From counterfactual simulation to causal judgment
Tobias Gerstenberg1 (tger@mit.edu), Noah D. Goodman2 (ngoodman@stanford.edu),David A. Lagnado3 (d.lagnado@ucl.ac.uk) & Joshua B. Tenenbaum1 (jbt@mit.edu)
1 Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139
2 Department of Psychology, Stanford University, Stanford, CA 94305
3 Cognitive, Perceptual and Brain Sciences, University College London, London WC1H 0AP

Abstract
In this paper, we demonstrate that people’s causal judgments
are inextricably linked to counterfactuals. In our experiments,
participants judge whether one billiard ball A caused anotherball B to go through a gate. Our counterfactual simulation
model predicts that people arrive at their causal judgments bycomparing what actually happened with the result of mentally
simulating what would have happened in the relevant counterfactual world. We test our model against actualist theories of
causation which aim to explain causation just in terms of what
actually happened. Our experimental stimuli contrast cases inwhich we hold constant what actually happened but vary thecounterfactual outcome. In support of our model, we find thatparticipants’ causal judgments differ drastically between suchcases. People’s cause and prevention judgments increase with
their subjective degree of belief that the counterfactual outcome would have been different from what actually happened.
Keywords: causality; attribution; counterfactuals; intuitive
physics.

Introduction
In American Football, pass interference is a foul whereby adefender hinders a receiver from catching the ball prior to theball’s arrival. However, a defender’s play is only penalized
when the referees rule that it would have been possible forthe receiver to catch the ball had the defender not interfered.
In other words, the defender’s action must have made a difference to the outcome to be called a foul.
The pass interference rule embodies a counterfactual criterion of causation. According to a counterfactual theory,
one event C is a cause of another event E if (i) both C and
E actually happened, and (ii) E would not have happened
if C had not happened. While the idea that at least part ofwhat it means to be a cause is to have made a difference tothe outcome is intuitively appealing, counterfactual theories
have been criticized on various grounds. On the one hand,counterfactuals seem to be doing too much: why consider
what would have happened in another possible situation andnot just look at what actually happened? On the other hand,counterfactuals seem to be doing too little: counterfactual dependence seems too abstract and indirect. It cannot capture
the sense of “oomph” that we get in paradigmatic cases ofphysical causation (Schaffer, 2005). For example, when we
see the collision of two billiard balls, it seems like we can directly perceive causality without considering counterfactuals(Michotte, 1946/1963).
These limitations of the counterfactual framework havemotivated a very different way of trying to analyze causation.
According to actualist theories of causation, causal relationships are determined just in terms of what actually happened.
For example, in Dowe’s (2000) conserved quantity theory,

what it means to say that one object caused a change to another object depends critically on the transfer of a physicalquantity (such as force or momentum) from the former to thelatter. Counterfactual considerations about what would have
happened in other possible worlds are irrelevant.
In psychology, the most comprehensive account of an actualist theory of causation has been developed by Wolff (2007).
In his force dynamics model, causality is reduced to configurations of forces that are present at the time of interaction between an agent and a patient. According to this view, an agent
caused a patient to reach a certain endstate if the following
conditions are met: (i) the patient did not have a tendency toward the endstate and (ii) the agent and patient force vectors
combined to a resultant force pointing toward the endstate.1
In our work, we combine what we see as the strengths ofthe counterfactual and actualist accounts of causation. Our
counterfactual simulation model maintains the view that people’s causal attributions are intrinsically connected to whether
the event of interest made a difference to the outcome. This
aspect of causation is well-captured by counterfactual theories. However, in line with Wolff’s (2007) force dynamics model, we acknowledge that people’s intuitive theories
are often much richer than what can be expressed in termsof the formal accounts of counterfactual reasoning to date(e.g. Halpern & Pearl, 2005). We assume that people use
their intuitive domain theories to simulate what would have
happened in the relevant counterfactual world (cf. Battaglia,
Hamrick, & Tenenbaum, 2013). Consider stepping into theshoes of a football referee to decide whether a given football was catchable. Being able to mentally simulate the counterfactual requires both a sophisticated understanding of how
people work (e.g., did the receiver try to catch the ball?) as
well as how the world works (e.g., was the distance between
the ball and the receiver such that it would have been physically possible for him to catch the ball?).
There is a rich literature on causal attribution which discusses the tension between counterfactual (e.g. Kahneman& Tversky, 1982) and actualist determinants (e.g. Mandel,
2003) of people’s causal judgments. Studies in this literature
1 Because the force dynamics model reduces causation to configurations of forces, a patient’s tendency is not defined counterfactually in terms of whether or not the patient would have reached theendstate in the absence of the agent. Rather, tendency is defined as
the direction in which the patient’s force points at the time of interaction between agent and patient. More recently, the counterfactual
concept of a virtual force has been incorporated into the force dynamics model to explain people’s causal judgments for situations
that involve omissions and/or chains of events (see Wolff, Barbey, &Hausknecht, 2010).

