Noisy Newtons:Unifying process and dependency accounts of causal attribution

Tobias Gerstenberg1 (t.gerstenberg@ucl.ac.uk), Noah Goodman2 (ngoodman@stanford.edu),David A. Lagnado1 (d.lagnado@ucl.ac.uk), Joshua B. Tenenbaum3 (jbt@mit.edu)

1Cognitive, Perceptual and Brain Sciences, University College London, London WC1H 0AP2Department of Psychology, Stanford University, Stanford, CA 94305

3Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139

AbstractThere is a long tradition in both philosophy and psychology toseparate process accounts from dependency accounts of cau-sation. In this paper, we motivate a unifying account that ex-plains people’s causal attributions in terms of counterfactualsdefined over probabilistic generative models. In our experi-ments, participants see two billiard balls colliding and indicateto what extent ball A caused/prevented ball B to go througha gate. Our model predicts that people arrive at their causaljudgments by comparing what actually happened with whatthey think would have happened, had the collision betweenA and B not taken place. Participants’ judgments about whatwould have happened are highly correlated with a noisy modelof Newtonian physics. Using those counterfactual judgments,we can predict participants’ cause and prevention judgmentsvery accurately (r = .99). Our framework also allows us tocapture intrinsically counterfactual judgments such as almostcaused/prevented.Keywords: causality; counterfactuals; physics.

IntroductionThere has been a longstanding divide in philosophy be-

tween two fundamentally different ways of conceptualizingcausality. According to dependency accounts of causation,what it means for A to be a cause of B is that B is in someway dependent on A. Dependence has been conceptualizedin terms of regularity of succession (A is regularly succeededby B; Hume, 1748), probabilities (the presence of A increasesthe probability of B; Suppes, 1970) or counterfactuals (if Ahad not been present B would not have occurred; Lewis,1970). For process accounts, in contrast, what it means forA to be a cause of B is that a physical quantity is transmittedalong a pathway from A to B (Dowe, 2000).

The psychological literature on causal learning and attribu-tion neatly maps onto the two major accounts in philosophy.On the one hand, people have been shown to use contingencyinformation when drawing inferences about whether and howstrongly two events are causally linked (Cheng, 1997). On theother hand, people display a preference to choose causes thatinfluence an effect via a continuous causal mechanism overcauses that are connected with the effect through mere depen-dence (Walsh and Sloman, 2011). A point that has been raisedin favor of process accounts is that they are capable of captur-ing the semantics of different causal terms. Whereas depen-dency accounts have mostly focussed on causation and pre-vention, Wolff (2007) has provided a process account whichnot only predicts when people use the terms cause and pre-vent but also enable and despite. Following a linguistic anal-ysis of causation by Talmy (1988) in terms of force dynamics,Wolff (2007) argues that the aforementioned causal terms can

be reduced to configurations of force vectors. For example,what it means for a patient (P) to have been caused by anaffector (A) to reach an endstate (E) is that P did not havea tendency towards E, A impacted on P in a way that theirforce vectors were not pointing in the same direction and Preached E. If, in contrast, the force vectors of both P andA point towards the endstate and P reaches the endstate, themodel predicts that people will say “A enabled (rather thancaused) P”. Wolff’s (2007) force dynamics model accuratelypredicted which causal terms people use to describe differentvideo clips depicting physical interactions.

While the force dynamics model has strong intuitive ap-peal for interactions between physical entities, it is question-able how it can be extended to capture causal attributionsin situations involving more abstract entities. For example,one might legitimately assert that the fall of Lehman Brotherscaused the financial crisis or that Tom’s belief that he forgothis keys caused him to turn around and go back home. Whileit is unclear how these causal relationships could be expressedin terms of force vectors, they do not pose a problem for themore flexible dependency accounts. For example, accordingto a counterfactual account, Tom’s belief qualifies as cause ofhis behaviour if it is true that his behavior would have beendifferent had the content of his belief been different. Hence,there appears to be a trade-off between the semantic richnessof process accounts on the one hand and the generality andflexibility of dependency accounts on the other hand.

