Theory-based Social Goal Inference
Chris L. Baker, Noah D. Goodman & Joshua B. Tenenbaum{clbaker,ndg,jbt}@mit.edu
Department of Brain and Cognitive Sciences
Massachusetts Institute of Technology
Abstract
Everyday human interaction relies on making inferences aboutsocial goals: goals that an intentional agent adopts in relation
to another agent, such as “chasing”, “fleeing”, “approaching”,“avoiding”, “helping” or “hindering”. We present a computational model of social goal inference that takes as input observations of multiple agents moving in some environmental context. The model infers a social goal for each agent that is most
likely to have given rise to that agent’s observed actions, under
an intuitive theory that expects agents to act approximately rationally. We provide evidence for our theory-based approachover a simpler bottom-up motion cue-based approach in a behavioral experiment designed to distinguish the two accounts.
Keywords: social cognition; theory of mind; action understanding; goal inference; inverse planning; Bayesian inference

Even the simplest everyday social interactions are supported by complex mental representations and processes.
Contingent on the environment, our prior knowledge and ourdesires, we form goals, beliefs, intentions and other mental
states with rich representational content. When we observeor participate in social interactions, we make joint inferences
about the mental states of multiple interacting agents. For
instance, watching a short film of two people moving inside a house, we might interpret it as follows: “X is trying to catch Y, and Y is trying to escape. X is sneaking
up on Y in the bedroom but doesn’t realize that Y has already seen him and is preparing to escape out the back.” Inferring such intentional descriptions of social interactions iscomputationally challenging. Other agents’ mental states are
not directly observable, and the space of possible beliefs andgoals that one agent could have with respect to another is effectively infinite. Yet these inferences are remarkably quick
and robust. Adults (Heider & Simmel, 1944) and even infants (Kuhlmeier, Wynn, & Bloom, 2003; Hamlin, Wynn,& Bloom, 2007; Gergely, Nádasdy, Csibra, & Biró, 1995)
can infer relational goals such as “chasing”, “helping”, and“harming” from brief animations of simple shapes moving ina two-dimensional environment.
To account for inferences like these, it has been proposedthat people draw on an intuitive theory of psychology, which
may already be present in infancy in some simpler form asa rational agent schema (Gopnik & Wellman, 1992; Gopnik& Meltzoff, 1997; Gergely et al., 1995). This approach casts
the interpretation of intentional action as a top-down process,drawing on a causal model of how agents’ beliefs, intentions
and environmental and social context jointly influence their
behavior. Agents’ mental states are inferred from observedsocial interactions through a process of “inference to the best
explanation”, positing the goals and beliefs under which theirobserved actions are most plausible in the observed context.

