Philosophy of Science, 74 (January 2007) pp. 1–27. 0031-8248/2007/7401-0005$10.00
Copyright 2007 by the Philosophy of Science Association. All rights reserved.

1

Evolution and the Explanation of
Meaning*

Simon M. Huttegger†‡

Signaling games provide basic insights into some fundamental questions concerning
the explanation of meaning. They can be analyzed in terms of rational choice theory
and in terms of evolutionary game theory. It is argued that an evolutionary approach
provides better explanations for the emergence of simple communication systems. To
substantiate these arguments, I will look at models similar to those of Skyrms (2000)
and Komarova and Niyogi (2004) and study their dynamical properties. My results
will lend partial support to the thesis that evolution leads to communication. In general,
states of partial communication may evolve with positive probability under standard
evolutionary dynamics. However, unlike states of perfect communication, they are
unstable relative to neutral drift.

1. Introduction. Signaling games model simple signaling interactions be-
tween a sender and a receiver. They emphasize the social aspects of lan-
guage. Thus, they are able to provide basic insights into some fundamental
questions concerning the explanation of how meaningful communication
can emerge. The importance of societal relations for the emergence of
communication was already pointed out by David Hume (1739), Jean-
Jacques Rousseau (1755), and Adam Smith (1761). It was further elab-
orated in David Lewis’s (1969) seminal work on conventions. One main

*Received June 2005; revised February 2007.

†To contact the author, please write to: Konrad Lorenz Institute for Evolution and
Cognition Research, Adolf Lorenz Gasse 2, A-3422 Altenberg, Austria; e-mail:
simon.huttegger@kli.ac.at.

‡I would like to thank Brian Skyrms for his advice and encouragement. I am also
grateful to Georg Dorn, Bill Harms, Josef Hofbauer, Gerhard Jäger, Natalia Koma-
rova, Hannes Leitgeb, Don Saari, Kevin Zollman, two anonymous referees, and the
editor of Philosophy of Science for many helpful suggestions and comments. Most of
this paper was written during a stay at the Department of Logic and Philosophy of
Science at the University of California at Irvine, whose members I wish to thank for
their hospitality and kindness. My research was supported by the Institut für Wissen-
schaftstheorie of the Internationales Forschungszentrum Salzburg and the Konrad
Lorenz Institute for Evolution and Cognition Research.



2 SIMON M. HUTTEGGER

idea underlying this view is that a basic function of language is to facilitate
coordinated behavior. Meaning is thus a consequence of pragmatic
factors.

Lewis (1969) was intended as an answer to Quine’s (1936) counterar-
guments against the logical positivist’s claim that conventions of meaning
form the basis of logical truth and logical inference. One way to express
Quine’s skepticism is to doubt that the truth of a statement is given by
a linguistic component and a factual component (Quine 1953). Analytical
statements have zero factual component. They are true regardless of how
the world looks. The linguistic component of the truth of a statement is
governed by conventions of meaning. But do we have a good explanation
of how conventions of meaning come about?

According to Quine and others, we do not have such an explanation
(see White 1950; a similar argument can already be found in Rousseau
[1755]). There is no noncircular explanation of conventions of meaning.
Conventions come about by agreement. To achieve agreement we always
have to presuppose some kind of rudimentary language, which is itself
left unexplained. One of Lewis’s basic insights—which can, in fact, be
traced back to Hume—is that conventions need not come about by agree-
ment. They might be stable outcomes of repeated nonverbal interactions.
Or they might be salient among their alternatives.

This paper continues Skyrms’s (1996, 2000, 2004) analysis of the ex-
planation of meaning conventions in evolutionary terms. I shall argue
that an evolutionary explanation avoids the difficulties that an explanation
in terms of rational choice (like Lewis’s account) faces at a very funda-
mental level. As such, the position underlying this study is the one men-
tioned above, to wit, that meaning is a consequence of pragmatic factors.
By this I mean that meaning emerges from the interactions of less than
fully rational agents (i.e., agents who are constrained in their computing
capacities and in the information they have about the structure of the
game). These agents may be less deliberate than their fully rational kin
in standard game theory. However, they might still be said to maximize
their utility within constraints that are not assumed in standard game
theory. In this sense, my analysis might be called pragmatic.

The arguments in this direction will be substantiated by reporting a
number of theorems on simple signaling games and signaling games that
allow probabilistic associations between states, signals, and acts. Both
kinds of signaling games allow for states with no communication, states
with partial communication, and states with perfect communication. My
main results show that, except for one special case, the latter two types
of states emerge with positive probability under selection dynamics. States
with no communication are always dynamically unstable. States of partial
communication can be destabilized by neutral drift. States of perfect com-



EVOLUTION AND THE EXPLANATION OF MEANING 3

TABLE 1. A STATE-ACT
COORDINATION PROBLEM

WITH a 1 0.

a1 a2

j1 a 0
j2 0 a

munication turn out to be the only states that are robust relative to
selection and drift.

I will start in Section 2 by defining simple signaling games and reviewing
some of their properties. In Sections 3 and 4 I shall discuss language
conventions with respect to evolution and rational choice. This discussion
will be a methodological background to the study of the dynamics of
simple and generalized simple signaling games in Sections 5 and 6, where
the main new results can be found. Section 7 concludes by reconsidering
the explanatory value of my results.

2. Simple Signaling Games. Simple signaling games are based on coor-
dination problems between states and acts. The simplest situation of this
kind consists of two states of the world, and , and two correspondingj j1 2
acts, and . Each act is a proper response to exactly one of the states.a a1 2
An individual who chooses the wrong act gets no positive payoff. This
payoff structure is illustrated in Table 1. Let us call a situation like this
one a “state-act coordination problem.”

Suppose that there are two individuals. The sender observes the state
while the receiver chooses an act. The latter cannot observe the state. The
sender can send two messages, and , to indicate which state hasm m1 2
occurred. The receiver might respond to each of the signals by choosing
a particular act. If the receiver chooses the right act, then both players
get the same payoff a ( ). Since the sender and the receiver alwaysa 1 0
get the same payoff, it is in the sender’s interest to communicate which
state has occurred. Likewise, the receiver has an interest to associate the
signal with the right act. So they need a common understanding about
the two signals in order to coordinate their actions.

The situation outlined above constitutes a simple signaling game with
two players, the sender and the receiver, two states of the world, two
messages, and two acts. A sender strategy specifies, for each state, what
signal to send. A receiver strategy specifies which act will be chosen as a
response to a message. Accordingly, there are four sender strategies and
four receiver strategies. The sender might send one of the two signals if

occurs and the other one if occurs, or she might always send thej j1 2
same signal regardless of which state occurs. The receiver might choose
one of the two acts as a response to and the other act as a responsem1



4 SIMON M. HUTTEGGER

TABLE 2. SENDER STRATEGIES AND RECEIVER
STRATEGIES.

s1 if , ifm j m j1 1 2 2 r1 a1 if , a2 ifm m1 2
s2 if , ifm j m j2 1 1 2 r2 a2 if , a1 ifm m1 2
s3 if , ifm j m j1 1 1 2 r3 a1 if , a1 ifm m1 2
s4 if , ifm j m j2 1 2 2 r4 a2 if , a2 ifm m1 2

to , or she might ignore the message and always choose the same actm 2
(see Table 2).

The state-act coordination problem underlying our simple signaling
game induces the players’ payoffs. For , these payoffs are illustrateda p 1
in Table 3. (I shall assume that throughout the paper since thisa p 1
just amounts to a choice of scale.) Notice that we have assumed that each
state occurs with probability . Each entry represents the signaler’s as1/2
well as the receiver’s payoff. Thus our simple signaling game is a pure
coordination game (Lewis 1969).

