X1 0;X 2 0](X3= 0) .
ThenX1=x1is a but-for cause of X3=x3in(M;~ u).
Proof: IfX2= 0 is the unique solution to the equations in the causal model
MX1 0in context~ uandX3= 0 in the unique solution to the equations in
MX1 0;X2 0in context~ u, then it is immediate that X3= 0in the unique solution
to the equations in MX1 0in context~ u. That is, (M;~ u)j= [X1 0](X3= 0) . It
follows from assumption (a) that (M;~ u)j=X1=x1. We must thus have x16= 0,
since otherwise (M;~ u)j=X1= 0^[X1 0](X3= 0) , so(M;~ u)j=X3= 0,
8
which contradicts assumptions (b) and (c). Thus, X1=x1is a but-for cause of
X3=x3, since the value of X3depends counterfactually on that of X1.
Although the conditions of Proposition 3.1 are clearly rather specialized, they
arise often in practice. Conditions (d) and (e) say that if X1remains in its default
state, then so will X2, and if both X1andX2remain in their default states, then so
willX3. (These assumptions are very much in the spirit of the assumptions that
make a causal network self-contained , in the sense deﬁned by Hitchcock [2007].)
Put another way, this says that the reason for X2not being in its default state is
X1not being in its default state, and the reason for X3not being in its default
state isX1andX2both not being in their default states. The billiard example can
be viewed as a paradigmatic example of when these conditions apply. It seems
reasonable to assume that if the expert does not shoot, then ball Adoes not move;
and if the expert does not shoot and ball Adoes not move (in the context of
interest), then ball Bdoes not move, and so on.
Of course, the conditions on Proposition 3.1 do not apply in either Example 1
or Example 2. The obvious default values in Example 1 are MT=TT= 0, but
the equations say that in all contexts ~ uof the causal model MBfor this example,
we have (MB;~ u)j= [MT 0](TT= 1) . In the second example, if we take
DB= 0andP= 0to be the default values of DBandP, then in all contexts ~ u
of the causal model MD, we have (MD;~ u)j= [DB 0](P= 1) .
While Proposition 3.1 is useful, there are many examples where there is no
obvious default value. When considering the body’s homeostatic system, even if
there is arguably a default value for core body temperature, what is the default
value for the external temperature? But it turns out that the key ideas of the proof
of Proposition 3.1 apply even if there is no default value. Suppose that X1=x1is
a but-for cause of X2=x2in(M;~ u)andX2=x2is a but-for cause of X3=x3
in(M;~ u). Then to get transitivity, it sufﬁces to ﬁnd values x0
1,x0
2, andx0
3such
thatx36=x0
3,(M;~ u)j= [X1 x0
1](X2=x0
2), and (M;~ u)j= [X1 x1;X 2
x0
2](X3=x0
3). The argument in the proof of Proposition 3.1 then shows that
(M;~ u)j= [X1 x0
1](X3=x0
3).3It then follows that X1=x1is a but-for
cause ofX3=x3in(M;~ u). In Proposition 3.1, x0
1,x0
2, andx0
3were all 0, but
there is nothing special about the fact that 0 is a default value here. As long as
3The analogous statement is also valid in standard conditional logic. That is, taking A > B to
represent “if Awere the case then Bwould be the case”, using standard closest-world semantics
[Lewis 1973], (A > B )^((A^B)> C ))(A > C )is valid. I thank two of the anonymous
reviewers of this paper for encouraging me both to note that this idea is the key argument of the
paper and to relate it to the Lewis approach.
9
we can ﬁnd some values x0
1,x0
2, andx0
3, these conditions apply. I formalize this as
Proposition 3.2, which is a straightforward generalization of Proposition 3.1.
Proposition 3.2: Suppose that there exist values x0
1,x0
2, andx0
3such that (a)
X1=x1is a but-for cause of X2=x2in(M;~ u), (b)X2=x2is a but-for cause
ofX3=x3in(M;~ u), (c)x36=x0
3, (d)(M;~ u)j= [X1 x0
1](X2=x0
2), and (e)
(M;~ u)j= [X1 x0
1;X 2 x0
2](X3=x0
3). ThenX1=x1is a but-for cause of
X3=x3in(M;~ u).