Rather than fostering the divide between process accountsand dependency accounts, we propose a theory of causal at-tribution that combines the best of both worlds. In the spiritof Pearl (2000), we model causal attributions in terms ofcounterfactuals defined over probabilistic generative models.However, we agree with Wolff (2007) that people’s causalknowledge is often richer than what can be expressed with acausal Bayes net. We aim to unify process and dependencyaccounts by showing that people have intuitive theories in theform of detailed generative models, and that causal judge-ments are made by considering counterfactuals over these in-tuitive theories. Here we demonstrate the superiority of ourapproach over existing models of causal attribution in a phys-ical domain. We show that people use their intuitive under-standing of physics to simulate possible future outcomes andthat their causal attributions are a function of what actuallyhappened and their belief about what would have happenedhad the cause not been present.

Overview of Experiments and ModelPredictions

Before discussing the predictions of our model and the sup-porting evidence from four experiments, we describe the do-main to which we applied our model. In all experiments, par-ticipants saw the same 18 video clips which were generatedby implementing the physics engine Box2D into Adobe FlashCS5. Figure 1 depicts a selection of the clips.1 In each clip,there was a single collision event between a grey ball (A) anda red ball (B) which enter the scene from the right. The blackbars are solid walls and the red bar on the left is a gate thatballs can go through. In some clips B went through the gate(e.g. clip 18) while in others it did not (e.g. clip 5). In the18 video clips, we crossed whether ball B went through thegate given that it collided with ball A (rows in Figure 1: ac-tual miss, actual close, actual hit) with whether B would gothrough the gate if A was not present in the scene (columnsin Figure 1: counterfactual miss, counterfactual close, coun-terfactual hit). Participants viewed two clips for each combi-nation of actual and counterfactual outcome.

In Experiments 1 and 2, the video clips stopped shortlyafter the collision event. Participants judged whether ball Bwill go through the gate (Experiment 1) or whether ball Bwould go through the gate if ball A was not present in thescene (Experiment 2). In Experiment 3, participants saw eachclip played until the end and then judged to what extent ballA caused ball B to go through the gate or prevented B fromgoing through the gate. Finally, in Experiment 4 participantschose from a set of sentences which best describes the clipthey have just seen.

In order to model people’s predictions of actual and coun-terfactual future outcomes, we developed the Physics Sim-ulation Model (PSM) which assumes that people make useof their intuitive understanding of physics to simulate whatwill or what might have happened. Hamrick et al. (2011)have shown that people’s stability judgments about towers ofblocks is closely in line with a noisy model of Newtonianphysics. While in their model, the introduced noise capturespeople’s uncertainty about the exact location of each block,the noise in our model captures the fact that people cannotperfectly predict the trajectory of a moving ball. Clip 1 inFigure 1 shows an illustration of our model. We introducenoise via drawing different degrees of angular perturbationfrom a Gaussian distribution with M = 0 and SD = {1, 2, ...,10} which is then applied to either B’s actual velocity vector(given that it collided with A, clip 1 bottom left) or B’s coun-terfactual velocity vector (given that A was not present in thescene, clip 1 top left) at the time that A and B (would have)collided.

We evaluate the probability that B would go through thegate when A was present, P(B|A), or absent P(B|¬A) by for-ward sampling from our noisy versions of Newtonian physics.For each clip and degree of noise (SD), we ran 1000 noisy

1All clips can be viewed here: http://www.ucl.ac.uk/lagnado-lab/experiments/demos/physicsdemo.html

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Figure 1: Selection of clips used in the experiment. Solidlines = actual paths, dashed lines = counterfactual paths. Clip1 depicts an illustration of the Physics Simulation Model andclip 7 of the Actual Force Model. Note: actual miss = Bclearly misses, actual close = B just misses/hits, actual hit =B clearly hits, counterfactual miss = B would have clearlymissed, counterfactual close = B would have just missed/hit,counterfactual hit = B would have clearly hit.repetitions of the original clip in which A was present andin which A was absent. We then simply counted the worldsin which B goes through the gate given that A was present,P(B|A), and the worlds in which B goes through the gategiven that A was absent, P(B|¬A).

Experiments 1 & 2: Intuitive PhysicsThe aim of Experiments 1 and 2 was to evaluate how well

people make use of their intuitive physical knowledge to pre-dict actual (Experiment 1) or counterfactual (Experiment 2)future states. Participants saw 18 video clips (see Figure 1 forsome examples) up to the point shortly after the two balls col-lided (0.1s). After having seen the clip twice, participants an-swered the question: “Will the red ball go through the hole?”(Experiment 1, N = 21) or “Would the red ball have gonethrough the goal if the gray ball had not been present?” (Ex-periment 2, N = 20). Participants indicated their answers ona slider that ranged from 0-100. The endpoints were labeled“definitely no” and “definitely yes”. The midpoint was la-beled “uncertain”. After having made their judgment, par-ticipants viewed the clip until the end either with both ballsbeing present (Experiment 1) or with ball A being removedfrom the scene (Experiment 2).