Simple signaling games based on state-act coordination problems can
easily be generalized to signaling problems involving more then two states,
acts, and messages. Let be an n-state-act coordinationP p AS, A, u*Sn
problem if is a set of n distinct states of the world,S p {j , . . . , j }1 n

is a set of n distinct acts, and is a function thatA p {a , . . . , a } u*1 n
determines the utility of each state-act pair such that (whereu*(j , a ) p di j ij
the Kronecker symbol if and otherwise). In addition,d p 1 i p j d p 0ij ij
let be a set of n distinct messages and be a prob-M p {m , . . . , m } !1 n
ability distribution over S such that the probability of each state is positive,

for .!(j ) 1 0 i p 1, . . . , ni

Definition 1 (simple signaling game). Let be an n-state-act coor-P n
dination problem, let M be a set of n distinct messages, and let be!
a probability distribution over S such that for!(j ) 1 0 i p 1,i

. A simple signaling game based on is a triplet. . . , n S Pn n
, whereAI, {S } , {u } Si i!I i i!I

1. is a set of two players, the sender, 1, and the receiver,I p {1, 2}
2;

2. , , is the set of strategies generated from as follows:S i p 1, 2 Pi n
is the set of sender strategies andS p {s Fs : S r M} S p1 k k 2

is the set of receiver strategies; and{a Fa : M r A}l l
3. the players’ utility functions are the same and are generated by

as follows: andP u : S # R r "n
n

u(s , r ) p !(j ) 7 u*(j , (r s )(j )),! !k l j j l k j
jp1

where denotes the operation of function composition.!



EVOLUTION AND THE EXPLANATION OF MEANING 5

TABLE 3. PAYOFFS IN A SIMPLE SIGNALING GAME.

r1 r2 r3 r4

s1 1 0 1/2 1/2
s2 0 1 1/2 1/2
s3 1/2 1/2 1/2 1/2
s4 1/2 1/2 1/2 1/2

The third condition states that the payoffs to each player are generated
by the underlying payoff function of the state-act coordination problem
by averaging each player’s payoff in each of the states according toji

’s probability of occurrence (the second argument of determines thej u*i
act chosen by the receiver as a response to the signaler’s message). Thus,
the payoff from a particular strategy combination is the expected value
of the payoffs associated with the state-act pairs that result from this
strategy combination relative to the probability distribution .!

Signaling systems are combinations of sender strategies and receiver
strategies that deserve special attention (Lewis 1969). They guarantee that
both players get the maximum payoff regardless of which state of the
world occurs. If the players employ a signaling system, they are fully
coordinated by virtue of the signals. In the example above (see Table 3),
there are two signaling systems, and .(s , r ) (s , r )1 1 2 2

Definition 2 (signaling system). Let be a simple signaling game.Sn
Then is a signaling system if .(s , r ) u(s , r ) p 1k l k l

Equivalently, we may call a signaling system if and only if(s , r ) u*(j ,k l j
for all .(r s )(j )) p 1 j p 1, . . . , n!l k j

Call (or ) a part of a signaling system if and only if there is ans r ri j k
(or ) such that (or ) is a signaling system. Then it is obviouss (s , r ) (s , r )l i k l j
that the following holds:

Proposition 3. Let be a simple signaling game. Then is part ofS sn i
a signaling system if and only if is one-to-one and is part of as ri j
signaling system if and only if is one-to-one.rj

According to Lewis (1969), Skyrms (1996), Vanderschraaf (1998), and
Young (1998), a behavioral regularity is conventional if everybody has
an interest to act in accordance with it and if it has an alternative (i.e.,
it involves some kind of arbitrariness). This intuition can be quite naturally
captured in game-theoretic terms. A Nash equilibrium is a combination
of strategies in which no player would gain by unilaterally deviating from
her part of the equilibrium. A strict Nash equilibrium is a Nash equilib-
rium in which each player would do worse by unilaterally choosing a
strategy different from her equilibrium strategy. In a pure coordination
game with at least two strict Nash equilibria, each of the two strict Nash



6 SIMON M. HUTTEGGER

equilibria is a candidate for a convention. In a simple signaling game, for
instance, each player does strictly better in a signaling system equilibrium
and no player has an interest that the other player deviates from it since
she would also be worse off in this case.1

In the example above (see Table 3) there are two strict Nash equilibria,
and , and four pure nonstrict Nash equilibria, ,(s , r ) (s , r ) (s , r )1 1 2 2 3 3

, , and . The two strict Nash equilibria are also the(s , r ) (s , r ) (s , r )3 4 4 3 4 4
two signaling systems and, hence, the two possible signaling conventions
of this game. This result holds with some generality (as has already been
pointed out in Skyrms [1996, 83]).

Proposition 4. Let be a simple signaling game. Then is aS (s , r )n i j
signaling system if and only if is a strict Nash equilibrium.(s , r )i j

Proof. If is a signaling system, then it is clear that unilateral(s , r )i j
deviation leads to a worse payoff. Conversely, if is not a sig-(s , r )i j
naling system, then there always exists at least one sender strategy
or one receiver strategy that yields no lower payoff (just rearrange
the mappings). The details are left to the reader. "

Thus, if conventions in signaling games are strict Nash equilibria, the
only candidates for conventions in simple signaling games are signaling
systems. In a population of individuals who are repeatedly playing a
simple signaling game, a signaling system is a simple conventional lan-
guage. The population could have adopted another signaling system that
would have done essentially the same job. But whatever signaling system
they have, they understand each other by virtue of a convention.

3. Stability and Emergence of Language Conventions. At this point two
questions become pressing: How is a conventional language maintained
in a population? And how might a conventional language be established
in the first place? These two questions concern conventions in general. It
is not enough to state what the candidates for conventions in a particular
game are if we want to explain why one of the possible conventions is in
fact a convention in a population. To do this, we have to explain how
this convention has emerged and why it is stable.

These questions become particularly pressing if the candidates for con-
ventions are entirely symmetric as they are in a simple signaling game.
There is no reason to choose one of the possible signaling conventions

1. For more on the definition of conventions in pure and impure coordination games,
see Vanderschraaf (1998). Adopting an evolutionary perspective would require us to
emphasize the historical process leading to a convention in its definition (see Harms
2004). This is, in some sense, achieved by the dynamical analysis below.



EVOLUTION AND THE EXPLANATION OF MEANING 7

rather than another since any signaling system does the same job. But if
there is no reason to choose one of the language conventions rather than
another, how will individuals decide on one of them without communi-
cation? And, once the decision is made, why do they not switch? This
scenario is one instance of what Skyrms (1996) calls the “curse of
symmetry.”

Lewis (1969) proposes an answer to these questions in terms of rational
choice theory and in terms of salience. According to Lewis, the stability
of conventions is guaranteed because the structure of the game, the pay-
offs, and the rationality of the players are common knowledge. A fact F
is common knowledge among agents A and if A knows that F, and A!A
knows that knows that F, and A knows that knows that A knows! !A A
that F, and so on. The same holds if we interchange A and . If sender!A
and receiver have common knowledge of the game structure and each
other’s rationality, they will stick to a signaling convention because there
is no room for doubting that the other player is going to stick to her part
of the signaling convention.

Following Schelling (1960), Lewis’s answer to the question of how
conventional languages are established in the first place is that one of the
strict Nash equilibria is a focal point or is salient. It stands out among
all equilibria in some psychologically significant respect. That is to say,
it is clear to all players that this one equilibrium should be chosen. Here
is Lewis’s (1969, 158–159) example for a salient signaling system: Suppose
that you come upon a patch of quicksand and you want to warn others
who might come there after you. A salient signal would be to put a
scarecrow up to the chest into the quicksand.

As Skyrms (1996) points out, both of Lewis’s answers face serious
problems. Concerning the first answer, Skyrms asks where all the common
knowledge comes from. It seems to be unclear to what extent players
must already understand each other to have this very demanding kind of
knowledge. It is possible that we have to assume preexisting communi-
cation or something equivalent to it in order to explain that the players
understand each other. This would put us back in the position where we
need a language to explain the emergence of another language. Hence,
without explaining where common knowledge comes from, Lewis’s ac-
count of why conventions are stable appears to be incomplete, to say the
least.

Concerning Lewis’s second answer, Skyrms asks where the salient equi-
librium is. Sometimes a salient equilibrium might be available, but some-
times not. The harder case is the latter one. For a fundamental investi-
gation a solution to the less hard cases is not enough. (This criticism is
a methodological one, but it does not mean that explanations in terms
of salience might not be useful in other contexts.) One of the reasons why



8 SIMON M. HUTTEGGER

we set up simple signaling games as involving symmetric strict Nash equi-
libria was that no one signaling system should stand out. So, to put
Skyrms’s second criticism in a slightly different way, if there would be a
signaling system that is salient to the players, then this should already be
expressed in the structure of the game.