To see how these ideas apply, suppose that a student receive an A+ in a course,
which causes her to be accepted at Cornell University (her top choice, of course!),
which in turn causes her to move to Ithaca. Further suppose that if she had re-
ceived an A in the course she would have gone to university U1and as a result
moved to city C1, and if she gotten anything else, she would have gone to uni-
versity atU2and moved to city C2. This story can be captured by a causal model
with three variables, Gfor her grade, Ufor the university she goes to, and Cfor
the city she moves to. There are no obvious default values for any of these three
variables. Nevertheless, we have transitivity here: The student’s A+ was a cause
of her being accepted at Cornell and being accepted at Cornell was a cause of her
move to Ithaca; it seems like a reasonable conclusion that the student’s A+ was
a cause of her move to Ithaca. And, indeed, transitivity follows from Proposi-
tion 3.2. We can take the student getting an Ato bex0
1, take the student being
accepted at university U1to bex0
2, and the student moving to C1to bex0
3(assuing
thatU1is not Cornell and that C1is not Ithaca, of course).
The conditions provided in Proposition 3.2 are not only sufﬁcient for causality
to be transitive, they are necessary as well, as the following result shows.
Proposition 3.3: IfX1=x1is a but-for cause of X3=x3in(M;~ u), then there
exist values x0
1,x0
2, andx0
3such thatx36=x0
3,(M;~ u)j= [X1 x0
1](X2=x0
2),
and(M;~ u)j= [X1 x0
1;X 2 x0
2](X3=x0
3).
Proof: SinceX1=x1is a but-for cause of X3=x3in(M;~ u), there must exist
valuesx0
16=x1andx36=x0
3such that (M;~ u)j= [X1 x0
1](X3=x0
3). Letx0
2
be such that (M;~ u)j= [X1 x0
1](X2=x0
2). Since (M;~ u)j= [X1 x0
1](X2=
x0
2^X3=x0
3), it easily follows that (M;~ u)j= [X1 x0
1;X 2=x0
2](X3=x0
3).
In light of Propositions 3.2 and 3.3, understanding why causality is so often
taken to be transitive comes down to ﬁnding sufﬁcient conditions to guarantee
10
the assumptions of Proposition 3.2. I now present another set of conditions suf-
ﬁcient to guarantee the assumptions of Proposition 3.2 (and thus, sufﬁcient to
make causality transitive), motivated by the two examples showing that causality
is not transitive. To deal with the problem in Example 2, I require that for ev-
ery valuex0
2in the range of X2, there is a value x0
1in the range of X1such that
(M;~ u)j= [X1 x0
1](X2=x0
2). This requirement holds in many cases of in-
terest; it is guaranteed to hold if X1=x1is a but-for cause of X2=x2ziand
X2is a binary variable (i.e., takes on only two values), since but-for causality re-
quires that two different values of X1result in different values of X2. But this
requirement does not hold in Example 2; no setting of DBcan forcePto be 0.
Imposing this requirement still does not deal with the problem in Example 1.
To do that, we need one more condition. Say that a variable Ydepends on Xif
there is some setting of all the variables in U[V other thanXandYsuch that
varying the value of Xin that setting results in Y’s value varying; that is, there is
a setting~ zof the variables other than XandYand valuesxandx0ofXsuch that
FY(x;~ z)6=FY(x0;~ z).
Up to now I have used the phrase “causal path” informally; I now make it
more precise. A causal path in a causal model Mis a sequence (Y1;:::;Y k)of
variables such that Yj+1depends onYjforj= 1;:::;k1. Since there is an edge
betweenYjandYj+1in the causal graph for Mexactly ifYj+1depends on Yj, a
causal path is just a path in the causal graph. A causal path from X1toX2is just
a causal path whose ﬁrst node is X1and whose last node is X2. Finally,Ylies on
a causal path from X1toX2ifYis a node (possibly X1orX2) on a directed path
fromX1toX2.
The additional condition that I require for transitivity is that X2must lie on
every causal path from X1toX3. Roughly speaking, this says that all the inﬂuence
ofX1onX3goes through X2. This condition does not hold in Example 1; as
Figure 1 shows, there is a direct causal path from MT toBMC that does not
include TT. On the other hand, this condition does hold in many examples of
interest. Going back to the example of the student’s grade, the only way that the
student’s grade can inﬂuence which city the student moves to is via the university
that accepts the student.
The following result summarizes the second set of conditions sufﬁcient for
transitivity.