Counterfactual Simulation Model
The counterfactual simulation model of causal attribution applies to any domain where people are able to simulate what
would have happened in the relevant counterfactual world.
Here, we will focus on judgments about the consequences ofcollisions between two billiard balls. The causal question ofinterest is whether ball A caused ball B to go through a gate(or prevented it from going through). Figure 1 shows four diagrammatic illustrations of clips used in the experiment. The
solid lines show the actual paths of ball A’s and B’s movement
up until the point of collision and the path that ball B actually
traveled after the collision. The dashed lines show the paththat B would have traveled if ball A had been removed fromthe scene.
We will now discuss how our general framework predicts
people’s causal and counterfactual judgments.

Causal judgments
Our counterfactual simulation model predicts that people’s
causal judgments are a function of their subjective degree ofbelief that the causal event made a difference to whether or
not the outcome would occur. People are predicted to compare the actual outcome with their belief about what the counterfactual outcome would have been. We capture the extent
to which a cause is believed to have made a difference to theoutcome in terms of Pearl’s (1999) counterfactual definitionof the probability that the cause event X was necessary for thecounterfactual misscounterfactual hit

actual miss

A

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usually ask participants to reach their causal verdicts basedon written vignettes which stipulate explicitly what would
have happened in the relevant counterfactual worlds. In ourwork, in contrast, we present participants with animated clips
of causal interactions in a physical domain. This allows us
to manipulate people’s uncertainty in the relevant counterfactual outcome in a quantitative way and thus test more rigorously the relationship between counterfactual and causal
judgments.
In previous work (Gerstenberg, Goodman, Lagnado, &Tenenbaum, 2012), we provided evidence that the counterfactual simulation model accurately predicts people’s causal
judgments. However, these experiments were not designed totest the broader claim that causal judgments are intrinsicallylinked to counterfactuals and that actualist theories cannot in
principle explain people’s judgments.
In this paper, we provide a stronger test of the role thatcounterfactuals play for causal attributions. We contrast
pairs of situations in which we match exactly what actually
happened but vary what would have happened in the relevant counterfactual world. Furthermore, we test situations
in which the counterfactual outcome is held constant but theactual outcome is brought about in different ways. The results support the counterfactual simulation model over actualist theories of causation: while participants’ causal judgments
are strongly affected by manipulating the relevant counterfactual, their judgments are much less influenced by the way inwhich the actual outcome was brought about.

A
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Figure 1: Diagrammatic illustrations of four stimuli. In each pair ofstimuli (top vs. bottom pair), what actually happened is held constant. The location of the brick only influences the counterfactual
outcome. Note: Clip 18 illustrates the noisy physics model.

effect event Y to occur. In our case, X denotes the event that
there was a collision between A and B which either occurred(x) or didn’t occur (¬x). Y denotes the event that B went
through the gate which, again, either occurred (y) or didn’t
occur (¬y).
The probability PNcaused that the collision event between A
and B was a necessary cause of B going through the gate isdefined in the following way:
PNcaused = P(¬ydo(¬x) |x, y)(1)

In words, PNcaused denotes the probability of the counterfactual that ball B would not have gone through the gate, if there
had been no collision between A and B (¬ydo(¬x) ), given thatin the actual world, A and B collided (x) and ball B went
through the gate (y).
Similarly, we can define the probability PN prevented that thecollision between A and B was a necessary cause of B not
going through the gate as:
PN prevented = P(ydo(¬x) |x, ¬y)(2)

