713053

research-article2017

PSSXXX10.1177/0956797617713053Gerstenberg et al.Eye-Tracking Causality

Research Article
Psychological Science﻿1­–14© The Author(s) 2017
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DOI: 10.1177/0956797617713053
https://doi.org/10.1177/0956797617713053
www.psychologicalscience.org/PS

Eye-Tracking Causality
Tobias Gerstenberg1, Matthew F. Peterson1,
Noah D. Goodman2, David A. Lagnado3, andJoshua B. Tenenbaum1

1
Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology; 2Departments of
Psychology and Computer Science, Stanford University; and 3Experimental Psychology,
University College London

Abstract
How do people make causal judgments? What role, if any, does counterfactual simulation play? Counterfactual theories
of causal judgments predict that people compare what actually happened with what would have happened if thecandidate cause had been absent. Process theories predict that people focus only on what actually happened, toassess the mechanism linking candidate cause and outcome. We tracked participants’ eye movements while theyjudged whether one billiard ball caused another one to go through a gate or prevented it from going through. Bothparticipants’ looking patterns and their judgments demonstrated that counterfactual simulation played a critical role.
Participants simulated where the target ball would have gone if the candidate cause had been removed from the scene.
The more certain participants were that the outcome would have been different, the stronger the causal judgments.
These results provide the first direct evidence for spontaneous counterfactual simulation in an important domain of
high-level cognition.
Keywords
causality, counterfactuals, mental simulation, intuitive physics, eye tracking, open data, open materials
Received 7/21/16; Revision accepted 5/11/17

In soccer, scoring an own goal is probably the worst
thing that can happen to a player. An own goal occurs
when a defender tries to prevent the other team fromscoring but instead deflects the ball into his or her own
team’s goal, thereby scoring a point for the other team.
Simply having touched the ball last, however, is not
sufficient to be “awarded” an own goal. Only when theball was not already on target does the play count as
an own goal. This kind of counterfactual reasoning ispervasive in sports. To take another example, in American football, pass interference is called when a defenderillegally interferes with a receiver and thereby prevents
him or her from making an attempt at catching a pass.
However, the pass-interference rule does not apply
when a referee deems the pass uncatchable. The
defender’s action must have made a difference to theoutcome. Of course, counterfactuals arise not only insports. We use them to make sense of history (Ferguson,
2000), to determine causation in the law (Hart &Honoré, 1959/1985), and to understand our own andother people’s actions and emotions (Alicke, Mandel,

Hilton, Gerstenberg, & Lagnado, 2015; Kahneman &Miller, 1986; Roese, 1997). We ponder over near misses(Kahneman & Varey, 1990) and regret decisions that
could have turned out better (Loomes & Sugden, 1982;
Zeelenberg et al., 1998).
The relationship between counterfactuals and causation has been a topic of long-standing debate inphilosophy (Beebee, Hitchcock, & Menzies, 2009;
Hiddleston, 2005; Paul & Hall, 2013), psychology (Lipe,
1991; Walsh & Sloman, 2011; Wolff, 2007), and the law(Hart & Honoré, 1959/1985; Schaffer, 2010; Stapleton,
2008). In philosophy, there are two broad theoreticalframeworks for thinking about causation: Process theories (e.g., Dowe, 2000) analyze causation in terms ofspatiotemporally continuous processes that link causes
to their effects. For example, when asked whether one
Corresponding Author:
Tobias Gerstenberg, Department of Brain and Cognitive Sciences,
Massachusetts Institute of Technology, Building 46-4053, 77
Massachusetts Ave., Cambridge, MA 02139
E-mail: tger@mit.edu

Gerstenberg et al.

2billiard ball (A) caused another ball (B) to move, a
process theorist would seek to establish whether aphysical quantity (such as momentum) was transferredfrom ball A to ball B. In contrast, counterfactual theories
capture causation by establishing whether the candidate cause made a difference to the outcome. A counterfactual theorist would say that A caused B to movebecause B would not have moved if A had not been
there (Lewis, 1973). Previous empirical work on how
people make causal judgments has yielded mixed results.
Sometimes, participants’ judgments are influenced byinformation about causal processes (Lombrozo, 2010;
Walsh & Sloman, 2011; Wolff, 2007), whereas at othertimes, participants care mostly about what would have
happened if the cause had been absent (Gerstenberg,
Goodman, Lagnado, & Tenenbaum, 2012, 2014).
Past research has shown that counterfactual thoughts
are triggered when experienced outcomes are negative,
unexpected, or close to alternative outcomes (Roese,1997). Researchers have employed different methods
to study spontaneous counterfactual thinking, such as
having participants list their thoughts (Sanna & Turley,1996) or measuring the speed with which they respondto different counterfactual statements (Roese & Olson,
1997). However, counterfactual thinking has never beenshown directly. In this article, we demonstrate how eye
tracking can be used to uncover evidence for a particular kind of counterfactual thinking, which we refer toas counterfactual simulation (Crespi, Robino, Silva, &de’Sperati, 2012; Hegarty, 1992; Johansson, Holsanova,& Holmqvist, 2006; Kahneman & Tversky, 1982). As acase study, we focus on how people make causal
judgments about physical interactions. We show thatpeople’s causal judgments mirror how own goals are
decided in soccer: People compare what actually
happened with what would have happened in the
counterfactual situation in which the candidate cause
was absent.
We first describe the experimental paradigm we usedin our study and then discuss model predictions separately for each experimental condition. In the Results
section, we report our analyses of participants’ judgments, their eye movements, and the relationshipbetween their eye movements and judgments. We conclude by discussing the implications our findings havefor theories of causal judgment specifically, and forcausal cognition more generally.

