Nonliteral understanding of number words
Justine T. Kaoa,1, Jean Y. Wub, Leon Bergenc, and Noah D. GoodmanaaDepartment of Psychology and bSymbolic Systems Program, Stanford University, Stanford, CA 94305; and cDepartment of Brain and Cognitive Sciences,
Massachusetts Institute of Technology, Cambridge, MA 02139

One of the most puzzling and important facts about communication is that people do not always mean what they say; speakers
often use imprecise, exaggerated, or otherwise literally false descriptions to communicate experiences and attitudes. Here, we focus on
the nonliteral interpretation of number words, in particular hyperbole(interpreting unlikely numbers as exaggerated and conveying affect)
and pragmatic halo (interpreting round numbers imprecisely). We
provide a computational model of number interpretation as social
inference regarding the communicative goal, meaning, and affective subtext of an utterance. We show that our model predicts
humans’ interpretation of number words with high accuracy. Ourmodel is the first to our knowledge to incorporate principles ofcommunication and empirically measured background knowledge
to quantitatively predict hyperbolic and pragmatic halo effects innumber interpretation. This modeling framework provides a unified
approach to nonliteral language understanding more generally.
pragmatics| computational modelingImagine a friend describing a new restaurant where she recently
dined. Your friend says, “It took 30 minutes to get a table.”
You are likely to interpret this to mean she waited ∼30 min.
Suppose she says: “It took 32 minutes to get a table.” You are
more likely to interpret this to mean exactly 32 min. Now, suppose she says: “It took a million years to get a table.” You willprobably interpret this to mean that the wait was shorter than amillion years, but importantly that she thinks it took much too
long. One of the most fascinating facts about communication isthat people do not always mean what they say—a crucial part ofthe listener’s job is to understand an utterance even when its literalmeaning is false. People’s ability to interpret nonliteral language
poses a critical puzzle for research on language understanding.
A rich body of literature in psychology and linguistics has
examined how people use and understand nonliteral language(1–4). However, most of the work has been qualitative, with little
focus on analyzing aspects of an utterance that predict thequantitative details of people’s figurative interpretations. Here,
we present a computational model that formalizes and integrates
three general principles of language and communication to
explain the basis of nonliteral language understanding. First,
speakers and listeners communicate with the assumption that
their interlocutors are rational and cooperative agents; second,
listeners assume that speakers choose utterances to maximize
informativeness with respect to their communicative goals; third,
speaker and listener use common ground—their shared knowledge of the world—to communicate effectively. The first principle has been formalized by a recent body of work on rational
speech act (RSA) models, which views pragmatic language understanding as probabilistic inference over recursive social
models and explains a range of phenomena in human pragmatic
reasoning (5–8). We go beyond the previous formal work andaddress the second principle by extending the RSA framework.
We first extend the space of potential interpretations to include
subjective dimensions such as affective opinion. We then assume
that the listener is uncertain about the speaker’s communicative
goal and jointly infers both the goal and the intended meaning.
Because the interpretation space has multiple dimensions, aspeaker’s goal may be to maximize the probability of successfully
conveying information along one dimension of meaning but not
another. This makes it possible for a literally false utterance towww.pnas.org/cgi/doi/10.1073/pnas.1407479111

be optimal as long as it is informative along the target dimension.
These elements of the model have important connections to
Gricean pragmatics (9, 10) and relevance theory (11), in particular the argument that listeners infer the meaning of metaphors as well as other forms of loose talk by assuming thatspeakers maximize relevance (12, 13). Finally, we address the thirdprinciple of communication by empirically measuring people’s
background knowledge to understand the interaction between
nonlinguistic and linguistic knowledge in shaping language understanding. By applying this computational approach to a casestudy on number words, we show that nonliteral interpretations
can arise from basic principles of communication without positing
dedicated processing mechanisms for nonliteral language.
At the core of RSA models, a listener and a speaker recursively
reason about each other to arrive at pragmatically enrichedmeanings. Given an intended meaning m, speaker S1 reasons
about a literal listener L0 and chooses utterance u based on theprobability that L0 will successfully infer the intended meaning (7):S1 ðujmÞ ∝ L0 ðmjuÞ · e−CðuÞ :[1]