Allow me to add a further criticism of salience as an equilibrium se-
lection device. As in the case of common knowledge, it seems unclear to
what extent the players must already know about each other in order to
view an equilibrium as salient. In terms of signaling systems this means
that I must already see the signals as somewhat meaningful or represen-
tational within my community. This is, however, the fact to be explained.
For instance, in the scarecrow example, one must understand a scarecrow
as a representation of ‘me’ (or ‘potentially me’) and the situation as rep-
resenting ‘me potentially sinking’.

Cubitt and Sugden (2003) criticize Skyrms’s (1996) and Vanderschraaf’s
(1998) reconstruction of Lewis’s account of convention. First, they show
that Lewis’s conception of common knowledge does not coincide with
the standard infinitely iterated knowledge conception (as described above).
Moreover, they try to reestablish salience as an explanation of how con-
ventions start.

Instead of assuming common knowledge, Cubitt and Sugden argue that
Lewis tries to derive common knowledge from the agents’ background
information and their inductive standards. As they show more formally,
“common knowledge in Lewis’ sense is possible only when individuals
have reason to believe that, in particular relevant respects, they have
common background information and common inductive standards”
(Cubitt and Sugden 2003, 185). Given this explanation of common knowl-
edge, we might again ask where the reasons to believe to have common
background information and common inductive standards come from,
and to what extent this presupposes a mutual understanding that must
itself be explained.2

As to the role of salience for the selection of strict Nash equilibria,
Cubitt and Sugden do not address the criticisms raised above, namely,
that assuming salience excludes the worst case scenario for the emergence
of conventions and that the prior extent of mutual understanding for
salience to be possible is unclear. Thus, it seems that Lewis’s account
of the emergence and stability of language conventions is deficient in
two important respects. This is the point where evolutionary theory
comes in.

2. It seems to be almost inconceivable to think of a solution concept for one-shot
games that does not presuppose some kind of common understanding of the situation.



EVOLUTION AND THE EXPLANATION OF MEANING 9

4. Evolution and Conventions of Language. Skyrms (1996) invites us to
approach the problem of the emergence and stability of language con-
ventions from an evolutionary standpoint. If an adaptive process governs
a population of individuals that are repeatedly confronted with a situation
that is similar enough to a signaling game, will one of the signaling systems
eventually become established in this population? Biological evidence sug-
gests that the answer to this question is, at least a cautious, yes. Signal
coordination is everywhere, from cells communicating via molecules and
the honeybee’s dance to predator alarm calls of monkeys and bird song
(for comprehensive treatments of animal signals, see Snowdon [1990],
Hauser [1997], and Maynard Smith and Harper [2003]).

The adoption of the evolutionary viewpoint implies that we do not
follow Lewis (1969, 58) and Vanderschraaf (1998) in invoking any com-
mon knowledge assumptions to define conventions. Instead, we shall say
that a population adopts a certain convention if it is a strict Nash equi-
librium in a game with at least two strict Nash equilibria and if every
individual in the population chooses her actions according to this strict
Nash equilibrium. By adopting the evolutionary viewpoint, we also escape
the criticisms raised in the previous section. We assume neither that the
individuals in the population reach a convention by explicit agreement,
nor that they have a preexisting language or common knowledge of the
game. Indeed, they may not have much knowledge at all.

The omnipresence of signaling in nature suggests that biological and
cultural evolution are likely to be responsible for these phenomena. To
get beyond such informal statements, we have to give the problem a clear
formulation. Studying the evolutionary dynamics of signaling games
seems to be a promising starting point for a first analysis. Skyrms (1996)
investigates a simple signaling game with two signals by simulating its
evolutionary dynamics. In addition, he provides some analytical results
in Skyrms (2000) for a simplified model of this game. In this model, a
signaling system is a global attractor for a population that consists of
three types, the signaling system type and two antisignaling types. On the
basis of these results one might conjecture that populations will develop
a signaling system under evolutionary dynamics for all simple signaling
games regardless of the initial state. The results presented in the following
section show that this is in general not true for the replicator equations,
which are the standard dynamics in formal models of selection.

5. Replicator Dynamics and Signaling Games. If we want to study one
population of individuals who can be senders or receivers, we have to
consider the symmetrized, or role-conditioned, version of (see, e.g.,Sn
Cressman 2003). Denote the role-conditioned version of by . GoingrS Sn n
from to amounts to assuming that each individual is sender orrS Sn n



10 SIMON M. HUTTEGGER

receiver with probability . This is a reasonable assumption as long as1/2
we do not impose more structure on the population. If some individuals
are in the role of the sender more often than others, then we must say
which individuals tend to be senders and which individuals tend to be
receivers. For a first analysis of the dynamics of signaling games, imposing
such a structure on the population does not seem to be reasonable.

An individual’s strategy consists of a sender part and a receiver partsi
. If a simple signaling game is given, then the strategies of are all pairsrr Sj n

of strategies of . (This implies that a player’s sender strategy need notSn
be consistent with her receiver strategy. She might not be able to “talk
to herself.”) Since has sender strategies and receiver strategies,n nS n nn

has strategies. For instance, in the example illustrated in Table 3r 2nS nn
there are 16 strategies. For , there are already 729 strategies. Then p 3
payoff to a strategy against is given by(s , r ) (s , r )i j k l

1
p((s , r ), (s , r )) p (u(s , r ) ! u(s , r )).i j k l i l k j2

This is just the expected payoff to a strategy relative to the uniform
distribution over the two roles of the game. This specifies completely.rSn
There are two players. Both can be sender or receiver. Their strategies
and payoffs are determined by . Unlike , is a symmetric game.rS S Sn n n
This means that the player positions are not distinguishable.

Suppose that a population consists of types of individuals in which
each type corresponds to a strategy of . Let be the expectedr !S p(s, s )n
payoff individuals in state s get when meeting individuals of type . Then!s
s is said to be evolutionarily stable if and only if for all!p(s, s) 1 p(s, s )

; or if for some , then! ! ! ! ! !s ( s p(s, s) p p(s, s ) s ( s p(s, s ) 1 p(s , s )
(Maynard Smith and Price 1973; Maynard Smith 1982). These conditions
guarantee that a large population will not move away from state s once
it has reached it. In other words, if a small proportion of individuals are
not of type s in a population consisting almost entirely of s, then selection
will carry the population back to a state in which only s is present.

By using a result of Selten (1980), Wärneryd (1993) shows that in
signaling games, signaling systems and evolutionarily stable states
coincide.

Proposition 5. Let be a simple signaling game. Then isS (s , r )n i j
evolutionarily stable if and only if is a signaling system.(s , r )i j

Notice that Proposition 5 is an answer to the first question raised in the
previous section: Why are language conventions stable? They are stable
in a model like that underlying the concept of evolutionary stability be-
cause they are evolutionarily stable. In addition, they are the only evo-
lutionarily stable states.



EVOLUTION AND THE EXPLANATION OF MEANING 11

But why should signaling systems emerge in the first place? To answer
this question, let us look at the replicator dynamics. The replicator dy-
namics were introduced in Taylor and Jonker (1978) to highlight the
dynamical considerations underlying the evolutionary stability concept
(Weibull [1995] and Hofbauer and Sigmund [1998] are comprehensive
treatments of the replicator dynamics).

Let the state of the population be represented by the relative frequencies
of the different types. For there are different types. Thusr 2nS f(n) p nn
the state of the population is represented as a vector x p (x , . . . ,1

, where is the simplex of :f(n) f(n) f(n)x ) ! D D "f(n)

f(n) f(n)D p x ! " : x p 1, 0 " x " 1, i p 1, . . . , f(n) .! i i{ }
i

The interior of is the part of where all types have positive fre-f(n) f(n)D D
quency. It is given by

f(n) f(n)int(D ) p x ! " : x p 1, 0 ! x ! 1, i p 1, . . . , f(n) .! i i{ }
i

The boundary of is the set where at least one type has zero frequency,f(n)D
that is, . We assume that the population is ef-f(n) f(n) f(n)bd(D ) p D \ int(D )
fectively infinite. Thus, the actual payoff of a type matches its expected
payoff. Let be the average payoff of type i when the currentp(x , x)i
population state is , and let be the average payoff of the wholex p(x, x)
population. The replicator dynamics is a system of differential equations
given by

ẋ p x (p(x , x) " p(x, x)) for i p 1, . . . , f(n), (1)i i i

where denotes the time derivative in the ith component. Hence theẋi
frequency of types with above-average payoff increases and the frequency
of types with below-average payoff decreases. Equation (1) guarantees
that evolution will occur in our population whenever there are fitness
differences between types.