Experimental Paradigm
In our experiment, participants watched video clips inwhich two balls, A and B, collided with each other (seeFig. 1). The clips differed in whether ball B subsequently

went through a gate (actual hit), barely went throughthe gate or missed the gate by a very small distance(actual close call), or missed the gate by a larger distance(actual miss). The clips also differed in what would have
happened if ball A had not been present in the scene;
specifically, they differed in whether ball B would haveclearly missed the gate (counterfactual miss), would
have just gone through or nearly gone through (counterfactual close call), or would have clearly gone throughthe gate (counterfactual hit). For example, in Clip 3,ball B clearly missed the gate in the actual situation(actual miss) and would have just missed the gate in the
counterfactual situation in which ball A had not beenpresent in the scene (counterfactual close call).
In a between-subjects design, we varied whether
participants answered (a) a counterfactual question
about what would have happened, (b) a causal question, or (c) a question about the actual outcome. Participants in the counterfactual condition rated theextent to which they agreed with the statement “ball B
would have gone through the gate if ball A had notbeen present in the scene.” In the causal condition,
participants judged the extent to which they agreed
with the statement “ball A caused ball B to go throughthe gate” (when ball B went through the gate) or thestatement “ball A prevented ball B from going throughthe gate” (when ball B did not go through the gate).
We made sure to avoid any reference to counterfactual
language in this condition. In the outcome condition,
participants were asked to indicate their agreementwith the sentence “ball B completely missed the gate”(when B did not go in) or the sentence “ball B went
through the middle of the gate” (when B went in).
Participants in all three conditions responded on a scalefrom 0 (not at all) to 100 (very much).

Theoretical Predictions
In previous work (Gerstenberg et al., 2012, 2014), we
developed the counterfactual simulation model, which
predicts causal judgments by simulating what would
have happened if the cause had been absent and thencomparing this counterfactual outcome with what actually happened. In the case of physical causation considered here, we assume that observers make use of
their intuitive understanding of physics to simulate
what would have happened in the relevant counterfactual situation. In this section, we discuss the model’s
predictions in the counterfactual and causal conditions,both for behavioral judgments and for eye movements.
We also discuss a model that we developed to makepredictions about participants’ judgments in the outcome condition.

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Fig. 1. Diagrams illustrating the 18 test clips shown to participants. Solid arrows indicate the motion paths of balls A and B prior to their collision and the path of B’s motion
after the collision. Dashed arrows indicate where ball B would have gone if ball A had not been present in the scene. From top to bottom, the rows show situations in which B
clearly missed the gate (actual miss), just missed the gate or only just went through (actual close call), and clearly went through the gate (actual hit). From left to right, the column
pairs show situations in which B would have clearly missed the gate if ball A had not been present (counterfactual miss), would have just missed the gate or would have only
just gone through (counterfactual close call), and would have clearly gone through the gate (counterfactual hit). Note that the font size of the balls’ labels has been increased
here for readability.

Actual Hit

Gerstenberg et al.

4Counterfactual condition
Our model assumes that participants make use of theirintuitive understanding of physics to simulate what
would have happened if ball A had not been present
in the scene and also that this intuitive understanding
takes the form of a runnable mental model (Craik,1943) that can be implemented to a first approximation
via the physics engine used to generate the stimulus
clips (Battaglia, Hamrick, & Tenenbaum, 2013; Gerstenberg et al., 2012, 2014; Gerstenberg & Tenenbaum,
2017; Goodman, Tenenbaum, & Gerstenberg, 2015;
Sanborn, Mansinghka, & Griffiths, 2013; Smith & Vul,
2013). Specifically, to capture participants’ uncertainty
in whether ball B would have gone through the gate,
the model generates noisy samples from the underlying
physics engine. In each sample, the model removes ball
A from the scene and introduces a small degree ofGaussian noise to ball B’s direction of motion at eachtime step in the physics simulation after the point at
which the two balls would have collided. It is at thispoint in time that the counterfactual situation diverges
from the actual situation that participants saw. For eachsample, the model records whether ball B went throughthe gate or missed it. We used the proportion of samples in which B went through the gate to predict participants’ counterfactual judgments. The black bars in
Figure 2a show the predictions of this approximate
physics simulation model for the counterfactual condition. The model predicts that counterfactual judgments
will be affected by how close the counterfactual outcome would have been (i.e., the columns in Fig. 1) but
that how close the actual outcome was (i.e., the rows
in Fig. 1) will make no difference.
Because participants were explicitly asked to judge
whether ball B would have gone through the gate ifball A had not been present in the scene, we expected
their eye movements to reveal attempts to extrapolate
where ball B would have gone if ball A had not beenpresent.