Results and DiscussionParticipants were very accurate in judging whether ball B

will go through the gate (Experiment 1) or would have gonethrough the gate (Experiment 2) with a mean absolute differ-ence from the deterministic physics model (which assigns avalue of 100 if B goes in and 0 if B does not go in) of 28.6 (SD= 29.9) in Experiment 1 and 25.1 (SD = 30.5) in Experiment

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Experiment 1: Will B go in?

Experiment 2: Would B have gone in?

Experiment 3: Did A cause/prevent B ...?

Experiment 4: Did A cause/help/almost cause B ...?

Figure 2: Correlation of the Physics Simulation Model withpeople’s judgments in all four experiments for different de-grees of noise.

2. Figure 2 shows the correlation of the PSM with partici-pants’ judgments in Experiment 1 (solid black line) and Ex-periment 2 (dashed black line) for different degrees of noise.While people’s judgments already correlate quite well witha deterministic Newtonian physics model (degree of noise =0◦), introducing small degrees of noise results in much highercorrelations with a maximum correlation of r = .95 in Exper-iment 1 and r = .98 in Experiment 2 (for SD = 5◦).

The results of Experiments 1 and 2 show that people arecapable of mentally simulating what will happen (Experiment1) or what would have happened (Experiment 2). Given thateach clip stopped very shortly after the collision event whichhappened M = 2.7s (SD = 0.5) after the start of the clip, par-ticipants’ accuracy in judging whether ball B will go in or notis quite impressive.

Experiment 3: Causation and PreventionIn Experiment 3, we wanted to investigate how people use

their intuitive understanding of physics to make judgmentsabout the extent to which one event caused or prevented an-other event from happening. Unlike in Experiments 1 and2, participants (N = 22) saw each clip played until the end.After having seen each clip twice, participants answered thequestion “What role did ball A play?” by moving a sliderwhose endpoints were labeled with “it prevented B from go-ing through the hole” and “it caused B to go through thehole”. The midpoint was labeled “neither”. The slider rangedfrom -100 (prevented) to 100 (caused). Participants were in-structed that they could use intermediate values on the sliderto indicate that ball A somewhat caused or prevented B.

Model PredictionsBefore discussing the results of Experiment 3, we will de-

scribe two different models that make different assumptionsabout how people arrive at their cause and prevention judg-ments.

Physics Simulation Model According to the PSM, peoplearrive at their cause and prevention judgments by compar-ing what actually happened with what they think would havehappened if the cause event had not taken place. More specif-ically, our model predicts that people compare P(B|A), theprobability that ball B will go through the gate given that itcollided with ball A, with P(B|¬A), the probability that Bwould have gone through the gate if A had not been presentin the scene. Since participants in Experiment 3 watch theclips until the end, the value of P(B|A) is certain: it is either1 when B goes through the gate or 0 when B misses the gate.In order to determine P(B|¬A) the PSM assumes that peopleuse their confidence in the result of their mental simulation ofwhat would have happened had A not been present based ontheir intuitive understanding of physics.

In general, if P(B|A)−P(B|¬A) is negative, participantsshould say that A prevented B from going through the gate.Intuitively, if it was obvious that B would have gone in had Anot been present (i.e. P(B|¬A) is high) but B misses the gateas a result of colliding with A (i.e. P(B|A) = 0), A shouldbe judged to have prevented B from going through the gate.Similarly, if the difference is positive, participants should saythat A caused B to go through the gate. If the chance that Bwould have gone through the goal without A was low but, as aresult of colliding with A, B goes through the gate, A shouldbe judged to have caused B to go through the gate. Clip 1 inFigure 1 shows an example for which our model predicts thatparticipants will say that A neither caused nor prevented B.P(B|A) is 0 since B does not go through the gate. However,P(B|¬A) is also close to 0 since it is clear that B would havemissed the gate even if A had not been present in the scene.

Actual Force Model The Actual Force Model (AFM) is ourbest attempt to apply Wollf’s (2007) force dynamics modelto our task.2 According to the AFM, participants’ cause andprevention judgments are a direct result of the physical forcesbeing present at the time of collision.

