From partners to populations: A hierarchical Bayesian account of coordination and convention.

From Partners to Populations: A Hierarchical Bayesian Account
of Coordination and Convention
Robert D. Hawkins1, Michael Franke2, Michael C. Frank3, Adele E. Goldberg1, Kenny Smith4,
Thomas L. Grif /uniFB01ths1, 5, and Noah D. Goodman3, 6
1Department of Psychology, Princeton University
2Institute for Cognitive Science, University of Osnabrück
3Department of Psychology, Stanford University
4Centre for Language Evolution, University of Edinburgh
5Department of Computer Science, Princeton University
6Department of Computer Science, Stanford University
Languages are powerful solutions to coordination problems: They provide stable, shared expectations about
how the words we say correspond to the beliefs and intentions in our heads. Yet, language use in a variable
and nonstationary social environment requires linguistic representations to be /uniFB02exible: Old words acquire
new ad hoc or partner-speci /uniFB01c meanings on the /uniFB02y. In this article, we introduce continual hierarchical
adaptation through inference (CHAI), a hierarchical Bayesian theory of coordination and convention
formation that aims to reconcile the long-standing tension between these two basic observations. We argue
that the central computational problem of communication is not simply transmission, as in classical
formulations, but continual learning andadaptation over multiple timescales. Partner-speci /uniFB01c common
ground quickly emerges from social inferences within dyadic interactions, while community-wide social
conventions are stable priors that have been abstracted away from interactions with multiple partners. We
present new empirical data alongside simulations showing how our model provides a computational
foundation for several phenomena that have posed a challenge for previous accounts: (a) the convergence to
more ef /uniFB01cient referring expressions across repeated interaction with the same partner, (b) the gradual
transfer of partner-speci /uniFB01c common ground to strangers, and (c) the in /uniFB02uence of communicative context on
which conventions eventually form.
Keywords: communication, learning, convention, generalization, coordination
To communicate successfully, speakers and listeners must share a
common system of semantic meaning in the language they are using.
These meanings are social conventions in the sense that they are
arbitrary to some degree, but sustained by stable expectations thateach person holds about others in their community ( Bicchieri, 2006 ;
Hawkins, Goodman, & Goldstone, 2019 ;Lewis, 1969 ). Impor-
tantly, these expectations extend to complete strangers. An English
speaker may order an “espresso ”at any café in the United States and
expect to receive (roughly) the same kind of drink. At the same time,
meaning can be remarkably /uniFB02exible and partner speci /uniFB01c(Clark,
1996 ) The same words may be interpreted differently by different
listeners, or take on new ad hoc senses over the course of a
conversation. Interactions between friends and colleagues are /uniFB01lled
with proper names, technical jargon, slang, shorthand, and inside
jokes, many of which are unintelligible to outside observers. The
tension between these two basic observations, between global
stability and local /uniFB02exibility, has posed a challenging and persistent
puzzle for theories of social coordination and convention.
Many in /uniFB02uential computational accounts explaining how stable
social conventions emerge in populations do not allow for partner-
speci /uniFB01city at all (e.g., Barr, 2004 ;Hurford, 1989 ;Shoham &
Tennenholtz, 1997 ;Skyrms, 2010 ;Steels, 2011 ;Young, 2015 ).
These accounts typically examine groups of interacting agents who
update their representations of language after each interaction.
While the speci /uniFB01c update rules range from simple associative
mechanisms (e.g., Steels, 1995 ) or heuristics (e.g., Young, 1996 )
to more sophisticated deep reinforcement learning algorithms (e.g.,
Graesser et al., 2019 ;Mordatch & Abbeel, 2018 ;Tieleman et al.,
2019 ), all of these accounts assume that agents update a single,This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Robert D. Hawkins
https://orcid.org/0000-0001-9089-8544
Michael C. Frank
https://orcid.org/0000-0002-7551-4378
Kenny Smith
https://orcid.org/0000-0002-4530-6914
Thomas L. Grif /uniFB01ths
https://orcid.org/0000-0002-5138-7255
This manuscript is based in part on nonarchival work presented at the
39th, 40th, and 42nd Conferences of the Cognitive Science Society
(Hawkins et al., 2017 ,2018 ;Hawkins, Goodman, et al., 2020 ). Materials
and code for reproducing all model simulations, behavioral experiments,
and analyses are open and available online at https://github.com/hawkrobe/
conventions_model .
Thanks to Herb Clark, Judith Degen, Natalia Vélez, Rosa Cao, Hyo
Gweon, Judith Fan, Dan Yamins, Chris Potts, Iris van Rooij, Mark Blokpoel,
Marten van Schijndel, and Josh Armstrong for helpful discussions. This
work was supported by NSF grant No. 1911835 to Robert D. Hawkins, Adele
E. Goldberg, and Thomas L. Grif /uniFB01ths.
Correspondence concerning this article should be addressed to Robert
D. Hawkins, Department of Psychology, Princeton University, Peretsman
Scully Hall, Prince ton, NJ 08540, United States. Email: rdhawkins@
princeton.eduPsychological Review© 2022 American Psychological AssociationISSN: 0033-295Xhttps://doi.org/10.1037/rev0000348
1
monolithic representation of language to be used with every partner,
and that agents do not (knowingly) interact repeatedly with the same
partner.
Conversely, accounts emphasizing rapid alignment ( Pickering &
Garrod, 2004 ) or the development of partner-speci /uniFB01c common
ground ( Clark & Marshall, 1981 ;Clark & Wilkes-Gibbs, 1986 )
across extended interactions with the same partner typically do not
specify mechanisms by which community-wide conventions arise
over longer timescales. The philosopher Donald Davidson articu-
lated one of the most radical of these accounts. According to
Davidson (1984 ,1986 ,1994 ), while we bring background expecta-
tions ( “prior theories ”) into interactions, it is the ability to coordinate
on partner-speci /uniFB01c meanings ( “passing theories ”) that is ultimately
responsible for communicative success:
In order to judge how he will be interpreted, [the speaker] uses a picture
of the interpreter ’s readiness to interpret along certain lines, [ ::: ] the
starting theory of interpretation. As speaker and interpreter talk, their
“prior”theories become more alike; so do their “passing ”theories. The
asymptote of agreement and understanding is when passing theories
coincide. Not only does it have its changing list of proper names and
gerrymandered vocabulary, but it includes every successful use of any
other word or phrase, no matter how far out of the ordinary [ ::: ] Such
meanings, transient though they may be, are literal. ( Davidson,
1986 , p. 261).
This line of argument led Davidson (1986) to conclude that “there
is no such thing as a language ”(p. 265), and to abandon appeals to
convention altogether (see Dummett, 1994 ;Hacking, 1986 ;Heck,
2006 ;Lepore & Ludwig, 2007 , for further discussion of Davidson ’s
view; Armstrong, 2016a ,2016b provides a philosophical foundation
for a synthesis). Here, we propose an account of coordination and
convention that aims to reconcile the emergence of community-level
conventions with partner-speci /uniFB01c common ground in a uni /uniFB01ed
cognitive model. This theory is motivated by the computational
problems facing individual agents who must communicate with one
another in a variable and nonstationary world. We suggest that three
core cognitive capacities are needed for an agent to solve this
problem:
C1: the ability to represent variability about what words will mean to
different partners,
C2: the ability to coordinate on partner-speci /uniFB01c meanings via /uniFB02exible
online learning , and
C3: the ability to appropriately generalize stable expectations about
meaning to new partners.
These capacities are naturally formalized in a hierarchical Bayes-
ian framework, which we call the Continual Hierarchical Adaptation
through Inference (CHAI) model. Indeed, one of our central theo-
retical aims is to ground the problem of convention formation —a
fundamentally interactive, social phenomenon —in the same
domain-general cognitive mechanisms supporting learning in other
domains where abstract, shared properties need to be inferred along
with idiosyncratic particulars of instances ( Berniker & Kording,
2008 ;Goodman et al., 2011 ;Kleinschmidt & Jaeger, 2015 ;
Tenenbaum et al., 2011 ).
Our argument is structured around a series of three key phenom-
ena in the empirical literature that have proved evasive for previous
theoretical accounts of coordination and convention:P1: the convergence to increasingly ef /uniFB01cient referring expressions over
repeated interactions with a single partner,
P2: the gradual transition from partner-speci /uniFB01c common ground to
conventions that are expected to be shared throughout a commu-
nity, and
P3: the in /uniFB02uence of communicative context on which terms eventually
become conventionalized.
We begin by introducing the repeated reference game para-
digm at the center of this literatu re and reviewing the empirical
evidence supporting each of these phenomena. We then introduce
the CHAI model in detail and highlight several important prop-
erties of our formulation. The remainder of the article proceeds
through each of the three phenomena ( P1–P3)i nt u r n .F o re a c h
phenomenon, we present computational simulations to evaluate
how CHAI explains existing data and present new behavioral
experiments to test novel predictions when existing data do not
suf/uniFB01ce. Finally, we close by discu ssing several broader conse-
quences of the theory —including the continuity of language
learning across development and the domain generality of dis-
course processes —as well as limitations related to scalability and
incrementality.
Three Lessons About Convention Formation From
Repeated Reference Games
A core function of language is reference : using words to convey
the identity of an entity or concept. Loosely inspired by
Wittgenstein (1953) , empirical studies of coordination and con-
vention in communication have predominantly focused on the
subset of language use captured by simple “reference games. ”
In a reference game, participants are assigned to speaker and
listener roles and shown a context of possible referential targets
(e.g., images). On each trial, the speaker is asked to produce a
referring expression —typically a noun phrase —that will allow the
listener to select the intended target object from among the other
objects in the context.
Critically, unlike typical studies of referring expression genera-
tion ( Dale & Reiter, 1995 ;Degen et al., 2020 ;van Deemter, 2016 ),
repeated reference games ask speakers to refer to the same targets
multiple times as they build up a shared history of interaction with
their partners (see Table A1 in Appendix, for a review of different
axes along which the design has varied). And unlike agent-based
simulations of convention formation on large networks (e.g., Barr,
2004 ;Centola & Baronchelli, 2015 ;Steels, 2011 ), which typically
match agents with a new, anonymous partner for each trial, repeated
reference games ensure that participants know their partner ’s iden-
tity and maintain the same partner throughout extended interactions.
This design allows us to observe how the speaker ’s referring
expressions for the same objects change as a function of interaction
with that particular partner. We highlight three /uniFB01ndings of particular
theoretical signi /uniFB01cance that emerge from the repeated reference
paradigm.
P1: Increasingly Ef /uniFB01cient Conventions
The most well-known phenomenon observed in repeated refer-
ence games is a dramatic reduction in message length over succes-
sive rounds ( Clark & Wilkes-Gibbs, 1986 ;Hawkins, Frank, &This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.2 HAWKINS ET AL.
Goodman, 2020 ;Krauss & Weinheimer, 1964 ); participants are able
to communicate much more ef /uniFB01ciently without sacri /uniFB01cing accuracy.
The/uniFB01rst time participants refer to a /uniFB01gure, they tend to use a lengthy,
detailed description (e.g., “the upside-down martini glass in a wire
stand ”) but with a small number of repetitions —between 3 and 6,
depending on the pair of participants —the description may be cut
down to the limit of just one or two words (e.g., “martini ”)1. These
/uniFB01nal messages are as short or shorter than the messages participants
produce when they are instructed to generate descriptions for
themselves to interpret in the future ( Fussell & Krauss, 1989 )
and are often incomprehensible to overhearers who were not present
for the initial messages ( Schober & Clark, 1989 ). This phenomenon
sets up a /uniFB01rst puzzle about ad hoc convention formation: What
allows a word or short description that would be largely ineffective
at the outset of a conversation to take on local meaning over mere
minutes of interaction?
P2: Partner-Speci /uniFB01c Conventions
Because meaning is grounded in the evolving common ground
shared with each partner, ad hoc conventions established over a
history of interaction with one partner are not necessarily transferred
to other partners ( Brown-Schmidt, 2009 ;Metzing & Brennan, 2003 ;
Weber & Camerer, 2003 )2. For example, Wilkes-Gibbs and Clark
(1992) paired participants for a standard repeated reference game, but
after six rounds, the listener was replaced by a naive partner. Without
partner-speci /uniFB01c representations, we would expect speakers to con-
tinue using the short labels they had converged on with their /uniFB01rst
partner; instead, speakers reverted to the longer utterances they had
initially used, and then coordinated on new ad hoc conventions with
their new partner (see Figure 1 ). These effects raise our second puzzle:
How do community-level conventions form in the presence of such
strong partner speci /uniFB01city? When are agents justi /uniFB01ed in transferring an
ad hoc convention formed with one partner to a new, unseen partner?
One important empirical clue was provided by Fay et al. (2010) ,
who examined the emergence of conventions in a lab experiment
where communities of eight people played a repeated graphical
communication game similar to Pictionary, where participants
produced drawings to allow their partner to identify a concept
from a list of possibilities. The eight participants in each network
interacted dyadically with every other member of the community, in
turn, for a series of seven repeated reference games. Strikingly,
participants behaved as observed by Wilkes-Gibbs and Clark (1992)
during the /uniFB01rst few partner swaps, consistent with partner speci /uniFB01c-
ity, but with subsequent partners, their initial drawings showed a
gradual convergence with the conventionalized drawings they had
settled upon with previous partners, indicating a slow gradient of
generalization within their community.
While intriguing, this work was limited by an extremely small
sample size ( N=4 groups) and technical challenges facing the
measurement of conventions in the graphical modality (see also
Hawkins, Sano, et al., 2019 ). More recent work has adopted a
similar design for an arti /uniFB01cial-language communication task ( Raviv
et al., 2019 ) but collapses across repeated dyadic interactions to
exclusively analyze network-level metrics, making it dif /uniFB01cult to
assess partner speci /uniFB01city. Given these limitations of existing data,
we evaluate our model ’s predictions using new data from a large-
scale, real-time web experiment directly extending Wilkes-Gibbs
and Clark (1992) to larger networks.P3: Context-Sensitive Conventions
Finally, while a degree of arbitrariness is central to
conventionality —there must exist more than one solution that
would work equally well —this does not necessarily imply that
all possible conventions for a given meaning are equally likely in
practice, or even any convention will form at all ( Hawkins &
Goldstone, 2016 ). Indeed, functional accounts of language have
frequently observed that lexical systems are well calibrated to the
needs of users under the statistics of their communicative environ-
ment ( Gibson et al., 2019 ). This Optimal Semantic Expressivity
hypothesis (OSE; Frank, 2017 ) has held remarkably well for the
lexical distributions found in natural languages across semantic
domains like color words and kinship categories ( Gibson et al.,
2017 ;Kemp et al., 2018 ;Kemp & Regier, 2012 ;Regier et al., 2015 ).This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure 1
Classic Phenomena in Repeated Reference Games
10203040
1 3 5 7 9 11
repetition # for speaker# wordspartner 1 partner 2
Note. Over multiple iterations with the same partner, speakers converge onincreasingly ef/uniFB01cient referring expressions (reps. 1–6). When the listener isreplaced by a new, naive partner, speakers display a key signature of partnerspeci/uniFB01city, reverting to longer utterances before converging again with theirnew partner (reps. 7–12). Comprehension failures tend to be rare (∼2.3%)throughout the experiment, indicating that speakers modulate their utteranceseffectively. These data were reconstructed from Table 3 inWilkes-Gibbs andClark (1992). See the online article for the color version of this/uniFB01gure.
1Of course, referring expressions are also lengthened for many reasonsother than pure references, such as politeness (Professor Davidson vs. Don),affection (Donny vs. Don), emphasis (the one and only Donald Davidson), orany number of manner implicatures (seeHorn, 1984;Levinson et al., 2000).However, the marked meanings of these longer forms are only obtainedagainst the backdrop of an unmarked or“default”form; repeated referencegames set these other functions aside to examine where unmarked expecta-tions come from and how they depend on discourse context (Grosz, 1974;Grosz & Sidner, 1986) This distinction may be seen by considering thenonreferential implicatures that may be triggered if a speaker suddenlyswitched from“martini”back to their original longer description at theend of a game.2We use the term“ad hoc convention”(inspired byBarsalou, 1983)interchangeably with the more common“conceptual pact”(Brennan &Clark, 1996;Ibarra & Tanenhaus, 2016) to emphasize the theoreticalrelationship between this construct and the usual sense of conventionreferring to longer-term communal knowledge.CONVENTIONS3
While such long-term, diachronic sensitivity to context has been
explained by abstract principles of optimality, such as the equilibria
concepts of evolutionary game theory ( Jäger, 2007 ;Jäger & Van
Rooij, 2007 ), it has not yet been grounded in a cognitive and
mechanistic account of the immediate, synchronic processes un-
folding in the minds of individual agents while they interact. In other
words, while there is abundant empirical evidence for context
sensitivity in the outcomes of convention formation processes,
our third puzzle concerns which cognitive mechanisms in indivi-
duals may be necessary or suf /uniFB01cient to give rise to such conventions
(seeBrochhagen, 2021 , which raises a similar linking problem).
Repeated reference games have emerged as a promising method
for probing these mechanisms in the lab. Such games allow re-
searchers to explicitly control the communicative context and
observe the resulting distribution of conventions that emerge
when participants communicate using arti /uniFB01cial languages ( Kirby
et al., 2015 ;Winters et al., 2015 ,2018 ). While these existing studies
are informative, it has remained challenging to directly evaluate
cognitive models against the full trajectories of convention forma-
tion on a trial-by-trial basis. In our /uniFB01nal section, we report new
empirical data from a dyadic repeated reference task manipulating
context, where simulated agents and human participants are shown
directly analogous trial sequences.
Convention Formation as Hierarchical
Bayesian Inference
In this section, we propose a uni /uniFB01ed computational account of ad
hoc coordination and convention formation that aims to address
these three empirical puzzles. We begin from /uniFB01rst principles: What
is the core computational problem that must be solved to achieve
successful communication? Classically, this problem has been
formulated in terms of coding and compression ( Shannon, 1948 ).
An intended meaning in the speaker ’s mind must be encoded as a
signal that is recoverable by the receiver after passing through a
noisy transmission channel. This transmission problem has since
been enriched to account for pragmatics —the ability of speakers
and listeners to use context and social knowledge to go beyond the
literal meaning of messages ( Rosenberg & Cohen, 1966 ;Sperber &
Wilson, 1986 ). We take the Rational Speech Act framework (RSA;
Frank & Goodman, 2012 ;Franke & Jäger, 2016 ;Goodman &
Frank, 2016 ) as representative of this current synthesis, formalizing
communication as recursive social inference in a probabilistic model
(see Appendix A ). In the next section, we review this basic
framework and raise two fundamental computational problems
facing it. These problems motivate the introduction of continual
learning in the CHAI model.
