J. theor. Biol. (1999) 200, 147}162
Article No. jtbi.1999.0981, available online at http://www.idealibrary.com on

The Evolutionary Language GameMARTIN A. NOWAK*, JOSHUA B. PLOTKIN AND DAVID C. KRAKAUER
Institute for Advanced Study, Olden ¸ane, Princeton, NJ 08540, ;.S.A.
(Received on 25 February 1999, Accepted in revised form on 2 June 1999)

We explore how evolutionary game dynamics have to be modi"ed to accomodate a mathematical framework for the evolution of language. In particular, we are interested in the evolution of
vocabulary, that is associations between signals and objects. We assume that successfulcommunication contributes to biological "tness: individuals who communicate well leave
more o!spring. Children inherit from their parents a strategy for language learning (a language
acquisition device). We consider three mechanisms whereby language is passed from one
generation to the next: (i) parental learning: children learn the language of their parents;(ii) role model learning: children learn the language of individuals with a high payo!; and(iii) random learning: children learn the language of randomly chosen individuals. We showthat parental and role model learning outperform random learning. Then we introduce
mistakes in language learning and study how this process changes language over time.
Mistakes increase the overall e$cacy of parental and role model learning: in a world witherrors evolutionary adaptation is more e$cient. Our model also provides a simple explanation
why homonomy is common while synonymy is rare.
 1999 Academic Press

1. Introduction
A unity of design and a diversity of implementations are de"ning features of evolved traits.
Common descent provides a set of shared characteristics upon which local adaptation andontogenetic mechanisms impress their modi"cations. For a small set of species learning has
opened up the possibility of almost limitless variety, to the extent that the underlying uniformity
has been obscured. This is particularly true forhuman language and to a lesser extent, for signal
evolution in animals.
Humans from di!erent language communities
employ di!erent sounds to denote the same object, action or property. Among the song birds,
dialects are highly variable and change within an* Author to whom correspondence should be addressed.
E-mail: kathei@ias.edu0022}5193/99/018147#16 $30.00/0

individual over the course of its life (Lemon, 1975;
Catchpole et al., 1984; Balaban, 1988; Kroodsma& Konishi, 1991). The common structures of human and some animal signals have in part been
elucidated (Chomsky, 1965, 1972, 1975, 1980;
Frisch 1967; Marler, 1970; Marler & Pickert,
1984; Marler et al., 1988). It is now thought thatfeatures of human language are &&universal'' andconstant across language communities: the principles of language are shared and have evolved,
while the parameters are free to change throughlearning (Pinker & Bloom, 1990; Pinker, 1994).
This begs the questions, why learn signals at all
and why not simply inherit them? Why whenthese signals have been learnt do they so oftenchange? Which members of a community should
serve as models for imitation? And which components of a strategy are inherited and which ones
learnt?
 1999 Academic Press

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M. A. NOWAK E¹ A¸.

Imitation like language is rare in nature (Galef,1988). Since language learning in humans requires imitation, a part of the explanation for
language's rarity might be ascribed to the di$culties of imitation. The problem is in part evolving
a means of associating an action with a goal anda reward without experiencing it oneself. Furthermore, which individual of a population isbest to imitate? A safe place to begin is with ones
parents. Human infants of a few weeks in age, andeven newborns, show a tendency to imitate thefacial gestures of those around them (Meltzo!,
1983; Meltzo! & Moore, 1994). Imitation is central to language acquisition but there is more tolearning a language than simple imitation. Language acquisition is not the same as learninga skill through guided trial and error because it
requires the formation of a shared representation
among signallers. Linguistic signals are referential and hence imitation need not imply understanding. Imitation of a signal when coupled to
an inappropriate behaviour will be selectedagainst.
There are numerous publications on the evolution of human language ranging from studies ofprimate or other animal behaviour (Smith, 1977;
Seyfarth et al., 1980; Burling, 1989; Cheney& Seyfarth, 1990; Poran & Coss, 1990; Green"eld, 1991; Hauser, 1996; Dunbar, 1997; Bradbury & Vehrencamp, 1998), to the developmentof language in young children (Brown, 1973;
Anis"eld, 1984; Sachs, 1985; Newport, 1990;
Hurford, 1991; Bates, 1992), the genetic and anatomic correlates of language competence (Wittelson & Pallie, 1973; Lieberman, 1984, 1991;
Newport, 1990; Nobre et al., 1994; Deacon, 1997)
and the diversity of existing human languages(Greenberg, 1971; Cavalli-Sforza & CavalliSforza, 1995; Cavalli-Sforza, 1997). Maynard
Smith & Szathmary (1995) described human language as a major transition in evolution (see also
Szathmary & Maynard Smith, 1995). Bickerton(1990) argues that this transition is likely to havebeen abrupt, while Pinker (1994) has argued thata gradual process is likely to have been su$cient.
Most studies of language evolution do not use
mathematical models. There are some exceptions(Aoki & Feldman, 1987, 1989; Hurford, 1991;
MacLennan, 1992; Hashimoto & Ikegami, 1995;
Hutchins & Hazelhurst, 1995; Oliphant, 1996;

