Shades of confusion: Lexical uncertainty modulates ad hoc coordination in an interactive communication task..

Cognition 225 (2022) 105152
Available online 20 May 2022
0010-0277/© 2022 Elsevier B.V. All rights reserved.Shades of confusion: Lexical uncertainty modulates ad hoc coordination in
an interactive communication task
Sonia K. Murthya,b, Thomas L. Griffithsa,c, Robert D. Hawkinsa,*
aDepartment of Psychology, Princeton University, Princeton, NJ, United States of America
bAllen Institute for Artificial Intelligence, Seattle, WA, United States of America
cDepartment of Computer Science, Princeton University, Princeton, NJ, United States of America
ARTICLE INFO
Keywords:
Communication
Concepts
Social representations
Word-color associations ABSTRACT
There is substantial variability in the expectations that communication partners bring into interactions, creating
the potential for misunderstandings. To directly probe these gaps and our ability to overcome them, we propose a
communication task based on color-concept associations. In Experiment 1, we establish several key properties of
the mental representations of these expectations, or lexical priors , based on recent probabilistic theories. Asso-
ciations are more variable for abstract concepts, variability is represented as uncertainty within each individual,
and uncertainty enables accurate predictions about whether others are likely to share the same association. In
Experiment 2, we then examine the downstream consequences of these representations for communication.
Accuracy is initially low when communicating about concepts with more variable associations, but rapidly in-
creases as participants form ad hoc conventions. Together, our findings suggest that people cope with variability
by maintaining well-calibrated uncertainty about their partner and appropriately adaptable representations of
their own.
1.Introduction
From jargon-filled scientific communication (Anderson-Cook, 2010;
Bullock, Col˘on Amill, Shulman, & Dixon, 2019; Martínez & Mammola,
2021) and medical consultations (Castro, Wilson, Wang, & Schillinger,
2007; Korsch, Gozzi, & Francis, 1968; McCabe & Healey, 2018) to the
linguistic battlefields of political discourse (Lakoff, 2006; Wodak, 1989), it
can often feel as if we are speaking different languages. Speakers not only
bring different perspectives, expertise, and background knowledge into a
conversation, they may even have different expectations about what words
mean (Labov, 1973; Marti, Piantadosi, & Kidd, 2019; McCloskey &
Glucksberg, 1978). For example, cognitive psychologists, doctors, and
lawyers all commonly use the word “trial” in professional settings but refer
to different concepts (a stimulus presentation, a clinical study, and a court
appearance, respectively). Such lexical variability is a troubling and
pervasive challenge, setting the stage for misunderstandings and other
communication breakdowns. How do we manage to understand each other
when we cannot be sure that we’re starting on the same page?
Modern theories have addressed this challenge by viewing communi -
cation not as a unitary act of transmission but as an ongoing collaborative
process where interlocutors must coordinate to reach mutual understanding over time (Clark, 1996; Davidson, 1986; Krauss & Fussell, 1996; Reddy,
1979; van Arkel, Woensdregt, Dingemanse, & Blokpoel, 2020). These
theories acknowledge idiosyncrasy, misunderstanding, and variation in
literal meaning across partners and communities as a fundamental and
unavoidable aspect of language use (Clark, 1998; Elman, 2004; Schuster &
Degen, 2020; Wilson & Carston, 2007). Still, a central open question has
concerned the underlying mental representations of signal meaning that
support this collaborative process. One hypothesis raised by recent prob-
abilistic theories of communication is that the ability to anticipate and
flexibly overcome misunderstandings depends on each speaker ’s initial
lexical uncertainty , reflecting prior expectations about what messages may
or may not mean to one’s partner (Bergen, Levy, & Goodman, 2016;
Brochhagen, 2020; Hawkins et al., 2022; Potts & Levy, 2015). This idea
builds on the classical construct of a mental lexicon containing signal-
meaning mappings, allowing words to be grounded in external referents.
For example, a speaker would consult their lexicon to evaluate the extent to
which a given word like “dog” applies to a given animal they’ve encoun -
tered in the world. Lexical uncertainty replaces a fixed dictionary of
mappings with a probability distribution over possible mappings that
different partners may be using. For example, there may be more uncer -
tainty about whether a given partner will share a given meaning for some
*Corresponding author.
E-mail address: rdhawkins@princeton.edu (R.D. Hawkins).
Contents lists available at ScienceDirect
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https://doi.org/10.1016/j.cognition.2022.105152
Received 13 May 2021; Received in revised form 7 February 2022; Accepted 26 April 2022
Cognition 225 (2022) 105152
2technical jargon than whether they will share a given meaning for “dog.1”
While lexical priors have played a key explanatory role in compu -
tational models of communication, they have been challenging to
measure and manipulate directly in experimental work. Classical studies
of coordination and communication have typically been restricted to
stimuli like ambiguous tangram shapes (Clark & Wilkes-Gibbs, 1986 ),
line drawings (Krauss & Weinheimer, 1964 ), or complex scenes (Weber
& Camerer, 2003 ) where lexical priors fall in a narrow, carefully-
calibrated band of uncertainty between completely random (e.g.,
white noise) and completely universal (e.g., a photograph of a proto -
typical dog). Despite the narrow range of these stimulus spaces, it has
been possible to isolate certain effects that are consistent with accounts
of lexical uncertainty. For example, there is evidence that the codeability
of tangrams (the number of distinct descriptions elicited; Hupet, Seron,
& Chantraine, 1991 ), and the shared expertise of speakers (whether
participants are equally familiar with New York landmarks; Isaacs &
Clark, 1987 ) both affect the time it takes for speakers to reach mutual
understanding. Still, there are many reasons to explore lexical uncer -
tainty in richer stimulus spaces and other communication modalities.
Richer stimulus spaces make it possible to observe and manipulate
lexical priors spanning a broader range of uncertainty, from the most idiosyncratic to the most universal. Other communication modalities
present a further opportunity to overcome inherent challenges associ -
ated with natural language, where probability distributions must be
estimated from sparse samples and poor coverage over the full (infinite)
spaces of possible utterances and meanings. For example, measures of
codeability for referential meaning (Hupet et al., 1991 ) are based on a
single description from each participant, where most descriptions
appear only once in the data set.
1.1. Color-concept associations as a window onto lexical uncertainty in
communication
To proceed in the face of these methodological challenges, we pro-
pose a communication paradigm based on color-concept associations.