Here, C(u) is the psychological cost of an utterance, potentiallydetermined by factors such as the utterance’s frequency, availability, and complexity. The exponential results from applying
a Luce choice rule to model utterance choice, which is used
extensively in models of decision making (14). A pragmatic listener L1 then reasons about S1 and uses Bayes’ rule to infer themeaning m given utterance u, where P(m) is the prior probabilityof a meaning (although in principle speaker and listener canrecurse to arbitrary depth, here we stop at recursive depth 1):
L1 ðmjuÞ ∝ PðmÞS1 ðujmÞ:[2]

Because the RSA framework operates under the assumption thatspeakers optimize informativeness, it predicts that choosing an
Significance
Human communication is rife with nonliteral language, ranging from metaphor to irony to hyperbole. How do people go so
far beyond the literal meaning of an utterance to infer the
speaker’s intended meaning? We present a computationalmodel that understands hyperbolic and other nonliteral usesof number words (e.g., “That watch costs 10,000 dollars”).
Our model integrates empirically measured backgroundknowledge, principles of communication, and reasoning aboutcommunicative goals to explain the computational basis ofnonliteral language understanding. This framework sheds lighton the nature of communication, marking a significant advancement in the flexibility and richness of formal models of
language understanding.
Author contributions: J.T.K., J.Y.W., L.B., and N.D.G. designed research; J.T.K., J.Y.W., andL.B. performed research; J.T.K. and J.Y.W. analyzed data; and J.T.K. and N.D.G. wrote
the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
Freely available online through the PNAS open access option.
1

To whom correspondence should be addressed. Email: justinek@stanford.edu.

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Edited by Barbara H. Partee, University of Massachusetts Amherst, Amherst, MA, and approved July 2, 2014 (received for review April 25, 2014)

utterance whose literal meaning directly contradicts the intended
meaning is never optimal. However, this contradictory use isprecisely the case in nonliteral language. For example, people
understand the utterance “It took a million years to get a table”
to mean that the wait time was long but not, in fact, a million years,
resulting in a contradiction between literal and interpreted meaning. This suggests that the basic RSA model is incomplete andrequires additional elements to explain nonliteral communication.
Previous work has examined people’s communicative reasonsfor using figurative language and suggested that certain goals,
such as conveying emotion and emphasis, are commonly satisfied
by nonliteral language (1). A natural extension is thus to add
an affective dimension to the meaning of utterances, which has
interesting connections to previous work on expressives (15).
However, simply adding this dimension is insufficient; it is still
unclear how people infer affect from an utterance whose literalsemantics is unconnected to affect (such as number terms).
Here, we additionally extend the RSA framework to represent
alternative communicative goals, such that a speaker can want toconvey information about one dimension but not another. We
show that the combination of these two extensions is sufficient togive rise to nonliteral understanding of language.
We explore the case where the interpretation space has twodimensions: the state of the world and the speaker’s affect or
opinion. In what follows, we describe the subtext dimension as“affect,” but it could be other kinds of speaker attitude, mutatis
mutandis. The speaker is now modeled as follows:X δgðs;aÞ=gðs′;a′Þ L0 s′; a′u · e−cðuÞ ;[3]
S1 ðujs; a; gÞ ∝s′;a′where the intended meaning includes two dimensions s (the stateof the world) and a (the speaker’s affect). The function g projects
the listener’s inferred meaning onto relevant dimensions, meaning the speaker’s communicative goal is to be informative (only)
along this “topic” dimension. A literal listener interprets utterances literally without reasoning about the speaker, whereas apragmatic listener performs joint inference on both the speaker’s
goal and her intended meaning:X
L1 ðs; ajuÞ ∝PS ðsÞPA ðajsÞPG ðgÞS1 ðujs; a; gÞ:[4]g