Clip 7 in Figure 1 illustrates how the AFM works. First,a goal vector (dotted arrow) is drawn from ball B’s locationat the time of collision to an end state (E), which we definedto be in the center of the gate. Second, the angle α betweenthe velocity vector that ball B has shortly after the collisionwith A (solid arrow) and the goal vector as well as the angleβ between the velocity vector that ball B has shortly beforecolliding with A (dashed arrow) are determined. Third, themodel predicts people’s cause and prevention judgments viacomparison of α and β. In general, if ball B goes in andβ−α is greater than 0, the model predicts people will saythat A caused B. Conversely, if ball B does not go in andβ−α is smaller than 0, the model predicts people will say Aprevented B. For situations in which β−α is greater than 0but B does not go in or β−α is smaller than 0 but B does goin, we fix the model prediction to 0. This constraint prevents

2While the force dynamics model only makes predictions aboutwhich out of several sentences participants should choose to de-scribe a situation, the AFM makes quantitative predictions about theextent to which an event is seen as causal/preventive.

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Figure 3: Mean cause (green) and prevention ratings (red)for the different clips. Black bars are the predictions of thePhysics Simulation Model which correlates r = .99 with par-ticipants’ judgments. The colors of the bars indicate whetherball B actually went through the gate (green) or not (red). Thenumbers on the x-axes denote the different clips. Error barsare ± 1 SEM.

the model from predicting, for example, that people will say“A caused B” in when B missed the gate.

Results and DiscussionFigure 3 shows participants’ mean cause (green bars) and

prevention judgments (red bars) for the 18 different clips to-gether with the predictions of the PSM (black bars). For theparticular implementation of the PSM depicted in Figure 3,we directly used participants’ judgments from Experiment2 in which they indicated whether ball B would have gonethrough the gate if ball A had not been present as the valuesfor P(B|¬A). The PSM predicts participants’ cause and pre-vention ratings very well with r = .99 and RMSE = 0.12. Ahigh median correlation across participants of r = .88 witha minimum of r = .61 and a maximum of r = .95 demon-strates that the good performance of the PSM is not due toa mere aggregation effect. Figure 2 shows that participants’judgments also correlate highly with the PSM when we gen-erate P(B|¬A) via angular perturbation with different degreesof noise. The AFM, in contrast, does not predict partici-pants’ judgments equally well with a correlation of r = .75and RMSE = 0.49.

People’s cause and prevention judgments were not affectedby the closeness of the actual outcome. That is, participants’cause ratings did not differ between situations in which B justwent through the gate (clips 8, 10, 12: M = .51, SD = .40)compared to situations in which B clearly went through (clips13 - 18: M = .49, SD = .42). Similarly, prevention judgmentswere not different between situations in which B just missed(clips 7, 9, 11: M = -.41, SD = .43) and situations in which Bclearly missed (clips 1-6: M = -.47, SD = .44).

People’s cause and prevention judgments were very wellpredicted by the PSM. In order to judge whether ball Acaused or prevented ball B, participants appear to comparewhat actually happened with what they think would have

happened had A not been present. This very high correla-tion is achieved without the need for any free parameters inthe model and holds when directly taking the counterfactualprobability P(B|¬A) from participants’ judgments in Experi-ment 2 or from the noisy Newtonian physics model.

The AFM which assumes that people arrive at their judg-ments via comparing instantaneous force vectors, rather thana mental simulation of the full physical dynamics cannot cap-ture people’s judgments equally well. Clip 14 (see Figure 1)gives an example in which the AFM gets it wrong. Whileparticipants indicate that A caused B to go through the gate(see Figure 3), the AFM model cannot predict this. In thissituation, the angle between the velocity vector of B shortlyafter the collision and the goal vector α is greater than theangle between the velocity vector of B shortly before the col-lision and the goal vector β. Hence, the model predicts thatA is preventing B but since B does in fact go in, the model’sprediction is fixed to 0.

Experiment 4: Almost Caused/PreventedThe results of Experiment 3 show that people’s cause and

prevention judgments are only influenced by their degree ofbelief about whether the event of interest would have hap-pened without the cause being present and not influencedby how close the outcome actually was. However, often thecloseness with which something happened clearly matters tous, such as when we almost missed a flight to Japan or onlyjust made it in time for our talk.