Parisi, 1997; Palacious, 1998; Steels, 1998;
Briscoe, 1999; see also Hurford et al., 1999). Inan early theoretical paper, Hurford (1989) discussed a mathematical model for the evolution ofSaussarian signs (Saussure, 1916). Hurford's
model is similar to our approach here and alsouses two matrices to describe communication.
Cavalli-Sforza & Feldman (1981) and Boyd
& Richardson (1985) provide alternative mathematical frameworks for cultural transmission in
evolution.
Our aim here is to expand the standard assumptions of evolutionary game theory and develop a general, but simple mathematical model
for understanding the evolution of language. In
particular, we are interested in the evolution of
a common vocabulary (that is a speci"c association between signals and objects). We do not
explore the evolution of syntax or grammar inthis paper. We extend an approach taken by
Nowak & Krakauer (1999) and study in moredetail how language is transmitted from one generation to the next. In Nowak & Krakauer(1999), we simply assumed that o!spring adopt
the language of their parents; here we specify
learning mechanisms. Section 2 de"nes themodel. Section 3 formalizes language learning.
Section 4 outlines the population dynamics anddiscusses three mechanisms whereby language
is transmitted from one generation to the next.
Section 5 takes advantage of mistakes. Section 6shows results on competition between the mechanisms of language transmission. Section 7 concludes. Appendix A classi"es some binary
matrices, which are the absorbing states of ourstochastic evolutionary processes. Appendix B
describes a paradoxical result: two individuals
may understand each other better than themselves.
2. De5ning the Model
Consider a group of individuals (animals or
early hominids) able to produce a number ofsignals (sounds). Information shall be transferred
about a number of objects; we use &&objects'' (orconcepts) in an extended sense to include otherindividuals, other animals, plants, inanimate objects, actions or events (all things that can bereferred to). Suppose there are m signals and149

EVOLUTIONARY LANGUAGE GAME

n objects. The matrix P contains the entries p ,GHwhich denote the probability that for a speakerobject i is associated with sound j. P is an n;mmatrix whose rows sum to 1:
K
H

p "1.
GH

(1)

The matrix Q contains the entries q , which deHG
note the probability that for a listener sound j isassociated with the object i. Q is an m;n matrix
whose rows sum to 1:

The passive matrix, Q, can be treated as completely independent of P. In this case, it is nottaken for granted that an individual who signals&&leopard'' when seeing a leopard will also conceive of a leopard when hearing &&leopard''; thissymbolic association is something which must be
formed by the evolutionary dynamics of the system. Alternatively, we could assume that individuals have an internal mechanism that links
their active and passive matrices.
2.1. SOME COMMENTS ON P AND Q

2.1.1. ¹he best possible Q
L
G

q "1.
HG

(2)

P describes speaking, whereas Q describes
listening (comprehension). We call P the &&active
matrix'' and Q the &&passive matrix''.
Imagine two individuals, I and I , who uselanguages ¸ (given by P and Q ) and ¸ (given
by P and Q ). For individual I , p denotes the GHprobability of making sound j when seeing object
i, whereas q denotes the probability of inferring
HG
object i when hearing sound j. For individual I ,these probabilities are given by p and q. SupGH
HG
pose I sees object i and signals, then I will infer
object i with probability K pq. A measure
H GH HG
of I 's ability to convey information to I is given
by summing this probability over all objects, n:LK pq. The overall payo! for communG H GH HG
ication between I and I is taken as the sum ofI 's ability to convey information to I , and I 's
ability to convey information to I . Thus,
1 L K
F(¸ , ¸ )"(pq#pq). (3)
 
GH HG
GH HG
2
G H
In this equation, both individuals are treatedonce as a listener and a speaker, which leads tothe intrinsic symmetry of the language game:F(¸ , ¸ )"F(¸ , L ). Language ¸ obtains
 
 

from ¸ the same payo! as ¸ from ¸ . If
two individuals use the same language, ¸, the payo! is
L

K

p q .
F(¸, ¸)"
GH HG
G H

(4)

An interesting question is: for a given P whatis the optimum Q that maximizes the payo!
F(¸, ¸)? We "nd that the optimum Q is always
a binary matrix such that q "1 if p is the
HG
GHlargest entry in a column of P; all other entries of
Q are 0. (In case there are 2 or more equally large
entries in a column of P one may be chosen
at random.) Thus for a given matrix P, themaximum payo! is given byK
F (¸, ¸)"
p ,
K?V
*H
H