The listener uses nonlinguistic background knowledge of theprobability of a state (PS) and the probability of having a particular affect given a state (PA), which we measure empirically(Experiment 3a: Price Prior and Experiment 3b: Affect Prior).
Based on the listener’s linguistic knowledge, the literal semantics
of utterance u conveys information about state s and nothing
about affect a. However, the common knowledge that affect isusually associated with certain states of the world allows thelistener to believe information about a given an assertion about s.
If it is known that the speaker’s goal is to convey affect, and not
the state, then the pragmatic listener will discount information
about s but retain information about a—a nonliteral interpretation is obtained. Even when the pragmatic listener is not certainof the speaker’s goal, a joint inference of goal, state, and affectcan also result in nonliteral interpretation. Common knowledgeof a domain and joint reasoning about communicative goals thus
allows the speaker to communicate additional dimensions of
meaning without explicitly describing these dimensions.
The incorporation of goal inference and multiple dimensionsof meaning is a major change to the existing RSA frameworkthat critically allows it to accommodate nonliteral language understanding. As a case study, we focus on the interpretation of
number words. We chose number words because they haveprecise literal meanings that can be easily modeled, and apply todomains (such as prices) that lend themselves to quantitative
measurement. We aim to capture two well-known phenomena
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Fig. 1. Model predictions of interpretations given utterances. Each bar in
the first three rows shows the probability of a type of interpretation given
an utterance. Exact interpretations are more likely given sharp rather thanround utterances; fuzzy interpretations are slightly more likely given roundutterances; hyperbolic interpretations are more likely given more extreme
utterances. The final row shows the probability of an affective interpretation.

regarding number interpretation: hyperbole and pragmatic halo.
Hyperbole is a figure of speech that uses exaggeration to convey
emphasis and emotion (16). Despite being literally false, hyperbolic
utterances are readily understood and serve purposes such as
establishing social closeness and expressing opinions (1, 16–18).
Pragmatic halo refers to people’s tendency to interpret roundnumbers such as 100 imprecisely and sharp numbers such as 103
precisely (19). The halo effect has been formalized in game-theoretic models as a rational choice given different utterance costs anda possibility of pragmatic slack (20, 21). Other research has shownthat speakers’ tendency to choose simple number expressions
decreases when more precise information is relevant to the listener(22), suggesting that higher-level pragmatic considerations such ascommunicative goals directly impact the production and interpretation of round versus sharp numbers. Our model uses alternative communicative goals coupled with differential utterance costs tomodel the pragmatic halo effect. We show that our framework for
pragmatic inference makes quantitative predictions for both hyperbole and pragmatic halo in the interpretation of number words.
Results
We tested our model on number words that refer to the prices ofthree types of everyday items: electric kettles, watches, andlaptops. We selected these items because they have distinct pricedistributions, PS, which we measured empirically by asking participants to rate the probability of various prices for the three
items (Experiment 3a: Price Prior). We also obtained an affectprior, PA, by asking participants to rate the probability ofa speaker thinking that an item is too expensive given a pricestate (Experiment 3b: Affect Prior). Using these priors, which
capture purely nonlinguistic knowledge, we aimed to model
people’s interpretations of utterances such as, “The electric
kettle cost u dollars.” A formal description of model assumptions
is in Materials and Methods.
Model Simulations. Using the price priors and affect priors measured for each of the three items, we obtained the meaning distributions predicted by the model for all utterances. Fig. 1
summarizes this distribution into different types of interpretations.
The first three are model interpretations regarding the pricestate: exact (e.g., “1,000” interpreted as 1,000), fuzzy (e.g.,“1,000” interpreted as 1,001), and hyperbolic (e.g., “1,000”
interpreted as 100). Round utterances (divisible by 10) such as“500” and “1,000” are interpreted less exactly and more fuzzily
than their sharp counterparts, which captures pragmatic halo.
Kao et al.