As mentioned above, one of the appeals of process ac-counts is that they acknowledge the semantic richness of theconcept of causation by making predictions about which outof several causal verbs people will choose to describe a par-ticular situation.

In this experiment, we will demonstrate that our frameworkis not only capable of capturing the difference between differ-ent causal verbs such as caused or helped but also predictswhen people make use of intrinsically counterfactual con-cepts such as almost caused or almost prevented. Currentprocess accounts (e.g. Wolff, 2007) cannot make predictionsin these situations as they aim to analyze causality withoutmaking reference to counterfactuals.

In Experiment 4, participants (N = 41) had to select froma set of seven sentences the one that describes the situationbest. The sentences were: A caused / helped / almost causedB to go in the hole; A prevented / helped to prevent / almostprevented B from going in the hole; A had no significant ef-fect on whether B went in the hole or not.

Model PredictionsTable 1 gives an overview of the model predictions which

are a function of the outcome, that is, whether B went inor not, and the probabilities P(B|¬A), P(almost B|A) andP(almost ¬B|A). For P(B|¬A) we can again use participants’judgments from Experiment 2 or the predictions of the PSM.

The model’s predictions for caused and prevented are iden-tical to the predictions in Experiment 3. According to our

Table 1: Predicted probability of choosing different sentencesin Experiment 4.

outcome probability

(1) caused hit 1−P(B|¬A)(2) helped hit 1−abs(0.5− caused)*

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(7) no effect hit/miss 1−max((1), ...,(6))* rescaled to range from 0 to 1

model, the difference between caused and helped is an epis-temic one. People are predicted to select helped when B wentin and when they were uncertain about what would have hap-pened had A not been present. However, when it was clearthat B would have gone in or would have missed, peopleare predicted to select no effect or caused, respectively, andnot helped. Similarly, when B missed and it was uncertainwhether B would have gone in, the model predicts that peo-ple select helped to prevent.

In order to predict when people select almost caused oralmost prevented, we first have to define the probabilitiesP(almost B|A) and P(almost ¬B|A). These probabilities ex-press the closeness of an alternative counterfactual outcometo the actual outcome. One way to get at the probabilitieswould be to have participants judge how closely B hit ormissed the gate. However, here we used a variation of thePSM to generate these probabilities. For each clip we rana set of 100 x 10 noisy simulations for different noise levelsfrom SD = 1◦ to 5◦, whereby the noise was again introduced atthe time of collision. If the outcome in the noisy simulationswas different from the original outcome in any of the 10 rep-etitions in each of the 100 simulated worlds, we counted thisas a positive instance. If the outcome in all 10 repetitions wasthe same as the original outcome, we counted this as a nega-tive instance. For example, a value of P(almost ¬B|A) = 87in a situation in which B goes through the gate in the originalclip, means that in 87 out of the 100 generated worlds, theball did not go through the gate in at least one the 10 repeti-tions of each of the worlds. For the remaining 13 worlds, theball did go in for all 10 repetitions. Intuitively, in situations inwhich the outcome was close, the chances that the outcomein the noisy simulation will be different from the outcome inthe original clip in at least one out of ten repetitions are high.However, if ball B clearly missed, for example, it is unlikelythat there will be a noisy simulation in which the introducedangular perturbation is sufficient to make B go in.

The model predicts that people will select almost causedwhen B just missed (which means that P(almost B|A) is high)and the probability that it would have gone in given that Awas absent is low. People should select almost prevented

when B just went in (P(almost ¬B|A) is high), and whenit was clear that B would have gone in had A been absent(P(B|¬A) is high). Finally, if none of these calculations re-sult in a high value, people are predicted to select that A hadno significant effect on whether B went through the gate.

Figure 4a shows the model predictions for the 18 differentclips. We used Luce’s choice rule (Luce, 1959) to transformthe different probabilities into predictions about the frequen-cies with which the different sentences will be selected. Themodel predicts that the modal choice in situations in which Bdoes not go in changes from prevented for clips in which itwas clear that B would have gone in (clips 5 & 6) to helped toprevent in situations in which it was unclear whether B wouldhave gone in (clips 3 & 4). The same switch of the modal re-sponse as a function about the certainty of what would havehappened is predicted for caused (clips 13 & 14) and helped(clips 15 & 16). If there was little uncertainty about the coun-terfactual outcome and it matches the actual outcome, peopleare predicted to select had no effect (clips 1, 2, 17, 18). Forclip 7 in which B just misses, people are predicted to selectalmost caused since it was clear that B would have missed butfor A. Conversely, for clip 12, the model predicts that peoplewill select almost prevented: B just goes in and it was clearthat it would have gone in without A being present.