A key challenge for theory-based accounts of action understanding is precisely specifying the nature of the causal
relation between intentions, context and behavior. Several
authors (Dennett, 1987; Gergely et al., 1995) have argued
that this relation embodies the principle of rationality: theassumption that intentional agents should act to achieve theirgoals as efficiently as possible, given their beliefs about theworld. The principle of rationality is appealing because it
applies very generally across agents, intentions, and contexts, providing a way to dynamically build causal models
of agents’ behavior in novel situations. However, the senseof rationality that people apply can be nuanced, particularlyin the case of social interaction, where the rational strategy
of an agent might depend on its expectations of others’ behavior, whose rational strategies depend on expectations of
its behavior, and so on ad infinitum.
An alternative approach (Blythe, Todd, & Miller, 1999;
Barrett, Todd, Miller, & Blythe, 2005; Zacks, 2004) to specifying the relation between mental states, context and behavior
emphasizes simple visual cues that can be computed throughbottom-up perceptual processes. Movements with a particular social intention, such as “chasing” or “fleeing”, are perceived as instances of categories defined by prototypical motion patterns. For example, by measuring the relative directions and velocities of two agents’ motions, we might be able
to identify if one is trying to catch the other.
A simple cue-based approach appeals in part because it
seems to be the kind of computation that can be done rapidly,robustly and in young infants without sophisticated cognitive
capacities. It can also be formalized precisely in computational models (Blythe et al., 1999). However, it is unlikely
to be sufficient to explain human social inference. The inferences of both adults and infants are highly sensitive to theenvironmental and task contexts in which actions take place,in ways that are hard to explain by a bottom-up motion analysis but suggest a deeper causal basis (Gelman, Durgin, &Kaufman, 1995). For instance, there are situations in which
Y may be trying to avoid X, but Y’s best action is to headtoward X rather than away from X; Y may be trapped in acorner with only one route of escape that runs directly past X.
Theory-based approaches seem to offer a deeper account ofsocial goal inference, but unlike bottom-up cue-based models, they have not been worked out computationally or tested
rigorously. These are the goals of our present paper. We givea Bayesian formulation of the theory-based approach to inferring social goals from observed action sequences. Our framework can capture a range of theory-based models, differing in
how richly they model agents’ mental states. We show howdifferent theory-based approaches can be tested against eachother and against simpler cue-based approaches.
Previously (Baker, Tenenbaum, & Saxe, 2006, 2007) we
showed how to model action understanding as inverse planning: Bayesian inference about the structure of a goal-based
Markov Decision Problem (MDP), a rational model for probabilistic planning. Here, we extend this work to model people’s inferences about social goals, such as approaching andavoiding, from observations of multiple agents interacting in
simple environments.
People’s representations of other minds might include notjust first-order content, such as agents’ intentions or beliefs
about the state of the world, but also second-order content:
representing agents’ strategies, contingent on their representation of other agents. Second-order mentalizing is particularly important in social interaction, where understanding social relations between agents such as helping, loving,
loathing, and empathizing require the ability to represent amind representing another mind. We show how to represent
both first- and second-order content in our framework.
The plan of the paper is as follows. We first introduce ourcomputational framework for theory-based social goal inference, and then describe several alternative heuristic motion
cue-based models which we consider. Next, we compare theaccuracy of both theory-based models and cue-based models in predicting how people categorize pairs of interacting
agents moving in simple environments. We close by discussing the insights gained into the representations and principles underlying people’s social goal inferences.

Computational framework
Our theory-based framework represents the structure of thecausal relation between agents’ environmental and social
context, their mental states, and their behavior. Fig. 1 illustrates how models of rational planning allow observers
to dynamically generate a conditional probability distribution
over agents’ actions (Ai ), given their goals (Gi ), their beliefs
about other agents’ goals (Bi (G j ); “social context”) and thestate of the world (W ). For simplicity, we consider situations
with two agents, who both have complete knowledge of theworld state W = hS1 , S2 , Ei, which includes both agent states
S1 , S2 and the environment E. The environment is assumed
to be static, while agent states are dynamic. In Fig. 1, agentschoose their actions independently of each other, so the conditional joint probability over actions factors into the productof the conditional probability of each agent’s action:
P(A1 , A2 |G1 , G2 , B1 (G2 ), B2 (G1 ),W ) =P(A1 |G1 , B1 (G2 ),W ) · P(A2 |G2 , B2 (G1 ),W ). (1)Inferring mental states given observed actions and theworld state is done by inverting this model using Bayes’ rule:P(G1 , G2 , B1 (G2 ), B2 (G1 )|A1 , A2 ,W ) ∝P(A1 |G1 , B1 (G2 ),W ) · P(A2 |G2 , B2 (G1 ),W )· P(G1 , B1 (G2 )) · P(G2 , B2 (G1 )), (2)called inverse planning. Equation 2 describes a kind of “interaction parse”, assigning social goals and beliefs about others’
goals to all agents by integrating bottom-up information fromobserved actions with top-down information from the context
and the prior over agents’ mental states, P(Gi , Bi (G j )).