Utterances whose literal meanings are less likely given the priceprior are more likely to be interpreted hyperbolically (e.g.,“1,000” is more likely to be interpreted hyperbolically for electric
kettles than laptops), which captures a basic feature of hyperbole. Affective interpretation refers to the probability that anutterance conveys the speaker’s opinion that the price is expensive. Utterances whose literal meanings are associated withhigher affect priors (such as “10,000” and “10,001”) are more
likely to be interpreted as conveying affect, which predicts theaffective subtext of hyperbole.
To build intuition for these predictions, consider a pragmatic
listener who reasons about a speaker and analyzes her choiceof utterance. The pragmatic listener hears “10,000 dollars” andknows that its literal meaning is extremely unlikely. Given that the
speaker reasons about a literal listener who interprets “10,000
dollars” literally and believes that the speaker very likely thinks it isexpensive, “10,000 dollars” is an informative utterance if the
speaker’s goal is to communicate an opinion that the kettle
is expensive (without concern for the actual price). Because
the pragmatic listener uses this information to perform joint

inference on the speaker’s communicative goal and the meaning of
the utterance, he infers that “10,000 dollars” is likely to mean lessthan 10,000 dollars but that the speaker thinks it is too expensive.
Behavioral Experiments. We conducted experiment 1 to evaluate
the model’s predictions for the interpreted price. Participants
read scenarios in which a buyer produces an utterance about theprice of an item he bought, for example: “The electric kettle cost
1,000 dollars.” Participants then rate the likelihood that the item
cost s dollars for s ∈ S (Experiment 1: Halo and Hyperbole). Participants were more likely to interpret utterances as hyperbolic when
their literal meanings have lower probabilities under the item’s priorprice distribution [F(1,10) = 44.06; P < 0.0001]. To examine thehalo effect, we computed the difference between the probability of an exact interpretation and the probability of a fuzzy
interpretation for each utterance. This difference is significantly smaller for round numbers than for sharp numbers [F(1,28) =18.94; P < 0.001], which indicates that round numbers tend
to be interpreted less precisely than sharp numbers. To quantitatively evaluate the model’s fit, we compared model and humanFig. 3. (A) Probability of hyperbolic interpretation given utterances. The leftmost panel shows human data (error bars are SEs). A full model that uses pricepriors measured in experiment 3a demonstrates similar hyperbole effects and distinguishes among item types; a model that uses uniform price priors doesnot. (B) Halo effect as measure by bias toward exact interpretation for round/sharp utterance types. Humans’ bias toward exact interpretation is significantly
higher for sharp numbers. A full model that assigns higher cost to sharp numbers captures this result; a model that uses uniform utterance cost does not.

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Fig. 2. (A) Model predictions vs. average human responses from experiment 1. Each point represents an utterance and price state pair (u, s). The x coordinateof each point is the probability of the model interpreting utterance u as meaning price state s; the y coordinate is the empirical probability. Correlation
between model and human interpretations is 0.968 (95% confidence region in gray). (B) Comparison of models with different communicative goals andhuman interpretations for the utterance: “The electric kettle cost 1,000 dollars.” A model that considers both affect and precision goals (full model) mostclosely matches human data.

Fig. 4. (A) Model predictions of affect vs. human responses from experiment 2. Each point represents an utterance and price state pair (u, s). For pairs whereu = s, the utterance is literal; for u > s, the utterance is hyperbolic. The x coordinate of each point is the model’s prediction of the probability that theutterance/price state pair conveys affect; the y coordinate is participants’ affect ratings (error bars are SE). Correlation between model and humans is 0.775(95% confidence region in gray). (B) Probability of interpreting a hyperbolic/literal utterance as conveying affect. For the same price state, humans infer
higher probability of affect given hyperbolic utterances than literal. A model that uses affect priors measured in experiment 3b captures this result; a modelthat uses uniform affect priors does not.