Causal condition
Given the results of our previous work (Gerstenberg
et al., 2012, 2014), we expected a close correspondence
between participants’ counterfactual and causal judgments. The black bars in Figure 2b show the predictionsof the counterfactual simulation model of causal judgment. The model specifies the probability that a candidate cause C (ball A in our case) caused a particularoutcome e (ball B going through the gate or ball B
missing the gate) as
P(C → e) = P(e′ ≠ e | S, remove(C )),(1)

where S represents the physical information about what
actually happened, and e′ denotes the counterfactual
outcome that would have happened if C had been
removed from the scene. According to this model, participants’ agreement with the causal statement should
increase the more certain they are that the outcome inthe counterfactual situation (e′) would have been different from the actual outcome (e). In the analyses
reported here, we used participants’ mean judgments inthe counterfactual condition to determine the probability that the outcome would have been different for eachclip, and then used these probabilities to predict participants’ mean judgments in the causal condition. 1
For example, in Clip 1 (see Fig. 1), ball B missed thegate, and participants in the counterfactual condition
believed that B would have missed the gate even if ball
A had not been present in the scene (see Fig. 2a).
Because the presence of ball A made no difference towhether ball B missed the gate, the model predicts thatparticipants should indicate that ball A did not preventball B from going through (Fig. 2b). In Clip 5, ball B
again did not go through the gate, but this time, participants were confident that it would have gone
through if ball A had not been present. Thus, the model
predicts that participants should indicate that ball A
prevented ball B from going through the gate in thiscase. In Clip 9, ball B missed the gate, but participantsin the counterfactual condition were less certain about
what would have happened if ball A had not beenpresent in the scene. Because it is unclear whether ball
B would have gone through the gate if ball A had notbeen there (i.e., the counterfactual probability in Equation 1 is close to .5), the model predicts an intermediate
rating.
The same relationship between participants’ confidence in ball A’s having made a difference to the outcome and their agreement ratings is also predicted forthe situations in which ball B ended up going throughthe gate. The model predicts that agreement with thestatement that ball A caused ball B to go through thegate should increase the more certain participants inthe counterfactual condition were that ball B would
have missed otherwise. In sum, we expected participants’ judgments in the causal condition to be influenced by the actual outcome (whether ball B went
through or missed the gate) and by how clear it waswhat would have happened if ball A had been removed.
Given the hypothesized connection between causal
judgments and counterfactual reasoning, we predictedthat participants in the causal condition would spontaneously engage in counterfactual simulation to gauge
whether the outcome would have been different if thecandidate cause had been removed from the scene.
Thus, we expected to observe eye movements to whereEye-Tracking Causality

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Predictions of the Approximate Physics Simulation ModelcCounterfactual
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B Went Through the Gate

Predictions of the Spatial-Distance Model
Fig. 2. Participant’s mean agreement judgments (0 = “not at all,” 100 = “very much”) and model predictions for each of the 18 video clips
in the (a) counterfactual condition, (b) causal condition, and (c) outcome condition. In the counterfactual condition, participants wereasked whether ball B would have gone through the gate if ball A had not been present; striped bars indicate situations in which ball B
would have missed the gate, and solid bars indicate situations in which ball B would have gone through the gate. In the causal condition, participants were asked whether ball A prevented ball B from going through the gate when ball B missed the gate (striped bars) orwhether ball A caused ball B to go through the gate when ball B went through (solid bars). In the outcome condition, participants wereasked whether ball B completely missed the gate when it did not go in (striped bars) or whether ball B went through the middle of thegate when ball B did go in (solid bars). Error bars indicate bootstrapped 95% confidence intervals.

Gerstenberg et al.

6ball B would have gone if ball A had been absent, asin the counterfactual condition.

Outcome condition
To predict participants’ agreement judgments in theoutcome condition, we developed a simple spatialdistance model that computes the euclidean distance
between the center of the gate and the point at whichball B hit the wall or went through the gate. The model
predicts that participants’ agreement ratings for situations in which ball B went through the gate should
increase the closer ball B was to the center of the gatewhen it went through (see Fig. 2c). Agreement ratings
for situations in which ball B missed the gate are predicted to increase the greater the distance was between
the point at which ball B hit the wall and the center ofthe gate. The spatial-distance model predicts that participants’ judgments should be influenced only by thecloseness of the actual outcome (i.e., the rows in Fig.
1) and not by the closeness of the counterfactual outcome (i.e., the columns in Fig. 1).
Judging how closely ball B was to the center of thegate when it went through or how far ball B was fromthe gate when it missed did not require simulating what
would have happened if ball A had been absent. Thus,
we predicted that participants in the outcome condition
would be less likely to engage in counterfactual simulation than participants in the other two conditions.
Accordingly, in the outcome condition, we expected tosee fewer eye movements in the direction of where ball
B would have gone if ball A had been absent.

loss after the removal of the additional participants was
9.39% (SD = 6.65) in the counterfactual condition,
7.31% (SD = 3.49) in the causal condition, and 8.27%(SD = 3.35) in the outcome condition. Thirty participants (mean age = 36 years, SD = 15.9; 9 female) were
included in the analyses reported here. 2

DesignParticipants were randomly assigned to the counterfactual, causal, or outcome condition (n = 10 participants
per condition). The conditions differed only in whatquestion participants were asked about ball B. Participants in all three conditions saw the same set of clips
illustrated in Figure 1. The clips varied in how close
the actual outcome was, from a clear miss (top row) toa close miss or hit (middle row) to a clear hit (bottomrow), as well as in how close the counterfactual outcome would have been, from a clear counterfactual
miss (leftmost two columns) to a close counterfactual
miss or hit (middle two columns) to a clear counterfactual hit (rightmost two columns). Thus, the experiment
had a 3 (condition: counterfactual, causal, outcome) ×3 (actual outcome’s closeness: miss, close call, hit) × 3(counterfactual outcome’s closeness: miss, close call,hit) design, with condition varied between participants
and closeness of the actual and counterfactual outcomes varied within participants. As Figure 1 shows,
we included two clips for each combination of theactual outcome’s closeness and the counterfactual outcome’s closeness.