World
State
Belief1(Goal2)

Goal1

Belief2(Goal1) Goal2

Rational planning(MDP)

Rational planning(MDP)

Action1

Action2

Figure 1: Causal schema for an intuitive theory of social interaction, specialized to two agents for simplicity.
We model first- and second-order content in people’s theories with agents of type L1 and L2, respectively. Agents ofboth types have social goals. We assume that L1 agents do
not represent other agents’ goals, and thus cannot anticipate
others’ actions. We model L1 agents’ predictions as being
maximally uncertain about other agents’ behavior, predicting they will follow a random walk. L2 agents are assumed
to represent other agents’ social goals, and to model others’
planning processes to predict their future behavior, allowing
L2 agents to pursue their own goals more efficiently.
For concreteness, we consider two kinds of social goals
for L1 and L2 agents: Approach and Avoid. The meaningof these goals roughly corresponds to the meaning of theirwords in English; however, their precise semantics is definedby their role in our theory-based models of social interaction.
We assume a uniform prior over goals, so that cooperationand competition are equally likely. Next, we describe how
Equation 1 is computed for these social goals.

(Inverse) planning with social goals
The principle of rationality states that agents will take efficient means to achieve their goals, given their beliefs about
the world. We assume that agents receive rewards that depend on their goals, their own states and the states of theirconspecifics, and that agents plan to maximize their expected
future rewards minus the costs they incur, potentially using a
predictive model of the other agent’s behavior.
Let R (W, Gi ) be an agent’s real-valued reward function,
defined over the state of the world and the agent’s goal. We
assume the reward function for each goal is a linear functionof the geodesic (shortest-path) distance geo dist(Si , S j ) between agent i and agent j:

R (W, Approach) = −α · geo dist(Si , S j ),R (W, Avoid) = α · geo dist(Si , S j ).

(3)
(4)

Let C (Sit , ait , E) be the cost of taking action ait from statet
Si in environment E, defined to be the Euclidean distance ofthe resulting movement. The rational behavior (or optimal
policy) π∗i for an agent is defined to be the rule of action thatmaximizes the expected sum of future rewards, while minimizing the sum of incurred costs. The value function of π∗i ,defined over the world state W , agent i’s goal Gi , and agent
i’s belief about agent j’s goal Bi (G j ) is:∗

V πi (W, Gi , Bi (G j )) =
 ∞
Eπ∗i




t
t
t t
0
R
(W
,
G
)
−
C
(S
,
a
,
E)
W
=
W
, (5)
γ
i
∑
i i

t=0

where the discount rate γ determines how much the agent values immediate rewards over future rewards.
To maximize their expected future rewards, agents mustchoose actions that incur a low cost and lead to states withhigh value. The value of action ait in world W t , given Gi and
Bi (G j ) is defined as the reward of the current state, minus thecost of the action, plus the expected discounted sum of future
rewards after the action:∗

Q πi (ait ,W t , Gi , Bi (G j )) = R (W t , Gi ) − C (Sit , ait , E)+ ∗
γ · EP(W t+1 |W t ,a t ) V πi (W t+1 , Gi , Bi (G j )) . (6)iWe assume that agents maximize rewards probabilistically,and sometimes only approximate the optimal sequence of actions. The probability that agent i will take action ait fromworld state W t , given Gi and Bi (G j ) is:∗P(ait |Gi , Bi (G j ),W t ) ∝ exp(β· Q πi (ait ,W t , Gi , Bi (G j ))). (7)Given Equation 7, Equation 1 is computed as the product ofthe probability of each agent’s actions over all time steps.
Computing the expectation involved in Equation 5 requires
averaging over all future state sequences, conditioned on W t ,Gi and Bi (G j ). For L1 agents, Bi (G j ) ≡ Null, and we assume
their model of other agents is given by a uniform distribution
over available actions, predicting a random walk through thestate space. For L2 agents, representing other agents’ goals
allows anticipation of their future actions. We assume that L2
agents represent the other agent as having type L1, and firstcompute the optimal policy π∗j for this agent given Bi (G j ). L2
agents then predict other agents’ behavior using Equation 7,
and plan their own actions contingent on these predictions.
For both L1 and L2 agents, Equation 5 can be efficiently
approximated using dynamic programming algorithms (Bertsekas, 2001).