The replicator equations (1) were originally introduced to capture bi-
ological phenomena. There are, however, a number of studies that show
how to make sense of the replicator equations in a cultural context (Bin-
more, Gale, and Samuelson 1995; Björnstedt and Weibull 1996; Schlag
1998; Harms 2004). For example, Björnstedt and Weibull show that a
model in which individuals imitate others who have adopted successful
strategies leads to a class of dynamics called “monotonic” by taking ap-
propriate limits. This class of dynamics is characterized by the property
that strategies yielding a higher payoff have a higher growth rate. The



12 SIMON M. HUTTEGGER

replicator dynamics (1) is monotonic in this sense and may be obtained
by choosing specific functional forms for the functions involved in general
monotonic dynamics. The system (1) may describe a learning process
governed by imitation. Thus (1) seems to be a good point of departure
for analyzing signaling interactions in the context of both biological and
cultural evolution.

The rest points of a system of ordinary differential equations are the
states at which vanishes. Rest points are fixed points of the flowdx/dt
corresponding to the differential equations. Basically, there are three dif-
ferent kinds of rest points: asymptotically stable, weakly stable, and un-
stable rest points. An asymptotically stable rest point is characterized by
attracting all nearby states. The set of all points that converge to an
asymptotically stable rest point is its basin of attraction. If a rest point
is weakly or Liapunov stable, then all nearby states stay nearby. (Note
that every asymptotically stable rest point is weakly stable, but not vice
versa.) Nearby solutions of an unstable state tend away from it (not
necessarily in all directions). For details, in particular, facts concerning
the relation between stability and eigenvalues, see Hirsch and Smale
(1974). There is another concept we will need. A subset F of state space
is called an “attractor” if all states sufficiently close to F converge to F.
Thus an asymptotically stable rest point is a singleton attractor.

If y is an asymptotically stable state, then the system will tend back to
y after a small perturbation. The same is true of any attractor F. If y is
Liapunov stable, small perturbations will result in a nearby state. If y is
unstable, then small perturbations will lead away from it. Thus asymp-
totically stable states are meaningful since we can expect the system to
be close to one of them after a sufficiently long time.

The rest of this section is devoted to studying the stability properties
of the rest points of signaling games under the replicator dynamics (1).
As pointed out at the end of the last section, signaling systems will not
almost surely emerge in the general case. I will show this by proving that
the set of points that does not converge to a signaling systemf(n)x ! int(D )
of has positive Lebesgue measure. I will identify the sets of points thatrSn
attract some significant portion of states as being on the boundary. Except
for signaling systems, these sets are not attractors, however. Thus it will
be important to be cautious when interpreting the results. After stating
the relevant theorems, I will try to clarify what they mean for evolutionary
explanations of signaling systems. All proofs may be found in the
Appendix.

The proof of the first part of Theorem 6 relies on the fact that all rest
points in for the evolutionary dynamics of signaling games aref(n)int(D )
(linearly) unstable. Notice that Theorem 6 holds for the replicator dy-
namics of any symmetrized simple signaling game.



EVOLUTION AND THE EXPLANATION OF MEANING 13

Theorem 6. Let (1) be the replicator dynamics for .rSn
1. Denote the set of points in that do not converge tof(n)int(D )

by S. Then S has Lebesgue measure zero.f(n)bd(D )
2. A state is asymptotically stable if and only if is af(n)p* ! D p*

signaling system.

Hence, generically interior rest points will not be observed under the
replicator dynamics. This situation is analogous to unstable states in sta-
tistical mechanics. For instance, the state in which all molecules in a gas
are at rest is an unstable equilibrium since any slight perturbation will
lead to a state that ultimately converges to mixing. If a population is at
an interior rest point for the replicator dynamics of , then slight per-rSn
turbations will carry it to some boundary state.

For a special kind of signaling game with two signals we can in fact
prove more:

Theorem 7. Let . Then the set of points that do not!(j ) p !(j )1 2
converge to a signaling system of for the replicator dynamics (1)rS 2
has Lebesgue measure zero.

Thus, almost all solutions will converge to a signaling system under the
assumptions of Theorem 7. The next theorem shows that the assumption

is necessary for obtaining this result. The intuition for this!(j ) p !(j )1 2
is quite simple. If , then the type in which the!(j ) p p 1 1/2 z p (s, r)1
sender strategy s maps both states on and the receiver strategy r mapsm1
both messages on gets a payoff of p when meeting itself. There area1
three other types that get p when meeting each other or when meeting z.
These types have r as receiver strategy and an arbitrary sender strategy.
The payoff of all other types against z is at most p. Although z is unstable
(it can be invaded by one of the signaling systems), a mixture of z and a
small amount of the other three types with r as receiver strategy will result
in a Liapunov stable state . The reason for this is that for any type w!z
that is not present at there is at least one type v that is present at! !z z
such that . (Details can be found in the Appendix.)p(w, v) ! p

Theorem 8. Let . Then there exist boundary faces ,m!(j ) ( !(j ) D1 2
, that contain sets of points R such that R is open in andmm ! 16 D

the set of points in converging to R for the replicator dynamics16D
(1) of has positive Lebesgue measure.rS 2

This result reflects the fact that the value of p characterizes the importance
of communication for a population playing . If , then bothS p p 1/22
states have equal weight. Thus coordination of both state-act pairs is
equally important to obtain a sufficiently high payoff. This no longer
holds for . In this case, coordination of the state-act pair wherep ( 1/2



14 SIMON M. HUTTEGGER

the state has higher probability is more important than coordination of
the other state-act pair. As (or, alternatively, as ), communi-p r 1 p r 0
cation becomes less important. If p is very close to 1, numerical simulations
suggest that the measure of the set of points converging to states that are
not signaling systems is nearly as large as the basins of attraction for the
signaling systems.

If , nonconvergence to signaling systems of does not dependn 1 2 Sn
on as in the case of . Moreover, for , not only subsets of boundary! S n 1 22
faces are attracting a significant number of initial states. Now all of the
interior of a boundary face may consist of Liapunov stable states.

Theorem 9. Let be a simple signaling game with . Then thererS n # 3n
exist boundary faces , , such that the set of points inmD m ! f(n)

converging to for (1) has positive measure.f(n) mD int(D )

Let us look at one example of such a boundary face. Consider the two
types and of , where and map and onrz p (s , r) z p (s , r) S s s j j1 1 2 2 3 1 2 1 2

, maps on , and maps on . r maps on and asm s j m s j m m a m1 1 3 2 2 3 3 1 2 2
well as on . Then with consists entirely ofm a ez ! (1 " e)z 0 ! e ! 13 3 1 2
Liapunov stable rest points, and the average payoff on this one-
dimensional simplex is . The points on this one-dimen-!(j ) ! !(j ) p r1 2
sional simplex are Liapunov stable because for any type ,w ( z , z1 2

and, in addition, for all w with , impliesp(w, z ) " r p(w, w) 1 r p(w, z ) p ri i
for and . Thus no mixture of strategies differentp(w, z ) ! r i ( j i, j p 1, 2j

from and can destabilize this part of the boundary as long asz z 0 !1 2
.e ! 1

We have to be cautious in interpreting the results stated in the previous
theorems. At first sight they might appear to subvert the thesis that evo-
lution facilitates the emergence of simple communication systems. States
of communication are represented by signaling systems. These states do
not almost always emerge, however, since a significant portion of initial
conditions in converges to states that are not signaling systems. Tak-f(n)D
ing a closer look at those suboptimal states reveals that they are not
attractors. To see this, suppose that , , is a set of restmO O D m ! f(n)
points open in such that the set of interior initial conditions convergingmD
to O has positive Lebesgue measure. Then is Liapunov stable.p* ! O
Each trajectory starting close to converges to it, stays close to it, orp*
converges to a nearby rest point in O. A basic proposition in evolutionary
game theory states that a point is a Nash equilibrium if it is Liapunov
stable (Hofbauer and Sigmund 1998). Hence is a Nash equilibrium.p*
Let e and be pure strategies that have positive probability at . We!e p*
may suppose that since O is open in . For the same reason,m mp* ! int(D ) D



EVOLUTION AND THE EXPLANATION OF MEANING 15

there exists such thate ! (0, 1)
!p(e, ee ! (1 " e)p*) p p(e , ee ! (1 " e)p*).