Models of Communication With Static Meaning
For concreteness, we restrict our scope to reference in a context C
containing a discrete set of objects o∈O, but the same formulation
aims to apply more generally. In this referential setting, the RSA
framework de /uniFB01nes a pragmatic speaker, denoted by S1, who must
choose an utterance uthat will allow their partner to choose a
particular target object o/C3∈C. They attempt to satisfy Gricean
Maxims ( Grice, 1975 ) by selecting utterances according to a utility
function U(u;o) that balances informativity to an imagined listener
against the cost of producing an utterance. Speci /uniFB01cally, S1choosesfrom a “soft-max distribution ”concentrating mass on the utterance
that maximizes U(u;o) to an extent modulated by a free parameter
αS∈[0,∞]:
S1ðujoÞ∝expfαS·Uðu;oÞg: (1)
ForαS=1, this decision rule corresponds to Luce ’s choice axiom
(Luce, 1959 ). Larger settings of αSconcentrate more probability on
the single utterance maximizing utility.
The basic speaker utility function in the RSA framework is
de/uniFB01ned as follows:
Uðu;oÞ=ð1−wCÞ· log L0ðojuÞ|ﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ}
informativity−wC·cðuÞ|{z}
cost, (2)
where c(u) is a function giving the cost of producing u, assuming
longer utterances are more costly, and wC∈[0, 1] is a second free
parameter controlling the relative weight of informativity and
parsimony in the speaker ’s production. Critically, the informativity
term in Equation 2 is de/uniFB01ned by how well utransmits the intended
target o*to an imagined listener. The simplest imagined listener, L0,
is typically called the “literal listener ”because they are assumed to
identify the target relying only on the literal meaning of the received
utterance, without appealing to further social reasoning about the
speaker. That is, the probability of the imagined listener choosing
object ois simply assumed to be proportional to the meaning of u
under some (static) lexical function L:
L0ðojuÞ∝Lðu,oÞ:
Throughout this article, we will take Lto be a traditional Boolean
function evaluating whether or not the expression uapplies to the
entity in question3:
Lðu,oÞ=/C26
1 if o∈/uni27E6u/uni27E7
0 otherwise:
Two Fundamental Problems for Static Meaning
The RSA framework and its extensions provide an account for a
variety of important phenomena in pragmatic language use (e.g.,
Kao et al., 2014 ;Lassiter & Goodman, 2017 ;Scontras et al., 2018 ;
Tessler & Goodman, 2018 ). Yet, it retains a key assumption from
classical models: that the speaker and listener must share the same
literal “protocol ”Lfor encoding and decoding messages. In this
section, we highlight two underappreciated challenges of commu-
nication that complicate this assumption.This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.3Due to the current limitations of representing lexical meaning in formal
semantics, it has not been straightforward to specify a truth-conditional
function explaining listener behavior for natural-language utterances (e.g.,
what makes one drawing belong in the literal extension of “upside-down
martini glass ”but not another, when neither of them are literally martini
glasses?) This representation is convenient for our simulations, where we
consider all possible discrete mappings between utterances and objects in the
context, but better representations of lexical meaning may be substituted (see
Potts, 2019 ). For example, Appendix B works out an example using a real-
valued, continuous function ( Degen et al., 2020 ) such as those learned by
multimodal neural networks ( Monroe et al., 2017 ;Achlioptas et al., 2019 ;
Hawkins, Kwon, et al., 2020 ).4 HAWKINS ET AL.
The/uniFB01rst problem arises from the existence of variability through-
out a language community ( Kidd et al., 2018 ;Wang & Bi, 2021 ).
Different listeners may recover systematically different meanings
from the same message, and different speakers may express the
same message in different ways. For example, doctors may /uniFB02uently
communicate with one another about medical conditions using
specialized terminology that is meaningless to patients. The words
may not be in the patient ’s lexicon, or common words may be used in
nonstandard ways. That is, being /uniFB02uent speakers of the same language
does not ensure agreement for the relevant meanings expressed in
every context. Different partners may be using different functions L.
The second problem arises from the nonstationarity of the world.
Agents are continually presented with new thoughts, feelings, and
entities, which they may not already have ef /uniFB01cient conventions to
talk about ( Gerrig & Gibbs, 1988 ). For example, when new tech-
nology is developed, the community of developers and early
adopters must /uniFB01nd ways of referring to the new concepts they
are working on (e.g., tweeting ,the cloud ). Or, when researchers
design a new experiment with multiple conditions, they must /uniFB01nd
ways of talking about their own ad hoc abstractions, often converg-
ing on idiosyncratic names that can be used seamlessly in meetings.
That is, any /uniFB01xedLshared by a group of speakers at one moment in
time can quickly become outdated (see Lazaridou et al., 2021 , for a
demonstration of the related problems posed by nonstationary for
large neural language models). We must have some ability to extend
our language on the /uniFB02y as needed.
CHAI: A Model of Dynamic Meaning
Rather than assuming a monolithic, universally shared language,
we argue that agents solve the fundamental problems posed by
variability and nonstationarity by attempting to continually, adap-
tively infer the system of meaning used by their current partner.
When all agents are continually learning in this way, and changing
their own behavior to best respond, we will show that they are not
only able to coordinate on local, ad hoc meanings or pacts with
speci /uniFB01c partners but also abstract away conventions that are ex-
pected to be shared across an entire community. We introduce the
CHAI model in three steps, corresponding to how it formalizes the
three core capacities C1–C3: hierarchical uncertainty about mean-
ing, online partner-speci /uniFB01c learning, and inductive generalization.
C1: Representing Variability in Meaning via
Structured Uncertainty
When an agent encounters a communication partner, they must
call upon some representation about what they expect different
signals will mean to that partner. We, therefore, replace the mono-
lithic, static function Lwith a parameterized family of lexical
meaning functions by Lϕ, where different values of ϕyield different
possible systems of meaning. To expose the dependence on a /uniFB01xed
system of meaning, Equation 2 can be rewritten to give behavior
under a /uniFB01xed value of ϕ:
L0ðoju,ϕÞ∝Lϕðu,oÞ
Uðu;o,ϕÞ=ð1−wCÞ· log L0ðoju,ϕÞ−wC·cðuÞ
S1ðujo,ϕÞ∝expfαS·Uðu;o,ϕÞg:(3)While we will remain agnostic fo rn o wt ot h ee x a c tf u n c t i o n a l
form of Lϕand the exact parameter space of ϕ,t h e r ea r et w o
computational desiderata we emp hasize. First, given the challenge
of variability raised in the previ ous section, these expectations
ought to be sensitive to the overall sta tistics of the population .
That is, an agent should know that more people will share the
meaning of some words than others, and should conversely expect
more consensus about how to refer to some concepts than others.
Second, expectations about which meanings will be evoked for a
given utterance and which utterances are expected to be used to
express a meaning should be sensitive to the social identity of
one’sp a r t n e r .T h e /uniFB01rst desideratum —the ability to represent
variability in the population —motivates a probabilistic formula-
tion. Instead of holding a single static function Lϕ,w h i c ha na g e n t
assumes is shared perfectly in common ground (i.e., one ϕfor the
whole population), we assume each agent maintains uncertainty
over the exact meaning of each word as used by different partners.
In a Bayesian framework, t his uncertainty is speci /uniFB01ed by a prior
probability distribution P(ϕ)o v e rp o s s i b l ef u n c t i o np a r a m e t e r s .
For example, imagine a doctor giving a diagnosis to a new patient.
Under some possible values of ϕ,ap i e c eo fm e d i c a lj a r g o nl i k e
“sclerotic aorta ”refers unambiguously to the patient ’sh e a r t
condition. Under other values of ϕ,i th a sal e s sc l e a rm e a n i n g .
Ad o c t o rw i t hg o o db e d s i d em a n n e rs h o u l da s s i g ns o m ep r o b a -
bility to each possibility rather than assuming everyone will share
the same precise meaning they learned in medical school. Impor-
tantly, this variability will be diffe rent for different words: likely
more people share the meaning of “dog”than “sclerotic aorta. ”
This core idea of introducing uncertainty over a partner ’sl e x i c a l
semantics has previously been explored in the context of one-shot
pragmatic reasoning, where it was termed lexical uncertainty
(Bergen et al., 2012 ,2016 ;Potts et al., 2016 ;Potts & Levy,
2015 ), as well as in the context of iterated dyadic interactions
(Smith et al., 2013 ).
Second, this representation should also, in principle, be sensitive
to the social identity of the partner: a doctor should be able to form
different expectations about a new colleague than a new patient
(Clark, 1998 ). This desideratum —sensitivity to partner-speci /uniFB01c
meanings —motivates a hierarchical model, where uncertainty is
represented by a multilevel prior. At the highest level of the
hierarchy is community-level uncertainty P(Θ), where Θrepresents
an abstract “overhypothesis ”about the overall distribution of all
possible partners. This level can be viewed as a representation of
long-term “communal lexicons ”about common knowledge based
on community membership ( Clark & Marshall, 1981 ).Θthen
parameterizes the agent ’spartner-speci /uniFB01cuncertainty PðϕkjΘÞ,
where ϕkrepresents the speci /uniFB01c system of meaning used by partner
k(seeFigure 2 ).ϕkcan be viewed as the “idiolect ”that has been /uniFB01ne-
tuned to account for partner-speci /uniFB01c common ground and concep-
tual pacts from previous interactions4.This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.
4We focus for simplicity on this basic two-layer hierarchy, but the model
can be straightforwardly extended to representing uncertainty at interme-
diate layers of social structure, including whether partners belong to
distinct subcommunities (e.g., represented by discrete latent variables
Gershman & Cikara, 2020 ;Gershman et al., 2017 ), which may explain
code-switching ( Auer, 2013 ;Hawkins et al., 2021 ) and other social
inferences based on language use ( Kinzler, 2021 ;Isaacs & Clark, 1987 ;
Roberts, 2010 ).CONVENTIONS 5
To integrate lexical uncertainty into our speaker and listener
models, we assume they each act in a way that is expected to be
successful on average , under likely values of ϕk(Smith et al., 2013 ).
In other words, they sample actions by marginalizing over their own
beliefs PS(ϕk) orPL(ϕk) about different meanings their partner kmay
be using:
LðojuÞ∝exp/C26
αL·ð
PLðϕkÞlogS1ðujo,ϕkÞdϕk/C27
SðujoÞ∝exp/C26
αS·ð
PSðϕkÞUðu;o,ϕkÞdϕk/C27
,(4)
where αS,αL∈[0,∞] control the speaker ’s and listener ’s soft-max
optimality, respectively5.
C2: Online Learning via Partner-Speci /uniFB01c Inference
The formulation in Equation 4 derives how agents ought to act
under uncertainty about the lexicon being used by their partner,
P(ϕk). But how do beliefs about their partner change over time?
Although an agent may begin with signi /uniFB01cant uncertainty about the
system of meaning their partner is using in the current context,
further interactions provide useful information for reducing that
uncertainty and therefore improving the success of communication.
In other words, ad hoc convention formation may be recast as an
inference problem. Given observations Dkfrom interactions with
partner k, an agent can update their beliefs about their partner ’s latent
system of meaning following Bayes rule:
Pðϕk,ΘjDkÞ∝PðDkjϕk,ΘÞPðϕk,ΘÞ
=PðDkjϕkÞPðϕkjΘÞPðΘÞ: (5)
This joint inference decomposes the partner-speci /uniFB01c learning
problem into two terms, a prior term PðϕkjΘÞPðΘÞand a likelihood
term PðDkjϕkÞ. The prior term captures the idea that, in the absence
of strong evidence of partner-speci /uniFB01c language use, the agent oughtto regularize toward their background knowledge of conventions:
The aspects of meaning that all partners are expected to share in
common. The likelihood term represents predictions about how a
partner would use language in context under different underlying
systems of meaning.
Importantly, the posterior obtained in Equation 5 allows agents to
explicitly maintain partner-speci /uniFB01ce x p e c t a t i o n s ,a su s e di n Equation 4 ,
by marginalizing over community-level uncertainty:
PðϕkjDkÞ=ð
ΘPðϕk,ΘjDkÞdΘ: (6)
We will show that when agents learn about their partner in this
way, and adjust their own production or comprehension accordingly
(i.e., Equation 4 ), they are able to coordinate on stable ad hoc
conventions.
C3: Generalization to New Partners via
Hierarchical Induction
The posterior in Equation 5 also provides an inductive pathway
for partner-speci /uniFB01c data to inform beliefs about community-wide
conventions. Agents update their beliefs about Θ, using data accu-
mulated from different partners, by marginalizing over beliefs about
speci /uniFB01c partners:
PðΘjDÞ=ð
ϕPðϕ,ΘjDÞdϕ, (7)
where D=∪N
k=1Dk,ϕ=ϕ1×::: ×ϕN,a n d Nis the number of
partners previously encountered. Intuitively, when multiple part-
ners are inferred to use similar systems of meaning, beliefs about Θ
shift to represent this abstracted knowledge: It becomes more
likely that novel partners in one ’s community will share it as
well. Note that this population-level posterior over Θnot only
represents what the agent has learned about the central tendency of
the group ’s conventions, but also the spread or variability, cap-
turing the notion that some word meanings may be more wide-
spread than others.
The updated Θshould be used to guide the prior expectations an
agent brings into a subsequent interactions with strangers. This
transfer is sometimes referred to as “sharing of strength ”or“partial
pooling ”because pooled data are smoothly integrated with domain-
speci /uniFB01c knowledge. This property has been key to explaining how
the human mind solves a range of other dif /uniFB01cult inductive problems
in the domains of concept learning ( Kemp et al., 2007 ;Tenenbaum
et al., 2011 ), causal learning ( Kemp et al., 2007 ,2010 ), motor
control ( Berniker & Kording, 2008 ), and speech perception
(Kleinschmidt & Jaeger, 2015 ). One consequence is the “blessing
of abstraction, ”(Goodman et al., 2011 ) where it is possible under
certain conditions for beliefs about the community ’s conventions inThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure 2
Schematic of Hierarchical Model
Note. At the highest level, denoted byΘ, is a representation of aspects ofmeanings expected to be shared across all partners. These conventions serveas a prior for the systems of meanings used by speci/uniFB01c partners,ϕk. Partner-speci/uniFB01c representations give rise in turn to predictions about language usePðDkjϕkÞ, whereDkrepresents observations in a communicative interactionwith partnerk. By inverting this model, agents can adapt to local, ad hocconventions and gradually update their beliefs about conventions intheir broader community. See the online article for the color version ofthis/uniFB01gure.
5We denoteLandSwithout a subscript because they are the only speakerand listener models we use in simulations throughout the article—thesubscripted de/uniFB01nitions are internal constructs used to de/uniFB01ne these models—but in the terminology of the RSA framework they representL1- andS1-levelpragmatic agents with lexical uncertainty. We found that higher levels ofrecursion were not necessary to derive the phenomena of interest, butLn- andSn-level lexical uncertainty models may be generalized by replacingS1in thelistener equation, andL0in the speaker’s utility de/uniFB01nition, with standard RSAde/uniFB01nitions ofn−1-level agents (see alsoZaslavsky et al., 2021).6HAWKINS ET AL.
general to outpace beliefs about the idiosyncracies of individual
partners ( Gershman, 2017 ).
Further Challenges for Convention Formation
The formulation in the previou ss e c t i o np r e s e n t st h ec o r eo f
CHAI. Here, we highlight sever al additional features addressing
more speci /uniFB01cc h a l l e n g e sr a i s e db yp r i o rw o r ko nc o m m u n i c a t i o n
and which we will encounter in the simulations reported in the
remainder of the article. Our organization of these details is
motivated by the analysis of Spike et al. (2017) ,w h oh i g h l i g h t e d
three common issues that all accounts of convention formation
must address: (a) the form of feedback available, (b) the in /uniFB02uence
of memory and temporal discounting, and (c) the form of
pragmatic reasoning being used. Finally, we set up the basic
simulation framework that will be used throughout the rest of the
article.
The Role of Social Observation
Learning and adaptation depend cri tically on the availability and
quality of social observations Dk(Equation 5 ). If the speaker has no
way of probing the listener ’s understanding, or if the listener has no
way of comparing their interpretation against the speaker ’s inten-
tions, however indirectly, they can only continue to rely on
their prior expectations, with no ground for conventions to form
(Garrod et al., 2007 ;Hupet & Chantraine, 1992 ). Communication
is empirically hindered under degraded observation conditions
(Krauss & Bricker, 1967 ;Krauss et al., 1977 ;Krauss &
Weinheimer, 1966 ;Schober & Clark, 1989 ), and we have all
been in situations where we thought we were on the same page
with a partner and only realized that we misunderstood much later,
when the consequences because clear. In principle, we expect that
Dkshould re /uniFB02ect all relevant sources of information that may
expose an agent ’s state of understanding or misunderstanding. Not
just ostensive signals like pointing ( van de Braak et al., 2021 ), but
verbal and nonverbal backchannels (e.g., mmhmm ,n o d so rq u i z -
zical looks), forms of self-initi ated and other-initiated repair ( Arkel
et al., 2020 ;Dingemanse et al., 2015 ;Schegloff et al., 1977 ), and
downstream actions taken in the world (e.g., attempts to follow
instructions).
While incorporating these richer sources of information presents
an exciting line of future work, we restrict our scope to the feedback
traditionally provided by the empirical repeated reference task,
where the speaker ’s intended target and the listener ’s response
are revealed at the end of each trial. Formally, this information
can be written as a set of tuples Dk=fo/C3,u′,o′gT
t=1, where o*denotes
the speaker ’s intended target, u′denotes the utterance they produced,
ando′denotes the listener ’s response, on each previous trial t. To
specify the likelihoods in Equation 5 for this referential setting, we
assume each agent should infer their partner ’s lexicon ϕkby
conditioning on their partner ’sprevious behavior. The listener
on a given trial should use the probability that a speaker would
produce uto refer to the target o*under different ϕk, that is,
PLðfo/C3,u′,o′gtjϕkÞ=S1ðut′jo/C3t,ϕkÞ, and the speaker should likewise
use the probability that their partner would produce response o′after
hearing utterance u,PSðfo/C3,u′,o′gtjϕkÞ=L0ðot′jut′Þ.This symmetry, where each agent is attempting to learn from the
other ’s behavior, creates a clear coordination problem6. In the case
of an error, where the agent in the listener role hears the utterance u′
and chooses an object o′other than the intended target o*, they will
receive feedback about the intended target and subsequently condi-
tion on the fact that the speaker chose u′to convey that target.