While color has commonly been used as the target of reference in
communication tasks (Caldwell & Smith, 2012 ; Monroe, Hawkins,
Goodman, & Potts, 2017 ; Morin, Müller, Morisseau, & Winters, 2022 ;
Winters & Morin, 2019 ), we instead ask participants to use a set of color
chips as their communication modality (e.g., Roberts & Clark, 2018 ) to
communicate the identity of a target concept in a context of distractors
(e.g. lemon , happiness ). While these color chips clearly differ in important
ways from natural language utterances as a communication modality
(see General Discussion), we argue that color-concept associations
nevertheless provide a number of advantages for examining the conse -
quences of lexical uncertainty. First, color-concept associations natu-
rally span a wide range of possible priors (Fig. 1). For example, some
concepts, like lemon , are expected to have strong, nearly universal color
associations, reflected in tight priors P(color|lemon ). Meanwhile, other
concepts, like fairness are expected to have more idiosyncratic and
distributed associations, reflected in looser and more spread-out priors
Fig. 1.Example color associations. We depict the response distribution of the 10 concepts with lowest and highest variability. The width of each color bar corresponds to the
proportion of a particular color response for a given word. Colors are presented in a fixed order across bars but colors that received no responses are not visible in the given bar.
We report two estimates of variability for each concept (see Methods; in both cases, higher is more variable) .
1 While lexical uncertainty is typically defined as a joint distribution over the
full set of signal-meaning tuples P(m,s) (see Appendix A), it can also be
expressed as the conditional uncertainty over meanings for a given signal, P(m|
s), or as the conditional uncertainty over signals for a given meaning P(s|m),
since the baseline probabilities P(m) and P(s) are of lesser interest. We will
loosely use lexical prior and lexical uncertainty to talk about this whole family of
expressions, but the final conditional expression (s|m) is particularly useful for
eliciting priors. S.K. Murthy et al.
Cognition 225 (2022) 105152
3(Hutchings, 2004 ). Prior studies eliciting such associations have found
systematic differences across different abstract concepts (Barchard,
Grob, & Roe, 2017 ; Guilbeault et al., 2020 ; Mohammad, 2011 ; Rathore,
Leggon, Lessard, & Schloss, 2019 ; Volkova, Dolan, & Wilson, 2012 ),
which may also vary cross-culturally (Hupka, Zaleski, Otto, Reidl, &
Tarabrina, 1997 ; Tham et al., 2020 ). We build on the elicitation methods
developed in prior work while also introducing several methodological
innovations to answer novel questions arising in the context of
communication. Second, while natural language utterances are typically
understood to be embedded in a complex and high-dimensional se-
mantic space (Jones & Mewhort, 2007 ; Pennington, Socher, & Manning,
2014 ), color signals are embedded in a much lower-dimensional space
with better-validated psychometric structure, which allows for denser
sampling and explicit measurements of variation. Finally, color remains
an important modality of natural communication in its own right (Riley,
1995 ), as evidenced by the deliberate choice of color palettes in graphic
design (Marcus, 1982 ) and marketing (Labrecque & Milne, 2011 ), or our
everyday metaphorical appeals to color when trying to convey complex
emotional states that are challenging to describe with words (Lakoff &
Johnson, 1980 ; Meier & Robinson, 2005 ; Van Leeuwen, 2011 ).
1.2. Three foundational questions about lexical uncertainty
We use the domain of color-concept associations to evaluate three
foundational hypotheses about lexical uncertainty raised by recent
theories of communication. First, and most fundamentally, do in-
dividuals actually maintain an internal probability distribution repre -
senting their uncertainty about associations (Fig. 2, top row), or do they
only represent a point estimate giving their strongest association (Fig. 2,
bottom row)? Second, we ask: is the population relatively homogeneous,
composed of individuals sharing similar representations (Fig. 2, right
column) or is the population actually more heterogeneous and idiosyncratic (Fig. 2, left column)? Third, when it comes time to use
these representations in a communicative context, is a given individual ’s
representation purely egocentric or do they maintain well-calibrated
expectations about whether their representation will be shared by
other agents?
While we unpack these hypotheses and operationalize their pre-
dictions more thoroughly in subsequent sections, it is worth noting here
that there is theoretical precedent for these questions not only in the
communication literature but also in the literature on color-concept
associations. For example, the color inference framework introduced by
Schloss (2018) proposes that individuals store and continually update
their color-concept associations from their experiences in the world.
Under this framework, every concept has a corresponding association
space with some weight placed on each possible color. These weights
effectively give rise to an internal probability distribution that can, for
example, be used to generate appropriately discriminative colors to
convey different meanings (Mukherjee, Yin, Sherman, Lessard, &
Schloss, 2022 ). Such resonances between models of meaning across the
domains of natural language communication and of color-concept as-
sociations provides a further theoretical motivation for using the color
domain to explore representations of lexical uncertainty.
In Experiment 1, we begin to explore basic theoretical properties of
lexical uncertainty by eliciting color associations for a variety of con-
cepts. By eliciting multiple responses from each participant, we were
able to compare the population-level variability of responses to the in-
ternal variability within each individual. And by asking participants to
estimate the extent to which others will share the same associations, we
were able to probe the extent to which any given individual represents
the population distribution, thus forming a basis for successful
communication. Next, in Experiment 2, we directly measure the
downstream effects of lexical priors on ad hoc coordination using an
interactive communication task. Pairs of participants were asked to
Fig. 2.Schematic of candidate hypotheses and corresponding predictions. We explore a 2 ×2 space of proposals about how color-concept associations are internally
represented by individuals (as a fully calibrated probability distribution representing uncertainty, or as a sparser point estimate) and how much these associations differ across
individuals in the population (from a fully homogeneous population to a more heterogeneous population). These hypotheses make distinct predictions about the relationship
between the variability of associations measured within an individual and across a population, which we evaluate in Experiment 1. S.K. Murthy et al.
Cognition 225 (2022) 105152
4communicate about sets of concepts using color chips as signals. Criti-
cally, we used the corpus of associations we elicited in our first experi -
ment to construct contexts that manipulated participants ’ initial priors:
some associations were strongly shared across the population while
others were more idiosyncratic. Taken together, our work suggests that
people maintain well-calibrated prior expectations about the potential
for miscommunication and use these flexible priors to rapidly adapt to
their partner.
2.Experiment 1: Eliciting color-concept association priors
We begin by eliciting color association priors for concepts spanning a
wide range of population variability. While there are a number of
existing datasets that may be used to assess variability at the population
level (e.g. Mohammad, 2011 ; Tham et al., 2020 ; Volkova et al., 2012 ),
there were three specific desiderata for our work that were not satisfied
by these existing datasets. First, we required participants to not only
choose the color they themselves most associated with a concept, but
also to predict whether that association would be shared by others. Such
an explicit query about expected agreement was not included in the
protocol of previous studies. Second, while previous studies revealed
differences in population-level variability, it remains unclear whether
such variability arises within individuals (e.g. from internal probability
distributions) or only at the population-level (e.g. from individuals with
different point estimates). To distinguish between these possibilities, we
required multiple blocks of judgements for each participant; previous
variants of the task only collected a single best exemplar per participant.
Finally, we required a set of stimuli that would span a wide range of
expected association variability from strong (e.g. “lemon ”) to weak (e.g.
“fairness ”). Previous studies have focused on smaller sets of concepts (e.
g. Tham et al., 2020 , used only 59 abstract concepts, screening out words
that corresponded to concrete objects like “lemon ”) or only collected
coarse-grained responses for basic-level colors (e.g. Mohammad, 2011 )
asked participants to choose from the 11 color terms from Berlin & Kay,
1969 ).