Proposition 3.4: Suppose that X1=x1is a but-for cause of X2=x2in the
causal setting (M;~ u),X2=x2is a but-for cause of X3=x3in(M;~ u), and the
following two conditions hold:
11
(a) for every value x0
22R(X2), there exists a value x0
12R(X1)such that
(M;~ u)j= [X1 x0
1](X2=x0
2);
(b)X2is on every causal path from X1toX3.
ThenX1=x1is a but-for cause of X3=x3.
The proof of Proposition 3.4 is not hard, although we must be careful to get
all the details right. The high-level idea of the proof is easy to explain though.
Suppose that X2=x2is a but-for cause of X3=x3in(M;~ u). Then there must
be some values x26=x0
2andx36=x0
3such that (M;~ u)j= [X2 x0
2](X3=x0
3).
By assumption, there exists a value x0
12R(X1)such that (M;~ u)j= [X1
x0
1](X2=x0
2). The requirement that X2is on every causal path from X1toX3
guarantees that [X2 x0
2](X3=X3)implies [X1 x0
1;X 2 x0
2](X3=X3).
Roughly speaking, X2“screens off” the effect of X1onX3, since it is on every
causal path from X1toX3. Now we can apply Proposition 3.2. I defer the formal
argument to the appendix.
It is easy to construct examples showing that the conditions of Proposition 3.4
are not necessary for causality to be transitive. Suppose that X1=x1causes
X2=x2,X2=x2causesX3=x3, and there are several causal paths from X1
toX3. Roughly speaking, the reason that X1=x1may not be a but-for cause
ofX3=x3is that the effects of X1onX3may “cancel out” along the various
causal paths. This is what happens in the homeostasis example. If X2is is on
all the causal paths from X1toX3, then, as we have seen, all the effect of X1on
X3is mediated by X2, so the effect of X1onX3on different causal paths cannot
“cancel out”. But even if X2is not on all the causal paths from X1toX3, the
effects ofX1onX3may not cancel out along the causal paths; and X1=x1may
still be a cause of X3=x3. That said, it seems difﬁcult to ﬁnd a weakening of the
condition in Proposition 3.4 that is simple to state and sufﬁces for causality to be
transitive.
A Proof of Proposition 3.4
To prove Proposition 3.4, I need a preliminary result, which states a key (and
obvious!) property of causal paths: if there is no causal path from XtoY, then
changing the value of Xcannot change the value of Y. Although it is intuitively
obvious, proving it carefully requires a little bit of work.
12
Lemma A.1 : IfYand all the variables in ~Xare endogenous, Y =2~X, and there is
no causal path from a variable in ~XtoY, then for all sets ~Wof variables disjoint
from~XandY, and all settings ~ xand~ x0for~X,yforY, and~ wfor~W, we have
(M;u)j= [~X ~ x;~W ~ w](Y=y)iff(M;u)j= [~X ~ x0;~W ~ w](Y=y)
and
(M;u)j= [~X ~ x](Y=y)iff(M;u)j=Y=y:
Proof: Deﬁne the maximum distance of a variable Yin a causal model M,
denoted maxdist (Y), to be the length of the longest causal path from an ex-
ogenous variable to Y. We prove the result by induction on maxdist (Y). If
maxdist (Y) = 1 , then the value of Ydepends only on the values of the exogenous
variables, so the result trivially holds. If maxdist (Y)>1, letZ1;:::;Z kbe the
endogenous variables on which Ydepends. These are the endogenous parents of
Yin the causal graph (i.e., these are exactly the endogenous variables Zsuch that
there is an edge from ZtoYin the causal graph). For each Z2fZ1;:::;Z kg,
maxdist (Z)<maxdist (Y): for each path from an exogenous variable to Z, there
is a longer path to Y, namely, the one formed by adding the edge from ZtoY.
Moreover, there is no path from a variable in ~Xto any ofZ1;:::;Z knor is any of
Z1;:::;Z kin~X(for otherwise there would be a path from a variable in ~XtoY,
contradicting the assumption of the lemma). Thus, the inductive hypothesis holds
for each of Z1;:::;Z k. Since the value of each of Z1;:::;Z kdoes not change
when we change the setting of ~Xfrom~ xto~ x0, and the value of Ydepends only
on the values of Z1;:::;Z kand~ u(i.e., the values of the exogenous variables), the
value ofYcannot change either.
I can now prove Proposition 3.4. I restate it here for the convenience of the
reader.