Meanwhile, the agent in the speaker role will subsequently condition
on the likelihood that the listener chose the object o′upon hearing
their utterance. In other words, each agent will subsequently condi-
tion on slightly different data leading to con /uniFB02icting beliefs. Whether
or not agents are able to resolve early misunderstandings through
further interaction and eventually reach consensus depends on a
number of factors.
The Role of Memory and Recency
One important constraint is imposed by the basic cognitive
mechanisms of memory. It is unrealistic to expect that memory
traces of every past interaction in the set of observations Dis equally
accessible to the agent. Furthermore, this may be to the agent ’s
advantage. Without a mechanism by which errors become less
accessible, early misunderstandings may interfere with coordination
much later in an interaction. One possible solution is to privilege
more recent outcomes. Especially if a partner is assumed to change
over time, then older data may provide less reliable cues to their
current behavior. Recency is typically incorporated into Bayesian
models with a simple decay term in the likelihood function
(Anderson & Schooler, 2000 ;Fudenberg & Levine, 2014 ;Kalm &
Norris, 2018 ;Yu & Cohen, 2009 ).
PðDkjϕkÞ=YT
τ=0Pðfo/C3,u′,o′gT−τjϕkÞβτ,
where τ=0 indexes the most recent trial Tand decay increases
further back through time. This decay term is motivated by the
empirical power function of forgetting ( Wixted & Ebbesen, 1991 ),
and can be derived by simply extending our hierarchical model
down an additional layer within each partner to allow for the
possibility that they are using slightly different lexicons at different
points in time; assuming a degree of auto-correlation between
neighboring time points yields this form of discounting7.
The Role of Pragmatics
While natural languages are rife with ambiguous and polysemous
terms, speaker and listeners must somehow resolve theseThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.6In some settings, agents in one role may be expected to take on more of
the burden of adaptation, leading to an asymmetric division of labor (e.g.,
Moreno & Baggio, 2014 ). This may be especially relevant in the presence of
asymmetries in power, status, or capability ( Misyak et al., 2014 ), but we
leave consideration of such asymmetries for future work.
7While this simple decay model is suf /uniFB01cient for our reference games, it is
clearly missing important mechanistic distinctions between working memory
and long-term memory; for example, explaining convention formation over
longer timescales may require an explicit model of consolidation or source
memory. It is also consistent with multiple algorithmic-level mechanisms;
for example, decay can be viewed as a form of weighted importance
sampling, where more recent observations are preferentially sampled
(Pearl et al., 2010 ), or a process where observations have some probability
of dropping out of memory at each time step.CONVENTIONS 7
ambiguities to be understood in context ( Piantadosi et al., 2012 )8.
For example, Brennan and Clark (1996) placed participants in a
context where the target object o*was easily distinguished from
other objects in the context Cby a referring expression like u=“the
shoe. ”In a second phase of the study, the context Cwas switched to
be a set of other shoes. Even though there was strong precedent for
referring to o*as“the shoe, ”this description was no longer
informative : The speaker recognized that ucould apply equally
well to all o∈Cleading to potential ambiguity about which shoe
they were referring to. As a result, the speaker switched to a more
speci /uniFB01c utterance like u′=“the pennyloafer ”which unambiguously
applied to o*in the new context. In a third and /uniFB01nal phase, the
context reverted back to the original one, C, but many speakers
continued to use the more speci /uniFB01c utterance u′even though uwould
have suf /uniFB01ced. This example emphasizes how ad hoc conventions or
pacts may be sensitive to the context in which they form.
CHAI solves this problem by the principles of pragmatic rea-
soning naturally instantiated in the RSA framework ( Frank &
Goodman, 2012 ), which plays two distinct roles. First, our Gricean
agents ’production and comprehension are guided by cooperative
principles ( Equation 4 ). They do not only make passive inferences
from observation, they participate in the interaction by using
language themselves. Second, our agents assume that their partner
is also using language in a cooperative manner, which strengthens
the inferences they may make about the underlying system of
meanings their partner is using. That is, we use the RSA equations
as the linking function in the likelihood PðDkjϕkÞ, representing an
agent ’s prediction about how a partner with meaning function ϕk
would actually behave in context ( Equation 5 ). This use of prag-
matic reasoning has been explicitly linked to principles like mutual
exclusivity in word learning ( Bloom, 2002 ;Frank et al., 2009 ;
Gulordava et al., 2020 ;Ohmer et al., 2020 ;Smith et al., 2013 ). For
example, upon hearing their partner use a particular utterance uto
refer to an object o, a pragmatic listener can not only infer that u
means oin their partner ’s lexicon, but also that other utterances u′
likely do notmean o: If they did, the speaker would have used them
instead.
Simulation Details
While our simulations in the remainder of the article each address
different scenarios, we have aimed to hold as many details as
possible constant throughout the article. First, we must be concrete
about the space of possible lexicons that parameterizes the lexical
meaning function, Lϕ. For consistency with previous models of
word learning (e.g., Xu & Tenenbaum, 2007 ), we take the space of
possible meanings for an utterance to be the set of nodes in a concept
taxonomy. When targets of reference are conceptually distinct, as
typically assumed in signaling games, the target space of utterance
meanings reduces to the discrete space of individual objects, that is,
/uni27E6u/uni27E7ϕ=ϕðuÞ∈Ofor all u∈U. For this special case, the parameter
space contains exactly jOj×jUjpossible values for ϕ, correspond-
ing to all possible mappings between utterances and individual
objects. Each possible lexicon can therefore be written as a binary
matrix where the rows correspond to utterances, and each row
contains one object. The truth-conditional function Lϕ(u,o) then
simply checks whether the element in row umatches object o. For
example, consider a simple reference game with two utterances and
two objects ( o1=ando2=). Then there are four possiblelexicons, corresponding to the four assignments of objects to
utterances:Second, having de/uniFB01ned the support of the parameterϕ,w ec a nthen de/uniFB01ne a lexical prior. We consider a partition-based simplicityprior based on the size of the lexicon (Carr et al., 2020;Frank et al.,2009):PðϕÞ∝expf−jϕjg,w h e r ejϕjis the number of lexical items.Again, for traditional signaling games, this reduces to a uniformprior because all possible lexicons are the same size:ϕðuiÞ∼UnifðOÞ. We can compactly write distributions overϕinterms of the same utterance-object matrix, where rowirepresentsthe marginal distribution over possible meanings of utteranceui.For example, the uninformative prior for two utterances and twoobjects can be written:
The partition-based prior becomes more important forP3, wherewe consider spaces of referents with more complex conceptualstructure. A single word may apply to multiple conceptually relatedreferents (e.g., all of the squares). In this case, the simplest lexicon isa single word that refers to everything and the most complex lexiconassigns a unique word for each object (seeAppendix C, fordiscussion of alternative priors).Finally, while the probabilistic model we have formulated in thissection is theoretically well motivated and mathematically wellde/uniFB01ned, it is challenging to derive predictions from it. Historically,interactive models like ours are not amenable to closed-formanalytical techniques and computationally expensive to studythrough simulation, likely contributing to the prevalence of simpli-/uniFB01ed heuristics in prior work. Our work has been facilitated by recentadvances in probabilistic inference techniques that have helped toovercome these obstacles (seeAppendix A, for further details of ourimplementation).SummaryIn this section, we formalized the computational problem facingagents who must communicate in a variable, changing world. Nostatic lexicon is appropriate for all partners and situations, requiringthem to update on the/uniFB02y. We proposed CHAI, a cognitive model ofhow people solve this problem through continual adaptation. CHAIinstantiates three core capacities in a hierarchical Bayesian frame-work: (C1) structured uncertainty over what words mean to differentpartners, (C2) social inference to back out likely latent systems ofThis document is copyrighted by the American Psychological Association or one of its allied publishers.This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.
8Indeed,Brochhagen (2020)has suggested that high degrees of lexicalambiguity and polysemy, that is, high degrees of uncertainty overΘinCHAI, are useful precisely because they allow much-needed/uniFB02exibilitysupporting partner-speci/uniFB01c adaptation.8HAWKINS ET AL.
meaning from a partner ’s observable behavior, and (C3) hierarchical
induction to generalize to the overall distribution of possible
partners. In the remainder of the article, we argue that CHAI
provides a new computational foundation for understanding coor-
dination and convention formation, focusing on three empirical
phenomena that have posed a challenge for previous accounts: (P1)
the increase in communicative ef /uniFB01ciency as a function of shared
history, (P2) the transfer of partner-speci /uniFB01c expectations to commu-
nal expectations, and (P3) the in /uniFB02uence of communicative context
on which conventions eventually form.
Phenomenon #1: Ad Hoc Conventions
Become More Ef /uniFB01cient
We begin by considering the phenomenon of increasing ef /uniFB01-
ciency in repeated reference games: Speakers use detailed de-
scriptions at the outset but converge to an increasingly
compressed shorthand while remaining understandable to their
partner. While this phenomenon has been extensively documen-
ted, to the point of serving as a proxy for measuring common
ground, it has continued to pose a challenge for models of
communication.
In this section, we argue that CHAI provides a rational expla-
nation for increasing ef /uniFB01ciency in terms of the inferences made by
speakers across repeated inter action. Given that this phenomenon
arises in purely dyadic settings, it also provides an opportunity to
explore more basic properties of the /uniFB01rst two capacities formalized
in our model (representing uncertainty andpartner-speci /uniFB01cl e a r n -
ing)b e f o r ei n t r o d u c i n gh i e r a r c h i cal generalization in the next
section.
In brief, we show that increasing ef /uniFB01ciency is a natural conse-
quence of the speaker ’s tradeoff between informativity and parsi-
mony ( Equation 4 ), given their inferences about the listener ’s
language model. For novel, ambiguous objects like tangrams, where
speakers do not expect strong referential conventions to be shared,
longer initial descriptions are motivated by high initial uncertainty in
the speaker ’s lexical prior PðϕkjΘÞ. Proposing multiple descriptors
is a rational hedge against the possibility that a particular utterance
will be misinterpreted and give the listener a false belief. As the
interaction goes on, the speaker obtains feedback Dkfrom the
listener responses and updates their posterior beliefs PðϕkjDkÞ
accordingly. As uncertainty gradually decreases, they are able to
achieve the same expected informativity with shorter, more ef /uniFB01cient
messages.
Simulation 1.1: Pure Coordination
We build up to our explanation of increasing ef /uniFB01ciency by /uniFB01rst
exploring a traditional signaling game scenario with only one-word
utterances. This simulation tests the most fundamental competency
for any model of ad hoc coordination: Agents are able to coordinate
on a communication system in the absence of shared priors.
We consider the simplest possible reference game with two objects,
O={,}, where the speaker must choose between two one-word
utterances U=fu1,u2gwith equal production cost.
We walk explicitly through the /uniFB01rst step of the simulation to
illustrate the model ’s dynamics (see Figure 3 ). Suppose the target
object presented to the speaker agent on the initial trial is . Bothutterances are equally likely to apply to either object under the
uniform lexical prior, hence each utterance is expected to be equally
(un)informative. The speaker ’s utility, therefore, reduces to sam-
pling an utterance at random u∼Sðuj). Suppose u1is sampled. The
listener then hears this utterance and selects an object according to
their own expected utility under their uniform lexical prior, which
also reduces to sampling an object at random o∼Lðoju1Þ. Suppose
they choose,, a correct response. Both agents may use the resulting
tuple D={*,u1,}, depicted in the top row of Figure 3 to update
their beliefs about the lexicon their partner is using:They then proceed to the next trial, where they use this updatedposterior distribution to produce or interpret language instead oftheir prior. To examine how the dynamics of this updating processunfold over further rounds, we simulated 1,000 such trajectories.The trial sequence was structured as a repeated reference game,containing 30 trials structured into 15 repetition blocks. The twoobjects appeared in a random order within each block, and agentsswapped roles at the beginning of each block. We show representa-tive behavior at soft-max optimality parameter valuesαL=αS=8and memory discounting parameterβ=0.8, but/uniFB01nd similarbehavior in a wide regime of parameter values (seeAppendixFigure A1).We highlight several key results from this simulation. First, andmost fundamentally, the communicative success of the dyad risesover the course of interaction: the listener is able to more accuratelyselect the intended target object (seeFigure 4A). Second, the initialsymmetry between meanings in the prior is broken by initial choices,leading toarbitrarybutstablemappings in future rounds. Becauseagents were initialized with the same priors in every trajectory,trajectories only diverged when different actions happen to besampled. This can be seen by examining the path dependenceof subsequent beliefs based on the outcome of the initial trial inFigure 3. Third, we observe the in/uniFB02uence of mutual exclusivity viaGricean pragmatic reasoning: Agents also make inferences aboutobjects and utterances that werenotchosen. For example, observingD={(*,u2,)} provides evidence thatu1likely does not mean(e.g., the third row ofFigure 3, where hearingu2refers toimmediately led to the inference thatu1likely refers to).Simulation 1.2: Increasing Ef/uniFB01ciencyNext, we show how our model explains speakers’gains inef/uniFB01ciency over multiple interactions. For ef/uniFB01ciency to change atall, speakers must be able to produce utterances that vary in length.For this simulation, we therefore extend the model to allow formultiword utterances by allowing speakers to combine togethermultiple primitive utterances. Intuitively, human speakers formlonger initial description by combining a collection of simplerdescriptions (e.g.,“kind of an X, or maybe a Y with Z on top”).This raises a problem about how the meaning of a multiwordutteranceLϕ(u1u2) is derived from its componentsLϕ(u1) andLϕ(u2). To capture the basic desideratum that an object shouldThis document is copyrighted by the American Psychological Association or one of its allied publishers.This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.CONVENTIONS9
be more likely to be chosen by L0when more components of the
longer utterance apply to it, we adopt a standard conjunctive
semantics:
Lϕðuiuj,oÞ=Lϕðui,oÞ×Lϕðuj,oÞ:
One subtle consequence of a conjunctive Boolean semantics is the
possibility of contradictions. For example, under a possible lexicon
where ϕ(u1)=andϕ(u2)=, the multiword utterance u1u2is not
only false of the particular referents in the current context, it is false
of all possible referents, re /uniFB02ecting a so-called truth-gap ( Strawson,
1950 ;Van Fraassen, 1966 ). We assume such an utterance is
uninterpretable and simply disregarded without changing the literal
listener L0’s beliefs. While this assumption is suf /uniFB01cient for our
simulations, we regard this additional complexity as a limitation of
classical truth-conditional semantics ( Degen et al., 2020 ) and show
inAppendix B that switching to a continuous semantics with
lexical values in the interval [0 ,1 ]m a yb e t t e rc a p t u r et h en o t i o n
of redundancy that motivates speakers to initially produce longer
utterances.
Now, we consider a scenario with the same two objects as in
Simulation 1.1, but give the speaker four primitive utterances
{u1,u2,u3,u4} instead of only two, and allow two-word utterances
such as u1u2. We established in the previous section that successful
ad hoc conventions can emerge even in a state of pure uncertainty,
but human participants in repeated reference games typically bringsome prior expectations about language into the interaction. For
example, a participant who hears “ice skater ”on the /uniFB01rst round of
the task in Clark and Wilkes-Gibbs (1986) may be more likely to
select some objects more than others while still having substantial
uncertainty about the intended target (e.g., over three of the 12
tangrams that have some resemblance to an ice skater). We thusThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure 4
Pairs of Agents Learn to Successfully Coordinate on Ef /uniFB01cient Ad
Hoc Conventions Over Repeated Interactions0.60.81.01 5 10 15repetition #% accuracy(A)1.001.251.501.752.00123456repetition ## words(B)
Note. (A) agents converge on accurate communication systems in Simula-tion 1.1, where only single-word utterances are available, and (B) convergeon shorter, more ef/uniFB01cient conventions in Simulation 1.2, where multiwordutterances were available. Error bars are bootstrapped 95% CIs across 1,000trajectories, computed within each repetition block of two trials.Figure 3Path Dependence of Conventionsinitialtargetobjectinitialspeaker utteranceinitiallistenerresponsesubsequenttrajectoryof agent 1 beliefs
123456 123456repetition #0.000.501.00
0.000.501.000.000.501.00
0.000.501.000.000.501.00
0.000.501.000.000.501.00
0.000.501.00probability assigned to meaning of subsequenttrajectoryof agent 2 beliefs
Note. The average trajectory of each agent’s beliefs about the meaning ofu1,ϕ(u1), is shown in blue andorange following all eight possible outcomes of the/uniFB01rst trial in Simulation 1.1. For each of the two possibletargets, the speaker could choose to produce either of the two utterances, and the listener could respond bychoosing either of the two objects. In the cases where the listener chose correctly (marked with a checkmark),agents subsequently conditioned on the same data and rapidly converged on a system of meaning consistentwith this feedback. For example, in the/uniFB01rst row, whenu1was successfully used to refer to the circle, bothagents subsequently believe thatu1means circle in their partner’s lexicon. In the cases where the listener failsto choose the target, the agents subsequently condition on different data, and they converge on a conventionthat is determined by later choices (lines represent the trajectories of individual agents). See the online articlefor the color version of this/uniFB01gure.10HAWKINS ET AL.
initialize both agents with weak biases δ(represented in compressed
matrix form in Figure 5 ):
ϕðu1Þ,ϕðu2Þ∼Categorical ð0.5+δÞ
ϕðu3Þ,ϕðu4Þ∼Categorical ð0.5−δÞ:
As in Simulation 1.1, we simulated 1,000 distinct trajectories of
dyadic interaction between agents. Utterance cost was de /uniFB01ned to be
the number of “words ”in an utterance, so c(u1)=1 and c(u1u2)=2.
As shown in Figure 4B , our speaker agent initially prefers longer
utterance (mean length ≈1.5 on /uniFB01rst block) but rapidly converges to
shorter utterances after several repetitions (mean length ≈1 on /uniFB01nal
block), qualitatively matching the curves measured in the empirical
literature.