(5)

where p is the maximum entry in column j of*H
the P matrix.
2.1.2. ¹he best possible P and Q
The maximum payo! for a P and Q pair is
obtained for binary active matrices P that have atleast one 1 in every column (if n*m) or in everyrow (if n)m), while Q is constructed as describedin Section 2.1.1. For n"m the maximum payo!
is obtained if P has exactly one 1 in every row andcolumn (while every other entry is 0) and Q"P2
(Q is the transposed matrix of P). Such a matrix iscalled a &&permutation matrix''. In general, themaximum payo! is given by
F (¸, ¸)"min+m, n,.
K?V

(6)

3. Learning a Language
We will assume that individuals acquire a language (a lexicon) by observing and imitatingother individuals. Speci"cally, we assume that

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M. A. NOWAK E¹ A¸.

each individual undergoes a &&learning phase''during which it constructs an &&association
matrix'', A. A is an n;m matrix whose entries, a ,GH
specify how often an individual has observed one
or several other individuals referring to object
i by producing signal j. Therefore, the entries
of A are nonnegative integers. The active andpassive matrices are then derived from the association matrix by normalizing rows and columns,
respectively:

 

p "a
GH
GH

K

J

a ,
GJ

q "a
HG
GH

 
L

J

a .
JH

(7)

Suppose individual I learns from I . We assumethat I samples the responses of I to every object
k times. In the limit kPR, I will exactly repro
duce I 's active matrix. For k"1, I 's association matrix, A , will be binary and P "A .

Thus, k"1 leads immediately to binary active
matrices.
Assume that all individuals of a population
speak the same language. Consider the stochastic
process
P PA PP PA PP P2.






(8)

A given active matrix, P , is sampled to give anGassociation matrix, A , which is then converted
G>
into P(and Q ) according to eqn (7). The
G>
G>
language changes over time because language
learning (constructing A from P) is a probabilistic
process. Without errors in language learning theprocess will ultimately reach an absorbing state.
The absorbing states are given by the set of allbinary P matrices where each object is associatedwith only one signal (but a given signal may referto several objects).
3.1. HOMONOMY AND SYNONYMY

There is a general understanding among
language scholars that homonomy is plentiful
in language (any common word in the dictionary
will have several meanings) whereas synonymy
is rare (it is hard to "nd two words thathave exactly the same meaning). Interestingly,
this observation is a direct consequence ofthe language learning mechanism described byeqn (9).

Homonomy refers to the case where we havemore than one non-zero entries in a column of
the active matrix of the language: thus two di!erent objects (concepts) can be associated with thesame word. This situation is stable. Consider theactive matrix

 

1 0
.
1 0

P"

If sampled by the child, it will lead to the association matrix

 

A"

k 0
,
k 0

which again leads to the same active matrix.
Homonomy is an absorbing state of our stochastic process.
Synonymy refers to the case where we havemore than one non-zero entries in a row of theactive matrix: thus, the same object (concept) is
associated with two di!erent signals. This situation is not stable. Consider the active matrix
P"(0.5 0.5).
Binomial sampling will lead to an association
matrix which is slightly asymmetric resulting inan asymmetric new P matrix, which most likely
gives rise to an even more asymmetric A matrix.
The only stable solutions are given by
P"(1 0) or P"(0 1),
where synonymy has disappeared.
4. Population Dynamics
Let us now consider a population of N individuals speaking languages ¸ to ¸ . Each in,
dividual talks to every other individual andreceives a total payo!,
F " F(¸ , ¸ ),
'
' (
(

(9)

where J"1,2, N, but JOI. We explore three
di!erent mechanisms for transmitting the language from one generation to the next. TheseEVOLUTIONARY LANGUAGE GAME

151

FIG. 1. A common language evolves if individuals with a high payo! in the language game produce more o!spring and if
children learn the language of their parents. There are N"100 individuals communicating about n"5 objects using (up to)m"5 signals. There are discrete generations. In one generation each individual is talking to every other individual. The
payo!s of all interactions are summed up. Individuals produce o!spring for the next generation proportional to their total
payo!. O!spring sample the active matrix from their parents thereby forming their own association matrix, A, between
objects and signs. For each object, k responses are sampled. Individuals derive their P and Q from their A. Initially, every
individual has a random A; a are taken from a uniform distribution on the interval (0, 1). The four panels show 20 runs eachGHfor k"1, 4, 7, 10. All simulations eventually settle to binary P matrices achieving payo!s 6, 8, or 10. The case k"1 has thefastest convergence to a common language but reaches on average a slightly lower "nal payo! than k"4, 7, or 10.