In words, PN prevented denotes the probability of the counterfactual that ball B would have gone through the gate in theabsence of the collision (ydo(¬x) ), assuming that in the actual
world, the balls collided (x) and B did in fact not go in (¬y).
Our counterfactual simulation model shares with Pearl(1999) the idea that people compute the relevant counterfactuals by manipulating an intuitive causal domain theory.
However, our model differs from Pearl’s account in both theform of the causal model and the nature of the manipulation.
First, while Pearl represents causal knowledge in terms ofgraphs and structural equations, we assume a richer representation akin to a physics engine (cf. Battaglia et al., 2013).
Second, whereas Pearl defines the counterfactual manipulation in terms of a formal operation on a graph, in our model,
we compute the probability of counterfactuals via removingball A from the world shortly before A and B would have collided and then simulate what would have happened.

Counterfactual judgments

counterfactual missactual missactual close

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We predict that people’s cause and prevention judgments
increase with PNcaused and PN prevented , respectively. People
are predicted to say that ball A caused ball B to go throughthe gate when B did in fact go in, and when they are confident that B would not have gone in had the collision not takenplace. Similarly, we predict that people give high prevention
ratings when B missed the gate and they are confident that it
would have gone in without the collision. We predict intermediate cause and prevention judgments when people are unsure
about what the counterfactual outcome would have been (i.e.
for intermediate values of PNcaused and PN prevented ). Finally,
we predict that people will say that A neither caused nor prevented B from going through the gate when they are confident
that the counterfactual outcome would have been the same as
the actual outcome (i.e. when PNcaused or PN prevented is low).

B

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Figure 2: Diagrammatic illustrations of nine stimuli used in the experiment. Note: If the teleport is active, ball B is teleported from theyellow rectangle to the blue circle (see, e.g. clip 10).
B

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Experimentsome clips also featured a brick and/or a teleport.2
While the location of the solid walls was fixed, the placement of the brick and teleport changed between the clips. Participants were instructed that the brick was solid and that theballs bounce off it in case of collision. Participants were also
instructed about how the teleport worked. The teleport only
affected ball B but not ball A. Ball B always exited the teleport through the blue circle in the same direction in which it
had entered through the yellow rectangle (see, e.g., Figure 2,
clip 10). Participants also learned that the teleport was sometimes deactivated which was marked by a red cross on topof the teleport’s entrance and exit. If the teleport was deactivated, the teleport’s entrance and exit had no influence on B’s
movement.
The stimuli were designed with two goals in mind: First, inorder to test the qualitative hypothesis that causal judgments
are intrinsically linked to counterfactual considerations, wecreated pairs of stimuli that held constant what actually happened and only varied what would have happened in the relevant counterfactual world. Figure 1 shows two pairs of clips
in which differences in the relevant counterfactual worlds
were achieved by varying the placement of a brick. For example, in both clips 1 and 5, the interaction between A and
B is exactly the same (as illustrated by the identical solidlines in both diagrams). However, whereas in clip 1, ball B
would have been blocked by the brick if ball A had not beenpresent, in clip 5, B would have gone through the gate in A’s
absence. Similarly, in both clips 14 and 18, A collides with B
and B goes through the gate after having bounced off the wall.
While in clip 14, B would have been blocked by the brick if
A hadn’t been present, in clip 18, B would have gone throughthe gate even in the absence of A.
While we had to vary the placement of the brick in order togenerate the desired contrast between counterfactual worlds,
the inclusion of the teleport allowed us to achieve the same
effect without having to change the placement of any compo-

In all of the clips used in the experiment, both balls enter
the scene from the right and collide once with each other. In
addition to the solid walls that mark the border of the scene,

2 The different clips as well as a demo of the experiment may beaccessed here:
http://web.mit.edu/tger/www/demos/teleport demos.html