To illustrate in detail how our model derives this behavior, we
walk step by step through a single trial ( Figure 5 ). Consider a
speaker who wants to refer to object.T h e ye x p e c tt h e i rp a r t n e r
to be slightly more likely to interpret their language using a
lexicon in which u1andu2apply to this object, due to their
weak initial biases. However, there is still a reasonable chance
(p=.45) that either u1oru2alone will be interpreted to mean,
giving their partner false beliefs. To see why our speaker model
initially prefers the longer utterance u1u2to hedge against this
possibility, despite its higher pr oduction cost, consider the ex-
pected informativity of u1u2under different possible lexicons. The
possibility with highest p robability is that both ϕ(u1)=ϕ(u2)=in the listener ’sl e x i c o n( p=.552≈0.3), in which case the listener
will correctly identify with high probability. The possibility
that both ϕ(u1)=ϕ(u2)=in the listener ’sl e x i c o ni so n l y p=
.452≈.2, in which case the listener will erroneously select .I n
the mixed cases, where ϕ(u1)=,ϕ(u2)=orϕ(u1)=,ϕ(u2)=in the listener ’sl e x i c o n( p=2·. 4 5 ×.55≈.5), the utterance would be
a interpreted as a contradiction and the listener would not change
their prior beliefs. Because the speaker ’s informativity is de /uniFB01ned
using the log probability of the listener ’s belief, the utility of giving
the listener a false belief, log( /uni03F5) is signi /uniFB01cantly worse than simply
being uninformative, that is, log(0.5), and the longer utterance
minimizes this harm.
Following the production of a conjunction, the speaker observes
the listener ’s response (say, ). This allows both agents to become
more con /uniFB01dent that the component utterances u1andu2mean in
their updated posterior over the listener ’s lexicon. This credit
assignment to individual lexical items is a consequence of the
compositional meaning of longer utterances in our simple grammar.
The listener knows a speaker for whom either u1oru2individually
means would have been more likely to say u1u2than a speaker for
whom either component meant ; and similarly for the speaker
reasoning about possible listeners. Consequently, the probability of
both mappings increases.
Figure 5 shows the trajectories of internal components of the
speaker utility as the interaction continues. We assume for illustra-
tive purposes in this example thatcontinues to be the target on
each trial and the same agent continues to be the speaker. As the
posterior probability that individual primitive utterances u1andu2
independently meanincreases (far left), the marginal gap in
informativity between the conjunction and the shorter components
gradually decreases (center left). As a consequence, production costincreasingly dominates the utility (center-right). After several trials
of observing a successful listener response given the conjunction,
theinformativity of the two shorter utterances reaches parity with the
conjunction but the cost makes the shorter utterances more attractive
(yielding a situation now similar to the outset of Simulation 1.1).
Once the speaker samples one of the shorter utterances (e.g., u1), the
symmetry collapses and that utterance remains most probable in
future rounds, allowing for a stable and ef /uniFB01cient ad hoc convention.
Thus, increasing ef /uniFB01ciency is derived as a rational consequence of
uncertainty and partner-speci /uniFB01c inference about the listener ’s lexi-
con. For these simulations, we used αS=αL=8,wC=0.24, β=0.8
but the qualitative reduction effect is found over a range of different
parameters (see Appendix Figure A2 ).
Discussion
The simulations presented in this section aimed to establish a
rational explanation for feedback sensitive increases in ef /uniFB01ciency
over the course of ad hoc convention formation. Speakers initially
hedge their descriptions under uncertainty about the lexical mean-
ings their partner is using, but are able to get away with less costly
components of those descriptions as their uncertainty decreases.
This explanation recalls classic observations about hedges (expres-
sions like sort of or morphemes like -ish) that explicitly mark
provisionality, such as a sort of silvery purple-colored car
(Fraser, 2010 ;Lakoff, 1975 ;Medlock & Briscoe, 2007 ).Brennan
and Clark (1996) counted hedges across repetitions of a repeated
reference game, /uniFB01nding a greater occurrence of hedges on early trials
than later trials and a greater occurrence under more ambiguous
contexts. While our model does not include hedges, it is possible to
understand this behavior as an explicit or implicit marker of the
lexical uncertainty construct in our account. Our account is also
broadly consistent with recent analyses of exactly what gets reduced
in a large corpus of repeated reference games ( Hawkins, Frank, &
Goodman, 2020 ). These analyses found that entire modifying
clauses are more likely to be dropped at once than would be
expected by random and independent corruption. In other words,
speakers apparently begin by combining multiple descriptive modi-
/uniFB01ers and collapse to retain only one of these “units”contingent on
evidence that their partner understands.
Why has this phenomenon remained outside the explanatory
scope of previous models? Our account differs in both level of
analysis and model complexity. For example, the in /uniFB02uential inter-
active alignment account proposes that speakers adapt and coor-
dinate on meaning through priming mechanisms that allow
phonetic or syntactic features associated with lexical items to
percolate up to strengthen higher levels of representation
(Pickering & Garrod, 2004 ,2006 ;Roelofs, 1992 ). While priming
mechanisms are certainly at play in repeated reference tasks,
especially when listeners engage in extensive dialogue and alter-
nate roles, it is not clear why priming alone would lead to
convergence on more ef /uniFB01cient descriptions as opposed to aligning
on the same longer initial description. Furthermore, priming cannot
explain why speakers still converge to shorter descriptions even
when the listener is prevented from saying anything at all and only
sparse, nonverbal feedback of success is provided, or why speakers
continue using longer descriptions when they receive nonverbal
feedback that the listener is repeatedly making errors ( Krauss &
Weinheimer, 1966 ; see also Hawkins, Frank, & Goodman, 2020 ).This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.CONVENTIONS 11
In these cases, there are no linguistic features available for priming
or alignment to act upon. To be clear, our computational-level
account is not mutually exclusive with these process-level prin-
ciples and does not in any way falsify or undermine them.Explaining when and why speakers believe that shorter descrip-
tions will suf /uniFB01ce, and how it depends on context, requires addi-
tional computational-level principles, which we hope will lead to
further enrichment of algorithms at the process level.This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure 5
Internal State of Speaker(A)
(B)
(C)
Note. (A) Schematic showing how lexical uncertainty is added to a simple pragmatic speaker utility; continualhierarchical adaptation through inference (CHAI) proposes that lexical expectations are adapted over time basedon social observations. (B) A single trial of Simulation 2.1. The speaker begins with uncertainty about themeanings in the listener’s lexicon (e.g., assigning 55% probability to the possibility that utteranceu1means objecto1). A targeto1is presented, and the speaker samples an utterance from the distributionSðujo1Þ. Finally, theyobserve the listener’s response and update their beliefs. Due to the compositional semantics of the utteranceu1u2,the speaker becomes increasingly con/uniFB01dent that both component primitives,u1andu2, apply to objecto1in theirpartner’s lexicon. (C) Each internal term of the speaker’s utility (Equation 4) is shown throughout the interaction.When the speaker is initially uncertain about meanings (far left), the longer utteranceu1u2has higher expectedinformativity (center-left) and therefore higher utility (center-right) than the shorter utterancesu1andu2, despiteits higher cost (far-right). As the speaker observes several successful interactions, they update their beliefs andbecome more con/uniFB01dent about the meanings of the component lexical itemsu1andu2. As a result, more ef/uniFB01cientsingle-word utterances gradually gain in utility as cost begins to dominate the utility. On trial 5,u1is sampled,breaking the symmetry between utterances. See the online article for the color version of this/uniFB01gure.12HAWKINS ET AL.
Another prominent account proposes that speakers coordinate on
meaning using a simpler update rule that simpler makes utterances
more likely to be produced after communicative successes and less
likely after communicative failures. This account has often been
implemented using a simple variant of reinforcement learning (RL)
such as Roth –Erev learning ( Barr, 2004 ;Erev & Roth, 1998 ;Steels,
1995 ;Young, 2015 ). While such minimal rules allow groups to
reach consensus, it is challenging to explain the full suite of
phenomena we have explored in this section. First, it is not clear
how simply reinforcing longer descriptions could lead them to get
shorter. In the rare cases that have allowed longer utterances to be
constructed compositionally from more primitive utterances, reduc-
tion has been hard-coded as a kind of /uni03F5-greedy exploration where the
speaker has a /uniFB01xed probability of dropping a random token at each
point in time ( Beuls & Steels, 2013 ;Steels, 2016 ). Such noisy
dropping, however, is inconsistent with studies by Hupet and
Chantraine (1992) where participants were asked to repeatedly refer
to the same targets for a hypothetical partner to see later, such that
any effects of familiarity or repetition on the part of the speaker
would be the same as the interactive task. No evidence of reduction
was found in this case, and in some cases utterances actually grew
longer (see also Garrod et al., 2007 ). Even if we /uniFB01xed this problem
by extending the update rule to be contingent on interaction, it is not
clear why a speaker would initially prefer to produce longer
utterances over shorter utterances.
Importantly, these limitations do not stem from the RL framework
itself, but from the simplifying assumption that the probability of
taking actions should be directly tied to the previous outcomes of
those actions. CHAI preserves a core idea from these accounts —the
ability to dynamically adapt one ’s behavior contingent on one ’s
partner ’s—but disentangles the inference problem (i.e., estimating a
partner ’s underlying lexicon) from the decision problem (i.e.,
deciding which action to take with these estimates in hand).
Introducing the latent variable of the lexicon increases the model ’s
complexity but is also more explanatory, as we show in the
subsequent sections. Importantly, more sophisticated model-based
RL algorithms make a similar distinction and may consequently be
/uniFB02exible enough to account for this phenomenon (see Gershman &
Niv, 2015 , for an explicit connection between hierarchical Bayes
and an RL algorithm known as TD-learning; but see Vélez &
Gweon, 2021 , for outstanding problems associated with bridging
these perspectives).
Finally, while our simulations captured several core features of
the reduction phenomenon, they have only scratched the surface of
its empirical complexity. First, our simulations only consider two-
word descriptions with homogenous uncertainty over the compo-
nents, while the semantic components of real initial descriptions
have more heterogeneity. It remains an open question as to how best
to instantiate more realistic priors in our model that can predict more
/uniFB01ne-grained patterns. For example, early hand-tagged analyses by
Carroll (1980) found that in three-quarters of transcripts from
Krauss and Weinheimer (1964) the conventions that participants
eventually converged upon were prominent in some syntactic
construction at the beginning, often as a head noun that was initially
modi /uniFB01ed or quali /uniFB01ed by other information. Second, gains in ef /uniFB01-
ciency associated with ad hoc conventions do not necessarily
translate into shorter utterances. Outside of the domain of reference
games, speakers often have control over what they want to convey
and may use the ef /uniFB01ciency afforded by their new conventions toexpress more information in the same number of words rather than
the same amount of information in fewer words ( Effenberger et al.,
2021 ). Once a convention is formed, it can be used as a new
primitive to bootstrap further conventions and convey ever-more
sophisticated meanings ( McCarthy et al., 2021 ).
Phenomenon #2: Conventions Gradually
Generalize to New Partners in Community
How do we make the inferential leap from ad hoc conventions
formed through interaction with a single partner to global conven-
tions expected to be shared throughout a community? Grounding
collective convention formation in the individual learning mechan-
isms explored in the previous section requires an explicit theory of
generalization capturing how people transfer what they have learned
from one partner to the next. One in /uniFB02uential theory is that speakers
simply ignore the identity of different partners and update a single
monolithic representation after every interaction ( Barr, 2004 ;Steels,
1995 ;Young, 2015 ). We call this a complete-pooling theory because
data from each partner are collapsed into an undifferentiated pool of
evidence ( Gelman & Hill, 2006 ). Complete-pooling models have
been remarkably successful at predicting collective behavior on
networks, but have typically been evaluated only in settings where
anonymity is enforced. For example, Centola and Baronchelli (2015)
asked how large networks of participants coordinated on conven-
tional names for novel faces. On each trial, participants were paired
with a random neighbor but were not informed of that neighbor ’s
identity, or the total number of different possible neighbors.
While complete pooling may be appropriate for some everyday
social interactions, such as coordinating with anonymous drivers on
the highway, it is less tenable for everyday communicative settings.
Knowledge about a partner ’s identity is both available and relevant
for conversation ( Davidson, 1986 ;Eckert, 2012 ). Partner speci /uniFB01city
thus poses clear problems for complete-pooling theories but can be
easily explained by another simple model, where agents maintain
separate expectations about meaning for each partner. We call this a
no-pooling model (see Smith et al., 2017 , which contrasted no-
pooling and complete-pooling models). The problem with no-
pooling is that agents are forced to start from scratch with each
partner. Community-level expectations never get off the ground.
In other words, complete-pooling and no-pooling models are
prima facie unable to explain partner speci /uniFB01city and network
convergence, respectively. CHAI is a hierarchical partial-pooling
account that offers a solution to this puzzle. We propose that social
beliefs about language have hierarchical structure. That is, the
meanings used by different partners are expected to be drawn
from a shared community-wide distribution but are also allowed
to differ from one another in systematic, partner-speci /uniFB01c ways. This
structure provides an inductive pathway for abstract population-
level expectations to be distilled from partner-speci /uniFB01c experience.
The key predictions distinguishing our model thus concern the
pattern of generalization across partners. Experience with a single
partner ought to be relatively uninformative about further partners,
hence our partial-pooling account behaves much like a no-pooling
model in predicting strong partner speci /uniFB01city and discounting out-
liers (see Dautriche et al., 2021 , which explores this prediction in a
developmental context). After interacting with multiple partners in a
tight-knit community, however, speakers should become increas-
ingly con /uniFB01dent that labels are not simply idiosyncratic features of aThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.CONVENTIONS 13
particular partner ’s lexicon but are shared across the entire commu-
nity, gradually transitioning to the behavior of a complete-pooling
model. In this section, we test this novel prediction in a networked
communication game. We then explicitly compare CHAI to
complete-pooling and no-pooling submodels that lesion the hierar-
chy, using only the top level or bottom level, to evaluate the
contribution of each component.
Model Predictions: Simulation 2.1
We/uniFB01rst examine the generalization behavior produced by each
model by simulating the outcomes of interacting with multiple
partners on a small network (see Figure 6A ). We used a round-
robin scheme ( Figure 6B ) to schedule four agents into a series of
repeated reference games with their three neighbors, playing eight
successive trials with one partner before advancing to the next, for a
total of 24 trials. These reference games used a set of two objects
{o1,o2} and four utterances { u1,u2,u3,u4} as in Simulation 1.2;
agents were randomized to roles when assigned to a new partner and
swap roles after each repetition block within a given interaction.
Consequently, all agents at a particular phase have interacted with
the same number of previous partners, allowing us to examine
network convergence (but see Hawkins, Goodman, et al., 2020 , for a
“/uniFB01rst-person ”version where each new partner is entirely fresh to the
task, /uniFB01nding similar speaker generalization).
Unlike our previous simulations with a single partner, where
hierarchical generalization was irrelevant, we must now specify the
hyper-prior P(Θ) governing the overall distribution of partners
(Equation 5 ). Following Kemp et al. (2007) , we extend the uniform
categorical prior over possible referents to a hierarchical Dirichlet-
Multinomial model ( Gelman et al., 2014 ), where the prior over the
partner-speci /uniFB01c meaning of u,P(ϕk(ui)=oj), is not uniform, but
given by a parameter Θthat is shared across the entire population.
Because Θis a vector of probabilities that must sum to 1 across
referents, we assume it is drawn from a Dirichlet prior:
ϕkðuÞ∼Categorical ðΘÞ
Θ∼Dirichlet ðλ·αÞ,(8)
where λ·αgives the concentration parameter encoding the agent ’s
beliefs, or “overhypotheses ”about both the central tendency and thevariability of lexicons in the population. The relative values of the
entries of αcorrespond to inductive biases regarding the central
tendency of lexicons, while the absolute magnitude of the scaling
factor λroughly corresponds to prior beliefs about the spread, where
larger magnitudes correspond to more concentrated probability
distributions across the population. We /uniFB01xλ=2 and assume the
agent has uncertainty about the population-level central tendency by
placing a hyper-prior on α(seeCowans, 2004 ) that roughly corre-
sponds to the weak initial preferences we used in our previous
simulations:
α∼/C26
Dirichlet ð1.0;1.5Þifu∈fu1,u2g
Dirichlet ð1.5;1.0Þifu∈fu3,u4g:
We may then de /uniFB01ne the no-pooling and complete-pooling models
by lesioning this shared structure in different ways. The no-pooling
model assumes an independent Θkfor every partner, rather than
sharing a single population-level parameter. Conversely, the
complete-pooling model assumes a single, shared ϕrather than
allowing different values ϕkfor different partners. We simulated
48 networks for each model, setting αS=αL=4,wC=.24 (see
Figure A3 in the Appendix for an exploration of other parameters).
Speaker Utterance Length Across Partners
We begin by examining our model ’s predictions about how a
speaker ’s referring expressions change with successive listeners.
While it has been frequently observed that messages reduce in
length across repetitions with a single partner ( Krauss &
Weinheimer, 1964 ) and sharply revert back to longer utterances
when a new partner is introduced ( Wilkes-Gibbs & Clark, 1992 ), the
key prediction distinguishing our model concerns behavior across
subsequent partner boundaries. Complete-pooling accounts predict
no reversion in number of words when a new partner is introduced
(Figure 7A ,/uniFB01rst column). No-pooling accounts predict that roughly
the same initial description length will reoccur with every subse-
quent interlocutor ( Figure 7A , second column).
Here, we show that a partial-pooling account predicts a more
complex pattern of generalization. First, unlike the complete-
pooling model, we /uniFB01nd that the partial-pooling speaker model
reverts or jumps back to a longer description at the /uniFB01rst partner
swap. This reversion is due to ambiguity about whether the behavior
of the /uniFB01rst partner was idiosyncratic or attributable to community-
level conventions. In the absence of data from other partners, a
partner-speci /uniFB01c explanation is more parsimonious. Second, unlike a
no-pooling model, after interacting with several partners, the model
becomes more con /uniFB01dent that one of the short labels is shared across
the entire community, and is correspondingly more likely to begin a
new interaction with it ( Figure 7A , third column).