2.1. Methods
2.1.1. Participants
We recruited 733 participants from Amazon Mechanical Turk,
restricting location to the United States. After implementing our pre-
registered exclusions criteria, we were left with data from 485 of these
participants. 129 participants were excluded for failing one of four
attention checks that were interspersed throughout the experiment.
These attention checks asked participants to provide color associations
for basic-level color words (e.g., “red”, “orange ”, “yellow ”). The first two
attention checks were embedded in a block of initial practice trials while
the other two were inserted randomly into the experiment (one in the
first half, and one in the second half). Participants who did not provide a
color within a (relatively permissive) set of valid responses were
immediately removed from the experiment for the base payment. We
also removed 5 additional participants who provided the same response
for more than three trials in a row, or consistently responded in less than
1000 ms, both of which indicated blind guessing without reading the
stimulus prompt. Finally, an additional 65 participants were excluded
due to colorblindness. To test color vision, we presented participants
with three Ishihara plates that detected common red-green deficiencies
or more extreme colorblindness. We excluded only those participants
whose responses indicated more extreme colorblindness (or
inattention). 2.1.2. Stimuli
We considered two factors when selecting our stimulus set. First, we
required a relatively large number of concepts to control for possible
item effects. Second, we needed these concepts to span a wide range of
different priors (i.e. different color associations, with different levels of
variability), but were unable to know these priors beforehand. We thus
considered a set of 5500 candidate concepts drawn from the Glasgow
Norms dataset (Scott, Keitel, Becirspahic, Yao, & Sereno, 2019 ), which
provides ratings for words on 9 different scales, including properties like
imageability and concreteness. Concreteness represents the degree to
which something can be experienced by the senses, while imageability
represents the extent to which a word invokes a mental image.2 We used
these measures as rough proxies for the level of variability in color as-
sociations we could expect for a concept and selected a balanced set of
200 concepts from this candidate set using the following procedure. We
began by imposing a familiarity threshold (familiarity ≽4.0) to ensure
that the majority of participants were likely to know the concept that
each stimulus word referred to. Next, we selected the 500 words with the
highest concreteness ratings (the concrete set) and the 500 words with
the lowest concreteness ratings (the abstract set). We then sampled 100
words from each set to obtain a roughly uniform distribution of
imageability. Lastly, we conducted a manual pass over the resulting 200
words to ensure that they were consistent in part of speech (e.g., con-
verting adjectives to their noun form) and to replace any that were
offensive, confusing, or redundant with one another. These 200 words
were randomly divided into 5 subsets, each containing 20 words from
the abstract set and 20 words from the concrete set.
2.1.3. Task, design, & procedure
Each participant was assigned one of the 5 distinct word sets. On each
trial, a single word was presented with a set of 88 (virtual) Munsell chips
sampling a wide range of color space.3 Participants were instructed to click
the color they most associated with the target word. To control for differ -
ences in individuals ’ color displays, participants were instructed to take the
experiment on a desktop or laptop computer and ensure that their screens
were set to their default brightness and color temperature (e.g. to turn off
programs like Flux). Participants were also screened through a pre-test
asking them to select color swatches for words like ‘blue ’ and ‘red’,
ensuring that any differences in color displays lay within tolerance of color
boundaries (see Supplemental Figs. S3-S5). To estimate internal variability,
we presented these words in a blocked sequence. After providing responses
for all 40 words in the set, presented in randomized order, participants
repeated the task a second time (Fig. 3A). On the second block, participants
were also asked: “How strongly do you expect others to share your color
association for this word? ” We presented a slider ranging from “not at all”
(most people will have a different color association than I do), to “very
strongly ” (most people will have the same color association as I do) with a
midpoint labeled “somewhat ” (roughly the same number of people will
have the same or different color associations as I do). This question allowed
us to compare the true proportion of shared responses to each participants ’
expectations.
2 Though these two measures are highly correlated with with each other (r
0.93), they are considered distinct aspects of a word ’s semantics (Paivio, Yuille,
& Madigan, 1968 ; Richardson, 1975 ). For example, emotion words like anger
may be rated low on concreteness but high on imageability; conversely, some
scientific or medical concepts like diabetes may be high on concreteness but low
on imageability
3 The World Color Survey (Kay, Berlin, Maffi, Merrifield, & Cook, 2009 ) used
the set of 320 chips (40 evenly spaced hues crossed with 8 levels of lightness)
proposed by Lenneberg and Roberts (1956) . Later studies down-sampled this set
to 160 (Heider, 1972 ) by removing hues at intermediate levels of 2.5 and 7.5,
and then further down-sampled to 80 (Gibson et al., 2017 ; Zaslavsky, Kemp,
Tishby, & Regier, 2019 ). Our 88-chip set was derived by adding 8 achromatic
chips to the 80-chip set from Gibson et al. (2017) , allowing participants to
select greyscale values. S.K. Murthy et al.
Cognition 225 (2022) 105152
52.1.4. Evaluation Metrics
We measured variability in two different ways: (1) the entropy of the
discrete response distribution and (2) the average pairwise similarity
between responses in color space.4 Our entropy measure was computed
on the distribution of response counts over the 88 Munsell chips, using
the Schurmann-Grassberger estimator (i.e. adding pseudo-counts of 1/
88 to each bin). Entropy is expected to be high when responses are
spread out across many different color chips, and low when participants
all concentrate their responses on a small number of colors. Our pairwise
distance measure is computed by taking the perceptual similarity ΔE
(where E stands for Empfindung , German for “sensation ”) between colors
in the psychometrically-validated CIELAB color space. It is close to
0 when colors are perceptually similar (colors with ΔE D1 are not able
to be discriminated by the human eye), and reaches values close to 100
for extremely dissimilar colors. We use the CIE2000 definition of ΔE,
which accounts for distortions in perceptual uniformity (Sharma, Wu, &
Dalal, 2005 ). For each word, we derived a measure of internal vari-
ability, internal ΔE, by taking the ΔE between the color chosen by a given
participant in the first block and the second block. We also obtained a
word-level measure of population variability, population ΔE, by taking
the average pairwise ΔE among every pair of color responses provided
by different participants.5 2.2. Results
2.2.1. Associations are more variable for abstract words
We begin by examining our use of concreteness and imageability as
proxies for constructing stimuli that elicit a wide range of associations.
These proxies were motivated by recent work suggesting that the extent
to which people share associations for a concept may be related to the
extent that they share the same sensory experiences for that concept, such
that abstract concepts have more variable associations than concrete
and imageable concepts. For example, recent analyses of Google Image
search results have found that raw color distributions found in the top
images retrieved for abstract terms are indeed more variable on average
than those measured for concrete terms (Desikan et al., 2020 ). At the
same time, however, the relationship between these measures is likely
more complex: recent work has also found coherent relationships among
abstract concepts (Guilbeault et al., 2020 ) and some abstract words may
nevertheless have strong associations (e.g. anger is red and sadness is
blue). To assess this relationship in our dataset, we constructed two
regression models predicting our two population variability measures
(entropy and ΔE) from the corresponding concreteness and imageability
ratings reported by (Scott et al., 2019 ) for each word (see Appendix B,
Fig. S1).