This implies that is a rest point for all . Hence,ee ! (1 " e)p* 0 ! e ! 1
consists entirely of rest points. Suppose that a trajectory converges tomD

some point . Once the state of the population corresponds to ,p* ! O p*
there is no selection pressure since each point in is a rest point of themD
replicator dynamics. Once at , the population may visit any state inp*

because of neutral drift. Thus a chain of small perturbations maymD
eventually carry the population to one of the m pure states on the bound-
ary of . A pure state that is not a signaling system is always dynamicallymD
unstable, however (see Wärneryd 1993). To be more specific, a pure state
that is not a signaling system can always be destabilized by a signaling
system. Hence, there is no neighborhood U of O such that all points in

converge to some point in O. This is equivalent to saying thatf(n)U ! D
O is not an attractor. Thus states in O are not robust relative to drift.

In a recent paper that states a number of important complementary
results to the theorems above, Pawlowitsch (2006) proposes another in-
terpretation of the fact that the replicator dynamics can generically con-
verge to suboptimal states. Pawlowitsch is able to show that Liapunov
stable states are states of partial communication and not states with no
communication at all. In particular, she shows that in the limit there can
be some cases of homonymy or synonymy in the population. For simple
signaling games this means that more than one state of the world may
be linked to one signal and more than one signal may be linked to one
action. The results on the evolutionary dynamics of signaling games may
thus be seen as counterparts of inconsistencies in natural languages. Com-
munication is not perfect in such states. But there is communication to
some degree.

We might conjecture that putting mutation to the replicator equations
(1) will change our results significantly. (See Hofbauer and Sigmund [1998]
for more on mutation-selection dynamics.) The reason for this is that
mutation often helps a population out of a local fitness maximum (which
gives less payoff than a global fitness maximum like, e.g., signaling sys-
tems). Numerical simulations suggest that this is true for simple signaling
games. An analytical treatment of signaling games under a mutation-
selection regime is left to future research.

6. Generalized Simple Signaling Games. Signaling games may be gener-
alized in various ways. One might, for instance, object that a more realistic
model has to include probabilistic strategies. By this I mean that a sender
chooses each message with some probability after the occurrence of a state
and a receiver chooses an act with some probability after getting a mes-



16 SIMON M. HUTTEGGER

sage. Variants of this generalization have previously been studied by Oli-
phant and Batali (1997), Nowak and Krakauer (1999), and Komarova
and Niyogi (2004).

Since not much is known about the replicator dynamics with infinite
strategy spaces, I will restrict considerations to finitely many probabilistic
strategies. Note that this is a crucial assumption for the subsequent anal-
ysis. It does not seem to be crucial for the result, however. A natural
extension of the construction of the strategy space for generalized simple
signaling games that allows a continuum of strategies would be desirable.

The sender strategies are now given by a probability distribution over
the set of messages M for each state in S. Similarly, a receiver strategy
is given by a probability distribution over the set of acts A for each message
in M. If individuals are allowed such probabilistic strategies, then a strat-
egy of a generalized simple signaling game is a mixed strategy of ; thatrSn
is, it corresponds to a point . Suppose thatf(n) f(n)x ! D x , . . . , x ! D1 m
are the mixed types under consideration. The payoffs of against arex xi j
given by , where A is the payoff matrix defined for . Sincerb p x 7 Ax Sij i j n
A is symmetric, . Hence, the payoff matrix for the generalizedb p bij ji
simple signaling game B is symmetric. This, in turn, implies that the results
presented in the previous section basically carry over to generalized simple
signaling games. (The symmetry of the payoff matrix B allows the same
analysis of the dynamics of generalized simple signaling games as the one
given in the Appendix for simple signaling games.) Suppose that hasS 2
two equiprobable states. If the two signaling systems of are presentrS 2
among the probabilistic strategies and if m is finite, then16x , . . . , x ! D1 m
the set of initial states that do not converge to one of the signaling systems
has Lebesgue measure zero. Suppose that has two states with unequalS 2
probabilities of occurrence. Then a significant number of states do not
converge to one of the signaling systems. If , then this result isn # 3
independent of the probability distribution over the set of states. For all
generalized simple signaling games, almost every initial state converges
to the boundary and signaling systems are the only attractors.

7. Conclusion. To what extent have studies like this one or Skyrms (1996)
been able to answer Quine’s skeptical doubts concerning conventional
meaning? There are several points to be noted.

In the first place, I chose to explain the spontaneous emergence of
meaning with the help of evolutionary theory. To do this in a specific way,
I worked with a particular model, the replicator dynamics. For signaling
games, the explanatory value of signaling system equilibria depends on
the stability properties of the corresponding rest points. I have shown
that signaling systems are the most robust states for the replicator dy-
namics. All the other states are not robust under selection or relative to



EVOLUTION AND THE EXPLANATION OF MEANING 17

neutral drift. This should not cover up the fact that there exist sets of
states that attract a significant part of state space. These sets consist of
states in which there is some, but not perfect, communication among
individuals. Thus it may be quite hard for a population to get to a state
of (almost) perfect communication under selection dynamics. There is
reason to suppose, however, that states of partial communication will be
unstable once mutation is explicitly introduced into the dynamics. States
of partial communication can also be seen as corresponding to persistent
imperfections of natural languages.

One way to improve the significance of these results is to study whether
they continue to hold in more general models. I have taken one step in
this direction by showing that the stability results still hold for a much
bigger strategy space with randomized strategies. One might object that
for addressing the question of how meaning emerges, an even bigger
strategy space has to be considered. More specifically, strategies in which
agents do not use any signal and never react to receiving a signal should
be included. This might be considered as a more realistic starting point
for an initial population state. In one sense, simple and generalized simple
signaling games include such a state. As long as signals are costless, being
silent can itself be regarded as a signal and thus be included in the set of
signals. Similarly, the act of just carrying on doing what one is doing at
the time of receiving a message can be a member of the set of acts. After
this particular signal and this particular act are specified, the strategies
of never using a signal and never reacting to signals are part of the strategy
space. But such a state is not stable in our models. This result might not
hold in more complex models, however. Signals may be costly, for in-
stance, because of the capacities an individual must possess in order to
be able to produce signals and to learn how to produce them. In this case,
sending no signal at all might be advantageous if many other individuals
never react to signals. The outcome will depend on the cost of being able
to send signals and on the benefits from communication.

Investigating the robustness of results by studying different models is
another important issue. Skyrms (2000) employs considerations of struc-
tural stability, where small perturbations in the differential equations are
studied, and by looking at a bigger class of dynamics, qualitatively adap-
tive dynamics, which includes the replicator dynamics. His results suggest
that we might be able to extend our general analysis in this direction.
Furthermore, there exist some simulation studies on signaling games
played in a spatial environment (Grim et al. 2001; Zollman 2005). The
general result in this direction is that signaling systems are likely to emerge.

Another point should be emphasized. Signaling games can solve only
part of the problem posed by Quine. After all, not only did Quine (1936,
1953) question the possibility of a noncircular account of the explanation



18 SIMON M. HUTTEGGER

of meaning. More generally, he questioned a conventionalist account of
logic in which logical truth and logical inference are based on conventional
meaning. These considerations lead to more sophisticated signaling games,
some of which where proposed in Skyrms (2004). A satisfactory conven-
tionalist approach to language cannot get around taking steps along these
lines.