Motion cue-based models
In addition to the theory-based models L1 and L2, we consider two models based on low-level motion cues, inspiredby a “simple heuristics” account of human social goal inference (Blythe et al., 1999; Zacks, 2004). Unlike theory-basedmodels, cue-based models do not represent other agents’mental states or use a model of rational planning to reason
about how other agents’ behavior depends on the context.
Instead, these models directly associate social goals such aschasing or fleeing with typical movement patterns that can becomputed efficiently from local information, like changes in
direction or velocity.
We denote our motion cue-based alternatives H1 and H2.
Both models assume that movements which decrease the distance to the other agent suggest the goal of approaching theother agent, while movements that increase the distance fromthe other agent suggest the goal to avoid. In H1, social goal
inferences are assumed to be independent of the environment,
and are computed solely based on movement relative to thelocation of the other agent. In H2, the environment is assumed to influence agents’ behavior in a local way: H2 is expected to move closer to or farther from the other agent along
the geodesic path, given its goal and subject to the constraintsof the environment, but without planning ways in which locally moving closer to or farther from the other agent may be
a rational strategy.
For the sake of comparison with L1 and L2, we implement
H1 and H2 as degenerate cases of theory-based models withno planning capacity. Let R (W t , Gi ) and C (Sit , ait , E) be defined as above. The probability of action ait , given the worldstate and goal Gi for an H1 or H2 agent is:P(ait |Gi , Bi (G j ),W t ) ∝
exp β EP(W t+1 |W t ,a t ) [R (W t+1 , Gi )] − C (Sit , ait , E) . (8)iCompared with L1 and L2, H1 and H2 could be considered tohave “zero-th order representational content”. Next, we describe an experiment designed to distinguish the predictionsof these models and provide evidence for higher-order representational content in human social goal inference.

Experiment
Our experiment was designed to distinguish the predictionsof theory-based models L1 and L2 from motion cue-basedmodels H1 and H2. We collected people’s judgments in atask of inferring two agents’ social goals toward one anothergiven short observations of their interactions in a simple, twodimensional maze-like environment. Subjects were told thatthe agents could have a variety of social goals, such as catching or meeting up with the other agent, or trying to flee, escape or avoid it. They were then asked to categorize eachagent in terms of its social goal.
Our experimental design varied the environmental andsocial context and the relative motion patterns of the two
agents to assess our models’ accuracy in predicting people’s
judgments. We hypothesized that only theory-based models
would be able to explain the interaction between people’s sensitivity to subtle environmental changes and the highly salient
motion cues provided by agents’ action sequences. In particular, by holding agents’ relative motion fixed while changing
the context in ways that affected the relative value of differ-ent actions for theory-based models, but not for motion cuebased models, we hypothesized that cue-based models would
fail to account for people’s social goal inferences across allconditions.