mechanisms represent di!erent strategies for language acquisition.
4.1. PARENTAL LEARNINGFirst, we assume that children learn the language of their parents. Figure 1 shows results ofcomputer simulations for this case. There isa population of N"100 individuals using (up to)m"5 signals to communicate n"5 objects. Initially, all individuals have random matrices. In
each generation, every individual &&talks'' to allothers and the payo!s are summed up. The total
payo! for each individual is given by eqn (10).
For the next generations, o!spring are produced
proportional to the total payo! of an individual:
the chance that a particular individual arises
from individual I is given by F / F . There is no' ( (sexual reproduction. Each individual has one
parent. The individuals of the new generation
learn the language of their parents by sampling
their responses to speci"c objects. Each indi-vidual forms its association matrix, A, by recording k responses of its parent to each object.
Figure 1 shows 20 runs for four di!erent valuesof k: 1, 4, 7, 10. Starting from a random initialcondition, the population converges to a common language. The evolutionary optimum isreached if every word is associated with exactly
one signal (and vice versa). This optimum (which
results in a payo! of 5 points) is reached often,
but not always. Sometimes the population settles
for a language where two di!erent signals areassociated with one object, whereas another signal is used for two di!erent objects. This suboptimum case achieves payo! 4. Payo! 3 isalso reached occasionally. In this case, either thesame signal is used for three di!erent objects, orthere are two pairs of objects each associatedwith one signal. There is no signi"cant di!erencein the average "nal outcome for di!erent valuesof k. For higher values of k, however, the population takes a longer time to converge to a common
language.

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M. A. NOWAK E¹ A¸.

4.2. ROLE MODEL LEARNING

A di!erent concept is that language may confera certain reputation in the group, which can beassociated with an increased chance of imitation
by members of the next generation. In this case, it
seems at "rst sight unimportant whether successful individuals also produce more genetic o!spring; they transmit their language culturally.
Therefore, we consider a scenario in which individuals with a high payo! have a higher chanceof being imitated. The probability that a memberof the new generation samples the active matrix
of individual I of the previous generation is given
by F / F . Each individual samples K indi' ( (viduals of the previous generation. For K"1,
this process leads to the identical algorithm as forthe case described in Section 4.1. Thus, parental
learning and role model learning can be described using the same formalism. The equivalence breaks down for K'1; there can be only

one genetic parent (without sexual reproduction),but several cultural parents. (Note the di!erence
between K and k: the parameter K denotes
the number of individuals that are sampled, while
k counts how often a given individual is sampled.)
Figure 2 shows computer simulations for
K"1, 4, 7, 10. For K"1, we have rapid convergence; the average payo! is equivalent to thatfound in Fig. 1. For K'1, convergence takes
much longer, but the average payo! is signi"cantly higher than for K"1. There is, however,no signi"cant di!erence between K"4, 7 and 10.
4.3. RANDOM LEARNING

Finally, we consider the situation where children learn the language of random members of
the previous generation, irrespective of their
payo!. There is no (direct) reward for high payo!
and consequently no selection on improving
language e$ciency. Nevertheless, the resulting

FIG. 2. A common language evolves if children learn their language from individuals in the previous generation who havea high payo!. There are N"100 individuals communicating about n"5 objects using (up to) m"5 signals. There are
discrete generations. In one generation each individual is talking to every other individual. The payo!s of all interactions
are summed up. Each individual of the next generation choses K individuals of the previous generation and samples their
responses (1 response per object per individual is sampled). P and Q are derived from A. The probability that an individual ischosen for being sampled is proportional to its payo! in the language game. Thus successful individuals have a higher chanceof passing on their language. The four panels show 20 runs each for K"1, 4, 7, 10. All simulations eventually settle to binary
P matrices achieving payo!s 6, 8, or 10. The case K"1 has the fastest convergence to a common language but reaches on
average a slightly lower "nal payo!. K"10 has slowest convergence, but reaches this maximum payo! in 20 of 20 cases. Thisis, however, a coincidence. In simulations using 100 runs per K-value, there is no signi"cant di!erence between K"4, 7 or 10,
although K"1 performs clearly worse.

EVOLUTIONARY LANGUAGE GAME

153

FIG. 3. A common language also evolves if children learn their language from randomly chosen individuals of the previous
generation irrespective of any payo!. In this case, however, there is no reward for e$cient communication. The active matrix,
P, is not at all optimized but drifts to an arbitrary binary matrix. (Note that the binary matrices are the absorbing states of thisrandom process). All parameter values are as for Fig. 2, but each individual samples K randomly chosen individuals of theprevious generation. Note that convergence is much slower than in Fig. 2 and leads on average to lower "nal payo!.

stochastic process can lead to fairly e$cient communication systems, because the absorbing statesof the high-dimensional &&random walk'' are random binary matrices, which on average achievea higher payo! than random matrices where theentries are numbers between 0 and 1. In Appendix A, we discuss the combinatorics of thesebinary matrices and calculate their average
payo! as well as the fraction of random binary
matrices that obtain maximum payo!: as m getslarge compared to n, the average "tness approaches the maximum "tness and the fraction ofbinary matrices that achieve maximum "tness
converges to 1.
Figure 3 shows computer simulations for thecase where children learn their language fromK random individuals of the previous generation,
irrespective of their payo!. Convergence to
a common language is slower for larger values of
K and in general, the convergence is slower thanin Figs 1 and 2. For all values of K, however, thesame average payo! is achieved. The distributionof payo! values simply re#ects the statistics ofrandom binary matrices. Note that the math-ematical algorithm for modelling random transmission in Fig. 3 does not require any payo!
evaluation as the payo! achieved in the language
game is totally irrelevant here.
5. Take Advantage of Mistakes
Let us add another fundamental feature to ourmodel: language acquisition should be errorprone. Assume that with probability, o, a learning individual mistakes the response of anotherindividual for a randomly chosen response. Thus
with probability 1!o the learning individual
makes the correct entry into its association
matrix, A, while with probability o it makesa random entry for this particular object. (Herewe consider errors in language learning; for the
e!ect of errors in communication we refer to
Nowak & Krakauer, 1999.)
Figure 4 shows the same simulation as Fig. 1(for k"1) but with four di!erent noise levels(o"0.0001, 0.001, 0.01 and 0.1). For low noiselevels (o"0.0001 and 0.001) the system behaves
similar to Fig. 1. For o"0.01, however, all154