In our experiments, we ask people to evaluate the counterfactual of whether ball B would have gone through the gate ifball A had not been present in the scene. We assume that people arrive at their belief about what would have happened in
the relevant counterfactual world by using their intuitive understanding of the domain. In our case, the domain of interest
comprises basic physical concepts such as velocity and force
as well as non-physical concepts such as teleportation.
Previous work has shown that people’s predictions in certain physical domains are well-explained as an approximation to Newtonian physics (Battaglia et al., 2013; Sanborn,
Mansinghka, & Griffiths, 2013). In line with this work, we
capture people’s uncertainty in their mental simulation of thecounterfactual by introducing Gaussian noise to ball B’s velocity vector at the time at which the collision between A andB would have taken place (see Figure 1, clip 18). It is at thispoint in time that the relevant counterfactual world diverges
from the actual world and participants are required to rely on
their mental simulation to predict whether or not ball B would
have gone in. By adding Gaussian noise to B’s velocity vector, we capture the dynamic uncertainty inherent in people’s
intuitive physical model (cf. Smith & Vul, 2012). In order topredict people’s counterfactual judgments, we first run a large
number of simulations for each clip in which we randomly
perturb B’s velocity vector as described above. We then compare the proportion of times in which ball B went through thegate in the sample of simulations to people’s judgments. For
example, if ball B went through the gate in all of the generated
samples, we predict that people should be very certain that B
would have gone in. We predict that participants should be
maximally uncertain about the counterfactual outcome when
the generated sample is exactly split between clips in which
B went in and didn’t go in.

Methods
Participants and materials 80 participants (43 female,
Mage = 33.94, SDage = 12.24) were recruited via Amazon
Mechanical Turk and paid $1.5 compensation. The experiment was programmed in Flash CS5 and the clips were generated using the physics engine Box2D.
Design and procedure Each participant saw two blocks oftrials: in the counterfactual block, the clips were paused at
the time of collision. After having seen the clip twice, participants were asked to judge whether ball B would have gone
through the gate if ball A had not been present in the scene.
Participants indicated their response on a slider whose endpoints and midpoint were labeled “definitely no” (0),“definitely yes” (100) and “unsure” (50), respectively. After having indicated their response, participants received feedback
by being shown the full clip in which ball A was removed
from the scene.
In the causal block, participants saw each clip played twice
until the end and then asked: “What role did ball A play?”
The endpoints were labeled “it prevented ball B from going
through the gate” (-100) and “it caused ball B to go throughthe gate” (100). The midpoint was labeled “neither” (0). Participants were instructed to use intermediate values to indicatethat A somewhat caused B to go through the gate (or somewhat prevented it from going through).
We counterbalanced the order of the two types of blocks.
Half the participants made counterfactual judgments before
causal judgments and vice versa. Each block started withtwo practice clips. The 18 test clips were randomized withinblocks. We also counterbalanced the vertical position of theballs and the other components. On average, the experiment
took 21.16 minutes (SD = 5.08) to complete.

Results and Discussion
Counterfactual judgments Since there was neither a significant main effect of block order (i.e. whether partici-

counterfactual misscounterfactual close

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nents. We simply contrasted cases in which the teleport waseither on or off. For example, while in both clips 13 and 17,
the collision event between A and B is identical, B would havenot gone through the gate in the absence of A when the teleport was off (clip 13), whereas it would have gone throughwhen the teleport was on (clip 17). The teleport further allowed us to test the flexibility of people’s counterfactual simulations.
Second, in order to test the quantitative hypothesis about
the relationship between people’s causal judgments and theirsubjective degree of belief about the counterfactual outcome,we crossed how closely ball B actually went through the gatewith how closely B would have gone through the gate if ball A
had not been present in the scene (see Figure 2). For example,
in clip 6, ball B actually missed the gate. However, if A had
not been present, B would have gone through the gate via theteleport. We generated two clips for each of the nine cells thatcross the closeness of the actual and counterfactual outcome.

50

0

Figure 3: Mean counterfactual judgments for situations in which
B would have missed (gray) or gone in (white) together with thepredictions of the noisy physics model (black bars). Labels (A) –(E) indicate pairs of clips which matched what actually happened(e.g. clips 1 and 5 in Figure 1). Error bars in all figures indicate
SEM.