MethodParticipants Participants were 20 members of the MIT
community, 13 female and 7 male.
Stimuli Our stimuli were designed following previous research showing that simple two-dimensional animations ofmoving shapes evoke strong impressions of animacy and inferences of mental states (Tremoulet & Feldman, 2000). All
experimental stimuli are shown in Fig. 2. Each stimulus displayed two agents, represented by red and green dots respectively. As agents moved along their trajectories, smaller dotsof the same color trailed behind them, recording their path
history. Agents’ paths were either 4 steps or 6 steps long.
The environment was a discrete grid of 23 squares wide by
23 squares high, with walls represented as solid black lines.
Agents’ movements were restricted to adjacent squares, withdirections {N,S,E,W,NE,NW,SE,SW}. Up-down orientationof the display was counterbalanced within subjects, yielding
56 stimuli in total. The side on which red and green agents
appeared, and the order of the “Avoiding/Approaching”
options were randomized between subjects (but constantthroughout the experiment for individual subjects). Stimuli
were presented in random order.
Our experimental design combined four different relative
movement conditions with 7 different environmental and social contexts. The four relative movement conditions were:→←, →→, ←←, and ←→. These movement conditions areshown across the rows of Fig. 2.
The 7 different contexts are shown across the columns ofFig. 2. Conditions 2-6 varied the environmental context by
modifying the pattern of gaps in the wall of the maze, andcontexts 3-5, marked with a star above in Fig. 2, displayed
identical motion patterns in all contexts to isolate the effectof the different contexts on people’s inferences. Conditions 1
and 7 varied the length of the agents’ paths, displaying two
more steps than the other conditions.
Procedure Subjects were first presented with a cover story
about intelligent aliens interacting in their natural environment. They were told that the aliens could not move throughwalls in the environment but that they could move throughgaps in walls, and that they could see past walls and knew thecomplete state of the world at all times. Several possible social goals, such as “trying to catch, ..., approach, ..., meet up
with, ..., and engage with” were given as reasons for trying
to get closer to the other alien, and “trying to flee, ..., escape,..., avoid, ..., or disengage” were given as reasons for trying
to get farther away. They were then told that each of the fourcombinations of social goals was equally likely. During theexperiment, subjects viewed short animations of agents’ interactions and made forced-choice decisions between the op-

tions “Approaching”, “Avoiding” and “Can’t tell” for both
agents in each stimulus.
Modeling We applied Equation 2 to model people’s judgments. For L2 agents, computing the posterior marginal overgoals was done by marginalizing over the agent’s beliefs
about the other agent’s goal, assuming a uniform prior over
Approaching and Avoiding.
For each stimulus, the model provided the posterior probability that each agent had goal Approaching (the probability that an agent had goal Avoiding was the complement ofthis). To compare people’s judgments to these model predictions, we first coded the Approaching rating as 1, theAvoiding rating as 0, and the “Can’t tell” rating as 0.5.
We then averaged over the counterbalanced orientation conditions within subjects, and averaged over ratings between
subjects and computed the standard errors of these means.
For L1 and L2, we used α = .5, β = 1, and γ = .9. α was set
by hand to yield Q values that were of comparable magnitude
to action costs, and β was set to 1 by default. We explored a
range of γ values, with γ = .9 providing the best fit to people’s judgments, although nearby values yielded comparable
results. For H1 and H2, we used β and α values of 1. L1and L2 made similar predictions for our stimuli at the chosenparameter values; we only display L2. H1 and H2 yielded
similar predictions as well; only H2 is shown.

Results
Results of the experiment are shown in Fig. 3. People’s judgments are shown in column 1, demonstrating significant sensitivity to environmental factors. For instance, in contexts
3-5, people’s ratings showed significant variability within the→←, →→, and ←← movement conditions, despite the factthat the paths were identical across the three environments.
L2 predicted this variability, while H2 predicted no variability in any of the contexts, basing its predictions on superficial
motion features every time.
In addition to predicting the sensitivity of people’s judgments to environmental factors very accurately across contexts 3-5, there are several specific phenomena that confirm
L2’s predictions as well. Consider L2’s predictions for contexts 3 and 5 for movement →←: in context 3, L2 rates Red’s
movement as being very ambiguous, as do people. However,in context 5, where the gap is located on the other side of theenvironment, L2 predicts that Red is probably trying to approach Green, in accord with people’s judgments again. A
replication of this effect occurs for Red between contexts 3
and 5 for the →→ movement as well.
Another example of a dramatic shift in judgments occurs
between contexts 2 and 3 for movements →← and →→. Inthese cases, in context 2, people rate the probability of Red
trying to approach Green very highly, presumably because if
it were trying to escape, it would have gone through the exit
behind it. However, in context 3, when this exit is eliminated,people are now uncertain about whether Red is approachingor avoiding. L2 predicts this effect very accurately.