M. A. NOWAK E¹ A¸.

FIG. 4. Errors during language acquisation favour the evolution of more e$cient languages. Here the simulations of Fig.
1 with k"1 are repeated but individuals make mistakes during language learning: with probability o the response ofanother individual to a given object is mistaken for a randomly chosen response. For small levels of noise, o"0.0001 and0.001, there is not much di!erence compared to the case without errors. For o"0.01 all simulations converge to a global
optimum. For o"0.1 the system does not evolve e$cient communication. Thus there seems to be an optimum error rate.

simulations converge to the maximum payo!
where each signal is associated with exactly one
object. Errors prevent the system from getting
trapped in sub-optimum situations where thesame signal would be used for two or more objects. For high probabilities of mistakes (o"0.1)
the optimization ability of the system is impaired.
Thus, without noise the system converges to
one of the absorbing states (given by all random,binary P matrices). As long as there is still heterogeneity in the population there is selection for
better adapted and more e$cient languages.
Once each individual speaks the same language,
there is no longer the possibility of change. Errors, however, introduce a constant degree ofvariation upon which natural selection for more
e$cient languages can act. Therefore, errors inlanguage learning can lead to the evolution ofa more e$cient language. There is an interesting
dependency on the error rate: small amounts ofnoise do not provide enough variability, large
amounts of noise introduce too much variability,
in between there is an optimum error rate.
The bene"t of noise for language evolution isreminiscent of other situations where noise can

improve the optimization properties of a system:in many evolutionary models there are optimum,
non-zero mutation rates; optimum levels of noise
can jiggle neural networks out of local optima;
noise is the basis of the optimization method
called simulated annealing.
In contrast, errors have only detrimental effects on the language acquisition mechanism forimitating randomly chosen individuals. Figure 5
shows computer simulations which repeat thesimulations of Fig. 3(a) with noise. Individuals
learn their language from one randomly chosen
individual from the previous generation (K"1).
Four levels of noise are studied. For o"10\ theevolution to a common language is essentiallyuna!ected. For o"10\ the system converges to
a common language, but there are occasional#uctuations that generally reduce the average
payo! of the population. For o"10\ convergence to a common language is still observed, butvariation and #uctuations in the average payo!
are there all the time and dominate the scenario.
For o"10\ (or greater) there is no longer any
convergence to a common language. Rememberthat for this level of error, o"10\, the system

EVOLUTIONARY LANGUAGE GAME

155

FIG. 5. Increasing amounts of noise make random sampling ine!ective. Here the simulations of Fig. 3 are repeated for
rho"0.00001, 0.0001, 0.001 and 0.01. For small values of o the system manages to converge to a common language (which isrepresented a randomly chosen binary active matrix). For larger values of o the population no longer converges to
a common language. Thus errors during language acquisition have very di!erent e!ects on systems where individuals
sample their parents (or other individuals with high payo! ) as opposed to systems where individuals sample randomly
chosen individuals. Five runs are shown for each value of o. For all runs we have K"1.

where individuals imitate their parents always
converged to optimum "tness (see Fig. 4).
Therefore, the quasi-accidental evolution of
a common language which emerges when individuals imitate other randomly chosen individuals is impaired by mistakes during language
acquisition. An interesting task here would be todevelop an analytic framework for understanding
the error-threshold phenomenon that operates
here: below a critical level of o the evolution of
a common language is possible; above this level
coherence is no longer achieved.
6. Competition among Di4erent Strategies
We have considered the performance of di!erent mechanisms of language acquisition anddescribed how e$ciently they can drive theevolution of a common language. Now, we analyse the competition among individuals using different language acquisition devices.
Figure 6 tests parental vs. random learning.
&&Random players'' (that learn the language fromrandomly chosen individuals of the previous gen-eration) compete with players that learn the language of their parents. The error rate is 0. The
payo! in the language game is interpreted as"tness. Individuals with a higher payo! leave
more o!spring which inherit the language acquisition strategy of their parents. Initially, 80%
of individuals are random players; 20% learn thelanguage of their parents. Initially, everyone
starts o! with random P matrices. As a common
language evolves (that is as the average payo!
increases) the fraction of random players declines.
Thus individuals who learn their parents' language seem to acquire an e$cient language more
rapidly. This is not unexpected, because these
individuals learn the language of someone who
was at least "t enough to reproduce. They havea higher chance of acquiring a better adapted
language. Once everyone speaks the same language, however, there is no longer selection
against the random players. Neutral drift determines the "nal outcome.
Figure 7 repeats the simulation of Fig. 6 withan error rate of o"0.001. In all 10 runs therandom players become extinct. Furthermore, in156