Finally, one might object that the meaning of signals in simple signaling
games remains unclear. If an animal gives an alarm call, does it mean
that a predator is present? Or does it mean that the proper response to
the presence of the predator should be performed? That is to say, a Qui-
nean skeptic might still question that we have explained conventional
meaning if our explanation is not based on a model of signaling in which
the meaning of signals is sufficiently close to what we commonly under-
stand by meaning in human languages. An answer to this criticism is
proposed in Huttegger (2007).

Appendix: Proofs

Let me first introduce some definitions and some notational conventions.
A gradient system on an open set is a system of differentialnU O "
equations of the form

dx
p $V(x).

dt

is assumed to be a function with continuous second-order par-V : U r "
tial derivatives and is called the “potential” of the gradient system. If the
gradient $V is defined with respect to the standard inner product for

, thenn"

!V !V
$V p , . . . , .( )!x !x1 n

If is equipped with an arbitrary inner product, the gradient $V cann"
straightforwardly be defined by considering the dual vector space of n"
(i.e., the space of all linear maps from to ; see Hirsch and Smalen" "
[1974] for details). The gradient systems for the replicator dynamics are
“Shashshahani gradients.” This means that they are defined with respect
to the following inner product:

n
1

Ay, hS p y h .!x i ixip1 i
Here, and y, , where is the tan-n n n nx ! D h ! " " p {y ! " : ! y p 0}0 0 ii



EVOLUTION AND THE EXPLANATION OF MEANING 19

gent space for every point in . A symmetric game G with payoff matrixnD
A is a “partnership game” if (where denotes the transpose ofT TA p A A
A). If are strategies of a game G, thens , . . . , s span(s , . . . , s )k k!m k k!m
denotes the set of all convex combinations of those strategies.

Before I prove the theorems stated in the main text, I will first prove
two lemmata that will be used frequently below.

Lemma 10. Let G be a partnership game with payoff matrixn # n
A. Then:
1. is evolutionarily stable if and only if is asymp-n nx ! S x ! S

totically stable under the replicator dynamics (1) generated by A.
2. The replicator dynamics (1) for G is a Shashshahani gradient

system with potential function .V(x) p (1/2)x 7 Ax

Proof. See Hofbauer and Sigmund (1998). "

The second part of Lemma 10 implies that all solutions converge to a
rest point (Akin and Hofbauer 1982). Moreover, it implies that there are
no circling solutions since the average payoff V is strictly increasing along
all nonstationary solutions.

Lemma 11. Let be a simple signaling game. Then the role-Sn
conditioned game based on where each player finds herself in theSn
sender position as well as in the receiver position with probability

is a partnership game.1/2

Proof. We have to show that the payoff matrix of the role-conditioned
game based on is symmetric. Consider two individuals of typeSn

and . Since we have supposed that any indi-!z p (s , r ) z p (s , r )i j l k
vidual finds herself in each of the positions with probability ,1/2

1!p(z, z ) p [u(s , r ) ! u(s , r )].i j l k2

It is easy to see that the payoff to must be the same. "!z

Proof of Theorem 6

The second part of Theorem 6 follows directly from Proposition 5, the
first part of Lemma 10, and Lemma 11. The first part of Theorem 6
follows from the three lemmata stated below together with the second
part of Lemma 10.

Lemma 12. Let be a simple signaling game. If is ar f(n)S p* ! int(D )n
rest point of (1), then is linearly unstable.p*



20 SIMON M. HUTTEGGER

Proof. Let be an interior rest point of (1). Let , wherep* p p p* ! yi i i
. Then the linearized vector field aroundf(n)y p (y , . . . , y ) ! "1 f(n) 0

is given byp*

ẏ p L y!i ij j
j

(see, e.g., Cressman 2003). is the jth partialL p p*(a " p* 7 Ae )ij i ij j
derivative of the ith equation of (1), and A is the symmetric payoff
matrix of with entries . The components constitute the Ja-rS a Ln ij ij
cobian matrix L evaluated at . Since is a partnership game byrp* Sn
Lemma 11, L is a self-adjoined linear operator relative to the Shash-
shahani inner product: (see Hofbauer and Sig-Ay, LhS p ALy, hSp* p*
mund 1998, 259). By a basic result in linear algebra, a self-adjoined
linear operator is similar to a symmetric matrix in an orthonormal
basis. This implies that L has at least one positive eigenvalue if and
only if L is not negative semidefinite. L is not negative semidefinite
if there exists some such that . Setf(n)y ! " Ay, LyS 1 0 y p p "0 p*

, wherep*

p p (1 " e)p* ! es, 0 ! e ! 1.

Then
f(n) f(n) f(n)

Ay, LyS p y a y " y p* 7 Ae y! ! !p* i ij j i j j
ip1 ip1 jp1

f(n)

p y a y p (p " p*) 7 A(p " p*) p p(p, p) " p(p*, p*).! i ij j
ip1

Since is an interior equilibrium,p*

p(e , p*) p p(e , p*) p p(p*, p*)i j

for all pure types , . At p the payoff to s ise ei j

p(s, p) p ep(s, s) ! (1 " e)p(s, p*)

and the average payoff is

p(p, p) p ep(p, s) ! (1 " e)p(p, p*) 1 p(p*, p*)

since . Thus L is not negative semidefinite. "p(s, s) 1 p(p*, p*)

A set of points S in is called path-connected if for all points x andn"
y in S there exists a continuous path f connecting x and y and lying
entirely in S.

Lemma 13. Let be a symmetrized simple signaling game and letrSn



EVOLUTION AND THE EXPLANATION OF MEANING 21

U be a connected set of rest points in . Then there exists atf(n)int(D )
most one such set U.

Proof. If are rest points, thenf(n) !p*, q* ! int(D ) p(e, p*) p p(e ,
and for all pure strategies (since all! !p*) p(e, q*) p p(e , q*) e, e

strategies are present at an interior rest point). Then ep* ! (1 "
is also a rest point for sincee)q* 0 " e " 1 p(e, ep* ! (1 " e)q*) p

by multilinearity of p. This shows that U is a!p(e , ep* ! (1 " e)q)
linear manifold. "

Lemma 13 shows that the set of interior rest points is indeed connected
since the line connecting two arbitrary interior rest points consists entirely
of interior rest points. The proof of the next lemma is based on results
from center manifold theory. Center manifold theory asserts that, at each
rest point, there exist invariant manifolds tangent to the (generalized)
eigenspaces spanned by eigenvectors corresponding to eigenvalues with
positive, negative, and zero real part. The center-stable manifold at some
rest point is the invariant manifold tangent to the union of the eigenspaces
given by eigenvalues with nonpositive real part. The center-stable manifold
thus contains all solutions converging to the rest point or staying suffi-
ciently close to it.

Lemma 14. Let U be a path-connected set of interior rest points for
the replicator dynamics of and let S be the set of points thatrSn
converge to U. Then has Lebesgue measure zero.U " S

Proof. From Lemma 12 we know that every is linearly un-p* ! U
stable. This together with the center-stable manifold theorem (see
Kelley 1967) implies that is contained in the center-stable! !U " S
manifold at , where is a set of rest points sufficiently close!M p* Up*
to and is the set of points that converge to a rest point close!p* S
to and are themselves sufficiently close to (from the secondp* p*
part of Lemma 10 we know that every solution converges to some
rest point). Locally a center-stable manifold exists for each in-Mp*
terior rest point . Since each is linearly unstable, the center-p* p*
stable manifold theorem implies that has Lebesgue measure zero.Mp*

is the union of all local center-stable manifolds. Because it isU " S
a subset of , Lindelöf’s theorem implies that any open coveringf(n)"
of has a countable subcovering. Since each center-stable man-U " S
ifold has Lebesgue measure zero and any countable union of measure
zero sets has again measure zero, this implies that has LebesgueU " S
measure zero. "



22 SIMON M. HUTTEGGER

Proof of Theorem 7

From Theorem 6 we know that solutions starting from almost every initial
condition converge to the boundary and that the two signaling systems
of are the only asymptotically stable states of the replicator dynamicsS 2
for . To analyze the stability properties of boundary rest points, werS 2
look at the following linear manifolds:

• and is not one-to-one}).D p span({z : z p (s , r ) r1 i j j
• and is not one-to-one}).D p span({z : z p (s , r ) s2 i k i
• Let be the union of all boundary faces spanned by strategiesD 3

z where at least one sender part is one-to-one, at least one receiver
part is one-to-one, and at least one type is not present.