Materials and procedure
MethodParticipants
Forty participants were recruited from the Massachusetts Institute of Technology’s participant pool and paid
for their participation. We determined the target numberof participants on the basis of our previous work
(Gerstenberg et al., 2012), which established verystrong behavioral effects in the counterfactual andcausal conditions using the same materials employedin this study. A power analysis conducted with G*Power
3.1 (Faul, Erdfelder, Lang, & Buchner, 2007) showedthat fewer than 10 participants were required for .95
power to detect the weakest effect (f = 1.01) found in
our previous work, given an α level of .05. To haveequal numbers of participants in the three conditions,
we kept the 10 participants with the least amount ofeye-tracking data loss (e.g., due to blinking, head
movement, or looking away from the monitor) in eachcondition. The mean percentage of eye-tracking data

The video clips were generated with the Flash implementation (http://www.box2dflash.org/) of the Box2D
physics engine (E. Cato, box2d.org). The experiment
was programmed in MATLAB (The MathWorks, Natick,MA) using the Psychophysics Toolbox (Brainard, 1997).
The video clips were presented at 1000 × 750 pixels,
centered on a screen with 1024 × 768 pixels; 75 pixels
on the screen corresponded to 1 m in the physics simulation. Each ball’s radius was 0.5 m.
The right eye of each participant was tracked using
an SR Research (Kanata, Ontario, Canada) EyeLink 1000
Desktop Mount sampling at 1000 Hz. Each 10-s video
clip was played at 30 frames per second. We recorded
the average eye position for each frame in each video.
Participants sat 50 cm from the monitor, and each pixel
subtended 0.036°. A nine-point calibration and a validation were run at the beginning of each session.
After completing the calibration and validation, participants received instructions about the task. They
were told that they would watch video clips of two

Eye-Tracking Causality
colliding billiard balls on a stage that featured walls
and a gate. They were also told what question theywould be asked on each trial, according to the experimental condition to which they had been assigned.
Prior to the 18 test clips (see Fig. 1), which were presented in randomized order, participants watched 2
practice clips. In one, ball B went through the gate but
would have clearly missed if ball A had not been present. In the other, ball B missed the gate but would haveclearly gone through if ball A had not been present.
Participants saw each clip twice before making their
judgment. On average, participants took 14 min (SD =0.91) to complete the experiment (excluding the timeit took to calibrate the eye tracker).

Results
We first discuss participants’ judgments separately forthe three experimental conditions. We then analyze
participants’ eye movements in each condition and lookat the relationship between eye movements and judgments in the causal condition.

Behavioral judgments
Counterfactual condition. The predictions of the ap­proxi­mate physics simulation model closely corresponded toparticipants’ counterfactual judgments, r = .92, rootmean-square error (RMSE) = 12.57 (see Fig. 2a).3 The
model that best explained participants’ counterfactual
judgments added Gaussian noise with a standard deviation of 0.7° to ball B’s direction of motion at each time
step after the collision.
As predicted by the approximate physics simulation
model, agreement ratings varied as a function of thecloseness of the counterfactual outcome (the columns
in Fig. 2a), F(2, 18) = 79.51, p < .001, ηp2 = .90. Participants tended to agree with the statement that “ball B
would have gone through the gate if ball A had notbeen present in the scene” for the counterfactual-hit
clips and tended to disagree with this statement for thecounterfactual-miss clips. For the cases in which thecounterfactual outcome was close, participants were lesscertain about whether ball B would have gone throughthe gate without ball A, and were somewhat biased tobelieve that ball B would have missed the gate. Agreement ratings did not vary significantly as a function of
how close the actual outcome was, F(2, 18) = 2.74, p =.091, ηp2 = .23.
Causal condition. The counterfactual simulation model
accurately predicted participants’ causal judgments, r =.92, RMSE = 10.8 (see Fig. 2b). As predicted by the model,
participants’ agreement with the statement that “ball A
prevented ball B from going through the gate” increased