interpretation probabilities across all utterances and showed thatmodel predictions are highly correlated with human interpretations of number words (r = 0.968; P < 0.0001) (Fig. 2A;Materials and Methods for details).
To show how each component of the proposed model isnecessary to capture effects observed in the human data, we
explore a series of simpler comparison models. For illustration,
Fig. 2B compares model interpretations of the utterance “Theelectric kettle cost 1,000 dollars” given inference over different
communicative goals. A model that does not consider alternative
goals interprets the utterance entirely literally. Note that eventhough such a model has information about the affect dimension(i.e., PA), without goal inference it is unable to produce nonliteral interpretations because it assumes that the speaker only
wants to maximize informativeness along the same dimension as
the utterance, i.e., the price state. A model that considers aspeaker whose goal may be to communicate precisely or imprecisely interprets the utterance as meaning either 1,000 or
1,001. A model that considers a speaker whose goal may be tocommunicate the price state or her affect prefers price stateswith higher prior probabilities. Finally, a model that considers
the full range of goals demonstrates hyperbole and halo effects

that closely match humans’ interpretations. To demonstrate that
our model is able to usefully incorporate nonlinguistic knowledge to infer the meaning of utterances, Fig. 3A shows the hyperbole effect as measured by the probability that an utterance u
is interpreted as price state s such that u > s. A full model thatuses empirically measured price priors captures humans’ interpretations, whereas a model that takes a uniform distribution
over price states does not. To demonstrate that our model is able
to use utterance costs and goal inference to capture pragmatic
halo, Fig. 3B shows the halo effect as measured by the bias toward exact interpretation for sharp versus round numbers. A fullmodel that assigns a higher utterance cost to sharp numbers
captures the significant difference in humans’ biases for sharp
versus round numbers, whereas a model where utterance costs
are uniform does not. These analyses suggest that extending the
RSA framework to include goal inference, incorporating empirically measured background knowledge, and including information
about utterance costs all contribute to the model’s ability tounderstand nonliteral language.
Does the model capture the rhetorical effect of hyperbole?
We conducted experiment 2 to examine humans’ interpretationof affect in hyperbolic versus literal utterances. Participants readFig. 5. (A) Posterior price state distributions predicted by the model given utterances. Each panel shows the interpretation distribution of an utterance. (B)Price state distributions rated by participants given utterances. Each panel shows the interpretation distribution of an utterance. Error bars are SEs.

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Discussion
We presented the first (to our knowledge) computational modelof nonliteral understanding that quantitatively predicts people’s
hyperbolic and imprecise interpretations of number words. Our
behavioral results show that complex patterns in number interpretation depend on common knowledge between speakerand listener, consideration of communicative efficiency, and,critically, reasoning about the speaker’s communicative goal.
Our model represents an explicit, computational-level hypothesis
about how these factors are integrated to give rise to theparticular, graded interpretations at which people arrive. The
model’s quantitative predictions closely match humans’ judgments, including cases of hyperbole, a complex phenomenon
previously beyond the scope of computational models.
The current approach has important connections to theories
of communication and linguistic meaning. Our speaker aims to
be informative, as in Gricean theories of communication, but
only with respect to a particular goal or topic—realizing a kind of
relevance principle. This relevance is critical for deriving nonliteral interpretations in our model. Although our model is currently limited to two dimensions of meaning and corresponding
goals, in future work we hope to capture dimensions central
to other figures of speech such as irony and metaphor, thus
extending our model to explain nonliteral language more broadly.
We believe that our framework significantly advances the flexibility and richness of formal models of language understanding,such that some day probabilistic models will explain everything(hyperbolically speaking).
Materials and Methods
Model. Let u be an utterance. The meaning of u has two dimensions: theactual price state s and the speaker’s affect a. We defined the set of pricestates S = {50, 51, 500, 501, 1,000, 1,001, 5,000, 5,001, 10,000, 10,001}. We
assumed that the set of utterances U is identical to S. We defined the set ofaffect states A = {0, 1} (0 means no affect and 1 means with affect—thisbinarization is purely for simplicity). Given S and A, the set of possible
meanings M is given by M = S × A. We denote each meaning as s, a, wheres ∈ S and a ∈ A.
The speaker S1 is assumed to be a planner whose goal is to be informative
about a relevant topic. We write the goal and its topic as g. S1 chooses
utterances according to a softmax decision rule that describes an approximately rational planner (14):Kao et al.