Since , the average payoff is on and . Hence!(j ) p !(j ) 1/2 D D D1 2 1 2 1
and consist entirely of rest points. Suppose . Let andD p* ! D z p (s, r)2 1

be the signaling systems of . Let and are! ! ! rz p (s , r ) S S p {(s , r ) : s r2 1 i j i j
not one-to-one}. Similarly, let and is not one-to-S p {(s , r ) : s p s r2 i j i j
one} and let and is not one-to-one}. Then!S p {(s , r ) : s p s r D p3 i j i j 1

. If , then . If , thenspan(S " S " S ) z ! S p(z, z ) p 1/2 z ! S p(z,1 2 3 i 1 i i 2
. And if , then . Define ,z ) p 3/4 z ! S p(z, z ) p 1/4 a p ! p* b pi i 3 i ii!S1

, and . Let L denote the Jacobian evaluated at .! p* g p ! p* p*i ii!S i!S2 3
The entries of the Jacobian for all types are of the forme " supp(p*)i

(see, e.g., Cressman 2003, 51). This implies thatd (p(e , p*) " p(p*, p*))ij i
the eigenvalue corresponding to z is given by

p(z, p*) " p(p*, p*).

If , then is linearly unstable. If ,p(z, p*) " 1/2 1 0 p* p(s, p*) " 1/2 p 0
then a simple computation of when perturbed toward z showsp(x, x)
that is second-order unstable. Suppose that . Thenp* p(s, p*) " 1/2 ! 0

. This implies, however, thatg 1 b

1 1 3 1!p(z , p*) p a ! b ! g 1 .2 4 4 2

Thus, is again linearly unstable. A similar argument can be given forp*
the case .p* ! D 2

Suppose now that is a rest point. Observe first that if at leastp* ! D 3
one signaling system is present on the boundary face under consideration,
then the same argument as in the proof of Lemma 12 shows that isp*
linearly unstable. Thus we may assume that no signaling system is in the
support of . Let , , and be the same as above. Define in additionp* S S S1 2 3

is not one-to-one and } and is notS p {(s , r ) : s r p r S p {(s , r ) : s4 i j i j 5 i j i
one-to-one and }. Let a, b, g, d, and e be the corresponding sums!r p rj



EVOLUTION AND THE EXPLANATION OF MEANING 23

of frequencies in , , at . ThenS i p 1, . . . , 5 p*i

1 3 1
p(z, p*) p a ! (b ! d) ! (g ! e)2 4 4

and

1 3 1!p(z , p*) p a ! (g ! e) ! (b ! d).2 4 4

Since , we may suppose without loss of generality that . Letp* ! D b 1 03
. Thenz ! Si 2

1 3 1
p(z , p*) p (a ! b ! g) ! d ! e.i 2 4 4

If , then . If , then there existsb 1 g p(z, p*) 1 p(z , p*) p p(p*, p*) g 1 bi
a such that . In both cases the!z ! S p(z , p*) 1 p(z , p*) p p(p*, p*)j 3 j
Jacobian at has a positive eigenvalue. If , then is second-p* b p g p*
order unstable, which can again be seen by computing the average payoff
when is slightly perturbed in the direction of one signaling system.p*

This shows that almost all trajectories will converge to one of the sig-
naling systems. To complete the proof of Theorem 7, observe that the
vector field (1) is invariant under permutation of the signaling systems
since the differential equations remain the same. Thus, the basins of at-
traction must be of equal size. "

Proof of Theorem 8

To prove the theorem, let us first suppose a boundary face consisting of
rest points for the replicator dynamics. Consider the following sender
strategies: maps both states on . and , where s and! !s m s p s s p s s1 1 2 3
are the two one-to-one sender strategies. And maps both states ons 4

. Define , , , and , wherem z p (s , r ) z p (s , r ) z p (s , r ) z p (s , r )2 1 1 1 2 2 1 3 3 1 4 4 1
is the receiver strategy that maps both messages on . Letr a D p1 1

and set , wherespan(z , z , z , z ) p* p az ! bz ! gz ! dz d p 1 "1 2 3 4 1 2 3 4
. Suppose without loss of generality that . Thena " b " g !(j ) p p 1 1/21

the average payoff on D is p. Hence all points are rest points.p ! D
Let . As in the proof of Theorem 7, the eigenvalues con-p* ! int(D)

cerning strategies are given by . I will showz " supp(p*) p(z , p*) " pi i
that, for some , all these eigenvalues are negative. Considerp* ! int(D)
strategies such that is many-to-one. These strat-z p (s , r ) " supp(p*) ri k l l
egies earn a payoff of against . Since , all ei-1/2( p ! q) p 1/2 p* p 1 1/2
genvalues with respect to these types are negative.

Consider the following sets: andS p {(s , r ) : r p r} S p {(s ,1 k l l 2 k
, where and are again the two signaling! ! ! !r ) : r p r } z p (s, r) z p (s , r )l l



24 SIMON M. HUTTEGGER

system strategies. Then for ,z ! Si i

1
p(z , p*) p (1 ! a(2p " 1) ! bp ! g( p " 1))1 2

and

1
p(z , p*) p (2p ! a(1 " 2p) " bp ! g(1 " p)).2 2

If both of these expressions are negative for certain values of a, b, and
g, then eigenvalues relative to strategies in and are also negative.S S1 2
Solving for the corresponding inequalities shows that the set of values
for a, b, and g is nonempty for some values of . The location1/2 ! p ! 1
of this set will depend on p. Suppose, for example, that

a ! g " 1
! p ! 1.

2a ! b ! g " 2

If , then the fraction on the left will be less than or equal tob " g # 0
. Hence this condition on p will hold. Thus points in the set given by1/2

, , and have negative eigenvalues relative0 ! a ! 1 0 ! b ! 1 " a 0 " g " b
to all types in and , as the reader may easily verify. Thus we haveS S1 2
only zero eigenvalues and negative eigenvalues at a suitably chosen point

. The zero eigenvalues correspond to the center manifold at . Thep* p*
center manifold coincides with D. This implies that is Liapunov stablep*
since nearby solutions either stay nearby (in the center manifold) or ap-
proach nearby points. From this we may conclude that a set of positive
measure converges to an open set on the boundary containing . "p*

Proof of Theorem 9

Let us first suppose that we have a signaling game with ,S !(j ) p p3 1
, , and . Consider the following!(j ) p q !(j ) p 1 " p " q p p q p 1/32 3

sender and receiver strategies:

1. and ; ( ) or ( );j .m j .m j .m s j .m s1 1 2 1 3 2 1 3 3 2
2. ( ) or ( ); and .m .a r m .a r m .a m .a1 1 1 1 2 2 2 3 3 3

(1) defines two sender strategies, and , while (2) defines two receivers s1 2
strategies, and . Let , , , andr r z p (s , r ) z p (s , r ) z p (s , r )1 2 1 1 1 2 1 2 3 2 1

. Let . A straightforward computationz p (s , r ) D p span(z , z , z , z )4 2 2 1 2 3 4
yields that the average payoff on D is . Hence D consists entirely of2/3
rest points. The corners can obviously be destabilized by signalingzi
systems (this follows again from a result in Wärneryd [1993]). Thus we
suppose that . I will show that is quasi-strict. A strategy qp* ! int(D) p*



EVOLUTION AND THE EXPLANATION OF MEANING 25

is quasi-strict if (a best response to q puts positiveBR(q) ! D(supp(q))
weight on a pure strategy if and only if q does). The quasi strictness of

implies that all real parts of the eigenvalues corresponding to statesp*
not on D are negative (this follows again from the fact that the entries of
the Jacobian at are of the form ifp* L p d (u(z , p*) " u(p*, p*)) z "ij ij i i

). This together with the fact that the zero eigenvalues correspondsupp(D)
to a manifold of rest points, D, implies that is Liapunov stable. Hence,p*
the open set attracts a set of positive measure from the interior ofint(D)