7
the more certain it was that B would have gone throughthe gate if ball A had not been present, F(2, 18) = 21.86,
p < .001, ηp2 = .71. How closely ball B missed the gate (actual
miss vs. actual close call) had no effect on participants’prevention judgments, F(1, 9) = 0.02, p > .250, ηp2 = 0.
Similarly, participants’ agreement with the statementthat “ball A caused ball B to go through the gate”
increased the more certain it was that ball B would havemissed the gate if ball A had not been present in thescene, F(2, 18) = 15.78, p < .001, ηp2 = .64. Again, howclose ball B had been to missing the gate (actual close
call vs. actual hit) had no effect on participants’ causal
judgments, F(1, 9) = 0.21, p > .250, ηp2 = 02.
Note that the tendency of participants in the counterfactual condition to believe that ball B would havemissed the gate (in the absence of ball A) when the
counterfactual outcome was close (see Fig. 2a, middle
column) was mirrored in the causal judgments (see Fig.
2b, middle column). Because participants believed thatball B would have missed in these cases, they tended
to agree that ball A caused ball B to go through thegate when ball B went through and to disagree that ballA prevented ball B from going through the gate whenball B missed.
Outcome condition. The spatial-distance model accounted
well for participants’ agreement ratings in the outcome
condition, r = .87, RMSE = 12.84 (see Fig. 2c). As predicted by the model, participants tended to agree with thestatement that “ball B completely missed the gate” when
the actual outcome was a clear miss (top row) and tended
to disagree with this statement when the actual outcome
was close (middle row), F(1, 9) = 40.89, p < .001, ηp2 = .82.
Participants’ judgments were unaffected by the closenessof the counterfactual outcome, F(2, 18) = 0.26, p > .250,
ηp2 = .03.
When ball B went through the gate, participants
agreed with the statement that “ball B went through the
middle of the gate” when the outcome was a clear hit(bottom row), but tended to disagree when B only
barely went in (middle row), F(1, 9) = 17.92, p = .002,
ηp2 = .67. In contrast to our predictions, participants’
judgments also differed as a function of the closenessof the counterfactual outcome, F(2, 18) = 7.53, p = .004,
ηp2 = .46. This effect was mostly driven by Clip 17, in
which ball B did not go right through the middle of thegate (despite the outcome being classified as an actual
hit; see Fig. 1).
Note that in Clip 10, ball B first bounced off the edgeof the gate and then went through the center. Because
the model considered only the final location at which
the ball went through the gate, it predicted a relatively
high agreement rating for this case. However, participants were less inclined to agree that B went throughthe middle of the gate than the model predicted.

Gerstenberg et al.

Ball B Goes
Through the Gate

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Causal Condition

Outcome Condition

Counterfactual Saccade (17%)
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Counterfactual Saccade (28%)
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Fig. 3. Endpoints of saccades that were made before the two balls collided in Clip 10 (top) and Clip 4 (bottom). The plot for eachcondition (counterfactual, causal, and outcome) includes the endpoints of all 10 participants’ saccades from the two trials on which
each participant saw the clip. The key at the top of each plot indicates the percentage of each type of saccade.

Eye movements
We expected that participants in the counterfactual andcausal conditions would engage more in counterfactual
simulation than would participants in the outcome condition. To test this prediction, we analyzed participants’eye movements in two complementary ways. 4 First, we
discuss the results of a static analysis that classified
where participants looked on the screen up until thetime at which balls A and B collided. Second, we show
the results of a dynamic analysis that used a probabilistic
model to classify participants’ gaze position on thescreen at each point in time. The static analysis had theadvantage that it was simple and required few assumptions to classify participants’ looks. The dynamic analysis allowed us to classify where participants looked at
any given moment and provides a more fine-grained
view of what participants were attending to over time.
However, the dynamic analysis contained a number offree parameters and thus required us to make moreassumptions about how the data were generated.
Static analysis: saccades prior to collision. In thisanalysis, we focused on the endpoints of saccades. We
defined a saccade as an eye movement with velocity
greater than 22°/s and acceleration exceeding 4000°/sec2.
We classified a saccade as counterfactual if its endpoint
was at least 50 pixels to the left of the point at which thetwo balls collided and within 100 pixels of the path thatball B would have taken if ball A had not been present in
the scene. We refer to the rest of the saccades simply asother saccades.
We constrained our analysis to saccades that weremade before the two balls collided because without
taking into account the temporal dynamics of theclip, it was unclear whether later saccades along ball
B’s counterfactual path were indeed counterfactual
saccades to ball B or simply saccades directed at ball
A’s current location. Figure 3 shows all participants’saccades for Clip 10 (top) and Clip 4 (bottom) prior
to the collision; results are separated by condition
and aggregated over both instances in which eachparticipant saw each clip. For these clips, the percentage of counterfactual saccades was much greaterin the counterfactual and causal conditions than in
the outcome condition. Across the 18 test clips, thepercentage of counterfactual saccades differed significantly between conditions, χ 2(2, N = 30) = 302.49,
p < .001. Participants in the causal condition (M =23%, SD = 11.24) and the counterfactual condition(M = 22%, SD = 9.75) made more counterfactual saccades than did participants in the outcome condition(M = 6%, SD = 4.78), β = −1.76, p < .001, and β =−1.78, p < .001, respectively. There was no evidence
for a difference between the counterfactual andcausal conditions, β = 0.02, p > .250.
Note that because we are focusing on saccadesthat were made prior to the collision, what we callEye-Tracking Causality
counterfactual saccades are strictly speaking saccades
to ball B’s position in a future hypothetical situation.
These saccades indicate participants’ attempts to simulate prospectively where ball B would go if ball A were
not present in the scene (rather than to retrospectively
simulate, after the balls collided, where ball B would
have gone had A not been present). We return to thispoint in the General Discussion.
Dynamic analysis: hidden Markov model of participants’ looks. In order to gain more insight into
what participants were attending to, we implemented ahidden Markov model (HMM) that dynamically classified
their eye movements.5 An HMM is a probabilistic modelthat specifies a hidden process that generates the observable data. In our case, we observed the x and y coordinates of participants’ gaze positions on the screen overtime and inferred the sequence of different types of looks
that each participant was making. At each point in time,a participant could simply have been looking at one of
the balls or could have been trying to predict where aball was going. The participant could also have been trying to predict where one ball would go if the other ball
had not been present in the scene. To classify participants’ eye movements, we defined seven different typesof looks, which we describe here informally:•• A look: gaze directed close to the actual position of ball A•• B look: gaze directed close to the actual position of ball B•• A prediction: gaze directed to where ball A
will go•• B prediction: gaze directed to where ball B
will go•• A counterfactual look: gaze directed to whereball A would go if ball B were not present in
the scene•• B counterfactual look: gaze directed to whereball B would go if ball A were not present in
the scene•• Other: gaze directed anywhere else on thescreenFigures 4a and 4b illustrate how a participant’s gaze
while viewing Clip 4 would be classified at two different time points (any gaze position outside the regions
shown would be classified as “other”). For each possible gaze position on the screen, the HMM yielded aposterior probability over the different types of looks.
Participants’ looks were classified in the same way forthe three different conditions.
Figure 4c shows the probability of participants making each type of look, separately for each condition
and averaged across all clips. We analyzed participants’