S1 ðujs,a,gÞ ∝ eU1 ðujs,a,gÞ :[5]

We want to capture the notion that the speaker aims to be informative about a
topic of discussion while minimizing cost. If the topic is represented by a projection g: M → X from the full space of meanings to a relevant subspace, thenthe speaker cares only about the listener’s distribution over the subspace,X

L0 ðxjuÞ = δx=gðs′,a′Þ L0 s′,a′u :

[6]

s′,a′

Following the RSA model, we formalize informativity of an utterance as
the negative surprisal of the intended meaning under the listener’s distribution; here, the listener’s distribution over the topical subspace X. Hence:
U1 ðujs,a,gÞ = log L0 ðgðs,aÞjuÞ − CðuÞ,[7]

where C(u) represents the utterance cost. Substituting into Eq. 5, this gives
the following:
S1 ðujs,a,gÞ ∝

X

 

δgðs,aÞ=gðs′,a′Þ L0 s′,a′u · e−CðuÞ :

[8]

s′,a′

In our situations, the speaker may have the goal to communicate along the pricedimension, affect dimension, or both. This gives three possible projections r:
rs ðs,aÞ = sra ðs,aÞ = a
rs,a ðs,aÞ = s,a:
The speaker may also want to communicate the price either exactly or approximately (we assume that no such distinction exists for affect, because we havealready binarized it). When the speaker wants to communicate the price approximately, she projects numbers to their closest round neighbors. For example,
such a speaker will represent the prices 51 and 1,001 as 50 and 1,000, respectively.
This gives two projections (exact and approximate), f, defined as follows:fe ðsÞ = s
fa ðsÞ = RoundðsÞ,where Round(s) denotes the multiple of 10 that is closest to s. The two typesof projections, f and r, can be composed to make the goal g of the speaker:g(s, a) = r(f(s), a), which results in 2 × 3 = 6 possible goals [although note thatra(fe(s), a) and ra(fa(s), a) are equivalent].
A literal listener L0 provides the base case for recursive social reasoning
between the speaker and listener. L0 interprets u literally without taking
into account the speaker’s communicative goals:L0 ðs,ajuÞ =PA ðajsÞ
0

if s = uotherwise:[9]

The pragmatic listener L1 performs Bayesian inference to guess the intended
meaning given the priors PS and PA and his internal model of the speaker. To
determine the meaning, the listener will marginalize over the possible goals
under consideration:
L1 ðs,ajuÞ ∝X

PS ðsÞPA ðajsÞPG ðgÞS1 ðujs,a,gÞ:

[10]

g

The prior probability of s is taken from an empirically derived price prior PS,
and the probability of a given s is taken from an empirically derived conditional affect prior PA (Experiment 3a: Price Prior and Experiment 3b: AffectPrior). The probability distribution P G is defined to be uniform. We used
C(u) = 1 when u is a round number (divisible by 10) and treated the
sharp/round cost ratio as a free parameter that we fit to data (Experiment 1:
Halo and Hyperbole). We obtained a posterior distribution for all possible
meanings s, a given an utterance u. Raw data for model predictions are at
http://stanford.edu/∼justinek/hyperbole-paper/data/model-predictions.csv.
Fig. 5A shows the full posterior distributions for all utterances.
Experiment 1: Halo and Hyperbole. A total of 120 participants was recruited
on Amazon’s Mechanical Turk. We restricted participants to those with IP
addresses in the United States (same for all experiments reported). Eachparticipant read 15 scenarios in which a person (e.g., Bob) buys an item (e.g.,a watch) and is asked by a friend whether the item is expensive. Bob
responds by saying “It cost u dollars,” where u ∈ {50, 50 ± k, 500, 500 ± k,
1,000, 1,000 ± k, 5,000, 5,000 ± k, 10,000, 10,000 ± k}, where k was randomly
selected from the set {1, 2, 3} for each trial. We refer to this set of utterances
as U. Given an utterance u, participants rated the probability of Bob thinking