.729D
First observe that for every sender strategy with ,s i ( 1, 2 u(s , r ) "i i j

for since coordination with each is possible only for two2/3 j p 1, 2 rj
state-act pairs and all three state-act pairs are equiprobable. For every

with , for since both sender strategies mapr j ( 1, 2 u(s , r ) " 2/3 i p 1, 2j i j
and on . Moreover, if , then forj j m u(s , r ) p 2/3 u(s , r ) ! 2/3 i (1 2 1 i j k j

. To see this, suppose without loss of generality that . Thenk u(s , r ) p 2/31 j
must map on and on or on . Then, for forr m a m a a r ( r k pj 2 3 1 1 2 j k

, it is necessary that does not map on . This implies that1, 2 r m aj 3 3
. Now let be an arbitrary strategy of withru(s , r ) ! 2/3 z p (s , r ) S i,2 j i j n

. Thenj ( 1, 2

1 2
p(z, z ) p (u(s , r ) ! u(s , r )) " .i i k l j2 3

Our arguments show moreover that there exists at least one z !r
such that . Since , it follows thatsupp(p*) u(z, z ) ! 2/3 p* ! int(D) p(z,r
for all .p*) ! 2/3 z " supp(p*)

Suppose . Then similar reasoning applies top 1 q p* ! int(span(z ,1
. now implies that for . Now is thez )) p 1 q u(s , r ) ! u(s , r ) i p 1, 2 r2 i 2 i 1 1

unique receiver strategy that is optimal for both and . This showss s1 2
that the conclusion of Theorem 9 also holds in the case in which the three
states are not equiprobable.

Suppose now that . Then similar reasoning applies to the strategiesn # 4
based on the following sender and receiver strategies:

1. and ; ( ) or ( );j .m j .m j .m s j .m s j .m , . . . , j .1 1 2 1 3 2 1 3 3 2 4 4 n
.mn

2. ( ) or ( ); and ;m .a r m .a r m .a m .a m .a , . . . ,1 1 1 1 2 2 2 3 3 3 4 4
.m .an n

The average payoff on defined by the four strategies based on (1) and!D
(2) will again be constant. for and!u(z, p*) ! u(p*, p*) p* ! D z "

by similar arguments as in the case of . "supp(p*) n p 3



26 SIMON M. HUTTEGGER

REFERENCES

Akin, Ethan, and Josef Hofbauer (1982), “Recurrence of the Unfit,” Mathematical Biosci-
ences 61: 51–62.

Binmore, Ken, John Gale, and Larry Samuelson (1995), “Learning to Be Imperfect: The
Ultimatum Game,” Games and Economic Behavior 8: 56–90.

Björnstedt, Johan, and Jörgen Weibull (1996), “Nash Equilibrium and Evolution by Imi-
tation,” in Kenneth J. Arrow, Enrico Colombatto, Mark Perlman, and Christian
Schmidt (eds.), The Rational Foundations of Economic Behaviour. New York: Macmillan,
135–171.

Cressman, Ross (2003), Evolutionary Dynamics and Extensive Form Games. Cambridge, MA:
MIT Press.

Cubitt, Robin P., and Robert Sugden (2003), “Common Knowledge, Salience and Conven-
tion: A Reconstruction of David Lewis’ Game Theory,” Economics and Philosophy 19:
175–210.

Grim, Patrick, Trina Kokalis, Ali Tafti, Nicholas Kilb, and Paul St. Denis (2001), Making
Meaning Happen. Technical Report 01-02. Stony Brook: SUNY, Stony Brook, Group
for Logic and Formal Semantics.

Harms, William F. (2004), Information and Meaning in Evolutionary Processes. Cambridge:
Cambridge University Press.

Hauser, Marc D. (1997), The Evolution of Communication. Cambridge, MA: MIT Press.
Hirsch, Morris W., and Stephen Smale (1974), Differential Equations, Dynamical Systems,

and Linear Algebra. Orlando, FL: Academic Press.
Hofbauer, Josef, and Karl Sigmund (1998), Evolutionary Games and Population Dynamics.

Cambridge: Cambridge University Press.
Hume, David (1739), A Treatise of Human Nature. London: John Noon.
Huttegger, Simon (2007), “Evolutionary Explanations of Indicatives and Imperatives,” Er-

kenntnis 66: 409–436.
Kelley, Al (1967), “The Stable, Center-Stable, Center, Center-Unstable, Unstable Mani-

folds,” Journal of Differential Equations 3: 546–570.
Komarova, Natalia, and Partha Niyogi (2004), “Optimizing the Mutual Intelligibility of

Linguistic Agents in a Shared World,” Artificial Intelligence 154: 1–42.
Lewis, David (1969), Convention: A Philosophical Study. Cambridge, MA: Harvard Uni-

versity Press.
Maynard Smith, John (1982), Evolution and the Theory of Games. Cambridge: Cambridge

University Press.
Maynard Smith, John, and David Harper (2003), Animal Signals. Oxford: Oxford University

Press.
Maynard Smith, John, and George Price (1973), “The Logic of Animal Conflict,” Nature

146: 15–18.
Nowak, Martin A., and David C. Krakauer (1999), “The Evolution of Language,” Pro-

ceedings of the National Academy of Sciences 96: 8028–8033.
Oliphant, Michael, and John Batali (1997), “Learning and the Emergence of Coordinated

Communication,” Center for Research on Language Newsletter 11. http://www.isrl
.uiuc.edu/amag/langev/paper/oliphant97learningAnd.html.

Pawlowitsch, Christina (2006), “Why Evolution Does Not Always Lead to an Optimal
Signaling System.” Working paper, University of Vienna.

Quine, Willard V. O. (1936), “Truth by Convention,” in Otis H. Lee (ed.), Philosophical
Essays for A. N. Whitehead. New York: Longmans, 90–124.

——— (1953), “Two Dogmas of Empiricism,” in From a Logical Point of View. Cambridge,
MA: Harvard University Press, 20–46.

Rousseau, Jean-Jacques ([1755] 1997), Discourse on the Origin and the Foundations of In-
equality among Men. Reprinted in V. Gourevitch (ed.), The Discourses and Other Early
Political Writings. Cambridge: Cambridge University Press, 111–222.

Schelling, Thomas (1960), The Strategy of Conflict. Oxford: Oxford University Press.
Schlag, Karl (1998), “Why Imitate, and If So How? A Bounded-Rational Approach to the

Multi-armed Bandit,” Journal of Economic Theory 78: 130–156.



EVOLUTION AND THE EXPLANATION OF MEANING 27

Selten, Reinhard (1980), “A Note on Evolutionary Stable Strategies in Asymmetrical Animal
Conflicts,” Journal of Theoretical Biology 84: 93–101.

Skyrms, Brian (1996), Evolution of the Social Contract. Cambridge: Cambridge University
Press.

——— (2000), “Stability and Explanatory Significance of Some Simple Evolutionary Mod-
els,” Philosophy of Science 67: 94–113.

——— (2004), The Stag Hunt and the Evolution of Social Structure. Cambridge: Cambridge
University Press.

Smith, Adam ([1761] 2000), Considerations Concerning the First Formation of Languages.
Reprinted in The Theory of Moral Sentiments. Amherst, NY: Prometheus, 505–538.

Snowdon, Charles T. (1990), “Language Capacities of Nonhuman Animals,” Yearbook of
Physical Anthropology 33: 215–243.

Taylor, Peter D., and Leo Jonker (1978), “Evolutionarily Stable Strategies and Game Dy-
namics,” Mathematical Biosciences 40: 145–156.

Vanderschraaf, Peter (1998), “Knowledge, Equilibrium and Convention,” Erkenntnis 49:
337–369.

Wärneryd, Karl (1993), “Cheap Talk, Coordination, and Evolutionary Stability,” Games
and Economic Behavior 5: 532–546.

Weibull, Jörgen (1995), Evolutionary Game Theory. Cambridge, MA.: MIT Press.
White, Morton (1950), “The Analytic and the Synthetic: An Untenable Dualism,” in Sidney

Hook (ed.), John Dewey: Philosopher of Science and Freedom. New York: Dial, 316–
330.

Young, H. Peyton (1998), Individual Strategy and Social Structure: An Evolutionary Theory
of Institutions. Princeton, NJ: Princeton University Press.

Zollman, Kevin J. S. (2005), “Talking to Neighbors: The Evolution of Regional Meaning,”
Philosophy of Science 72: 69–85.