Results and DiscussionFigure 4b shows the proportions with which the different

sentences were selected for each clip. The model predictsthe frequencies with which participants select the differentsentences very well, r = .86. Figure 2 shows the correla-tion when we generate P(B|¬A) through noisily perturbingthe vector rather than taking participants’ ratings. It predictsthe modal choice correctly in 12 out of 18 clips. While par-ticipants’ modal response does not change between clips 5 &6 and clips 3 & 4 as predicted by the model, the proportion ofhelped to prevent selections clearly increases. A similar shiftis observed between clips 13 & 14 and 15 & 16 for whichparticipants’ selection of helped increases as a function of theuncertainty over what would have happened.

As predicted by the model, participants’ modal response inclip 7 is almost caused and in clip 12 almost prevented. Thevariance in responses within a clip is greater for the clips inwhich the actual outcome was close (middle row) comparedto when it clearly missed (top row) or clearly hit (bottomrow). For example, in clip 10 in which B just goes in and thecounterfactual outcome is close the majority of participantsselected caused or helped while a minority of participantsselected almost prevented. This pattern closely matches thepredictions of our model. Whether a participant is expectedto select caused or almost prevented depends on the partici-pant’s subjective belief about the counterfactual outcome. Ifa participant thought that B would have missed she will say Acaused or helped it. However, if a participant thought that Bwould have gone in but for A he will select almost preventedbecause B barely goes in.

The close fit between our model prediction and partici-

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Figure 4: Predicted (left) and empirical (right) frequency of sentence selection in Experiment 4, r = .86. The color of the clipnumber indicates whether the ball went in (green) or not (red). Note: c = caused, hc = helped (to cause), ac = almost caused, p= prevented, hp = helped to prevent, ap = almost prevented, n = no significant effect.pants’ selection of sentences demonstrates that our model iscapable to capture some of the richness of people’s causal vo-cabulary. Our model not only allows to distinguish cases ofcausing/preventing from helping but also accurately predictspeople’s almost caused/prevented judgments. Process theo-ries that analyze different causal concepts without the use ofcounterfactuals (e.g. Wolff, 2007) cannot make predictionsabout when people will say that something almost happened.

General DiscussionIn this paper, we developed a framework for understand-

ing causal attributions that aims to break the longstandingdichotomy between process accounts and dependency ac-counts of causation. We showed that people’s quantitativecause and prevention judgments (Experiment 3) as well aspeople’s use of different causal verbs (Experiment 4) can bevery well predicted by assuming that people compare whatactually happened when the cause was present with whatthey think would have happened in the absence of the cause.We provided evidence that people use their intuitive under-standing of physics (Experiments 1 & 2) to simulate possi-ble outcomes. Understanding causal attributions in terms ofcounterfactuals defined over probabilistic generative modelssidesteps the trade-off between flexibility and richness de-scribed above. Our model retains the generality of depen-dency accounts while the use of a rich generative model basedon Newtonian physics allows us to capture some of the rich-ness of people’s concept of causation.

We are currently developing our approach in two direc-tions: First, having established the close fit of our model forrelatively simple interactions between two balls, we will lookat more complex interactions between three balls. This willallow us investigate situations that are considered problematicfor a counterfactual analysis of causation such as overdeter-mination, preemption and double prevention. Furthermore,these situations will allow us to look systematically at differ-

ences between cause and prevent. While our current frame-work conceptualizes causation and prevention symmetrically,there is evidence that this is not true for people (Walsh andSloman, 2011). Second, we are interested in looking at peo-ple’s actual cause judgments for interactions between inten-tional agents with beliefs and desires. By developing richermodels of people’s intuitive understanding of domains suchas physics and psychology, we will better understand howpeople make sense of their world through attributing causalityand responsibility.

AcknowledgmentsWe thank Tomer Ullman, Peter Battaglia and Christos

Bechlivanidis for very helpful discussions. This work wassupported by a doctoral grant from the AXA research fund(TG), a John S. McDonnell Foundation Scholar Award (NG),a ONR grant N00014-09-1-0124 (NG, JT), an ESRC grantRES-062- 33-0004 (DL) and ONR MURI grants W911NF-08-1-0242 and 1015GNA126 (JT).

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