It is possible, however, that these two predictions only distinguish
our partial-pooling model at a few parameter values; the no-pooling
and complete-pooling could produce these qualitative effects else-
where in parameter space. To conduct a more systematic model
comparison, then, we simulated 10 networks in each cell of a large
grid manipulating the optimality parameters αS,αL, the cost param-
eterwC, and the memory discounting parameter β. We computed a
“reversion ”statistic (the magnitude of the change in P(u1u2) imme-
diately after a partner swap) and a “generalization ”statistic (the
magnitude of the change in P(u1u2) from the initial trial with theThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure 6
Simulations and Behavioral Experiment(A) (B)
Note.Participants were (A) Placed in fully connected networks of 4 and (B)Paired in a round-robin schedule of repeated reference games with eachneighbor. See the online article for the color version of this/uniFB01gure.14HAWKINS ET AL.
agent ’s/uniFB01rst partner to the initial trial with the /uniFB01nal partner) and
conducted single-sample t-tests at each parameter value to compare
these statistics with what would be expected due to random varia-
tion. We found that only the partial-pooling model consistently
makes both predictions across a broad regime. The complete-
pooling model fails to predict reversion nearly everywhere while
the no-pooling model fails to predict generalization nearly every-
where. Detailed results are shown in Figure A4 in the Appendix.
Network Convergence
Because all agents are simultaneously making inferences about
the others, the network as a whole faces a coordination problem. For
example, in the /uniFB01rst block, agents 1 and 2 may coordinate on using
u1to refer to o1while agents 3 and 4 coordinate on using u2. Once
they swap partners, they must negotiate this potential mismatch in
usage. How does the network as a whole manage to coordinate? We
measured alignment by examining the intersection of utterances
produced by speakers: If two agents produced overlapping utter-
ances to refer to a given target (i.e., a nonempty intersection), we
assign a 1, otherwise we assign a 0. We calculated alignment
between currently interacting agents (i.e., within a dyad) and those
who were not interacting (i.e., across dyads), averaging across the
target objects. Alignment across dyads was initially near chance,
re/uniFB02ecting the arbitrariness of whether speakers reduce to u1oru2.
Under a complete-pooling model ( Figure 7B ,/uniFB01rst column), agents
sometimes persist with mis-calibrated expectations learned from
previous partners rather than adapting to their new partner, and
within-dyad alignment deteriorates, re /uniFB02ected by a sharp drop from99% to 85%. Under a no-pooling model ( Figure 7B , second
column), convergence on subsequent blocks remains near chance,
as conventions need to be renegotiated from scratch. By contrast,
under our partial-pooling model, alignment across dyads increases
without affecting alignment within dyads, suggesting that hierar-
chical inference leads to emergent consensus ( Figure 7B , third
column).
Behavioral Experiment
To evaluate the predictions derived in our simulations, we de-
signed a natural-language communication experiment following
roughly the same network design as our simulations. That is, instead
of anonymizing partners, as in many previous empirical studies of
convention formation (e.g., Centola & Baronchelli, 2015 ), we
divided the experiment into blocks of extended dyadic interactions
with stable, identi /uniFB01able partners (see Fay et al., 2010 ;Garrod &
Doherty, 1994 , for similar designs). Each block was a full repeated
reference game, where participants had to coordinate on ad hoc
conventions for how to refer to novel objects with their partner. Our
partial-pooling model predicted that these conventions will partially
reset at partner boundaries, but agents should be increasingly willing
to transfer expectations from one partner to another.
Participants
We recruited 92 participants from Amazon Mechanical Turk to
play a series of interactive, natural-language reference games using
the framework described in Hawkins (2015) .This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure 7
Simulation Results and Empirical Data for (A) Speaker Reduction and (B) Network Convergence Across
Three Partners
1 3 5 7 9 111 3 5 7 9 111 3 5 7 9 111.001.251.501.752.00repetition blockutterance length
(# words)
10.000.250.500.751.00alignment
(% match)2.04.06.08.0
1 3 5 7 9 11# words
(B)
0.000.250.500.751.00% match(A)partial
poolingcomplete
poolingno
poolingempirical
data
partner # 23 123 123 1 2 3
partner # repetition block
across pairswithin pair
Note. In (A) vertical boundaries mark time points when new partners were introduced, and the dotted gray line
represents what would be produced for a stranger at each point in time. In (B), dashed line represents alignment
between partners who are currently paired while solid line represents alignment across partners who are not paired.
Error bars represent bootstrapped 95% con /uniFB01dence intervals. See the online article for the color version of this /uniFB01gure.CONVENTIONS 15
Stimuli and Procedure
Each participant was randomly assigned to one of 23 fully
connected networks with three other participants as their neighbors
(Figure 6A ). Each network was then randomly assigned one of three
distinct contexts containing abstract tangram stimuli taken from
(Clark & Wilkes-Gibbs, 1986 ). The experiment was structured into
a series of three repeated reference games with different partners,
using these same four stimuli as referents. Partner pairings were
determined by a round-robin schedule ( Figure 6B ). The trial
sequence for each reference game was composed of four repetition
blocks, where each target appeared once per block. Participants were
randomly assigned to speaker and listener roles and swapped roles
on each block. After completing 16 trials with one partner, parti-
cipants were introduced to their next partner and asked to play the
game again. This process repeated until each participant had part-
nered with all three neighbors. Because some pairs within the
network took longer than others, we sent participants to a temporary
waiting room if their next partner was not ready.
Each trial proceeded as follows. First, one of the four tangrams in
the context was highlighted as the target object for the speaker. They
were instructed to use a chatbox to communicate the identity of this
object to their partner, the listener. The two participants could
engage freely in dialogue through the chatbox but the listener
must ultimately make a selection from the array. Finally, both
participants in a pair were given full feedback on each trial about
their partner ’s choice and received bonus payment for each correct
response. The order of the stimuli on the screen was randomized on
every trial to prevent the use of spatial cues (e.g., “the one on the
left”). The display also contained an avatar for the current partner
representing different partners with different colors as shown in
Figure 6 to emphasize that they were speaking to the same partner
for an extended period. On the waiting screen between partners,
participants were shown the avatars of their previous partner and
upcoming partner and told that they were about to interact with a
new partner.
Results
We evaluated participants ’generalization behavior on the same
metrics we used in our simulations: utterance length and network
convergence.
Speaker Utterance Length
Now we are in a position to evaluate the central prediction of our
model. Our partial-pooling model predicts (a) gains in ef /uniFB01ciency
within interactions with each partner and (b) reversions to longer
utterances at partner boundaries, but (c) gradual shortening of the
initial utterance chosen with successive partners. As a measure of
ef/uniFB01ciency, we calculated the raw length (in words) of the utterance
produced on each trial. Because the distribution of utterance lengths
is heavy-tailed, we log-transformed these values. To test the /uniFB01rst
prediction, we constructed a linear mixed-effects regression pre-
dicting trial-level speaker utterance length. We included a /uniFB01xed
effect of repetition block within partner (1, 2, 3, 4), along with
random intercepts and slopes for each participant and each tangram.
We found that speakers reduced utterance length signi /uniFB01cantly oversuccessive interactions with each individual partner, b=−0.19,
t(34) =−9.88, p<.001.
To test the extent to which speakers revert to longer utterances at
partner boundaries, we constructed another regression model. We
coded the repetition blocks immediately before and after each
partner swap and included it as a categorical /uniFB01xed effect. Because
partner roles were randomized for each game, the same participant
did not always serve as listener in both blocks, so in addition to
tangram-level intercepts, we included random slopes and intercepts
at the network level (instead of the participant level). As predicted,
we found that utterance length increased signi /uniFB01cantly at the two
partner swaps, b=0.43, t(22) =4.4,p<.001.
Finally, to test whether ef /uniFB01ciency improves for the very /uniFB01rst
interaction with each new partner, before observing any partner-
speci /uniFB01c information, we examined the simple effect of partner
number at the trials immediately after the partner swap (i.e., t=
{1, 5, 9}). We found that participants gradually decreased the length
of their initial descriptions with each new partner in their network,
b=−0.2,t(516.5) =−6.07, p<.001 (see Figure 7A ,/uniFB01nal column),
suggesting that speakers are bringing increasingly well-calibrated
expectations into interactions with novel neighbors. The partial-
pooling model is the only model predicting all three of these effects.
Network Convergence
Now, we examine the content of conventions and evaluate the
extent to which alignment increased across the network over the
three partner swaps. Speci /uniFB01cally, we extend the same measure of
alignment used in our simulations to natural-language data by
examining whether the intersection of words produced by different
speakers was nonempty. We excluded a list of common stop words
(e.g., “the,”“both”) to focus on core conceptual content. While this
pure overlap measure provides a relatively weak notion of similar-
ity, a more continuous measure based on the sizeof the intersection
or the string edit distance yielded similar results.
As in our simulation, the main comparison of interest was
between currently interacting participants and participants who
are not interacting: The partial-pooling model predicted that
within-pair alignment should stay consistently high while (tacit)
alignment between noninteracting pairs will increase. To test this
prediction, we constructed a mixed-effects logistic regression
including /uniFB01xed effects of pair type (within vs. across), partner
number, and their interaction. We included random intercepts at
the tangram level and maximal random effects at the network level
(i.e., intercept, both main effects, and the interaction). As predicted,
we found a signi /uniFB01cant interaction ( b=−0.85, z=−5.69, p<.001;
seeFigure 7B ,/uniFB01nal column). Although different pairs in a network
may initially use different labels, these labels begin to align over
subsequent interactions.
This /uniFB01nding is consistent with the primary prediction of interest
for both the complete-pooling and partial-pooling models. These
two models only pull apart for a secondary prediction concerning the
transition from the /uniFB01rst to second partner. The complete-pooling
model predicts a signi /uniFB01cant drop in within-pair convergence from
the/uniFB01rst to second partner, due to the continued in /uniFB02uence of the /uniFB01rst
partner, while the partial-pooling model predicts no drop. We found
no evidence of such a drop in the empirical data ( z=−0.66, p=
.511), providing further evidence in favor of the full partial-pooling
structure.This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.16 HAWKINS ET AL.
Discussion
Drawing on general principles of hierarchical Bayesian infer-
ence, CHAI suggests that conventions represent the shared struc-
ture that agents “abstract away ”from partner-speci /uniFB01c learning. In
this section, we evaluated the extent to which CHAI captured
human generalization behavior in a natural-language communication
experiment on small networks. Unlike complete-pooling accounts, it
allows for partner-speci /uniFB01c common ground to override community-
wide expectations given suf /uniFB01cient experience with a partner, or
in the absence of strong conventions. Unlike no-pooling accounts,
it results in networks that are able to converge on shared
conventions.
Partner speci /uniFB01city and generalization present an even steeper
challenge for previous accounts than P1. It is not straightforward for
previous interactive alignment or RL accounts to explain patterns
across partner boundaries without being augmented with additional
social information. If a particular representation has been primed
due to precedent in the preceding dialogue, then the shifting identity
of the speaker should not necessarily alter its in /uniFB02uence ( Brennan &
Hanna, 2009 ;Ferreira et al., 2012 ;Ostrand & Ferreira, 2019 ). More
sophisticated hierarchical memory retrieval accounts that represent
different partners as different contexts (e.g., Brown-Schmidt et al.,
2015 ;Polyn et al., 2009 ) may allow priming to be modulated in a
partner-speci /uniFB01c way, but such an account would presuppose that
social information like partner identity is already a salient and
relevant feature of the communicative environment. Indeed, an
account assuming socially aware context reinstatement for
partner-speci /uniFB01c episodic memories, and slower consolidation of
shared features into population-level expectations, may be one
possible process-level candidate for realizing our hierarchical
computational-level model.
A frequent concern in prior work using repeated reference games
is that improvements in communication over time are due to generic
effects of task familiarity and repetition rather than interactive
adaptation to a partner ’s language use ( Hupet & Chantraine,
1992 ). As they get more practice with the task, speakers may simply
get better overall at describing images and listeners may learn how to
better identify target images. The effects we observe at partner
boundaries show that something is being learned beyond pure
familiarity with the task: If speakers and listeners were just learning
to better describe and identify targets regardless of who their partner
is, we would not expect these reversions. These partner-speci /uniFB01city
effects clearly rule out the complete-pooling model, but cannot rule
out a no-pooling model combined with a practice effect. Under this
alternative possibility, partner-speci /uniFB01c adaptation would be genu-
ine, but the general decrease in utterance length and increase in
accuracy with new partners would be due to practice rather than
inductive generalization. Our best current evidence against this
practice-based explanation lies in our network convergence results:
networks as a whole converge to similar short descriptions across
partners, and different networks converge to different descriptions,
indicating some gradual degree of transfer across partners. Future
work may further address these concerns by including /uniFB01ller trials or
by manipulating the length of interaction with each partner.
Our account also predicts that similar inductive learning mechan-
isms would operate not only across different partners but across
different contexts containing different referents. By holding the
partner constant across different contexts, rather than holding thecontext constant across different partners, it would be possible to test
the extent to which additional experience along one axis of gener-
alization would affect generalization along the other axis. Finally,
one subtler point, which we believe is a rich direction for future
research, is how generalization may still depend on the speaker ’s
beliefs about how partners are sampled , manifested in their induc-
tive biases at the community level ( Equation 8 ). If they believe they
are in a tight-knit community where different partners are experts
with the domain and have likely interacted with one another before,
they may generalize differently than if they believe their community
has higher turnover and many novices, brand-new to the task
(Isaacs & Clark, 1987 ).
Phenomenon #3: Conventions Are Shaped by
Communicative Context
In the previous two sections, we evaluated a model of rapid,
partner-speci /uniFB01c learning that allows agents to form stable but
arbitrary ad hoc conventions with partners that gradually generalize
to their entire community. The /uniFB01nal phenomenon we consider is the
way that ad hoc conventions are shaped by the communicative needs
of the context in which they form. This phenomenon is most
immediately motivated by recent /uniFB01ndings that more informative
or diagnostic words in the local referential context are signi /uniFB01cantly
more likely to become conventionalized ( Hawkins, Frank, &
Goodman, 2020 ). For example, consider an initial description
like“the guy that looks like an ice skater with a leg up in front. ”
A word like “skater, ”which is distinctive of that single referent, is
empirically more likely to persist in the resulting convention than
words like “guy”or“leg”which are used in descriptions for multiple
referents. Our broader theoretical aim is to suggest that context
sensitivity in the synchronic processes at play when individual
dyads coordinate on ad hoc meanings may help to explain dia-
chronic balance of ef /uniFB01ciency and expressivity in the long-term
evolution of a community ’s lexicon, as highlighted by functionalist
accounts like the OSE hypothesis (Frank, 2017) .
Brie/uniFB02y, when there is already a strong existing convention that is
expected to be shared across the community, our model predicts that
speakers will use it. New ad hoc conventions arise precisely to /uniFB01ll
gaps in existing population-level conventions, to handle new situa-
tions where existing conventions are not suf /uniFB01cient to accurately and
ef/uniFB01ciently make the distinctions that are required in the current
context. A corollary of this prediction is that ad hoc conventions
may only shift to expectations at the population level (and ultimately
to population-level convergence) when those distinctions are con-
sistently relevant across interactions with different partners9. For
example, while most English speakers have the general term “tree”
in their lexicon, along with a handful of subordinate-level words like
“maple ”or“/uniFB01r,”we typically do not have conventionalized labels
exclusively referring to each individual tree in our yards —we are
rarely required to refer to individual trees. Meanwhile, we dooften
have shared conventions (i.e., proper nouns) for individual peopleThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.
9This follows by induction from the hierarchical generalization mechan-
isms evaluated for P2, which provide the pathway by which ad hoc
conventions become adopted by a larger community over longer time scales.
Many ad hoc conventions never generalize to the full language community
simply because the contexts where they are needed are rare or variability
across partners is too high. They must be renegotiated with subsequent
partners on an ad hoc basis.CONVENTIONS 17
and places that a community regularly encounters and needs to
distinguish among. Indeed, this logic may explain why a handful of
particularly notable trees do have conventionalized names, such as
the Fortingall Yew, the Cedars of God, and General Sherman, the
giant sequoia.
As a /uniFB01rst step toward explaining these diachronic patterns in
which conventions form, we aim to establish in this section that our
model allows a single dyad ’s ad hoc conventions to be shaped by
communicative context over short timescales. Speci /uniFB01cally, our
model predicts that people will form conventions at the highest
level of abstraction that is able to satisfy their communicative needs.
That is, when the local environment imposes a communicative need
to refer to particular ad hoc concepts (e.g., describing a particular
tree that needs to be planted), communicative partners are able to
coordinate on ef /uniFB01cient lexical conventions for successfully doing so
at the relevant level of abstraction (e.g., “the mossy one ”).
We begin by showing that this form of context sensitivity
naturally emerges from our model, as a downstream consequence
of recursive pragmatic reasoning. When a particular partner uses a
label to refer to an object in a context, we can infer that they do not
believe it ambiguously applies to distractors as well; otherwise, they
would have known it would be confusing and chosen a different
label. We then empirically evaluate this prediction by manipulating
which distinctions are relevant in an arti /uniFB01cial-language repeated
reference game building on Winters et al. (2015 ,2018 ), allowing us
to observe the emergence of ad hoc conventions from scratch. In
both the empirical data and our model simulations, we /uniFB01nd that
conventions come to re /uniFB02ect the distinctions that are functionally
relevant for communicative success.
Model Predictions: Simulation 3.1
To evaluate the impact of context on convention formation, we
require a different task than we used in the previous sections. Those
tasks, like most reference games in the literature on convention
formation, used a discrete set of unrelated objects in a /uniFB01xed context,
{o1,::: ,ok}. In real referential contexts, however, targets are
embedded in larger conceptual taxonomies, where some objects
are more similar than others ( Bruner et al., 1956 ;Collins & Quillian,
1969 ;Xu & Tenenbaum, 2007 ). Here, we therefore consider a spaceof objects embedded in a three-level stimulus hierarchy with shape
at the top-most level, color/texture at the intermediate levels, and
frequency/intensity at the /uniFB01nest levels (see Figure 7A ). While we
will use the full stimulus set in our empirical study, it is suf /uniFB01cient for
our simulations to consider just one of the branches (i.e., just the four
squares). We populate the space of possible utterance meanings
P(ϕ) with four meanings at the subordinate level (one for each
individual object, e.g., ϕ(u)=“light blue square ”), 2 meanings at the
center level (e.g., ϕ(u)=“blue square ”), 1 meaning at the super-
ordinate level (e.g., ϕ(u)=“square ”). We then populate the utter-
ance space with eight single-word labels ( Figure 8B ). The partition-
based simplicity prior restricts the space of possible lexicons to only
those which partition the space of objects (i.e. each object is
contained in the extension of exactly one word), and prefers lexicons
with fewer words.