9
gaze positions up until the time when the outcome
event happened, that is, when ball B went throughthe gate or missed the gate by hitting one of the
nearby walls. Again, we focused our analysis on theendpoints of saccades. We conducted a separate analysis of variance for each type of look using a Bonferroni-adjusted alpha level of .007 (α = .05/7). There
was a marginally significant influence of experimental
condition on the probability that participants lookedat ball B, F(2, 27) = 5.59, p = .009, η p2 = .29. Experimental condition significantly affected the probabilities that participants looked predictively at where ball
A would go, F(2, 27) = 6.46, p = .005, ηp2 = .32, andthat they made counterfactual looks to where ball B
would have gone, F(2, 27) = 17.63, p < .001, η p2 = .57.
None of the other types of looks were significantly
affected by condition.
We used post hoc tests with Bonferroni-adjustedalpha levels of .017 (α = .05/3) to compare probabilities
between each pair of conditions. There was no significant difference between the probabilities in the counterfactual and the causal conditions for the looks thatdiffered significantly between conditions. However, thelooking pattern differed between those two conditions
and the outcome condition. As predicted, participantsin the counterfactual and causal conditions were more
likely to make counterfactual looks to where ball B
would have gone (B counterfactual look) than wereparticipants in the outcome condition, t(27) = 5.33, p <.001, d = 2.85, and t(27) = 4.94, p < .001, d = 2.26,
respectively. Furthermore, compared with participantsin the outcome condition, participants in the counterfactual and causal conditions were more likely to predict where ball A would go (A prediction), t(27) = 3.14,
p = .004, d = 1.57, and t(27) = 3.08, p = .005, d = 1.50,
respectively. The finding that participants were more
likely to predict where ball A would go in the counterfactual and causal conditions is likely due to the factthat ball A’s actual path after the collision overlappedwith ball B’s counterfactual path in many of the clips(ball A’s postcollision paths can be extrapolated from
Fig. 1). Thus, the HMM assigned some probability to
predictive looks to ball A when participants may havebeen making counterfactual looks to ball B (and viceversa; cf. Fig. 4b).

Relationship between eye movements
and causal judgments
The counterfactual simulation model predicts that participants make causal judgments by comparing what
actually happened with what would have happened if
the candidate cause had been removed from the scene.
In the previous section, we reported that participantsin the causal condition looked significantly more atGerstenberg et al.

10

a

b
HMM Classification Regions
for Clip 4 at an Early Time Point

HMM Classification Regions
for Clip 4 at a Later Time Point
Gaze
Position

Gaze
PositioncClassification of Looks Across
All Clips Separated by Condition
Counterfactual Condition

Causal Condition

Outcome Condition

Probability

40%
30%
20%
10%
0%
A LookB LookA Prediction

A Counterfactual

B Counterfactual

Other

B Prediction

Fig. 4. Illustration of how the hidden Markov model (HMM) dynamically classified participants’ eye movements andresults of this analysis. The images in (a) and (b) illustrate, for Clip 4, where in the scene a participant’s gaze would be
expected to be if the participant looked directly at ball A or B, predicted where ball A or B would be, or simulated thecounterfactual of where ball A or B would be if the other ball had not been present in the scene. The two images show
how the classification regions changed dynamically as the clip progressed from an earlier time point to a later time point.
For the gaze position in (a), the HMM assigned a high probability that the participant was looking at where ball B would
have gone if ball A had not been present in the scene (B counterfactual look). For the gaze position in (b), the HMM
assigned some probability that the participant was predicting where ball B would go (B prediction) and some probability
that the participant was looking at where ball B currently was (B look). The bar graph (c) shows the probability of eachtype of look in each experimental condition, averaged over all participants and clips. The data bars for counterfactual
looks to B are highlighted by a thick outline. Error bars indicate bootstrapped 95% confidence intervals.