We found that both concreteness and imageability are independently
correlated with entropy (Spearman rank correlation ρ 0.51, p D
0.001 and ρ 0.54, p D0.001, respectively), suggesting that color
associations for more abstract words were much more variable. How-
ever, in a combined model including both predictors we only found a
main effect of imageability (b 0.31, t(191) 3.19, p 0.002),
with no independent contribution of concreteness (b 0.03, p 0.74),
suggesting that the information provided by concreteness may be
redundant. Similarly, both concreteness and imageability are indepen -
dently correlated with ΔE (ρ 0.53, p D0.001 and ρ 0.52, p D
0.001, respectively), but we found the opposite relationship in the
combined model: concreteness was a weak but significant predictor of
ΔE (b 1.22, t(191) 2.07, p 0.039), while imageability was not
(b 0.8, p 0.26). Together, these findings support a strong negative
relationship between abstractness and variability, regardless of how we
measure them, but we were unable to distinguish the unique contribu -
tions of concreteness from those of imageability due to their high
collinearity in the combined model. Importantly, while they were useful
for choosing stimuli, neither measure was a particularly good proxy for
population variability in absolute terms. We therefore use our own
direct estimates of variability (i.e. entropy and ΔE) to construct high and
Fig. 3.Design and results of Experiment 1. (A) We elicit two associations from each participant for a given word (1st block and 2nd block). On the second block, participants
additionally estimate the probability that others would provide the same association. Internal ΔE is measured as the similarity between the two responses provided by the same
individual, while population ΔE is measured between different individuals. (B) Internal ΔE of color associations for each concept tracks population variability ΔE, although a
single participants ’ responses are somewhat closer together than expected from sampling from the population distribution. (C) Expectations are well-calibrated to the true
agreement odds. Error bars for each concept are bootstrapped 95% CIs.
4 We also pre-registered a procedure to estimate parametric variability by
fitting a multi-dimensional Gaussian distribution over color space, but chose to
replace this measure by the ΔE measure. It is highly correlated with the
Gaussian measure, r 0.861, but better accounts for metric distortions in color
space and the existence of multi-modal response distributions.
5 We may define population ΔE in several ways, depending on where the
different participants ’ responses are drawn from. For example, block 1 ΔE
compares responses from different participants in the first block Δ(c1i,c1j),
block 2 ΔE restricts to the second block, Δ(c2i,c2j), and a “cross-block ” ΔE
compares responses in one block to those obtained from other participants in
the other block, Δ(c1i,c2j). These different ways of measuring population ΔE are
highly correlated (r 0.994 between the block 1 and cross-block versions; r
0.993 between the block 2 and cross-block versions; r 0.973 between the
block 1 and block 2 versions), and results are invariant to this choice. While
Fig. 3A shows the block 1 variant to make it clear that the axis of comparison is
across different participants, we report results using the version that pools
together all responses across both blocks to get the most highly powered
estimate. S.K. Murthy et al.
Cognition 225 (2022) 105152
6low variability conditions in Experiment 2, rather than relying on
coarser concreteness or imageability norms.
2.2.2. Population variability reflects individual uncertainty
Given that we successfully elicited associations with different levels
of variability in the population, we now proceed to ask how this vari-
ability is related to the internal representation of color-concept associ -
ations within each individual. First, we ask whether individuals
maintain a single point estimate representing a single strongest associa -
tion or instead maintain a full internal probability distribution over
different associations. Second, we ask whether the population is rela-
tively homogeneous , where all individuals maintain roughly the same
representations, or whether it is more heterogeneous , containing in-
dividuals with somewhat differing representations (see Fig. 2). To
evaluate this space of candidate hypotheses, we leveraged our blocked
design to compare internal variability against population variability.
In particular, we draw on the notion of the “crowd within ” (Fiechter
& Kornell, 2021 ; Hourihan & Benjamin, 2010 ; Rauhut & Lorenz, 2011 ;
Steegen, Dewitte, Tuerlinckx, & Vanpaemel, 2014 ; van Dolder & van
den Assem, 2018 ; Vul & Pashler, 2008 ) that has recently been proposed
for phenomena like forecasting and estimation in the decision-making
literature (see Herzog & Hertwig, 2014 , for a review). The “crowd
within ” views a judgement as a sample from an (implicit) probability
distribution, explaining why the same participant may make different
judgements at different times, and why averaging together multiple
judgements from the same participant may yield better predictions (an
analogue of the “wisdom of the crowd ” where judgements are averaged
across different participants). Thus, while it is unrealistic to perfectly
reconstruct any single individual ’s prior distribution from only two re-
sponses per concept, a key insight of this literature is that we do not need
to reconstruct the full distribution maintained by any single participant.
It is statistically sufficient to compare the distance between a small
number of samples obtained from a given participant for a given concept
(internal ΔE) against the expected null distribution of distances derived
from different participants for that concept (population ΔE; see Fig. 3A).
Each hypothesis predicts a different pattern of relationships between
internal ΔE and population ΔE (see Fig. 2). The distinction between
different internal representations concerns whether internal and popu -
lation variability are related across concepts. The point estimates repre -
sentation allows for some response noise but assumes such noise is
identical across all concepts, hence internal variability should be inde-
pendent of population variability and we would expect similar internal
ΔE across blocks for all concepts.6 Meanwhile, the internal probability
distribution account predicts that individuals maintain different distri -
butions for different concepts, which vary in their internal variability,
hence there should be a systematic relationship between internal ΔE and
population ΔE (regardless of whether the population is more homoge -
neous or heterogeneous). The distinction between population homoge -
neity and heterogeneity, on the other hand, concerns the extent to which
population variability is higher or lower overall than internal vari-
ability. In the extreme case where every participant i shared the same
distribution of color associations P(r|c) in a perfectly homogeneous
population, then the entire dataset of responses rij would be drawn from
the same distribution rij ~ P(r|c) and the distances between two samples
taken from the same participant i, Δ(ri1,ri2), would be the same, in
expectation, as the distances between samples taken from distinct par-
ticipants, Δ(r1j,r2j). In other words, we may obtain a distribution of the
expected variation in any pair of sampled responses, under the null
hypothesis of a homogeneous population, and compare the extent to
which the actual pairs of samples we obtained from the same participant are more or less similar than expected under this null.7
Our findings are shown in Fig. 3B. We found that the average ΔE
between an individual participant ’s responses was strongly correlated
with the overall population ’s ΔE, r 0.81, t(192) 19.04, p D0.001.