Another important feature of real environments is that speakers do
not have the advantage of a /uniFB01xed context; the relevant distinctions
change from moment to moment as different subsets of objects are in
context at different times. This property poses a challenge for models
of convention formation because the relevant distinctions cannot be
determined from a single context; they must be abstracted over time.
We, therefore, only displayed two of the four possible objects on a
given trial. Distractors could differ from the target at various levels of
the hierarchy, creating different types of contexts de /uniFB01ned by the
/uniFB01nest distinction that had to be drawn (e.g., Figure 8C ).
Critically, we manipulated the prevalence of different kinds of
contexts, controlling how often participants are required to make
certain distinctions to succeed at the task. In the /uniFB01necondition, every
context contained a subordinate distractor, requiring /uniFB01ne low-level
distinctions to be drawn. In the coarse condition, contexts never
contained subordinate distractors, only distractors that differed at the
central level of the hierarchy (e.g., a blue square when the target is a
red square). For comparison, we also include a mixed condition,
where targets sometimes appear in /uniFB01necontexts with subordinate
distractors and other times appear in coarse contexts without them;
the context type is randomized between these two possibilities on
each trial. We constructed the trial sequence identically for the three
conditions. On each trial, we randomly sampled one of the four
possible objects to be the target. Then we sampled a distractor
according to the constraints of the context type. As before, the agentsThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure 8
Context-Sensitivity Experiment
(A) (B) (C)
Note. (A) Targets are related to one another in a conceptual taxonomy. (B) Speakers choose between labels, where the label“niwa”has beenselected. (C) Examples of/uniFB01ne and coarse contexts. In the/uniFB01ne context, the target (marked in black) must be disambiguated from a distractor(marked in gray) at the same subordinate-level branch of the taxonomy. In the coarse context, the closest distractor belongs to a differentbranch of the center level of the taxonomy (i.e., a spotted circle) such that disambiguation at the subordinate level is not required. See theonline article for the color version of this/uniFB01gure.18HAWKINS ET AL.
swapped roles after each trial. We ran 400 distinct trajectories with
parameter settings of αL=8,αS=8 and memory discounting
parameter of β=0.8 (see Figure A6 , for results at other parameter
values).
Partners Successfully Learn to Communicate
First, we compare the model ’s learning curves across context
conditions ( Figure 9A ). In a mixed-effects logistic regression, we
/uniFB01nd that communicative accuracy steadily improves over time
across all conditions, b=0.72, z=16.9, p<.001. However,
accuracy also differed across conditions: adding a main effect of
condition signi /uniFB01cantly improves model /uniFB01t,χ2(2)=9.6,p=.008.
Accuracy is signi /uniFB01cantly higher in the coarse condition than the /uniFB01ne
condition b=−0.71, z=9.3,p<.001 and marginally higher than
the mixed condition.
Lexical Conventions Are Shaped by Context
As an initial marker of context sensitivity, we examine the
effective vocabulary sizes used by speakers in each condition.
We operationalized this measure by counting the total number of
unique words produced within each repetition block. This measure
takes a value of 8 when a different word is consistently used for
every object, and a value of 1 when exactly the same word is used for
every object. In a mixed-effects regression model including inter-
cepts and random effects of trial number for each simulated trajec-
tory, we /uniFB01nd an overall main effect of condition, with agents in the
/uniFB01ne condition using signi /uniFB01cantly more words across all repetition
blocks ( m=4.7 in coarse ,m=6.5 in /uniFB01ne, t =4.5,p<.001).
However, we also found a signi /uniFB01cant interaction: The effective
vocabulary size gradually dropped over time in the coarse condition,
while it stayed roughly constant in the /uniFB01ne condition, b=0.18,
t=8.1,p<.001, see Figure 9B .
Next, we examine more closely the emergence of terms at
different levels of generality. We have access not only to the
signaling behavior of our simulated agents, but also their internal
beliefs about their partner ’s lexicon, which allows us to directly
examine the evolution of these beliefs from the beginning of the
interaction. At each time point in each game, we take the single
meaning with a highest probability for each word. In Figure 10 , we
show the proportion of words with meanings at each level of
generality, collapsing across all games in each condition. Qualita-
tively, we observe that agents begin by assuming null meanings (i.e.,
with an effectively empty vocabulary) but quickly begin assigning
meanings to words based on their partner ’s usage. In both condi-
tions, basic-level meanings and subordinate-level meanings are
equally consistent with the initial data, but the simplicity prior
prefers smaller effective vocabulary sizes. After the /uniFB01rst repetition
block, however, agents in the coarse condition begin pruning out
some of the subordinate-level terms and become increasingly
con/uniFB01dent of basic-level meanings. Meanwhile, agents in the /uniFB01ne
condition begin to disaggregate these basic-level terms into a greater
number of subordinate-level meanings.
By the /uniFB01nal trial, the proportion of basic-level versus subordinate-
level terms is signi /uniFB01cantly different across the coarse and /uniFB01ne
conditions. Only 9% of words had subordinate-level meanings
(green) in the coarse condition, compared with 79% in the /uniFB01ne
condition, χ2(1)=436, p<.001. At the same time, 45% of wordshad basic-level meanings (blue) in the coarse condition, compared
with only 8% in the /uniFB01ne condition, χ2(1)=136, p<.001. The
remaining words in each condition were assigned the “null”mean-
ing (red), consistent with an overall smaller effective vocabulary
size in the coarse condition. The diverging conventions across
contexts are driven by Gricean expectations: Because the speaker
is assumed to be informative, only lexicons distinguishing betweenThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure 9
Comparison of Simulation Results to Empirical Data
0255075100
0 25 50 75 100
trial #% correctcoarse
mixed
fine(A) (B)
(D)
# unique words
repetition block% correct
45678
123456
repetition blockfine
mixed
coarse# unique words 0 5 10 15 206080100
mixed
finecoarse
0122.02.53.03.54.0
trial #fine
mixed
coarse
(C)
Note. (A) Agents in our simulation learn to coordinate on a successful
communication system, but converge faster in the coarse condition than the
/uniFB01ne condition. (B) The number of unique words used by agents in each
repetition block stayed roughly constant in the /uniFB01ne condition but decreased
over time in the coarse condition. (C –D) The same metrics computed on our
empirical data, qualitatively matching the patterns observed in the simula-
tions. Each point is the mean proportion of correct responses by listeners;
curves are nonparametric /uniFB01ts and error bars are bootstrapped 95% CIs. See
the online article for the color version of this /uniFB01gure.
Figure 10
Dynamics of Lexical Beliefs Over Time in Model Simulations
coarse fine
024
trial #terms in
lexiconlevel 1 object
0 5 10 15 20 0 5 10 15 20 05 1 0 1 5 2 0mixed2 objects
Note. Regions represent the average number of words at each level of
generality in an agent ’s beliefs about the lexicon. Level of generality is
determined by taking the maximum a posteriori (MAP) meaning. In the /uniFB01ne
and mixed conditions, agents shift toward more subordinate terms. See the
online article for the color version of this /uniFB01gure.CONVENTIONS 19
subordinate level objects can explain the speaker ’s behavior in the
/uniFB01necondition.
Experimental Method
In this section, we evaluate our model ’s qualitative predictions
about the effect of context on convention formation using an
interactive behavioral experiment closely matched to our simula-
tions. We use a between-subjects design where pairs of participants
are assigned to different communicative contexts and test the extent
to which they converge on meaningfully different conventions.
Participants
We recruited 278 participants from Amazon Mechanical Turk to
play an interactive, multiplayer game. Pairs were randomly assigned
to one of three different conditions, yielding n=36 dyads in the
coarse condition, n=38 in the /uniFB01necondition, and n=53 in the
mixed condition after excluding participants who disconnected
before completion10.
Procedure and Stimuli
Participants were paired over the web and placed in a shared
environment containing an array of four objects at a time ( Figure 8A )
and a “chatbox ”to choose utterances from a /uniFB01xed vocabulary by
clicking-and-dragging ( Figure 8B ). On each trial, one player (the
“speaker ”) was privately shown a highlighted target object and
allowed to send a single word to communicate the identity of this
object to their partner (the “listener ”), who subsequently made a
selection from the array. Players were given full feedback, swapped
roles each trial, and both received bonus payment for each correct
response.
We randomly generated distinct arrays of 16 utterances for each
pair of participants (more than our model, which was restricted by
computational complexity). These utterances were created by string-
ing together consonant –vowel pairs into pronounceable two-
syllable words to reduce the cognitive load of remembering previous
labels (see Figure 8B ). These arrays were held constant across trials.
However, as in our simulations, the set of referents on each trial was
manipulated in a between-subjects design to test the context sensi-
tivity of the resulting conventions. The trial sequence consisted of
six blocks of 16 trials, for a total of 96 trials. Each of the eight
possible objects shown in Figure 8A appeared as the target exactly
twice per block and was prevented from being shown twice in a row.
In addition to behavioral responses collected over the course of the
game, we designed a posttest to explicitly probe players ’/uniFB01nal lexica.
For all 16 words, we asked players to select all objects that a word
can refer to (if any), and for each object, we asked players to select
all words that can refer to it (if any). This bidirectional measure
allowed us to check the internal validity of the lexica reported
checking mismatches between the two directions of the lexicon
question (e.g., if they clicked the word “mawa ”when we showed
them one of the blue squares, but failed to click that same blue
square when we showed “mawa ”). We conservatively take a
participant ’s/uniFB01nal lexicon to be the intersection of their word-to-
object and object-to-word responses.Behavioral Results
Partners Successfully Learn to Communicate
Although participants in all conditions began with no common
basis for label meanings, performing near chance on the /uniFB01rst trial
(proportion correct =0.19, 95% CI [0.13, 0.27]), most pairs were
nonetheless able to coordinate on a successful communication
system over repeated interaction (see Figure 9C ). A mixed-effects
logistic regression on listener responses with trial number as a /uniFB01xed
effect, and including by-pair random slopes and intercepts, showed a
signi/uniFB01cant improvement in accuracy overall, z=14.4, p<.001.
Accuracy also differed signi /uniFB01cantly across conditions: Adding an
additional main effect of condition to our logistic model provided a
signi/uniFB01cantly better /uniFB01t,χ2(2)=10.8, p=.004. Qualitatively, the
coarse condition was easiest for participants, the /uniFB01necondition was
hardest, and the mixed condition was in between. These effects track
the most important qualitative feature of our simulations —our
arti/uniFB01cial agents were also able to successfully coordinate in both
conditions, and did so more easily in the coarse condition than the
/uniFB01ne condition. However, we found that the speed of coordination in
the mixed and /uniFB01ne conditions was slower than predicted in our
simulations. The additional dif /uniFB01culty participants experienced in the
/uniFB01necondition may be due to additional motivational constraints,
memory constraints, or other factors not captured in our model.
Contextual Pressures Shape the Lexicon
We predicted that in contexts regularly requiring speakers to
make /uniFB01ne distinctions among objects at subordinate levels of the
hierarchy, we would /uniFB01nd lexicalization of speci /uniFB01c terms for each
object (indeed, a one-to-one mapping may be the most obvious
solution in a task with only eight objects). Conversely, when no such
distinctions were required, we expected participants to adaptively
conventionalize more general terms that could be reused across
different contexts. One coarse signature of this prediction lies in the
compression of the resulting lexicon: Less speci /uniFB01c conventions
should allow participants to achieve the same communicate accu-
racy with a smaller vocabulary. We calculated the same measure of
effective vocabulary size that we used in our simulations ( Figure 9D ):
the number of unique words produced in each repetition block. We
then constructed a mixed-effects regression model predicting effec-
tive vocabulary size, including /uniFB01xed effects of condition and six
repetition blocks, with random intercepts and repetition block slopes
for each dyad. First, we found an overall main effect of condition,
with signi /uniFB01cantly fewer words used in the coarse condition ( m=5.5)
than the mixed, m=7.9,t(95) =13.2, p<.001, or /uniFB01ne,m=8.3,
t(95)=13.3, p<.001, conditions. Consistent with our simulations,
we also found a signi /uniFB01cant interaction between block and condition,This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.
10This experiment was preregistered at https://osf.io/2hkjc/ with a target
sample size of roughly 40 games per condition. We planned to include all
participants for our accuracy analyses but then exclude participants who were
still below 75% accuracy on the /uniFB01nal quarter of the task ( n=29 pairs) for our
analyses of the lexicon, to ensure posttest measurements could be interpreted
as“converging ”lexicons (as opposed to pairs who had lost interest or given
up). We were later concerned that this exclusion could lead to spurious
differences because convergence rates differed across conditions, but no
results substantially changed depending on the exclusion criteria. All
statistical tests in mixed-effects models reported in this section use degrees
of freedom based on the Satterthwaite approximation ( Luke, 2017 ).20 HAWKINS ET AL.
with the coarse condition decreasing more over time than mixed, b=
0.39, t(95)=3.4,p<.001, or /uniFB01ne,b=0.36, t(95)=2.6,p=.009;
seeFigure 9D .
What allowed participants in the coarse condition to get away with
fewer words in their lexicon while maintaining high accuracy? We
hypothesized that each word had a larger extension size. To test this
hypothesis, we turned to our posttest survey. We counted the numbers
of“speci /uniFB01c”terms (e.g., words that refer to only one object) and
“general ”terms (e.g., words that refer to two objects) in the posttest.
We found that the likelihood of lexicalizing more general terms
differed systematically across conditions ( Figure 11 ). Participants in
the coarse condition reported signi /uniFB01cantly more general terms ( m=
2.3) than in the mixed, m=0.47, t(91.8) =8.8,p<.001, or /uniFB01ne,m=
0.04, t(90.2) =9.2,p<.001, conditions, where lexicons contained
almost exclusively speci /uniFB01c terms. Using the raw extension size of
each word as the dependent variable instead of counts yielded similar
results. Indeed, the modal system in the /uniFB01ne condition was exactly
eight speci /uniFB01c terms with no more general terms, and the modal system
in the coarse condition was exactly four general terms (red, blue,
striped, spotted) with no speci /uniFB01c terms. However, many individual
participants reported a mixture of terms at different levels of general-
ity (see Figure A6 ).
Discussion
There is abundant evidence that languages adapt to the needs of
their users. Our model provides a cognitive account of how people
coordinate on ad hoc linguistic conventions that are adapted to their
immediate needs. In this section, we evaluated predictions about
these context effects using new data from a real-time communica-
tion task. When combined with the generalization mechanisms
explored in the previous section, such rapid learning within dyadic
interactions may be a powerful contributor allowing languages to
adapt at the population level over longer time scales.
Previous studies of convention formation have addressed context
sensitivity in different ways. In some common settings, there is noexplicit representation of context at all, as in the task known as the
“Naming Game ”where agents coordinate on names for objects in
isolation ( Baronchelli et al., 2008 ;Steels, 2012 ). In other settings,
communication is situated in a referential context, but this context is
held constant, as in Lewis signaling games ( Lewis, 1969 ) where
agents must distinguish between a /uniFB01xed set of world states ( Bruner
et al., 2018 ;Skyrms, 2010 ). Finally, in the more sophisticated
Discrimination Game ( Baronchelli et al., 2010 ;Steels &
Belpaeme, 2005 ), contexts were randomly generated on each trial,
but have not been manipulated to assess context sensitivity of the
convention formation process.
In other words, to the extent that context sensitivity has been
addressed by existing models, it has been implicit. Models using
simple update rules have accounted for local referential context with
alateral inhibition heuristic used by both the speaker and listener
agents ( Franke & Jäger, 2012 ;Steels & Belpaeme, 2005 ). If
communication is successful, the connection strength between
the label and object is not only increased, the connection between
the label and competing objects (and, similarly, between the object
and competing labels) is explicitly decreased by a corresponding
amount. This lateral inhibition heuristic is functionally similar to our
pragmatic reasoning mechanism, in terms of allowing the agent to
learn from negative evidence (i.e., the speaker ’s choice notto use a
word, or the listener ’s choice notto pick an object). Under our
inferential framework, however, this form of statistical preemption
emerges as a natural consequence of normative Gricean principles of
pragmatic reasoning rather than as a heuristic (see also Appendix C
andFigure A5 , for similar results using alternative priors).
General Discussion
Communication in a variable and nonstationary landscape of
meaning creates unique computational challenges. To address these
challenges, we advanced a hierarchical Bayesian approach in which
agents continually adapt their beliefs about the form-meaning
mapping used by each partner, in turn. We formalized this approach
by integrating three core cognitive capacities in a probabilistic
framework: Representing initial uncertainty about what a partner
thinks words mean ( C1), partner-speci /uniFB01c adaptation based on
observations of language use in context ( C2), and hierarchical
structure for graded generalization to new partners ( C3). This
uni/uniFB01ed model resolves several puzzles that have posed challenges
for prior models of coordination and convention formation: why
referring expressions shorten over repeated interactions with the
same partner ( P1), how partner-speci /uniFB01c common ground coexists
with the emergence of conventions at the population level ( P2), and
how context shapes which conventions emerge ( P3).
We conclude by raising three broader questions that arise from the
perspective of our model, each suggesting pathways for future work:
(a) to what extent is ad hoc convention formation in adults the same
as word learning in children and how is it different? (b) to what
extent do the proposed mechanisms depend on the communication
modality? and (c) which representations are involved in adaptation
at a process level?
Continuity of Language Learning Across Development
CHAI aims to shift the central computational problem of com-
munication from transmission to learning and adaptation. AlthoughThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure 11
Different Lexicons Emerge in Different Contexts
0481216
coarsefine# words with
post-test meaningone object
two objectsnot used
mixedNote. Mean number of words, out of a word bank of 16 words, that humanparticipants reported giving more speci/uniFB01c meanings (black; applying to 1object) or less speci/uniFB01c meanings (dark gray; applying to two objects) in thepost-test. See the online article for the color version of this/uniFB01gure.CONVENTIONS21
it is intended as a theory of adult communication among mature
language users, our emphasis on learning has much in common with
theories of language acquisition in development. Could the basic
cognitive mechanisms allowing adults to coordinate on conventions
be the same as those supporting learning in children? In other words,
is it possible that adults never stop learning language and simply
develop better-calibrated priors? In this section, we discuss three
possible implications of viewing language acquisition in terms of
social coordination and convention formation, which may help to
further unify models of adult communication with those of language
learning (e.g., Bohn & Frank, 2019 ;Frank et al., 2009 ;Xu &
Tenenbaum, 2007 ).