M. A. NOWAK E¹ A¸.

FIG. 6. It is better to learn from your parents than fromrandom members of the population. The simulation compares the performance of two di!erent strategies: (i) individuals who learn the language of their parents; and (ii)individuals who learn the language of an arbitrary memberof the community (so called &&random players''). The population size is 200. Initially 80% of the population are random
players. The "gure shows the increase in the average payo!
as the community evolves a common language and thedecrease of the fraction of random players. Note that thefrequency of random players decline as long as the population is in the state of improving its language; once everyone
speaks the same language there is no longer a di!erence inthe performance of the two strategies (and random drift will
eventually drive one of them to extinction). Parameters:n"5, m"5, every individual samples one other individual
once (i.e. K"1 and k"1), 10 runs are shown.

all 10 cases the population converges to the maximum "tness, where each object is associated withexactly one word. Errors make a di!erence.
In Fig. 8, we expand this theme and analyse
competition among strategies of language acquisition that are described by integers M, suchthat an individual playing strategy M learns its
language by sampling its parent together with
M randomly chosen individuals. We consider the
strategies M"0, 1, 2, 3 and 4. M"0 denotes thestrategy where individuals only learn from theirparents. Starting with equal proportions of all

FIG. 7. Take advantage of mistakes. The possibility oferrors during language acquisition enables the system toreach the global optimum. Furthermore, such errors introduce a consistent selection pressure against the random
players. The "gure shows 10 simulation for exactly the same
parameter values as in Fig. 4. The only di!erence is that
individuals can make mistakes while learning the language:
with probability o, the response of another individual toa particular object is mistaken for another (randomly
chosen) response, We use o"0.01. In 10 of 10 runs, therandom players become extinct and the population converges to the maximum payo! were each object is associatedwith exactly one signal and vice versa.

strategies, we "nd that M"0 wins in all 10 runs.
All other strategies become extinct. Thus learningfrom randomly chosen individuals in addition to
your parents seems to be disadvantegeous.
Figures 6}8 can also be interpreted as analysing the competition between role model and random learning. (Remember that for K"1 thealgorithms for biological and cultural transmission are identical.) Thus, learning the language
from individuals with a high payo! in the language game is better than learning from random
individuals. Hence, both role model and parental
learning outperform random learning.
But what about parental vs. role model learning? Clearly, there is no di!erence in our model

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EVOLUTIONARY LANGUAGE GAME

FIG. 8. In the presence of errors, is it better only toacquire the language from the parents or acquire the language from the parents and some other, randomly chosen
individuals? Strategy, M, denotes the number of individuals
that are sampled in addition to the parents. That is, every
individual samples its parents and M other individuals of
the previous generation. In our simulation, M varies from0 to 4. In 10 out of 10 runs the average strategy of the
population eventually converges to 0. In all runs the average
payo! eventually converges to its maximum value. Sameparameters as in Fig. 5.

between learning from the parent or one otherindividual chosen proportional to its payo!, but
is it better to imitate only the parent or the parent
and some other players proportional to their
payo! ? The question is equivalent to whether it
is better to have only one or several role models.
In Section 4.2 (and Fig. 2), we saw that learningfrom one role model led to a faster optimization,
but learning from more than one role model
usually led to a higher average "tness. Here we
study the direct competition among these strategies and include errors. We performed the same
simulations as for Fig. 8, analysing the competition among strategies that learn from their parents and M other individuals of the previous
generation which are selected proportional totheir payo!. Again using the concept of errors
during language acquisition (with o"0.001), we
performed 100 runs starting with equal proportion of all strategies, M ranging from 0 to 4.
The outcome is inconclusive. Fifty-"ve runs
converged to a homogeneous population with
M"0, whereas 36 runs converged to homogeneous populations with M either 1, 2, 3 or 4. (Nineruns still contained a mixture of strategies after
1000 generations.) Hence, it does not seem tomake much di!erence whether the language islearned from one relatively successful individual(the parent) or several. A clearer understanding of
the di!erential e!ects of such strategies means
considering repeated rounds of learning andcommunication during each generation.
An interesting issue in linguistics is the di!erence between learning the language from the parents vs. learning from other individuals. While
early language acquisition is clearly from theparents, later on children seem to adopt the language and accent of their peers, not their parents.
A possible explanation is that language serves as
an important tool of identi"cation between andwithin peer groups. While it should be possible toexplore this fascinating question with the mathematical framework developed here, it is beyond
the scope of the current paper.
7. Conclusions
We presented a general mathematical framework for studying the evolution of a simple communication system. Starting from random initialconditions we demonstrated how a population
can evolve a common language, where speci"cwords are associated with speci"c objects. The
words that evolve in our simulations are arbitrary, discrete and referential.
We analysed the evolutionary properties ofthree mechanism of language acquisition: (i) Parental learning implies that children learn the language of their parents. (ii) Role model learning isbased on the concept that the payo! in the language game is related to reputation. Children
learn the language of individuals with a highreputation. (iii) Random learning describes thesituation where children learn the language fromrandomly chosen individuals from the previous
generation.