pants answered the counterfactual or causal questions first),
F(1, 78) = 0.59, p = .45, nor an interaction effect of block order and clip number, F(17, 1326) = 0.31, p = .99, we aggregated the counterfactual judgments across both groups of participants. In order to model participants’ counterfactual judgments, we created different noisy physics models by varying
the degree of Gaussian noise (from SD = 1◦ to 20◦ ) that wasapplied to perturb B’s velocity vector in the counterfactual
world (see Figure 1, clip 18).
Figure 3 shows participants’ mean counterfactual judgments together with the predictions of the best-fitting noisy
simulation model (r = .98, RMSE = 14.64 for SD = 4◦ ). The
model accurately captures that participants are certain about
the counterfactual outcome for those cases in which ball
B would clearly have missed (left column) or clearly gone
through the gate (right column). It also captures participants’uncertainty for those cases in which the counterfactual outcome would have been close (middle column). For comparison, the correlation between participants’ counterfactual
judgments and the simulation model for SD = 0◦ (i.e. nonoise), 10◦ , and 20◦ was r = .89, .89 and .65, respectively. A
model without noise fails to capture people’s uncertainty inthose cases in which the counterfactual outcome was close.
The high correlation between the noisy physics modeland participants’ counterfactual judgments demonstrates thatwithin our domain, people’s counterfactual predictions arewell approximated by a noisy version of Newtonian physics.
The finding that people had no trouble in predicting the counterfactual outcome in situations that involved the teleportdemonstrates the flexibility of people’s mental simulations(see Figure 2, clips 6, 12, and 17).
Causal judgments There was a significant interaction effect of block order and clip number on participants’ causal
judgments, F(17, 1326) = 3.29, p < .001. However, since
the qualitative pattern of results was very similar between

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Figure 4: Mean cause (white) and prevention (gray) judgments together with the predictions of the counterfactual simulation model (blackbars) separated according to the block presentation order. Labels (A) – (E) indicate pairs of clips which matched what actually happened.

the two conditions, we will first focus on the commonalities
between the conditions before discussing the differences between them.
The counterfactual simulation model predicts a close coupling between people’s counterfactual and causal judgments.
Cause and prevention judgments are predicted to increase
with people’s beliefs that the counterfactual outcome would
have been different from the actual outcome. Figure 4 shows
participants’ mean cause and prevention judgments together
with the predictions of the counterfactual simulation model
separated by block order. We reverse coded the prevention
ratings so that both cause and prevention ratings are on ascale from 0 to 100. As predicted, participants’ causal judgments differed significantly between situations in which weheld constant what actually happened but varied what would
have happened in the counterfactual world via changing theplacement of the brick (Figure 1) or turning the teleport on oroff (Figure 2). Across both conditions, prevention judgments
were significantly higher for clips in which ball B would havegone in compared to clips in which B would have missed (seepairs of clips with the labels A and B in Figures 4a and 4b),t(79) = 16.87, p < .001, d = 1.89. Cause judgments were significantly higher for the clips in which B would have missedcompared to the matched clips in which it would have gonein (see pairs C–E), t(79) = 12.87, p < .001, d = 1.44.
We used participants’ own counterfactual judgments
to determine the values of PNcaused and PN prevented forthe different clips. The counterfactual simulation model
explains participants’ causal judgments very accurately,r = .96, RMSE = 8.33. The model achieves this highexplanatory fit without the need for any free parameters.
The correlation between the model and participants’ causal
judgments is similarly high if we determine PNcaused andPN prevented based on the best-fitting noisy physics simulation
model with SD = 4◦ , r = .93, RMSE = 18.95.
Our model accurately predicts that participants’ prevention judgments increase from situations in which the counterfactual outcome was a clear miss (clips 1, 2, and 7), close(clips 3, 4, and 9), to where ball B would clearly have gonein (clips 5, 6, and 11), F(2, 158) = 160.4, p < .01. Conversely, participants’ cause judgments decreased from situations in which B would have clearly missed (clips 8, 13, and14) to where it would have been close (clips 10, 15, and 16)
to where B would clearly have gone in (clips 12, 17, and 18),
F(2, 158) = 93.05, p < .01.
Taken together, these results support both our qualitativeand quantitative hypotheses about the relationship between
counterfactuals and causal judgments. Causal judgments differed significantly between clips in which what actually happened was held constant. Furthermore, causal judgments
were closely linked to people’s uncertainty in the counterfactual outcome. Cause and prevention ratings were low when
the counterfactual outcome was believed to be identical tothe actual outcome (e.g., clips 1, 2 and 17, 18). Participants’gave intermediate judgments when they were unsure about
what the counterfactual outcome would have been (e.g., clips
3, 4 and 15, 16). Judgments were highest when participants
were confident that the counterfactual outcome would havebeen different from the actual one (e.g., clips 5, 6 and 13, 14).
Finally, the results also show that differences in how the actual outcome came about had almost no effect on participants’
judgments. This can be seen by comparing cases in which thecloseness of the actual outcome was varied but the closenessof the counterfactual outcome held constant (cf. clips withthe same outcome between different rows in Figure 4, suchas clips 13 and 14 vs. clip 8).
Order effects As can be seen by contrasting Figures 4a and4b, the overall differences between the two block orders were
small with a mean absolute difference between clips of 10.47(SD = 7.65). The counterfactual simulation model predicted
participants’ causal judgments better when they answered thecounterfactual questions first (r = 0.98, RMSE = 6.03) compared to when the order of judgments was reversed (r =0.91, RMSE = 14.26). In the causal-judgments-first condition, the correlation between counterfactual and causal judgments was significant on the individual participant level for22 out of the 40 participants with a median correlation overall 40 participants of r = .49. In the counterfactuals-first condition, the correlation was significant for 29 out of 40 participants with a median correlation of r = .64.
The differences between the two order conditions werestrongest for the situations in which B went through the gate
and it was clear that it would have gone through even in theabsence of A (i.e. clips 12, 17, and 18). Participants’ causal
judgments who experienced the causal block first were significantly higher (M = 44, SD = 33.56) compared to those who
answered the counterfactual questions first (M = 23.86, SD =30.71), t(78) = 2.8, p < .01, d = 0.57. Overall, these ordereffects suggest that there was a stronger influence of counterfactual judgments on causal judgments than vice versa.