where ball B would have gone if ball A had not beenpresent than did participants in the outcome condition.
In this section, we focus on the relationship between
individual participants’ eye movements and their causal
judgments. The counterfactual simulation model predicts that participants’ causal judgments should be more
extreme in situations in which the outcome in the relevant counterfactual situation is clear compared withsituations in which what would have happened is lessclear (cf. the left and right columns in Fig. 2b with the
middle column). If participants were motivated toassess whether ball A made a difference to whether ball
B went through the gate, they would also likely havemade more counterfactual looks to where ball B would
have gone when the counterfactual outcome was
unclear than when it was clear. Thus, the modelEye-Tracking Causality
predicted a relationship between counterfactual looks
and certainty in causal judgments. The more uncertain
the counterfactual outcome was, the more we expectedparticipants to look at whether ball B would have gone
through the gate had ball A not been present, and theless extreme we expected their causal judgments to be.
Indeed, the probability that participants made counterfactual looks to ball B (as determined by the HMM
model) differed significantly as a function of how closethe counterfactual outcome would have been, F(2, 18) =9.50, p = .002, ηp2 = .51. Participants were more likely tomake counterfactual looks to ball B when the outcome
would have been close than when ball B would haveclearly missed the gate, t(18) = −3.37, p = .003, d = 0.82,
or clearly gone through, t(18) = 4.08, p = .001, d = 1.05.
And, as predicted, there was a negative correlation
between the overall probability per clip that participants
looked at where ball B would have gone and the certainty
in their causal judgment, r = −.31, p < .001. The moreparticipants made counterfactual looks to ball B, the lessextreme their causal judgment was.
Another way of assessing the relationship between
eye movements and causal judgments is by looking at
how well the counterfactual simulation model explainsparticipants’ causal judgments as a function of theirlooking patterns. Specifically, we expected the model
to predict participants’ causal judgments well only tothe extent that they actually engaged in counterfactual
simulations. To test whether this was the case, we first
calculated how well the counterfactual simulation
model fitted each participant’s causal judgments. We
then looked at the relationship between each participant’s model fit and the extent to which he or she
engaged in counterfactual simulation (again, as determined by the HMM). As predicted, the causal judgmentsof participants who made more counterfactual looks
were better explained by the counterfactual simulation
model, r = .79, p = .007.

General Discussion
When participants watch dynamic collision events
unfold, they look at what happens and anticipate what
will happen in the near future. Participants who are
asked to make causal judgments do more than that:
They use their intuitive understanding of physics tomentally simulate what would have happened if thecandidate cause had been removed from the scene. In
our paradigm, participants extrapolated the target ball’s
counterfactual motion path in an attempt to establish
whether the candidate cause made a difference to theoutcome. Would the target ball have gone through thegate even if the candidate cause had not been there?
The more certain participants were that the counterfactual outcome would have been different from the actual

11
outcome, the more they agreed with the statement thatone ball caused the other to go through the gate, or
prevented it from going through.
Although the claim that counterfactual reasoning andcausal judgments are related is not new, our results
demonstrate, for the first time, how close this relationship actually is. First, as predicted by the counterfactual
simulation model, there was a very high quantitative
correspondence between participants’ counterfactual
judgments in one condition and participants’ causal
judgments in another condition (cf. Gerstenberg et al.,2012, 2014). Second, by tracking participants’ eye movements, we saw that participants in the causal condition
spontaneously anticipated where the target ball would
have gone in the counterfactual situation in which thecandidate cause was absent. These counterfactual looks
happened much less frequently when participants wereasked to evaluate the actual outcome (for additional
evidence of task-related effects on eye movements, see
Castelhano, Mack, & Henderson, 2009; Peterson &Eckstein, 2012). Overall, we found a remarkable similarity in looking patterns between the causal and counterfactual conditions, and a very different pattern oflooks in the outcome condition (see Figs. 3 and 4).
But do our results really demonstrate counterfactual
simulation? The finding that participants’ eye movements were extremely similar in the causal and counterfactual conditions suggests that participants in thesetwo conditions may have engaged similar cognitive
processes. However, as mentioned earlier, participantsin the causal condition often tried to simulate whereball B would go before the two balls collided. One
might argue that participants’ eye movements are thus
better characterized as future-directed hypotheticalsimulations than as counterfactual simulations. We
believe that these eye movements can be understoodas counterfactual simulations, and indeed may havebeen the best means participants had to judge the relevant counterfactual probabilities. Note that participants could answer the causal question only after theoutcome had actually occurred. By simulating the outcome on-line, rather than waiting until the end of theclip, participants were better able to acquire the information they needed in order to answer the causal question they would be asked later. At that later point, therelevant information they had computed earlier provided the counterfactual contrast they needed in orderto make their causal judgment. It is plausible that participants continued to mentally simulate what would
have happened even after having seen a clip.
How do the results of our experiment speak to thedebate about whether causal judgments are better
explained by process theories or counterfactual theories? Our results show that the counterfactual simulation
model adequately captures people’s causal judgments,

Gerstenberg et al.