158

M. A. NOWAK E¹ A¸.

Only for parental learning do we directly assume that language confers increased biological"tness (better communicators have more o!spring). For role model learning, language confers reputation (better communicators have more
immitators), while for random transmission the
payo! in the language game appears to be totallyirrelevant. Our results demonstrate that parental
learning and role model learning are e!ectively
indistinguishable, and are in fact identical when
individuals learn from a single role model. (However, our model does not consider deceit, and thiswill be minimized by either kin selection acting
on parental learning, or in structured populations, where limited mobility favours honesty.)
Random learning consistently underperforms
both parental learning and role model learning.
In fact, random transmission can be modelled
without any payo! evaluation; the individuals
only have to imitate others. It is important, however, to emphasize that for the evolutionary stability of language there must always be an implicit
assumption that language confers biological"tness. Otherwise, individuals who opt not toparticipate in language learning and communication, would be neutral and therefore would
not be selected against. This in e!ect states that
individuals who speak a sort of gibberish, could
eventually come to dominate the population.
While it is very di$cult to relate the complexitiesof language to biological "tness, it is not so di$cult to "nd examples where linguistic imperfections exclude speakers from dominant (cultural)
positions. It becomes a separate issue to relate
status to long-term reproductive success (inhumans).
Without errors during language acquisition,
all three mechanisms can lead to a common language. The "nal state is reached when each object
is associated with exactly one signal. The same
signal, however, may be associated with several
objects. Thus our models provide one simple
account for why true synonyms should be rare,
while homonyms are plentiful. This discrepancy
arises out of the dynamics of the learning process,in particular, the fact that synonyms are expected
to disappear over several learning trials.
For parental and role model learning, the "nalstate is optimized for performance; in the processof convergence toward the "nal state there isselection for languages with higher payo!. For
random learning, this is not the case, and the "nalstate is a completely randomly chosen binary
matrix. There is, however, some accidental
optimization for random learning, because arandomly chosen binary P matrix leads on anaverage to higher payo! than a random P matrix
whose entries are real numbers between 0 and 1.
(Hurford, 1989, used random transmission to
study the acquisition of signs and hence his simulations are to a large extent based on this optimization property of the stochastic process.)
By allowing errors during language acquisition
there is strictly speaking no "nal state and evolution will continue inde"nitely. For parental androle model learning, there is an optimum error
rate that maximizes the performance of the system: the chance of reaching a global optimum isincreased as the system does not remain at local
optima. This is of course analogous to an annealing process, whereby stochasticity can prevent"xation on local optima. The interesting questionis whether children employ low levels of error, inan e!ort to make learning more e$cient. Thus
mistakes during acquisition would be viewed as
an essential, rather than incidental component oflanguage learning. In contrast, random learning
is impaired by the presence of errors: if the error
rate is above a small threshold value random
learning does not work at all.
The interesting complications, which lead todi!erent possibilities for formulating a language
game dynamics, arise here because the languageof an individual (P and Q), determines its "tness,but what is inherited from the parents is a mechanism for acquiring a language*a language acquisition device. In standard evolutionary gametheory (Maynard Smith, 1982; Sigmund, 1995;
Hofbauer & Sigmund, 1998), o!spring inherita strategy from their parents and this strategy
determines the individual's (frequency-dependent) "tness. In the language game, children inherit a device for acquiring a language fromothers in the population. Thus, the language thatthey learn depends on what others are doing andis therefore frequency dependent. In the evolutionary language game, not only the "tness of
individuals, but also their language is frequency
dependent. Thus there is the possibility for decoupling cultural and biological "tness. ThisEVOLUTIONARY LANGUAGE GAME

could lead to the emergence of speci"c linguistic
practises that were biologically deleterious,
nevertheless maintained by biological selection
on the acquisition strategy.
We thank the two anonymous referees for theirconstructive suggestions. Support from The Florence
Gould Foundation, J. Seward Johnson, Sr. Charitable Trusts, The Ambrose Monell Foundation and
The Alfred P. Sloan Foundation is gratefully acknowledged.
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APPENDIX ACounting Random Binary Matrices
In this appendix, we describe how to count andclassify all binary P matrices, which are the absorbing states for the dynamical systems discussed in Section 4. In other words, we focus ourattention on n;m binary matrices with a single
1 in each row. There are clearly mL such matrices.
The more di$cult combinatorics of the classi"ca-tion occur in the case when n)m, which we will
now consider.
We would like to classify these matrices by the"tness which they yield. Let us denote the number of matrices yielding "tness i by Z . The maxG
imum possible "tness is, of course, 2n. There areclearly Z "m!/(m!n)! such maximal-"tness
Lmatrices. However, we also want expressions forthe number of sub-optimal matrices, i.e. for Z,
L\Z ,2, Z . Notice that if we count each Z
L\