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scenarios in which a speaker bought an item that cost s dollars
and says it cost u dollars, where u ≥ s. They then rate how likely itis that the buyer thinks the item was too expensive (Experiment 2:
Affective Subtext). We focused on the affect of an item being too
expensive because previous findings suggest that hyperbole ismore often used to communicate negative rather than positive
attitudes (1, 16). Results showed that utterances u where u > s
are rated as significantly more likely to convey affect than utterances where u = s [F(1,25) = 12.57; P < 0.005]. This suggests that
listeners infer affect from hyperbolic utterances above and beyond
the affect associated a priori with a given price state. Quantitatively, we compared model and human interpretations of affectfor each of the 45 utterance and price state pairs (u, s) where u ≥ s.
Although there is a significant amount of noise in the humanjudgments (average split-half correlation is 0.833), the model
predicts human interpretations of the utterances’ affective subtext significantly better than chance (r = 0.775; P < 0.00001),
capturing most of the reliable variation in these data (Fig. 4A).
To demonstrate how our model explains this effect, Fig. 4Bshows probabilities of affect given a price state and a literal or
hyperbolic utterance. The human data show that higher actual
price states are associated with higher probabilities of affect.
Within the same price state, hyperbolic utterances are interpreted as conveying more affect than literal utterances. These
effects are replicated by the full model, but not by a model thattakes in a uniform affect prior. This analysis suggests that therhetorical effect of hyperbole is driven in part by people’s sharedknowledge about prices and associated affect.