Proposition 3.4: Suppose that X1=x1is a but-for cause of X2=x2in the
causal setting (M;~ u),X2=x2is a but-for cause of X3=x3in(M;~ u), and the
following two conditions hold:
(a) for every value x0
22R(X2), there exists a value x0
12R(X1)such that
(M;~ u)j= [X1 x0
1](X2=x0
2);
(b)X2is on every causal path from X1toX3.
ThenX1=x1is a but-for cause of X3=x3.
13
Proof: SinceX2=x2is a but-for cause of X3=x3in(M;~ u), there must exist
x0
26=x2andx0
36=x3such that (M;~ u)j= [X2 x0
2](X3=x0
3). By assumption,
there exists a value x0
1such that (M;~ u)j= [X1 x0
1](X2=x0
2). I claim that
(M;~ u)j= [X1 x0
1](X3=x0
3). This follows from a more general claim. I show
that ifYis on a causal path from X2toX3, then
(M;~ u)j= [X1 x0
1](Y=y)iff(M;~ u)j= [X2 x0
2](Y=y). (1)
Although it is not obvious, this is essentially the argument sketched in the main
part of the text. Literally the same argument as that given below for the proof of
(1) also shows that
(M;~ u)j= [X1 x0
1](Y=y)iff(M;~ u)j= [X1 x0
1^X2 x0
2](Y=y).
Deﬁne a partial order on endogenous variables that lie on a causal path from
X2toX3by takingY1Y2ifY1precedesY2on some causal path from X2to
X3. SinceMis a recursive model, if Y1Y2, we cannot have Y2Y1(otherwise
there would be a cycle). I prove (1) by induction on the ordering. The least
element in this ordering is clearly X2;X2must come before every other variable
on a causal path from X2toX3. By assumption, (M;~ u)j= [X1 x0
1](X2=x0
2),
and clearly (M;u)j= [X2 x0
2](X2=x0
2). Thus, (1) holds for X2. Thus
completes the base case of the induction.
For the inductive step, let Ybe a variable that lies on a causal path from X2
andX3, and suppose that (1) holds for all variables Y0such thatY0Y. Let
Z1;:::;Z kbe the endogenous variables that Ydepends on in M. For each of
these variables Zi, either there is a causal path from X1toZior there is not. If
there is, then the path from X1toZican be extended to a directed path Pfrom
X1toX3, by going from X1toZi, fromZitoY, and fromYtoX3(sinceY
lies on a causal path from X2toX3). Since, by assumption, X2lies on every
causal path from X1toX3,X2must lie on P. Moreover, X2must precede Y
onP. (Proof: Since Ylies on a path P0fromX2toX3,X2must precede Y
onP0. IfYprecedesX2onP, then there is a cycle, which is a contradiction.)
SinceZiprecedesYonP, it follows that ZiY, so by the inductive hypothesis,
(M;~ u)j= [X1 x0
1](Zi=zi)iff(M;~ u)j= [X2 x0
2](Zi=zi).
Now if there is no causal path from X1toZi, then there also cannot be a
causal path PfromX2toZi(otherwise there would be a causal path from X1
toZiformed by appending Pto a causal path from X1toX2, which must exist
since, if not, it easily follows from Lemma A.1 that X1=x1would not be a
cause ofX2=x2). Since there is no causal path from X1toZi, by Lemma A.1,
14
we must have that (M;~ u)j= [X1 x0
1](Zi=zi)iff(M;~ u)j=Zi=ziiff
(M;~ u)j= [X2 x0
2](Zi=zi).
Since the value of Ydepends only on the values of Z1;:::;Z kand~ u, and
I have just shown that (M;~ u)j= [X1 x0
1](Z1=z1^:::^Zk=zk)iff
(M;~ u)j= [X2 x0
2](Z1=z1^:::^Zk=zk), it follows that (M;~ u)j= [X1
x0
1](Y=y)iff(M;~ u)j= [X2 x0
2](Y=y). This completes the proof of
the induction step. Since X3is on a causal path from X2toX3, it follows that
(M;~ u)j= [X1 x0
1](X3=x0
3)iff(M;~ u)j= [X2 x0
2](X3=x0
3). Since
(M;~ u)j= [X2 x0
2](X3=x0
3)by construction, we have that (M;~ u)j= [X1
x0
1](X3=x0
3), as desired. Thus, X1=x1is a but-for cause for X3=x3.
15
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