*Movement condition

®¬

®®

¬¬

¬®
1

2

3

4

Context

5

6

7

Figure 2: Experimental stimuli. The experimental design combined four different relative movement conditions with 7 different
contextual conditions. In contexts 3-5, marked above by an “*”, contexts and movement patterns varied independently, allowing
the effect of the different contexts to be isolated.
The L2 model also provides a plausible explanation for
why people make the inferences they do. For instance, formovement →← in context 3, Red’s action appears to be anequally sensible option whether it is approaching or avoiding,which the model confirms. In the cases where L2 strongly favors one interpretation over the other, both people and themodel see the agent’s actions as unambiguously supporting
this interpretation.
In contrast, because H2 does not incorporate planning into
its predictions, it assumes that agents with the avoid goal willhead directly away from the other agent, and that agents withthe approach goal with head directly for the other agent. Because of this, H2 makes the same predictions in every context
for the movement condition, failing to account for the effectof context on people’s judgments, as originally hypothesized.

a very general set of contexts, but can be quite difficult tointerpret in a precise, computational way.

Conclusion

Further work is needed to better distinguish between firstand second-order content in social goal inference, and to understand the factors that affect people’s use of these various
representational abilities. More generally, the inverse planning approach can be applied to a much wider range of social
goals than we have considered here. We are particularly interested to model goals such as “helping” or “hindering”, inwhich agents take as their reward function some function ofanother agent’s reward. Inference to these goals has recently
been shown to be within the reach of human infants. Though

How can we reason about the social goals of other people,
effortlessly inferring that “she’s chasing him” and “he wants
her to notice him”? On one side, bottom-up approaches focus on simple, easily computed cues such as the relative motion between two agents. On the other side, theory-based approaches cast these inferences as top-down processes drawing on abstract knowledge and sensitive to context and background knowledge. The theory-based approach is attractive
for its promise of capturing more realistic interactions acrossThis paper presented a family of theory-based models ofsocial goal inference. These models all embody the principleof rationality and use inverse planning as their core inferential
engine, but vary in the sophistication of representations required. At one end, the simplest of these models (L1) allows
agents to represent only the properties of other agents (such as
location), but not their goals. The most sophisticated of thesemodels (L2) realizes second-order social goal inference: reasoning about agents’ representations of other agents’ goals.
We distinguished simple cue-based models from our theorybased models using an experiment based on simple motion
patterns of pairs of agents, but both first-order and secondorder models accounted for people’s judgments.

®®

¬¬

¬®

P(Approach) P(Approach) P(Approach) P(Approach)

Movement condition®¬

People*1: Approach
0.5: Can’t tell

1

L2
*

1

0.5

0.5

Avoid

0

0

1: Approach

1

1

0.5: Can’t tell

0.5

0.5

Avoid

0

0

1: Approach

1

1

0.5: Can’t tell

0.5

0.5

Avoid

0

0

1: Approach

1

1

0.5: Can’t tell

0.5

0.5

0:

0:

0:

0:

Avoid

1 2 3 4 5 6 7

Context

0

1 2 3 4 5 6 7

Context

0

H2*

1 2 3 4 5 6 7

Context

Figure 3: Experimental results. Column 1: average subject ratings with standard error bars for all stimulus conditions. Column
2: predictions of L2 for all stimuli. L2 closely matches the trends of people’s judgments. Column 3: Predictions of H2 for all
stimuli. H2 makes the same predictions for every context, failing to account for the differences in people’s judgments due tochanges in the environment.
computationally challenging, inferring these goals still falls
under the scope of our inverse planning framework, and is an
exciting direction for future research.
Acknowledgments: This work was supported by AFOSR MURI
contract FA9550-05-1-0321, the James S. McDonnell Foundation
Causal Learning Collaborative Initiative and the Department ofHomeland Security Graduate Fellowship (CLB).

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