First, developmental paradigms have typically focused on vari-
ability and generalization across referential contexts (e.g., in cross-
situational word learning) rather than variability and generalization
across speakers ( Regier, 2005 ;Siskind, 1996 ;Smith et al., 2014 ;
Yurovsky & Frank, 2015 ). Yet, it is increasingly apparent that
children are able to track who produced the words they are learning
and use this information when generalizing. For example, bilingual
children learn to expect different languages to be used by different
speakers and even infants are sensitive to coarse social distinctions
based on foreign versus native language ( Kinzler et al., 2007 ), or
accent ( Kinzler et al., 2009 ). Children are also sensitive to the
reliability of individual speakers. For example, young children may
limit the generalizability of observations from speakers who use
language in idiosyncratic ways, such as a speaker who calls a ball a
“dog”(Koenig & Woodward, 2010 ;Luchkina et al., 2018 ), and may
even retrospectively update their beliefs about earlier evidence from
a speaker after observing such idiosyncracies ( Dautriche et al.,
2021 ). Such discounting of idiosyncratic speakers may be under-
stood as an instance of the same inductive problem that convention
formation poses for adults in P2. Unlike complete-pooling models,
which predict that all observations should be equally informative
about a community ’s conventions, CHAI predicts that children
should be able to explain away “outliers ”without their
community-level expectations being disrupted. One novel predic-
tion generated by our account is that children should be able to
accommodate idiosyncratic language within extended interaction
with the same speaker (e.g., continue to pretend the ball is called
“dog,”given partner-speci /uniFB01c common ground) while also limiting
generalization of that convention across other speakers.
Second, CHAI emphasizes the importance of representing lexical
uncertainty (C1), capturing expected variability in the population
beyond the point estimates assumed by traditional lexical represen-
tations. But how do children calibrate their lexical uncertainty? The
number of distinct speakers in a child ’s environment may play a key
role, by analogy to the literature on talker variability ( Clopper &
Pisoni, 2004 ;Creel & Bregman, 2011 ). Exposure to fewer partners
may result in weaker or mis-calibrated priors (e.g., Lev-Ari, 2017 ).
If an idiosyncratic construction is overrepresented in the child ’s
environment, they may later be surprised to /uniFB01nd that it was speci /uniFB01c
to their household ’s lexicon and not shared by the broader commu-
nity (see Clark, 2009 , Chap. 6). Conversely, however, hierarchical
inference predicts a blessing of abstraction ( Goodman et al., 2011 ):
under certain conditions, reliable community-level conventions may
be inferred even with relatively sparse observations from each
partner. To resolve these questions, future work will need to develop
new methods for eliciting children ’s expectations about partner
speci /uniFB01city and variability of meanings.Third, our work suggests a new explanation for why young
children struggle to coordinate ad hoc conventions with one another
in repeated reference games ( Glucksberg et al., 1966 ;Krauss &
Glucksberg, 1969 ,1977 ;Matthews et al., 2007 ). Early explanations
appealed to rigidity in the child ’s perspective that prevented adap-
tation. Yet subsequent /uniFB01ndings that children could not even interpret
their own utterances after a delay ( Asher & Oden, 1976 ) suggest that
the challenge may instead stem from production quality and the lack
of coordinating “signal. ”Children may either be unable to anticipate
how much information is required for their partner to discriminate
the referent, or struggle to access those more complex formulations.
In terms of our model, children ’s lexical priors may be weaker than
adults ’: Without existing conventions for describing the novel
objects in their vocabulary, their utterances are dispersed widely
over easier-to-access “good-enough ”formulations ( Goldberg,
2019 ). Indeed, when children are paired with their caregivers rather
than peers, they easily coordinate on new conventions ( Leung et al.,
2020 ). Adults helped to interactively scaffold the conventions, both
by proactively seeking clari /uniFB01cation when in the listener role (e.g.,
Anderson et al., 1994 ) and by providing more descriptive labels
when in the speaker role, which children immediately adopted11.
From this perspective, ad hoc conventions may not be so different
from other settings where children look to adults for guidance
and rapidly adopt new conventions to talk about new things
(e.g., Carey & Bartlett, 1978 ;Heibeck & Markman, 1987 ).
The Role of Communication Modality
One of our core claims is that the basic learning mechanisms
underlying coordination and convention formation are domain
general. In other words, we predict that there is nothing inherently
special about spoken or written language: Any system that humans
use to communicate should display similar ad hoc convention
formation dynamics because in every case people will be trying
to infer the system of meaning being used by their partners. Directly
comparing behavior in repeated reference games across different
modalities is therefore necessary to determine which adaptation
effects, if any, are robust and attributable to modality-general
mechanisms. In fact, there has been signi /uniFB01cant progress in under-
standing the dynamics of adaptation during communication in the
graphical modality ( Garrod et al., 2007 ;Hawkins, Sano, et al., 2019 ;
Theisen et al., 2010 ), the gestural modality ( Bohn et al., 2019 ;Fay et
al., 2014 ;Motamedi et al., 2019 ), and other de novo modalities
(Galantucci, 2005 ;Kempe et al., 2019 ;Roberts et al., 2015 ;
Roberts & Galantucci, 2012 ;Verhoef et al., 2015 ,2016 ).
CHAI views the similarities and differences between modalities
through the lens of the hierarchical priors we have built up across
interactions with different individuals. For example, in the verbalThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.
11It may be observed that agents in our simulations were still able to
quickly coordinate despite being initialized with weak priors, but they had
the bene /uniFB01t of using feedback from the referential task, as well as small,
shared vocabularies. In the paradigms used by Krauss and Glucksberg
(1969) , young children did not have access to such information and may
have struggled to search their vocabulary for better candidates even if they
did, especially under time pressure (e.g., Glucksberg & Krauss, 1967 ). This
kind of accessibility consideration has previously been instantiated in
computational models via the cost term c(u), but further work on convention
formation in developmental samples may bene /uniFB01t from a more /uniFB01ne-grained,
process model of production.22 HAWKINS ET AL.
modality, the tangram shapes from Clark and Wilkes-Gibbs (1986)
are highly “innominate ”(meaning empirically dif /uniFB01cult to name;
Hupet et al., 1991 ;Zettersten & Lupyan, 2020 )—most people do not
have much experience naming or describing them with words, so
relevant priors are weak and local adaptation plays a greater role. In
the graphical modality, where communication takes place by draw-
ing on a shared sketchpad, people can be expected to have a stronger
prior rooted in assumptions about shared perceptual systems and
visual similarity ( Fan et al., 2018 ). Drawing a quick sketch of the
tangram ’s outline may suf /uniFB01ce for understanding. Other referents
have precisely the opposite property: To distinguish between natural
images of dogs, people may have strong existing conventions in the
linguistic modality (e.g., “husky, ”“poodle, ”“pug”) but making the
necessarily /uniFB01ne-grained visual distinctions in the graphical modality
may be initially very costly for novices ( Fan et al., 2020 ), requiring
the formation of local conventions to achieve understanding
(Hawkins, Sano, et al., 2019 ). The gestural modality also has its
own distinctive prior, which also allows communicators to use time
and the space around them to convey mimetic or depictive meanings
that may be dif /uniFB01cult to encode verbally or graphically ( Clark, 2016 ;
Goldin-Meadow & McNeill, 1999 ;McNeill, 1992 ). We, therefore,
suggest that differences in production and comprehension across
modalities may be understood by coupling modality-speci /uniFB01c priors
with modality-generic learning mechanisms.
Process-Level Mechanisms for Adaptation
Finally, while we have provided a computational-level account of
coordination and convention formation in terms of hierarchical
inference, there remain many possible process-level mechanisms
that may perform this computation. In this section, we discuss two
interlocking process-level questions which emphasize current lim-
itations and areas of future work: (a) exactly which representations
should be adapted? and (b) what is required to scale models of
adaptation to more naturalistic language?
Which Representations Are Adapted?
While our model formulation focused on adaptation at the level of
lexical meaning (i.e., inferences about ϕ, representing different
possible lexical meanings), this is only one of many internal
representations that may need to be adapted to achieve successful
coordination. Three other possible representational bases have been
explored in the literature.
First, it is possible that adaptation takes place upstream of the
lexicon, directly implicating perceptual or conceptual representa-
tions ( Garrod & Anderson, 1987 ;Healey et al., 2007 ) That is, there
may be uncertainty about how a particular partner construes the
referent itself, and communication may require constructing a
shared, low-dimensional conceptual space where the relevant re-
ferents can be embedded ( Stolk et al., 2016 ). This is particularly
clear in the classic maze task ( Garrod & Anderson, 1987 ) where
giving effective spatial directions requires speakers to coordinate on
what spatial representations to use (e.g., paths, coordinates, lines, or
landmarks).
Second, it is possible that adaptation takes place even further
upstream, at the level of social representations ( Jaech & Ostendorf,
2018 ). Rather than directly updating beliefs about lexical or con-
ceptual representations, we may update a holistic representation ofthe partner themselves (e.g., as a “partner embedding ”in a low-
dimensional vector space) that is used to retrieve downstream
conceptual and lexical representations. Under this representational
scheme, the mapping from the social representation to particular
conventions is static, and ad hoc adaptation is limited to learning
where a particular partner belongs in the overall social space.
Third, expectations about other lower-level features may also be
adapted through interaction, such as a partner ’s word frequencies
(Louwerse et al., 2012 ), syntax ( Gruberg et al., 2019 ;Levelt &
Kelter, 1982 ), body postures ( Lakin & Chartrand, 2003 ), or even
informational complexity ( Giles et al., 1991 ), or even informational
complexity ( Abney et al., 2014 ). This level of adaptation may lead
some forms to become more accessible or entrenched in memory
over time, possibly allowing partner identity to be used as a retrieval
cue (e.g., Horton, 2007 ;Horton & Gerrig, 2005 ,2016 ; but see
Brown-Schmidt & Horton, 2014 ).
Computational Tractability and Scalability
While a fully Bayesian formulation elegantly formalizes the
computational-level inference problem at the core of the CHAI
account, this formulation faces a number of limitations. For one, it is
clearly intractable ( Van Rooij, 2008 ;Van Rooij et al., 2019 ): The
posterior update step in Equation 5 grows increasingly intensive as
the space of possible utterances and meanings grows ( Woensdregt
et al., 2021 ). The intractability problem also raises a scalability
problem: Does CHAI provide any guidance toward building arti /uniFB01-
cial agents that are actually able to adapt to human partners as
humans do with one another? Through this applied lens, a number of
recent efforts have focused on developing algorithms for state-of-
the-art neural networks that tractably scale to arbitrary natural
language (e.g., referring expressions using the full vocabulary of
an adult language user) and arbitrary visual input (e.g., sensory
impressions of novel objects such as tangrams).
For example, building on recent formal connections between
hierarchical Bayes and gradient-based metalearning approaches in
machine learning ( Grant et al., 2018 ), the algorithm proposed by
Hawkins, Kwon, et al. (2020) (a) relaxes the full community-level
prior over Θto a point estimate and (b) replaces the dif /uniFB01cult integral
in the posterior update with a /uniFB01xed number of (regularized) gradient
update steps. Another recent proposal builds on connections to
classical exemplar-based algorithms ( Nosofsky, 1984 ): An agent ’s
lexical expectations at time tmay be determined via weighted
similarity to memory traces of lexical items used by different
partners in the past ( Shi et al., 2010 ), where similarity is computed
by a neural network. While such algorithms cannot /uniFB01x the intracta-
bility of the Bayesian formulation ( Kwisthout et al., 2011 ), and their
precise correspondence to constraints on the computational-level
theory remain unexplored, they nonetheless provide promising
algorithmic instantiations of the CHAI account. When lexical
meaning is represented by the parameters of a neural network,
conventions can be interpreted as (meta-)learned initializations used
for new partners and coordination is partner-speci /uniFB01c/uniFB01ne-tuning or
domain adaptation of vector representations.
Neural network instantiations also provide a possible pathway
toward addressing the lack of incrementality in the fully Bayesian
formulation. As more scalable implementations of pragmatic rea-
soning have proliferated in machine learning ( Andreas & Klein,
2016 ;Monroe et al., 2017 ;Shen et al., 2019 ;Takmaz et al., 2020 ;This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.CONVENTIONS 23
Vogel et al., 2013 ), it has been natural to use incremental archi-
tectures ( Augurzky et al., 2019 ;Cohn-Gordon et al., 2018 ,2019 ;
Waldon & Degen, 2021 ). However, there remain a number of
limitations to address in future work, including how to incorporate
incremental feedback into lexical updates (e.g., backchannel re-
sponses or interruptions), how to de /uniFB01ne a more satisfying notion of
compositional semantics for incrementally constructed utterances,
and how to maintain representations of partner-speci /uniFB01c parameters
alongside community-wide parameters in memory.
Conclusion
How do we manage to understand one another? We have argued
that successful communication depends not just on transmission but
on continual learning across multiple timescales. We must coordi-
nate on meaning through common ground with individual partners
but also abstract these experiences away to represent stable con-
ventions and norms that generalize across our communities. Like
other socially grounded knowledge, language is not a rigid dictio-
nary that we acquire at an early age and deploy mechanically for the
rest of our lives. Nor do languages only change over the slow time
scales of intergenerational drift. Language is a means for
communication —a shared interface between minds —and as new
ad hoc concepts arise, new ad hoc conventions must be formed to
solve the new coordination problems they pose. In other words, we
are constantly learning language. Not just one language, but a family
of related languages, across interactions with each partner.
Let us conclude not that ‘there is no such thing as a language ’that we
bring to interaction with others. Say rather that there is no such thing as
the one total language that we bring. We bring numerous only loosely
connected languages from the loosely connected communities that we
inhabit. ( Hacking, 1986 )
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Appendix A
Details of RSA Model
Our setting poses several technical challenges for the Rational
Speech Act (RSA) framework. In this Appendix, we describe these
challenges in more detail and justify our choices.
Action-Oriented Versus Belief-Oriented Listeners
First, both agents are “action oriented, ”in the sense that they
behave proportional to the utility of different actions, according to a
soft-max normalization σðUðzÞÞ=eUðzÞ=PeUðzÞ. This contrasts
with some RSA applications, where the listener is instead assumed
to be “belief oriented, ”simply inferring the speaker ’s intended
meaning without producing any action of their own ( Qing &
Franke, 2015 ).
Placement of Uncertainty
Second, our instantiation of lexical uncertainty differs subtly from
the one used by Bergen et al. (2016) , which placed the integral over
lexical uncertainty at a single level of recursion (speci /uniFB01cally, within
a pragmatic listener agent). Instead, we argue that it is more natural
in an interactive, multi-agent setting for each agent to maintain
uncertainty at the highest level, such that each agent is reasoning
about their partner ’s lexicon regardless of what role they are
currently playing.
Handling Degenerate Lexicons
Finally, when we allow the full space of possible lexicons ϕ, we
must confront degenerate lexicons where an utterance uis literally
false of every object in context, that is, where Lϕ(o,u)=0 for all
o∈C. In this case, the normalizing constant in Equation 3 is zero,and the literal listener distribution is not well de /uniFB01ned. A similar
problem may arise for the S1distribution.
Several solutions to this problem were outlined by Bergen et al.
(2016) . One of these solutions is to use a “softer ”semantics in the
literal listener, where a Boolean value of false does not strictly rule
out an object but instead assigns a very low numerical score, for
example,
Lϕðo,uÞ=/C26
1 if o∈ϕðuÞ
/uni03F5 o:w:
Whenever there is at least one o∈Cwhere uis true, this
formulation will assign negligible listener probability to objects
where uis false, but ensures that the normalization constant is
nonzero (and speci /uniFB01cally, that the distribution is uniform) when uis
false for all objects.
While this solution suf /uniFB01ces for one-shot pragmatics under lexical
uncertainty, where /uni03F5may be calibrated to be appropriately large, it
runs into several technical complications in an iterated setting. First,
due to numerical over /uniFB02ow at later iterations for some parameter
values, elements may drop entirely out of the support at higher levels
of recursion (e.g., L1), leading the normalization constant to return to
zero. Second, this “soft”semantics creates unexpected and unin-
tuitive consequences at the level of the pragmatic speaker. After
renormalization in L0, an utterance uthat fails to refer to any object
in context is also by de /uniFB01nition equally successful for all objects (i.e.,
evaluating to /uni03F5for every object), leading to a uniform selection
distribution. However, this assumption has the unintuitive conse-
quence that S1’s utility of using an utterance known to be false of the
target may be the same as an utterance known to be true.This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.
(Appendices continue )30 HAWKINS ET AL.
Instead of injecting /uni03F5into the lexical meaning, we ensure that the
normalization constant is well de /uniFB01ned by adapting another method
suggested by Bergen et al. (2016) . First, we add a “null”object to
every context so that, even if a particular utterance is false of every
real object in context, it will still apply to the null object, assigning
the true target a negligible probability of being chosen. Intuitively,
this null object can be interpreted as recognizing that the referring
expression has a referent but it is not in context, that is, a “failure to
refer, ”and effectively prevents L0from assigning belief to a referent
for which the utterance is literally false. Note that this case is distinct
from the case of a contradiction , which arises when de /uniFB01ning the
meaning of multiword utterances in Section P1.
Second, we add an explicit noise model at every level of recursion.
That is, we assume every agent has a probability /uni03F5of choosing a
random element of their support, ensuring a /uniFB01xed nonzero /uniFB02oor on the
likelihood of each element that is constant across levels of recursion.
Formally this corresponds to a mixture distribution, for example,
L/uni03F5
0ðoju,ϕÞ=/uni03F5·PunifðoÞ+ð1−/uni03F5Þ·L0ðoju,ϕÞ
S/uni03F5
1ðujo,ϕÞ=/uni03F5·PunifðuÞ+ð1−/uni03F5Þ·S1ðujo,ϕÞ
Marginalizing Over ϕk
Another theoretical question arises about exactly how speaker and
listener agents ought to marginalize over their uncertainty about ϕk
when selecting actions ( Equation 4 ). In our formulation, theexpectation is naturally taken over the entire utility each agent is
using to act, that is, if the speaker and listener utilities are de /uniFB01ned
to be
ULðo;u,ϕkÞ=logS1ðujo,ϕkÞ
USðu;o,ϕkÞ=ð1−wCÞlogL0ðoju,ϕkÞ−wC·cðuÞ
then the expectation is taken as follows:
LðojuÞ∝exp/C26
wLð
PLðϕkjDkÞ·ULðu;o,ϕkÞdϕk/C27
SðujoÞ∝exp/C26
wSð
PSðϕkjDkÞ·USðu;o,ϕkÞdϕk/C27
This formulation may be interpreted as each agent choosing an
action proportional to its expected utility across different possible
values of ϕk, weighted by the agent ’s current posterior beliefs about
the lexicon their partner is using.