That is, for concepts where the population as a whole most disagreed
with one another, each individuals ’ own responses across blocks also
tended to disagree with one another. This pattern was only predicted by
the internal distribution account and not the point estimates account,
which cannot explain why there would be such systematic differences in
internal ΔE across concepts. At the same time, we found that internal
variability was lower than population variability overall; the estimated
intercept in a linear regression significantly differed from zero, b
17.3, t(192) 20.8, p D0.001, consistent with a more heterogeneous
account where not all participants shared exactly the same internal
distribution. A homogeneous population would be closer to the line of
unity.
2.2.3. Expectations about others are well-calibrated
Our results so far support the view that each individual implicitly
maintains a full probability distribution or prior for each concept, which
tracks the true population variability but which tends to be somewhat
narrower on average. However, it is unclear whether this representation
has a social component. In principle, communicative success depends on
each individual ’s expectations about how their partner will understand
(or misunderstand) a given message, not just whether individuals
represent similar meanings. This distinction is crucial for understanding
the consequences of lexical priors for resolving misunderstandings in
communication. If individuals are unable to represent whether others
will share the same prior (whether it is a point estimate or a distribu -
tion), they might over- or under-estimate agreement and inhibit suc-
cessful communication.8 To assess whether participants maintain well-
calibrated social expectations, we turned to the expected agreement
measure collected on the second block. For each expected agreement
rating in our dataset, we calculated the log odds that other participants ’
color responses actually matched that participants ’ reported color as-
sociation, log(p/(1 p)), representing ground-truth agreement. We
found a strong correlation between expected agreement and true
agreement (Pearson ’s correlation r 0.88, t(197) 25.5, p D0.001 and
Spearman ’s rank correlation, ρ 0.87, p D0.001), suggesting that in-
dividuals ’ expectations about the others ’ were remarkably well-
calibrated to the true statistics (see Fig. 3C).9
6 More formally, under this hypothesis, we could suppose an individual i’s
responses j are expected to be drawn from rij⊃NciCεwhere ci is their point
estimate for the concept and ε captures a fixed probability that they may click
on nearby color chips rather than the precise value ci. 7 There exist other possible explanations for a smaller internal ΔE, although
we have taken measures to minimize them; for example, if participants ’
response was strongly influenced by their response on the first block, then their
samples may violate the assumption of independent draws from the distribu -
tion. However, such influence is unlikely given the number of intervening
stimuli in each block and the relatively long delay between samples. We return
to this concern in the discussion.
8 Of course, there are multiple ways for well-calibrated social expectations to
be achieved. One possibility is that individuals maintain their own idiosyncratic
associations, based on their unique experiences, but also track the degree of
divergence from the overall population and correct for it in social settings using
theory of mind (i.e. they are aware that their associations are idiosyncratic).
Another possibility is that agents are egocentric (i.e. maintain a single internal
distribution of associations) but have tuned their own internal distribution over
time to match the population distribution. Distinguishing between these
representational possibilities is outside the scope of this paper.
9 While the log-odds linking function is orthogonal to our question of interest,
we note that it is consistent with previous observations about how participants
spontaneously map sliders to logarithmic rather than linear scales (e.g. Griffiths
& Tenenbaum, 2005, 2007 ; Landy, Silbert, & Goldin, 2013 ). S.K. Murthy et al.
Cognition 225 (2022) 105152
73.Experiment 2: The consequences of lexical priors for
communication
Experiment 1 established several key properties of word-color asso-
ciations: more abstract words have more variable associations, this
variability is represented within individuals, and individuals can accu-
rately predict whether an association is likely to be shared with others.
These properties are precisely those that probabilistic models of
communication have associated with lexical uncertainty , the recognition
that particular word meanings or concepts may or may not be shared
with others. Such uncertainty may allow for more flexible adaptation
than simple point estimates: individuals may be able to anticipate po-
tential confusions ahead of time, and possess rich enough knowledge
about likely alternative associations to rapidly adjust their expectations
on the fly. Here, we examine the downstream effects of these priors in a
Pictionary-like communication task where participants sent color
swatches as messages that allowed their partner to guess a target word.
We hypothesized that target words with nearly universal color associ -
ations, reflected in strong, tightly overlapping priors, would provide a
common foundation and allow for instant communicative success.
Meanwhile, words with more variable or uncertain color associations
would be more difficult to communicate about. In either case, individual
pairs should still be able to adaptively coordinate on mutually agreeable
solutions given the flexibility provided by their initial uncertainty.
3.1. Methods
3.1.1. Participants
We recruited 234 participants on Amazon Mechanical Turk and
paired them up to form 117 dyads. After removing 6 dyads that
disconnected before completing the task and 4 additional dyads where
at least one participant failed our attention checks, we were left with
107 dyads in our sample. Participants were screened for comprehension
and color vision before being paired with a partner (see Supplemental
Fig. S6).
3.1.2. Stimuli
We constructed 100 four-word contexts from the 200 words we used
in Experiment 1. We designed these contexts to span a broad range of
priors based on the population variability we estimated in the prior experiment. We iteratively sampled contexts to satisfy to two main
constraints, both derived from possible pragmatic context effects which
may mask the effects of priors: (1) we aimed to group words with similar
variability while (2) minimizing the extent to which different words in
the same context have overlapping priors. To satisfy these constraints,
we first ordered words by their estimated response entropy and greedily
sampled from the list of words to build an initial context. To check the
extent to which these words had overlapping priors, we computed the
Jensen-Shannon (JS) divergence between the Experiment 1 response
distributions for each pair of words in the proposed context. We imposed
a minimum divergence threshold of 0.3: when the context exceeded this
threshold for a word, we replaced it with the next in the list until a
satisfactory set was formed. We repeated this procedure to obtain 50
contexts where each word appeared in exactly one context and overlap
within each context was low. To obtain a distinct alternative set of
contexts, we repeated the same procedure with the additional criterion
of rejecting pairs of words that had previously appeared together in the
first set. These 50 contexts were divided into a “high prior variability ”
condition (the top 25) and a “low prior variability ” condition (the bot-
tom 25).
3.1.3. Procedure
Participants were paired into dyads to play an interactive reference
game using color as the communication medium (Fig. 4). Participants
were randomly assigned to speaker and listener roles and placed in an
environment containing a context of 4 concept words and 88 Munsell
chips, both shared in common ground. At the beginning of each trial,
one of the 4 words was privately shown to the speaker as the target
word. The speaker was then instructed to choose a color chip from the
set of Munsell chips that would best allow the listener to select that
target from the distractors. After the listener received the speaker ’s
message and clicked on one of the words, both participants received
feedback: the listener was shown the true target and the speaker was
shown the listener ’s selection. Participants were awarded a performance
bonus of $0.03 for each correct response.
3.1.4. Design
We tested the effect of lexical priors by manipulating the target
words in a within-dyad design: each dyad was assigned two 4-word
contexts, one “high prior variability ” context and one “low prior
Fig. 4.Design of Experiment 2. Participants are paired into dyads and assigned to speaker and listener roles. Both participants are shown the full set of Munsell chips and the
same context of 4 words. The speaker is additionally shown which one of these 4 words is the target word and asked to select a color to send to the listener. The listener then
guesses the word they believe is the target. Participants independently complete a color association elicitation task before and after the reference game for the test set of eight
words appearing in the communication game and also a control set of eight additional words . S.K. Murthy et al.