that the item was expensive. They then rated the probability of the item
costing the following amounts of money: 50, 50 ± k, 500, 500 ± k, 1,000,
1,000 ± k, 5,000, 5,000 ± k, 10,000, 10,000 ± k, where k was randomly selected from {1, 2, 3} for each trial. We refer to this set of prices as S. Ratingsfor each price state were on a continuous scale from “impossible” to “extremely likely,” represented as real values between 0 and 1. There are a totalof 30 possible trial configurations (3 Items × 10 Utterances). We randomized
the order of the trials as well as the names of the buyers (same for all
experiments). See stimuli for experiment 1 at http://stanford.edu/∼justinek/
hyperbole-paper/materials/experiment1.html.
We normalized participants’ ratings across price states for each trial to sum
up to 1. The average normalized ratings across participants for each item/utterance pair is shown in Fig. 5B, and the data can be found at http://stanford.
edu/∼justinek/hyperbole-paper/data/experiment1-normalized.csv. To adjust
for humans’ biases against using the extreme ends of the slider bars, we
performed a power-law transformation on the model’s distribution: we
multiplied the predicted probability for each meaning by a free parameter λand renormalized the probabilities to sum up to 1 for each utterance. We
jointly fit λ and the model’s cost ratio C to optimize correlation with the behavioral data. The best fit was with λ = 0.36 and C = 1.3, resulting in a correlation of r = 0.974 (95% confidence interval = [0.9675, 0.9793]). The range of
cost ratios that produces correlations within this confidence interval is [1.1,3.7], which is quite broad, suggesting that the overall model fit is not very
sensitive to the cost ratio. To further capture the details of the halo effect, we
jointly fit λ and C within this range to a measure that is more sensitive toutterance cost: we computed the difference between the probabilities of exact
versus fuzzy interpretations for each utterance, which gives us each utterance’s bias toward exact interpretation. We then computed the difference inthis bias for sharp versus round numbers, which gives us a “halo” score foreach sharp/round pair. We fit λ and C to minimize the mean squared error
between the model and humans’ halo scores. We found that the cost ratiothat best captures the magnitude and pattern of the halo effect found inparticipants’ data are 3.4, whereas λ = 0.25. This produces an overall correlation of 0.9677 with human data from experiment 1. All figures and analyses
that we report in the main text are with these parameter values.
For the analysis reported in Fig. 3A, we computed the probability of aparticipant interpreting an utterance u as hyperbolic by summing up ratingsfor each interpreted price state s where u > s. Because our analysis of hyperbole does not involve utterance costs, we collapsed across round andsharp versions of utterances and price states. For example, “1,001” interpreted as 1,000 does not count as hyperbole. Because 50 and 51 are thelowest available price states, the probabilities for hyperbolic interpretationof utterances “50” and “51” are 0. We computed the average probabilityof a hyperbolic interpretation across subjects for each utterance. We thenshowed the hyperbole effect with a linear regression model, using priorprobabilities for the utterances’ literal meanings as predictor and probabilities for hyperbolic interpretation as response. Results indicated that participants were more likely to interpret utterances as hyperbolic when theirliteral meanings have lower prior probabilities [F(1,10) = 44.06; P < 0.0001].
For Fig. 3B, we analyzed the pragmatic halo effect by computing eachsubject’s bias for interpreting an utterance u exactly versus fuzzily. Bias wasmeasured by subtracting the probability of a fuzzy interpretation from theprobability of an exact interpretation. We then obtained the average bias
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exact interpretation is significantly higher for sharp utterances than forround utterances [F(1,28) = 18.94; P < 0.001].
Experiment 2: Affective Subtext. A total of 160 participants was recruited on
Amazon’s Mechanical Turk. Each participant read 30 scenarios in which a person (e.g., Bob) buys an item that costs s dollars and is asked by a friend whether
the item is expensive. Bob responds by saying “It cost u dollars,” where u ∈ U
and u ≥ s. Participants then rated how likely Bob thinks the item was expensive
on a continuous scale ranging from “impossible” to “absolutely certain,” represented as real values between 0 and 1. There is a total of 180 trial configurations (3 Items × 60 {u, s} pairs, where u ≥ s). The stimuli for experiment 2
can be found at http://stanford.edu/∼justinek/hyperbole-paper/materials/experiment2.html; the raw data at http://stanford.edu/∼justinek/hyperbole-paper/
data/experiment2-raw.csv. Because our analysis of affective subtext does not
involve utterance cost, for the analyses reported in Fig. 4 A and B, we collapsed
round and sharp versions of each utterance and price state such that there are atotal of 45 utterance/price state pairs under consideration. Utterances u forwhich u = s are considered literal; utterances u for which u > s are hyperbolic.
For the analysis reported in Fig. 4B, we obtained average ratings of affect foreach utterance given that it is literal or hyperbolic. A linear regression model
showed that hyperbolic utterances are rated as having significantly higher affect than literal utterances across price states [F(1,25) = 12.57; P < 0.005].
Experiment 3a: Price Prior. To obtain people’s prior knowledge of the pricedistributions for electric kettles, laptops, and watches, 30 participants wererecruited from Amazon’s Mechanical Turk. Each participant rated theprobability of someone buying an electric kettle, laptop, and watch that costs dollars (s ∈ S), without any linguistic input from the buyer. Ratings for each
price state were on a continuous scale from “impossible” to “extremely
likely,” represented as real values between 0 and 1. The stimuli for experiment 3a can be found at http://stanford.edu/∼justinek/hyperbole-paper/materials/experiment3a.html. We normalized participants’ ratings acrossprice points for each trial to sum up to 1. The average normalized ratings foreach item were taken as the prior probability distribution of item prices.
These price distributions were used in the model as PS to determine the priorprobability of each price state. The normalized ratings can be found at http://
stanford.edu/∼justinek/hyperbole-paper/data/experiment3a-normalized.csv.
Experiment 3b: Affect Prior. To obtain people’s prior knowledge of the probability of affect given a price state, 30 participants were recruited from Amazon’s Mechanical Turk. Each participant read 15 scenarios where someone hadjust bought an item that cost s dollars (s ∈ S) without any linguistic input fromthe buyer. They then rated how likely the buyer thinks the item was expensive
on a continuous scale from “impossible” to “absolutely certain,” represented
as real values between 0 and 1. The stimuli for experiment 3b is at http://
stanford.edu/∼justinek/hyperbole-paper/materials/experiment3b.html. The average ratings for each price state were taken as the prior probability of anaffect given a price state and used in the model as PA. The data can be found at
http://stanford.edu/∼justinek/hyperbole-paper/data/experiment3b-raw.csv.
ACKNOWLEDGMENTS. This work was supported in part by National Science
Foundation Graduate Research Fellowships (to J.T.K. and L.B.), by a John S.
McDonnell Foundation Scholar Award (to N.D.G.), and by Office of Naval
Research Grant N000141310788 (to N.D.G.).

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