L\I
correctly then we will have the equality
IL\ Z "mL.
L\IIWorking from the de"nition of Z , we derive
L\I
the following expression for the number of matrices yielding "tness n!k:



n
Z "
L\I
t #1
R R,2, RK 







n!(t #1)

2
t #1




n! K\ (t #1)(m!a)!
G G,
t #1(m!n#k)!
K

where we sum over all partitions t ,2, t satis
Kfying:t *0,
G
t #t #2t "k,Kt )n!1,Gk#a)n,
where a"C+t "t '0,.
G G
Moreover, in each term of the expression above
we only want to include the factors involving t
Gand t '0. We can rewrite our expression for
G
Z
as
L\I
n!
Z "
L\I (m!n#k)!
(m!a)!
,+ ,2, , (t #1)!(t #1)!2(t #1)!(n!k!a)!
KR RK ;where the sum is taken over the same set ofpartitions t as before.
G
Table 1 shows numerical results for n"4 andm"4, 5,2, 15. For each value of m we report

161

EVOLUTIONARY LANGUAGE GAME

the average "tness,
2 L
i)Z ,
fM "
G
mL
G

For Z , we must consider the four partitions
L\2"2#0#0#0,
2"0#2#0#0, etc.

as well as the proportion of matrices which havemaximum "tness,Z
P " L.
L K mLWe see that the average "tness, fM , approaches the
maximum value 2n for increasing values of m.
There is a simple, analytic explanation for thistrend. When n is "xed, notice thatZ
lim P " lim L
LKKK mLand also the () partitions
2"1#1#0#0,
2"1#0#1#0, etc.
to obtain
Z

m!
" lim
K (m!n)!(mL)



 

4!
4
2!
3!
"
4)
#
"84.
L\ 2!
2 2!2!0!
3!1!

For Z , we consider all the partitions
L\(m)(m!1)(m!2)2(m!n#1)
" lim
mL
K
mL#lower powers of m
" lim
mL
K
"1.

3"3#0#0#0, etc.
3"2#1#0#0, etc.
3"1#1#1#0, etc.

In other words, as m gets large, the fraction of
matrices which yield maximum "tness approaches 1. As a result, the average "tness approaches the maximum "tness, n. Thus, learning
the language from random individuals irrespective of their payo! in the language game can leadto a fairly e$cient language especially if m issigni"cantly larger than n.
As a speci"c example, we consider n"m"4.
In this case,

Finally, we check that Z #Z #Z #
L
L\
L\
Z "256"mL, as desired.
L\Z "4!"24.
L

APPENDIX B

For Z
we consider the four partitions
L\
1"1#0#0#0,
1"0#1#0#0, etc.

But notice that only the partitions of 3 in whicha"1 satisfy k#a)n, so that according to ourformula





4! 3!
"4.
Z "4 )
L\
3! 4!0!

A Paradoxical Result on Mutual Understanding
If P and Q are unrelated in an individual, thenit is easily possible to construct 2 languages ¸and ¸ that understand each other better than
themselves. Thus, we may haveto obtain





4! 3!
"144.
Z "4 )
L\
1! 2!2!

F(¸ , ¸ )(F(¸ , ¸ )"F(¸ , ¸ )'F(¸ , ¸ ).
 
 
 
 
(7)

162

M. A. NOWAK E¹ A¸.

A simple example is

 

 

1 0
0 1
P "
, Q "
,


0 1
1 0
and

 

 

0 1
1 0
P "
, Q "
.


1 0
0 1
Here, F(¸ , ¸ )"F(¸ , ¸ )"0 whereas F(¸ ,
 
 

¸ )"F(¸ , ¸ )"2.

 
An interesting question is how closely related
P and Q must be such that this paradoxicalsituation is not possible?

Note that the coupling of P and Q via an
association matrix, see eqn (9), is not strong
enough to exclude this possibility. An example isgiven by the 2 association matrices





23 77
A "

66 34





1 99
and A ".

34 66

If P , Q and P , Q are the constructed accord  
ing to eqn (9) then we "nd F(¸ , ¸ )"2.3743.., F(¸ , ¸ )"F(¸ , ¸ )"2.3894..; and F(¸ , ¸ )"
 
 
 
2.3771... Hence ¸ and ¸ understand each otherbetter than themselves. In evolutionary game dynamics the two languages are in equilibrium.
Both languages have an advantage when rare!