Cognition 225 (2022) 105152
8variability ” context. The trial sequence was constructed from 6 repeti -
tion blocks, allowing us to observe the trajectory of behavior as each
target is referred to multiple times. The four target words from each
context were randomly interleaved in each block, for a total of 48 trials
(six blocks of eight words). Participants switched roles between repeti -
tion blocks. Finally, we included two blocks of the prior elicitation task
used in Experiment 1. At the beginning and end of the reference game,
we asked participants to provide associations for the 8 test words in the
contexts they were assigned as well as 8 control words not encountered
during the reference game task – four with high prior variability and
with low prior variability.
3.2. Results
3.2.1. Shared priors facilitate communicative success
Our first prediction concerned the effect of priors on communicative
success, which we operationalized as the probability that the listener
correctly selects the target (chance is 25%). We hypothesized that par-
ticipants would initially struggle more to communicate in the high prior
variability condition, compared to the low prior variability condition,
where they could take advantage of stronger expectations and more
closely overlapping priors. At the same time, we expected participants to
improve through interaction, as they built common ground across
repeated appearances of the same targets. To test these hypotheses, we
constructed a logistic regression to predict correctness at the trial-by-
trial level, including fixed effects of condition (high prior variability
context vs. low prior variability context), repetition block number, and
their interaction, with the maximal random-effect structure that
converged (random intercepts and both main effects at the dyad-level).
We found a significant main effect of condition (b 0.44, z 4.8, p D
0.001) with higher accuracy for the low prior variability condition
throughout the task. We also found a significant improvement in accu-
racy across the task in both conditions (b 0.43, z 12.0, p D0.001),
reflecting ad hoc coordination. Finally, these effects were clarified by a
significant interaction (b 0.06, z 2.6, p 0.009), likely reflecting
ceiling effects for the low-variability conditions. On the first round,
dyads were at approximately 54% accuracy in the high variability
condition, compared to 67% accuracy in the low variability condition.
By the final round, they achieved approximately 78% and 91% accuracy, respectively.10 Because each condition contained a wide va-
riety of items spanning different priors (reflected in high variances for
random effect estimates), we also probed these effects using a more fine-
grained, individualized measure. We computed the ΔE distance between
the two participants ’ pre-test responses for each word in each game, and
found that quintiles of this continuous measure followed the same trend
(dashed lines in Fig. 5A).
3.2.2. Rapid convergence to shared meanings
To evaluate the extent to which participants flexibly shifted their
associations for target concepts over the course of interaction, we
compared pre-test and post-test responses. We operationalized the
similarity between partners ’ associations as the ΔE between their re-
sponses at each phase. For example, as a manipulation check, we found
that participants ’ responses were indeed more similar in the pre-test for
words in the low variability condition than the high variability condi -
tion, d 12.78, t(117.3) 9.1, p D0.001), implying that we successfully
constructed separable context sets from the population variability esti-
mates obtained in Experiment 1. Critically, however, we hypothesized
that participants ’ responses in the post-test would become significantly
closer for the words that were repeated in the reference game, compared
to a comparable set of control words that only appeared in the pre-test
and post-test. We tested these predictions in a mixed-effects linear
regression model including fixed effects of phase (pre-test vs. post-test),
word set (repeated vs. control), and condition (high vs. low variability)
as well as all interactions, including random intercepts and slopes for all
main effects). All variables were effect-coded to facilitate interpretation
of interactions. Because the three-way interaction in this model is
complex to reason about, we begin by considering a sub-model restricted
to the repeated set only. We found a significant main effect of condition,
b 0.25, t(110.8) 10.2, p D0.001, with responses to low prior vari-
ability words being more similar at both phases, as well as a main effect
of phase b 0.38, t(120.0) 16.49, p D0.001, with more similar
responses in the post-test for all words. We also find a significant
interaction, b 0.08, t(355.1) 3.87, p 0.0001, with high prior
variability words experiencing an even larger shift, likely reflecting a
floor effect. We evaluate the null hypothesis that this convergence
simply reflects additional practice with the task using our control words,
which appeared only in the pre-test and post-test. We found evidence of
Fig. 5.Results of Experiment 2. (A) Communicative success increases across repeated interaction. Dashed lines represent quintiles of initial similarity between partners ’
associations in the pre-test (a finer-grained measure than condition). (B) Speakers ’ associations begin closer in color space for low prior variability words than for high prior
variability words, but converge to closer associations in the post-test, relative to a control set that did not appear in the reference game. Error bars are bootstrapped 95% CIs.
10 A Bayesian logistic regression model with the maximal random-effect
structure at both the dyad- and item-level yielded similar effects (see Appen -
dix Figure S2 for full results) S.K. Murthy et al.
Cognition 225 (2022) 105152
9a three-way interaction, where the interaction between phase and con-
dition reported above is significantly different from the relationship
found for control words, where similarity between partners remained
relatively unchanged between the pre- and post-test (Fig. 5B).
3.2.3. The dynamics of adaptation
While we established strong effects of adaptation from the pre-test to
the post-test, it remains unclear how, exactly, adaptation unfolds
throughout the communicative interaction. Our final exploratory analysis
examines how each participant ’s choices as speaker change from round to
round as they observe one another ’s behavior. We took advantage of our
experimental design, which required participants to alternate between the
speaker and listener roles at the outset of each repetition block. We
measured two distances at each block i (see Fig. 6): (1) the distance be-
tween the chosen color chip chosen by the speaker on that block and their
own initial choice as measured in the pre-test (which we call “distance from
own initial ”) and (2) the distance between the speaker ’s chosen color chip
and their partner ’s initial choice (which we call “distance from other ’s
initial ”; note that the partner ’s initial choice is unobserved). We tested the
effects of speaker turn (first vs. second), initial variability condition (high
vs. low), and block number in mixed-effects regression models predicting
these distances, including random intercepts at the subject-level and item-
level as well as random effects for speaker turn at the subject level.
First, overall, participants tend to shift away from their initial asso-
ciations over time, b 1.2, t(4713) 5.5, p D0.001, while also moving
closer to their partner ’s initial associations, b 1.1, t(4702) 4.7,
p D0.001. These shift suggests that partners tend to settle somewhere in
the middle of their own initial associations and their partner ’s initial
associations. Second, however, we find a significant asymmetry in how
much each participant adapts. The first participant to take the role of the
speaker shifts away from their initial association to a lesser extent than
the second participant that takes the role of the speaker, b 4.9,
t(111.7) 6.03, p D0.001 and also shifts closer to their partner to a
lesser extent, b 3.7, t(113.8) 5.1, p D0.001. This asymmetry is
consistent with a “first mover advantage ” where the first participant in
the speaker role only has their own lexical prior to go on, while the
second participant has additional information from observing those
initial choices and is able to use that information to anchor their choices
closer to their partner ’s. The asymmetry created by this alternation is
solidified over the remainder of the game: the second participant actu-
ally ends up adopting a convention that is closer to their partner ’s initial
association than to their own. Third, we find that the initial variability
condition appears to simply shift this overall pattern up or down without qualitatively changing the effect. In the low-variability condition, par-
ticipants begin closer together but adaptation proceeds similarly, b
10.0, t(211.4) 10.8, p D0.001 and b 10.3, t(216.9) 9.7,
p D0.001 for one’s own associations and one’s partner ’s associations,
respectively.