12
and we have shown in other work that this kind of
simulation forms a necessary component of people’s
causal judgments (Gerstenberg et al., 2014). But we also
know from previous work that a simple counterfactual
contrast between what actually happened and what
would have happened if the candidate cause had notbeen present is not sufficient (Gerstenberg & Lagnado,
2010; Halpern, 2016; Lagnado, Gerstenberg, & Zultan,
2013; Wolff, 2007; Zultan, Gerstenberg, & Lagnado,2012). Many outcomes, such as the outcomes of elections, are causally overdetermined, so that the absenceof an individual cause would not have made a difference to the outcome.
The situations we have focused on here featured asingle candidate cause. Cases with multiple candidate
causes have been especially challenging for simple
counterfactual accounts (Paul & Hall, 2013; Walsh &Sloman, 2011). In other work (Gerstenberg, Goodman,
Lagnado, & Tenenbaum, 2015, 2017), we have shownhow people’s causal judgments in more complex settings that involve multiple candidate causes can be
explained by considering not only whether the outcome would have been different if the cause had beenabsent, but also how the outcome actually came about.
Whereas counterfactual theories have traditionallyfocused on the whether, and process theories havefocused on the how, we have shown that counterfactual
theories can capture both aspects of causation by considering not only what would have happened if thecause had been absent, but also what would have
happened if the cause had been slightly perturbed (cf.
Woodward, 2011).

Conclusion
Psychologists have long argued that counterfactual
thoughts play an important role in how people makesense of the world. Although we all have a rich innerexperience with counterfactual thoughts, this study isthe first to show direct evidence for spontaneous counterfactual simulation as it happened. When asked tomake causal judgments, people compare what actually
happened with their mental simulation of what would
have happened if the candidate cause had not beenpresent.
Action Editor
Marc J. Buehner served as action editor for this article.

Author Contributions
T. Gerstenberg and J. B. Tenenbaum developed the study
concept. All the authors contributed to the study design. Testing and data collection were performed by T. Gerstenberg and
M. F. Peterson. T. Gerstenberg performed the data analysis andinterpretation under the supervision of J. B. Tenenbaum andN. D. Goodman. T. Gerstenberg drafted the manuscript. M. F.
Peterson, D. A. Lagnado, N. D. Goodman, and J. B. Tenenbaum
provided critical revisions. All the authors approved the final
version of the manuscript for submission.

Acknowledgments
We thank Eliza Kosoy for help with running the study; Nori
Jacoby, Max Kleiman-Weiner, Peter Krafft, Kevin Smith, andTomer Ullman for helpful discussions; and the members ofMIT’s Computational Cognitive Science lab for constructive
feedback.

Declaration of Conflicting Interests
The authors declared that they had no conflicts of interest
with respect to their authorship or the publication of thisarticle.

Funding
This work was supported by the Center for Brains, Minds &Machines (CBMM), which is funded by the National Science Foundation’s Science and Technology Center (Award CCF-1231216),
and by an Office of Naval Research grant (N00014-13-1-0333).

Supplemental Material
Additional supporting information can be found at http://
journals.sagepub.com/doi/suppl/10.1177/0956797617713053

Open Practices

All data and materials, including analyses, videos, and additionalplots, have been made publicly available via the Open Science
Framework and can be accessed at https://osf.io/du5jc/. The
complete Open Practices Disclosure for this article can be foundat http://journals.sagepub.com/doi/suppl/10.1177/095679761771
3053. This article has received badges for Open Data andOpen Materials. More information about the Open Practices
badges can be found at http://www.psychologicalscience.org/
publications/badges.

Notes
1. We would have also been able to use the predictions of
the approximate physics simulation model, which captures participants’ counterfactual judgments, to get the counterfactual
probabilities required by the counterfactual simulation model
as detailed in Equation 1. However, because we asked participants in the counterfactual condition to judge whether they
thought ball B would have gone through the gate if ball A
had not been present in the scene, we instead directly mapped
these ratings onto participants’ prevention judgments for theclips in which ball B missed the gate. The model predicts thatprevention judgments should increase the more certain participants are that ball B would have gone through the gate. To
predict participants’ causal judgments for situations in whichEye-Tracking Causality
ball B went through the gate, we subtracted participants’ counterfactual judgments in these situations from 100 (the maximum of the scale) to capture their belief that ball B would havemissed the gate if ball A had not been present. Participants’causal judgments when ball B went through the gate were predicted to increase the more certain they were that B would havemissed the gate if ball A had not been present. More generally,
whereas the approximate physics simulation model yields theprobability of a particular counterfactual outcome, the counterfactual simulation model of causal judgment captures whether
the counterfactual outcome would have been different fromwhat actually happened.
2. Because of a misunderstanding, in the initial protocol underwhich this experiment was run, the use of the eye tracker was
not approved by the university’s institutional review board(IRB) and was not mentioned in the approved consent form
signed by participants. Retrospectively, the IRB acknowledgedthat participants had given verbal consent for use of the eye
tracker before beginning the experiment and approved the eyetracking data for inclusion in this publication.
3. Note that for all three conditions, we used a linear transformation (α0 + α1 × prediction) to map the model’s predictions
onto participants’ response scale.
4. Example videos of participants’ eye movements in the counterfactual (Video S1), causal (Video S2), and outcome (Video
S3) conditions are provided in the Supplemental Material available online. Note that the videos play at half speed.
5. Details about how the model was implemented may be foundin the Supplemental Appendix, in the Supplemental Material
available online. An example video (Video S4) shows how themodel classified a participant’s eye movements. Note that thevideo plays at one-third speed.

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