General Discussion
The results of our experiment demonstrate that people’s
causal judgments are inextricably linked to counterfactuals.
As predicted by our counterfactual simulation model, people
make causal judgments by comparing what actually happened
with what they think would have happened in the counterfactual world in which the causal event of interest hadn’t takenplace. They use their intuitive understanding of the domain
in order to simulate what would have happened in the relevant counterfactual world. These counterfactual simulations
are not limited to physical interactions but may also include
more abstract interactions such as teleportation.
We capture people’s uncertainty in the counterfactual outcome by assuming that their mental simulations of what
would have happened are somewhat noisy (cf. Smith & Vul,2012). The counterfactual simulation model accurately predicts the close relationship between people’s uncertainty inthe counterfactual outcome and their causal judgments. People’s cause and prevention judgments increase with their subjective degree of belief that the causal event of interest madea difference to the outcome.
By contrasting situations in which we held constant what
actually happened and only varied what would have happenedin the relevant counterfactual world, our experiments constitute the strongest possible test between actualist and counterfactual theories of causal judgment. The fact that participants’ judgments differed dramatically between these cases
provides strong evidence for our counterfactual simulation
model and against the possibility of giving an adequate account of people’s causal judgments within an actualist framework.
Most participants’ causal judgments were highly correlatedwith their own counterfactual judgments and the correlation
was particularly strong for the group of participants who was
asked to make counterfactual judgments first. We take this asevidence that while most people seem to naturally consider
counterfactuals when making their causal judgments, someparticipants might have been prompted to do so via having
been asked explicitly.
Finally, let us discuss one limitation of our current model.

We explain causal judgments in terms of the subjective degree of belief that the event of interest made a difference tothe outcome. However, there are some situations in which
participants are sure that the collision between A and B madeno difference as to whether or not B would go through thegate but they still give a relatively high causal rating. In clip
17, for example, participants are certain that ball B would
have gone through the gate via the teleport if there had been
no collision with A (Figure 3). Nevertheless, participants still
say that A somewhat caused B to go through the gate. This isparticularly the case for participants who made causal judgments first (cf. Figures 4a and 4b).
One way to capture what’s going on here is to say thatpeople not only care about whether the cause was necessary
to bring about the outcome in the given situation but also
whether it was sufficient. In clip 17, the collision event was
sufficient because it would have caused B to go through thegate even if the teleport had been off. Alternatively, people
might not only consider whether an event of interest made a
difference to the outcome but also care about how it did so. In
future work, we will explore ways of incorporating the notionof sufficiency into our counterfactual simulation model.

Acknowledgments
TG and JBT were supported by the Center for Minds, Brains andMachines (CBMM), funded by NSF STC award CCF-1231216 andby an ONR grant N00014-13-1-0333.

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