4.Discussion
Communication is a continual challenge. Even when we speak the
same language, our vocabularies are full of words that may mean
different things to different partners. In this paper, we investigated the
mental representations of meaning that people use to overcome this
challenge. Using the domain of color-concept associations, where pop-
ulation variability can be carefully measured and manipulated, we first
established that variability does not take people by surprise: individuals
represent internal uncertainty about associations rather than point es-
timates and these agreement were well-calibrated to the actual
population-wide statistics. We then used these elicited distributions to
systematically manipulate the degree of variability in the lexical priors
of partners in an interactive communication game. Although commu -
nication was initially difficult for words with more variable associations,
participants were able to quickly adapt their expectations based on
common ground accumulated within the game, leading their priors to
become more similar over time. Taken together, these findings suggest
that partners enter communicative settings with well-calibrated but
flexible priors about the likely difficulty of communication. Our work
provides new support for recent probabilistic accounts of lexical uncer -
tainty in communication and raises a number of new directions.
First, most prominently, translating our findings from the relatively
low-dimensional domain of colors back to the much larger lexical priors
for natural language expressions will require further methodological
advances. While the gap between these domains may seem to limit
generalization and any potential differences in signaling medium need
to be evaluated empirically, we suggest that colors may have more in
common with words as a signaling medium than is initially apparent
(Schloss, 2018 ). Just as the meanings of discrete word “tokens ” are
typically thought to lie in a continuous space (such that we can mean -
ingfully talk about the semantic similarity between words or sentences),
participants were presented with a discrete set of 88 color “tokens ”
whose semantic similarity is also grounded in a continuous underlying
space (i.e. LAB space). Just as natural-language speakers cannot directly
convey meanings from the continuous semantic space and must pass
through the bottleneck of discrete tokens, our participants could not
Fig. 6.Dynamics of adaption in Experi -
ment 2. Dotted lines represent distance from
one’s own initial choice while solid lines
represent the distance from one’s partner ’s
initial choice choice. The red and blue lines
track each participant ’s identity as they
alternate roles. Both participants tend to shift
away from their initial associations over time
(dotted lines rise) while also moving closer to
their partner ’s initial associations (solid lines
fall). However, there is significant asymmetry
in the directionality of adaptation: the first
participant to take the role of the speaker
shifts from their initial association to a lesser
extent than the second participant that takes
the role of the speaker. Horizontal black lines
are provided for easier comparison to pre-test
distances; error bars represent bootstrapped
95% CIs. (For interpretation of the refer-
ences to color in this figure legend, the
reader is referred to the web version of this
article.)   S.K. Murthy et al.
Cognition 225 (2022) 105152
10directly access the continuous color space; they had to pass through the
bottleneck of discrete chips. Perhaps the most significant difference
between domains is not discreteness but that the underlying similarity of
the chips is exposed visually, whereas the semantic similarity of word
tokens is typically not exposed visually (e.g. words that are nearby in
meaning often appear very different when written out orthographi -
cally). Especially because we organized the discrete set of color tokens
according to their visual similarity in the task interface, we may have set
up an easier pathway for speakers to compare similar signals. But we
expect that this difference would primarily affect search and retrieval
from the space of discrete tokens rather than the representation of un-
certainty about a given token ’s meaning or the choice to use that token.
Second, our evidence for lexical uncertainty raises deeper mechanistic
questions about how, exactly, variability is encoded and learned by in-
dividuals. There are a number of different computational models for
probabilistic meanings that have arisen in natural language processing
(NLP), including Gaussian embeddings and Gaussian mixtures (Athiwar -
atkun, Wilson, & Anandkumar, 2018; Bra¯zinskas, Havrylov, & Titov, 2018;
Vilnis & McCallum, 2015) as well as a number of different proposals for
how these distributions may be learned over time depending on an in-
dividual ’s own idiosyncratic experiences (Johns & Jamieson, 2018; Johns,
Jones, & Mewhort, 2019; Kleinschmidt, 2019; Kraljic, Samuel, & Brennan,
2008). A related direction is to better characterize the mechanistic pro-
cesses allowing participants to align their associations to communicate
better (as we found in Experiment 2), and whether updates to an in-
dividual ’s associations are long-lasting or transient. One lower-level
explanation for alignment is that participants are simply priming one
another and making certain associations more salient or accessible (Pick-
ering & Garrod, 2006). A higher-level explanation, not mutually exclusive
with priming, is that speakers update their beliefs to form partner-specific
common ground (Hawkins et al., 2022). This hierarchical account predicts
that the extent to which local adaptation will persist in longer-term updates
depends on sustained use of that association over time, across different
partners. For example, as a slang term is more consistently and widely used
throughout a language community, it may eventually supplant whatever
initial associations individuals had with that word and persist as a longer-
term update. However, such persistence would likely require more than a
single interaction.
A third direction for future work is to examine how sources of vari-
ability in color associations arise in the first place. While we confirmed a
coarse relationship between the abstractness of a concept and the vari-
ability of its associations, it remains unclear how this relationship arises.
On one hand, some portion of this relationship may be driven by sensory
aspects of word representations. We are all exposed to roughly the same
visual imagery statistics for concrete concepts such as “tree, ” suggesting
strong associations with greens and browns. Meanwhile, abstract con-
cepts such as “justice ” lack concrete referents and associations may be
driven by more idiosyncratic semantic properties that vary depending
on each individuals ’ own history with the concept. These terms may be
more susceptible to cultural variation across different languages and
latent social groups (Tham et al., 2020 ), which would be interesting to
measure in broader cross-cultural samples.