This formulation contrasts with the one suggested by Bergen et al.
(2016) , which assumes the expectation takes place at a single level
of recursion, say the L1, as above, and then derives the other agent ’s
behavior by having them reason directly about this marginalized
distribution, for example,
Ualt1ðu;oÞ=ð1−wCÞ· log LðojuÞ−wC·cðuÞ
Salt1ðujoÞ∝expfwS·Ualt1ðu;oÞgThis document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Table A1
Proposed Parameterization for Repeated Reference Game
Category Parameter Example parameter settings
Partner design What feedback is provided? –No feedback at all
–Only correct/incorrect
–Real-time responses from partner
Are you playing with the same partner? –Same partner for whole game
–Swap out partners every round
–Swap after krounds
What do you know about your partner? –Anonymous stranger
–Stranger with perceptual information
–Close friend
How consistent are roles across repetitions? –Consistent director/matcher
–Alternate roles each round
Stimulus design How familiar are targets? –Very familiar: colors, household objects
–Not at all familiar: tangrams, novel line drawings
How complex are targets? –Very complex: busy visual scenes, clips of music
–Not at all complex: geometric drawings
How consistent are targets across repetitions? –Exact same image of object
–Different pose/view of same object
–Different objects from same neighborhood
Context design How similar are distractors to the target? –Very similar: same basic-level category
–Not at all similar: other categories
What is the size of context? –Between 2 and 21
How consistent is context across repetitions? –Exact same context each round
–Randomized context (sometimes far, sometimes close)
Repetition design How many repetitions per target? –Between 3 and 100
What is spacing between repetitions? –Block structure
–Sequential structure with interspersed contexts
Modality design What medium is used for communication? –Text
–Audio
–Gesture
–Drawing
(Appendices continue )CONVENTIONS 31
where LðojuÞis de /uniFB01ned as above. This formulation may be
interpreted as an assumption on the part of the speaker that the
listener is already accounting for their own uncertainty, and best
responding to such a listener. Isola ting lexical uncertainty over ϕ
to a single level of recursion is a natural formulation for one-shot
pragmatic phenomena, where additional layers of recursion can
build on top of this marginal distribution to derive implicatures.
However, the interpretation is mes sier for the multi-agent setting,
since it (a) induces an asymmetry where one agent considers the
other ’su n c e r t a i n t yb u tn o tv i c ev e r s aa n d( b )r e q u i r e st h es p e a k e r
to use their own current posterior beliefs to reason about the
listener ’sm a r g i n a l i z a t i o n .
A third possible variant is to place the expectation outside the
listener distribution but inside the speaker ’s informativity term, that is,
Lavg=ð
PðϕkjDkÞ·L0ðoju,ϕkÞdϕk
Ualt2ðu;oÞ=ð1−wCÞ· log LavgðojuÞ−wC·cðuÞ
Salt2ðujoÞ∝expfwS·Ualt2ðu;oÞg
The interpretation here is that the speaker /uniFB01rst derives a distribu-
tion representing how a listener would respond on expectation and
then computes their surprisal relative to this composite listener.
While this variant is in principle able to derive the desired phenom-
ena, it can be shown that it induces an unintuitive initial bias under a
uniform lexical prior, since the logarithm cannot distribute over the
integral in the normalization constant. This bias is most apparent in
the case of context sensitivity (Simulation 3).
Mathematically, the difference between these alternatives is whether
the speaker ’su n c e r t a i n t ya b o u t ϕkgoes inside the renormalization of
LðojuÞ(as in Salt1), outside the renormalization but inside the logarithm(as in Salt2), or over the entire utility (as in our chosen formulation).
While other formulations are conceivable, we argue that marginalizing
over the entire utility is not only the most natural but also normatively
correct under Bayesian decision theory. When an agent is uncertain
about some aspect of the decision problem, rational choice requires the
agent to optimize expected utility ma rginalizing over subjective uncer-
tainty, as in our formulation.
Inference Details
We have implemented our simulations in the probabilistic program-
ming language WebPPL ( Goodman & Stuhlmüller, electronic ). All of
our simulations iterate the following trial-level loop: (a) sample an
utterance from the speaker ’sd i s t r i b u t i o n ,g i v e nt h et a r g e to b j e c t ,( b )
sample an object from the listener ’so b j e c td i s t r i b u t i o n ,g i v e nt h e
utterance produced by the speaker, (c) append the results to the list of
observations, and (d) update both agents ’posteriors, conditioning on
these observations before continuing to the next trial. To obtain the
speaker and listener distributions (Steps 1 –2;Equation 3 ), we always
use exhaustive enumeration for exact inference. We would prefer to use
enumeration to obtain posteriors over lexical meanings as well (Step 4;
Equation 5 ), but as the space of possible lexicons ϕgrows, enumeration
becomes intractable. For simulations related to P2andP3,w et h e r e f o r e
switch to Markov Chain Monte Carlo (MCMC) methods to obtain
samples from each agent ’sp o s t e r i o r s ,a n da p p r o x i m a t et h ee x p e c t a t i o n s
inEquation 4 by summing over these samples. Because we are
emphasizing a set of phenomena where our model makes qualitatively
different predictions than previous models, our goal in this article is to
illustrate and evaluate these qualitative predictions rather than provide
exact quantitative /uniFB01ts to empirical data. As such, we proceed by
examining predictions for a regime of parameter values ( wS,wL,
wC,β)t h a th e l pd i s t i n g u i s ho u rp r e d i c t i o n sf r o mo t h e ra c c o u n t s .This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.(Appendices continue )32 HAWKINS ET AL.
This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure A1
Coordination Success (Simulation 1.1) Across a Range of Parameter Values0.70.81 2 4 8 16 32
0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 150.50.60.70.80.91.0
0.50.60.70.80.91.0
0.50.60.70.80.91.0
0.50.60.70.80.91.0
0.50.60.70.80.91.0
0.50.60.70.80.91.0
repetition #% accu racy0.6 0.9 1.0soft-max temperature
Note. Columns represent memory discount parameter β, and rows represent the agents ’soft-max
optimality parameters, where we set αS=αL. Communicative success is achieved under a wide range
of settings, but convergence is limited in some regimes. For example, at high values of β, with no ability to
discount prior evidence, accuracy rises quickly but asymptotes below perfect coordination; at low α,
inferences are slightly weaker and agent actions are noisier, slowing convergence; /uniFB01nally, at low values of
β, when prior evidence is forgotten too quickly, convergence interacts with α: The latest evidence may
overwhelm all prior evidence, preventing the accumulation of shared history. The agent noise model is set
to/uni03F5=0.01 in all simulations. See the online article for the color version of this /uniFB01gure.
(Appendices continue )CONVENTIONS 33
This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure A2
Speaker Ef /uniFB01ciency (Simulation 1.2) Across a Range of Parameter Values Representing Different
Weights on Informativity and Cost0 0.12 0.24 0.36 0.48 0.612481632640246024602460246024602461.001.251.501.752.001.001.251.501.752.001.001.251.501.752.001.001.251.501.752.001.001.251.501.752.001.001.251.501.752.001.001.251.501.752.001.001.251.501.752.00repetition ## words0.6 0.8 1 (no forgetting)soft-max temperatureproduction cost
Note. Rows represent agent soft-max optimalityαS=αL, columns represent costswC, and different memorydiscount factorsβshown in different colors. Agents converge on more ef/uniFB01cient ad hoc conventions for a wideregime of parameters. When utterance production costwCis more heavily weighted relative to informativity, thespeaker is less likely to produce longer utterances, even at the beginning of the interaction; when optimalityαS,αLis higher, and the speaker maximizes utility, we observe faster reduction and more categorical behavior. Notethat asα→∞, utterances only become shorter atwC=0 in the absence of forgetting. In this case, the shorterutterances approach the exact same utility as the longer utterance, and the speaker reaches equilibrium simplysampling among them at random (i.e., choosing the longer utterance with 1/3 probability and each of the shorterutterances with 1/3 probability). See the online article for the color version of this/uniFB01gure.(Appendices continue)34HAWKINS ET AL.
This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure A3
Speaker Ef /uniFB01ciency Simulations for P2 Across a Larger Parameter Regime1248160 5 1015200 5 1015200 5 1015200 5 1015200 5 1015200 5 1015200.000.250.500.751.000.000.250.500.751.000.000.250.500.751.000.000.250.500.751.000.000.250.500.751.00trial #0 0.1 0.2 0.3 0.4 0.51248160.000.250.500.751.000.000.250.500.751.000.000.250.500.751.000.000.250.500.751.000.000.250.500.751.00P(long utterance)1248160.000.250.500.751.000.000.250.500.751.000.000.250.500.751.000.000.250.500.751.000.000.250.500.751.000.6 0.8 1 (no forgetting)production cost soft-max temperaturesoft-max temperaturesoft-max temperatureP(long utterance)P(long utterance)partial poolingcompletepooling  no pooling
Note. We examine the behavior of complete-pooling, no-pooling, and partial-pooling models, where rows represent agent soft-max optimalityαS=αL, columns represent cost weightwC, and colors represent memory discount parameterβ. See the online article for the color version of this/uniFB01gure.(Appendices continue)CONVENTIONS35
This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure A4
Qualitative Predictions of Our Three Models for P2complete poolingno poolingpartial pooling
1 2 4 8 161 2 4 8 16 1 2 4 8 16 1 2 4 8 16 1 2 4 8 16 1 2 4 8 1600.10.20.30.40.5
00.10.20.30.40.5
00.10.20.30.40.5drop jump drop jump drop jump1 3 5 7 9 111.001.251.501.752.00# words
1 3 5 7 9 11 1 3 5 7 9 11drop1.001.251.501.752.00# wordsjumpdropjumpjumpdrop0.60.81.0
−1 0 1 2 3
log(t)
Note. Across a wide range of parameter values, only the hierarchical model consistently produces both qualitative phenomena of interest: reversion
to the prior at partner boundaries (i.e., a “jump”) and gradual generalization across partners (i.e., a “drop”). Approximately N=10 simulations used to
compute t-statistic in each cell. Cells marked with black boxes are signi /uniFB01cantly different from a null effect of 0 change, p<.005. See the online article
for the color version of this /uniFB01gure.
(Appendices continue )36 HAWKINS ET AL.
This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure A5
Simulation Results for P3 Using Different Values of αandβfor (A) Partition-Based Simplicity Prior, and (B) Alternative
Unconstrained Simplicity Prior
2480.60.70.80122.02.53.03.54.0
2.02.53.03.54.0
2.02.53.03.54.0block ## unique words used2480.60.70.80 5 10 15 200 5 10 15 200 5 10 15 20406080100
406080100
406080100trial #% correctcoarsemixedfine
012 012soft-max temperaturesoft-max temperature
0 5 10 150 5 10 150 5 10 15406080100
406080100
406080100trial #% correct
01010234
234
234block #number unique words used
10.60.7 0.8
0.60.70.8(A)
(B)
Note. Simulations were run for fewer trials in (B). Overall, we observe similar qualitative predictions for the difference between thecoarse and/uniFB01ne condition, although the mixture condition is more sensitive to parameters and priors. See the online article for the colorversion of this/uniFB01gure.(Appendices continue)CONVENTIONS37
Appendix B
Alternative Lexical Representations
In this section, we reconsider two speci /uniFB01c choices we made about
how to represent lexical meanings.
First, for simplicity and consistency with earlier models of Bayesian
word learning, we adopted a traditional truth-conditional representa-
tion of lexical meaning throughout the article. Each word in the lexicon
is mapped to a single “concept ”to, for example, w1=‘bluesquare, ’
where this utterance is true of objects that fall under the given concept,
and false otherwise. The inference problem over lexicons, therefore,
requires searching over this discrete space of word-concept mappings.
However, it is important to emphasize that our model is entirely
consistent with alternativ e lexical representations.
For example, for some settings, a continuous, real-valued repre-
sentation may be preferred, or a higher-dimensional vector repre-
sentation. Rather than assigning each word a discrete concept in the
lexicon, we may simply assign each word –object pair ( wi,oj)a
scalar meaning representing the extent to which word wiapplies to
object oj, such that ϕis a real-valued matrix:
ϕ=2
6664ϕð11Þϕð12Þ··· ϕð1iÞ
ϕð21Þϕð22Þ··· ϕð2jÞ
............
ϕðj1Þϕðj2Þ··· ϕðijÞ3
7775andLϕ(wi,oj)=ϕ(ij). In this case, rather than discrete categorical
priors over meanings, we may place Gaussian priors over the entries
of this matrix:
ΘðijÞ∼Nð0, 1Þ
ϕðijÞ∼NðΘðijÞj1Þ
We have previously achieved similar results using this alterna-
tive lexical representations in earl ier iteration of this manuscript
(Hawkins et al., 2017 ;Hawkins, Goodman, et al., 2020 ), although
deriving predictions required variational inference techniques
rather than Markov Chain Monte Carlo. Such optimization-based
inference techniques may also provide the most promising path
for extending our adaptive model to larger language models,
including neural networks that operate over continuous spaces
of image pixels and natural-language embeddings, where ϕ
simply represents the weights of the network ( Hawkins, Kwon,
et al., 2020 ).This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure A6
Empirical Mixtures of Terms Reported by Participants in P3coarse mixed fine
0 2 4 6 8 10 0 2 4 6 8 10 02468 1 0012345
# words referring to single object# words referring
 to multiple objects
10 20 30 40count
Note. While the modal lexicon in the coarse condition contained 0 speci /uniFB01c terms and 4 more general terms (32% of participants) and the modal lexicon in the
mixture and /uniFB01ne conditions contained eight speci /uniFB01c terms and 0 more general terms (42% and 38% of participants, respectively), many participants reported a
mixture of abstract and speci /uniFB01c terms.
(Appendices continue )38 HAWKINS ET AL.
Appendix C
Alternative Lexical Priors
A variety of priors have been proposed for probabilistic models of
language learning and convention formation, which build in stronger
or weaker assumptions about the structure of the lexicon. We used a
simplicity prior over lexicons that strictly partition the set of referents,
ensuring that every object must be in the extension of exactly one
word (e.g., Carr et al., 2020 ). To explore the robustness of our results
inP3to alternative choices of lexical priors, we also considered a
weaker prior where the space of possible lexicons allows any
denotation to be assigned to any word, including highly redundant
and overlapping lexicons (where every object is in the extension of
every word), and highly degenerate lexicons (where every word has a
completely empty extension). Given this more unconstrained space of
lexicons, the simplest way to penalize complexity is to de /uniFB01ne the size
of the lexicon jϕjas the total extension size , that is, the summed
extensions of all terms. In this case, favoring simpler word meanings
also necessarily favors smaller lexicons.
To illustrate this property of the weaker prior, consider a
reference game with a /uniFB01xed set of two referents: a blue square
and a red square. Then the meaning of a given utterance uican
either have extension size two (applying both objects, effectively
meaning “square ”), size one (applying only to “blue square ”or
only to “red square ”), or size zero (applying to neither of the
objects). Now suppose there are 16 possible utterances. Then the
lexicon with highest prior probability is the one where every
utterance has an empty extension ( jϕj=16⋅0=0 ), which is
also the smallest lexicon (an effective vocabulary size of 0).
The lexicon with lowest prior probability is the one where every
utterance has the maximal extension ( jϕj=16⋅2=32), which is
also the largest lexicon (an effective vocabulary size of 16).
Removing a single word from this maximal lexicon would reduce
the size of the lexicon (e.g., 15 words instead of 16) and also reduce
the total size of the words ’extensions (e.g., taking a word with the
maximal extension of size two and replacing it with the minimalextension of size zero: jϕj=15⋅2+1⋅0=30), which would
slightly increase its prior probability. Of course, this scheme also
makes it possible to have a lexicon with smaller word meanings but
not a smaller lexicon (e.g., for a lexicon where all 16 words have an
extension size of 1, we have jϕj=16⋅1=16, but the effective
vocabulary size is still 16). So this prior straightforwardly encodes
a preference for lexicons with fewer words, but gives partial credit
when the words have “simpler ”meanings, breaking ties between
lexicons with the same number of words. Simulation results using
this prior are shown in Figure A5 .
Note that other choices are possible, such as those only enforcing
mutual exclusivity, or only the principle of contrast ( Clark, 1987 ). In
the context of P3, we expect these choices to primarily affect choices
on early trials. Some priors may predict an agglomerative form of
learning where all conditions begin using /uniFB01ne-grained language and
then the coarse condition gradually collapses down to a more
minimal lexicon, while others predict a divisive form where all
conditions begin using coarse-grained language and then the /uniFB01ne
condition gradually introduces words with more re /uniFB01ned meanings.
For example, one qualitative feature of our empirical results in P3
(see Figure 9D ) is that participants apparently begin using more
unique terms at the outset, and remain constant at that large number
of unique terms in the /uniFB01neandmixed conditions but gradually
whittle away their vocabulary size in the coarse condition. Because
we noticed that neither of the priors considered in Figure A5
displayed this pattern, we considered a third possible prior. This
prior enforced full coverage over all meanings (i.e., disallowed
“degenerate ”lexicons where some objects are not in the extension of
any words at all), unlike the unconstrained prior, but otherwise
allowed redundancy (i.e., some objects were in the extension of
multiple words), unlike the partition-based prior. This prior gave rise
to a qualitatively more similar pattern of lexical convergence (see
Figure C1 ).This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.
(Appendices continue )CONVENTIONS 39
Received April 12, 2021
Revision received October 18, 2021
Accepted November 4, 2021 ▪This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.Figure C1
Simulation Results for P3 Using a Full-Coverage Lexical Prior That Disallows Degenerate Lexicons ( αS=αL=6,β=0.6)
0.0 1.0 2.02.02.53.03.54.0
block #number unique words usedcoarsemixedfine
Note. See the online article for the color version of this/uniFB01gure.40HAWKINS ET AL.

Hawkins
R
D
Franke
M
Frank
M
C
Goldberg
A
E
Smith
K
Griffiths
T
L
& Goodman
N
D