It is worth noting several potential limitations of our data. For one,
there are intrinsic challenges associated with accurately sampling re-
sponses from color space. While we evenly sampled color chips from the
Munsell color chart (Landa & Fairchild, 2005 ; Munsell, 1905 ), following
standard practice for color elicitation (Berlin & Kay, 1969 ; Brown &
Lenneberg, 1954 ; Gibson et al., 2017 ; Sturges & Whitfield, 1995 ), there
are known distortions created by this set of chips (Zaslavsky, Kemp,
Regier, & Tishby, 2018 ). Most noticeably, discrepancies in the relative
number of green-blue and red-pink chips compared to yellow-ish chips
may have biased responses towards better-represented hues. We ex-
pected this distortion to have the biggest effect on entropy-based mea-
sures of variability, which is more sensitive to the support of the
response distribution then our ΔE measure. Second, our online data
collection setup prevented us from ensuring perfectly consistent color calibration across participants ’ screens, raising concerns that variability
in associations is simply due to presentation noise. While we did our best
to minimize such sources of variability and bias, our primary compari -
sons were importantly relative comparisons between different words
(e.g. between words with more or less variability). Because the display
of colors on a given participant ’s screen was likely to be fixed across the
experiment, any noise or bias arising from the color display should
contribute equally across all words and conditions. Third, it is possible
that our use of within-participant designs, both when eliciting multiple
responses across blocks in Experiment 1 and when interleaving high and
low variability contexts in Experiment 2, may have resulted in spillover
or “self-priming ” effects that reduce our estimates of internal variability.
Generally, these possible effects would work against our hypotheses: in
Experiment 1, self-priming would have favored the point estimate hy-
pothesis rather than the internal uncertainty hypothesis, and in Exper -
iment 2, it would have reduced our estimate of differences across
conditions. More broadly, it will be important to reproduce our findings
using longer delays between blocks.
More broadly, color associations are of substantial interest for
communication in their own right. The very properties that we high-
lighted in Experiment 1 may be responsible for the prevalence and
usefulness of color in communicating about abstractions, relative to
more concrete modalities (Gass, 1975 ; Johansson, Anikin, & Aseyev,
2020 ; Schloss, Witzel, & Lai, 2020 ; Winter, 2019 ). When a speaker says
“love is blue ” they draw attention to different semantic dimensions than
“love is bright red,” which may be difficult to reach with other meta-
phorical expressions. Thus, characterizing uncertainty in communica -
tion with color associations is not only useful for practical visual
communication like the choice of color in data visualization (Lin, For-
tuna, Kulkarni, Stone, & Heer, 2013 ; Schloss, Leggon, & Lessard, 2021 ;
Setlur & Stone, 2015 ), or for design elements like signage (Mahnke,
1996 ; Schloss, Lessard, Walmsley, & Foley, 2018 ), but is core to a more
general theory of multi-modal communication. The meanings of colors
may have much in common with the meanings of words we use in
everyday conversations. Rather than rigid dictionaries, well-calibrated
probability distributions allow us to better anticipate mis-
understandings and coordinate with one another to achieve mutual
understanding.
Author contributions
All authors conceived of the project and contributed to the study
design. SM and RH implemented the experiments, collected data, per-
formed all analyses, and wrote the manuscript. TG provided feedback on
the manuscript. All authors approved the final version of the manuscript
for submission.
Credit Authorship Contribution Statement
Sonia Murthy: Conceptualization, Methodology, Investigation,
Formal analysis, Software, Writing - Original Draft, Visualization.
Thomas Griffiths: Conceptualization, Writing- Reviewing and
Editing.
Robert Hawkins: Conceptualization, Methodology, Investigation,
Formal analysis, Software, Writing - Original Draft, Writing- Reviewing
and Editing, Visualization, Supervision, Project administration.
Author Note
Materials and code for reproducing all behavioral experiments and
analyses are open and available online at s��|�>22r t�s�l1!
m{y2skÐv~ {lo2m{w{~/ ~oq. The pre-registration for Experiment 1 is
at https://osf.io/v86xz and the pre-registration for Experiment 2 is at
s��|�>22{ �q1t{2ry}� �. Correspondence should be addressed to
Robert Hawkins (~nskÐvtz� E|~tzmo�{z 1on� ). S.K. Murthy et al.
Cognition 225 (2022) 105152
11Acknowledgements
We are grateful for helpful conversations with Bill Thompson, Christiane Fellbaum, Josh Peterson, and Ken Norman. This work was
supported by NSF Grant #1911835 to RH.
Appendix A.What is a lexical prior?
While the present work focuses on basic qualitative properties of lexical priors, and a formal model is not strictly necessary to interpret our findings,
an overview of previous formal definitions nonetheless helps make this construct more precise (see Bergen et al., 2016 ; Hawkins et al., 2022 , for a
more extensive treatment). We begin by defining a lexicon L, which specifies a specific set of form-meaning mappings, assign semantic meanings to all
utterances in a language.
Definition 1.A lexicon L : (u,m) → {0,1} is a function assigning a Boolean truth value11 to every pair of utterances u∃U and meanings m ∃M.
Traditionally, all speakers of a language are assumed to learn a single fixed lexicon specifying the meaning of every utterance in the language.
Example 1.Consider a simple referential language game where there are 2 possible utterances u ∃{u1,u2} and 2 possible meanings m ∃{m1,m2}.
Then a single lexicon L can be represented as a binary ∣U∣×∣M∣ matrix with 2 rows and 2 columns, where each entry represents whether utterance u
has meaning m or not, and L(u,w) simply looks up the specified entry:
L[
10
11]
Now, it is straightforward to define a lexical prior as a probability distribution over such matrices, representing an agent ’s uncertainty over exactly
which lexicon is being used by their partner.
Definition 2.A lexical prior is a probability distribution over the support of possible lexicons, denoted by P(L).
Note that the lexical prior P(L is distinct from P(u) or P(m), which simply represent the background probability of a given utterance or meaning
independently popping up in the environment.
Example 2.Let Lij be the matrix entry representing whether utterance ui has meaning mj. Then
P
Lij)
⊃Bernoulli5
defines a maximally uninformative prior where every utterance is equally likely a priori to have every meaning.
Example 3.If we additionally assume that lexicons satisfy the constraint that entries sum to one for each column of the matrix (a weak form of
mutual exclusivity, where every meaning is assumed to have exactly utterance that expresses it), then the lexical prior may be specified even more
compactly in conditional form as
P
Lij)
P
ui†mj)
⊃Categorical
θj)
C
where every meaning mj is associated with a vector θj giving a distribution over utterances. We are now prepared to consider how these theoretical
constructs correspond our setting of color-concept signaling games, where the meanings m are a discrete set of concept words (e.g. lemon , randomness )
and the utterances u are a discrete set of 88 color chips. We could in principle try to elicit the full latent object L by showing participants every color-
concept pair (u,m) and collecting a slider response or 2AFC judgement about whether that participant (or that participant ’s partner) would endorse
that specific mapping. This kind of elicitation procedure might be the most direct instantiation of the formal definition but is expensive to collect (e.g.
requiring 50 ×88 4400 judgements from a single participant to cover just a quarter of the concepts we use). Instead, we elicit the prior by taking
samples from the conditional distribution P(u|m), which also happens to be a well-vetted method for eliciting color associations. In other words, we
query a column of the lexicon (i.e. conditioning on a given concept) and ask participants to draw a sample from the induced distribution over color
chips. For this reason, it is convenient to define the lexical prior for a given concept m as the elicited distribution over colors given m, although it is a
slight abuse of the term.
Appendix B.Supplementary data
Supplementary data to this article can be found online at https://doi.org/10.1016/j.cognition.2022.105152 .
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