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7/29/2019 Br J Philos Sci 2012 Wagner 547 75
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Deterministic Chaos and theEvolution of Meaning
Elliott O. Wagner
ABSTRACT
Common wisdom holds that communication is impossible when messages are costless
and communicators have totally opposed interests. This article demonstrates that
such wisdom is false. Non-convergent dynamics can sustain partial information transfer
even in a zero-sum signalling game. In particular, I investigate a signalling game in
which messages are free, the state-act payoffs resemble rockpaperscissors, and senders
and receivers adjust their strategies according to the replicator dynamic. This system
exhibits Hamiltonian chaos and trajectories do not converge to equilibria. This persistent
out-of-equilibrium behaviour results in messages that do not perfectly reveal the senders
private information, but do transfer information as quantified by the KullbackLeibler
divergence. This finding shows that adaptive dynamics can enable information trans-
mission even though messages at equilibria are meaningless. This suggests a new explan-
ation for the evolution or spontaneous emergence of meaning: non-convergent adaptive
dynamics.
1 Introduction
2 Lewis Signalling Games and Information Transfer
3 Evolution and Lewis Signalling Games
4 Signalling Games with Opposing Interests
5 Dynamics of Zero-Sum Signalling Games
6 Deterministic Chaos and Information Transfer
7 Conclusion
1 Introduction
Is communication possible when messages are free and the interests of the
communicators are opposed? According to one common line of reasoning,
perhaps not. Consider a sender with private information about the world and
an opportunity to convey this information to some receiver. If these two
parties have different preferences over the receivers possible actions in each
Brit. J. Phil. Sci. 63 (2012), 547575
The Author 2011. Published by Oxford University Press on behalf ofBritish Society for the Philosophy of Science. All rights reserved.
For Permissions, please email: journals.permissions@oup.comdoi:10.1093/bjps/axr039
Advance Access published on December 16, 2011
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state of the world, then why should the sender bother communicating her
information to the receiver? And likewise why should the receiver believe
any messages she receives from the sender? As Franke et al. ([2009]) put
things, it is easy to see that under conditions of extreme conflict (a zero-sum
game), no informative communication can be sustained. For why should we
give information to the enemy, or believe what the enemy tells us.
This questionwhether or not communication can be sustained when
interests opposeis not an idle one. On the contrary, an understanding of
the strategic foundations of communication is of importance to at least four
disciplines: philosophy, linguistics, economics, and biology. Starting with
Lewis ([1969]), philosophers have used the tools of game theory to explain
how terms can gain semantic meaning and thus how language can be the
product of convention (see also Millikan [1984]; Skyrms [1996]; Harms[2004]). Linguists have employed game theory to explicate pragmatics and,
in particular, Grices conversational implicatures (Parikh [2001]; van Rooij
[2003]). Economists are interested in understanding when so-called cheap talk
can influence strategic decision-making (Crawford and Sobel [1982]; Farrell
and Rabin [1996]). Theoretical biologists also turn to game theory to under-
stand how animal signalling systems can evolve (Maynard Smith and Harper
[2003]; Searcy and Nowicki [2005]).
Many researchers from these disciplines have endorsed the common-senseconclusion that cheap talk cannot convey information when the interests of
the sender and receiver are sufficiently opposed. As an example from phil-
osophy, consider:
If the kind of intention that Grice uses to analyze speaker meaning is
really essential to genuine communication, then it will be essential to the
possibility of communication that there be a certain pattern of common
interest between participating parties. (Stalnaker [2005])
And from economics:
A misinformed listener will do something that is not optimal for himself
and, if their interests are sufficiently aligned, this is bad for the speaker
too. In a nutshell, this is how cheap talk can be informative in games,
even if players ruthlessly lie when it suits them. (Farrell and Rabin [1996])
Or that once interests diverge by a given, finite amount, only no commu-
nication is consistent with rational behaviour (Crawford and Sobel [1982]).
Informative cheap talk is held to be impossible when interests oppose.1
But these researchers have generally relied upon standard equilibrium
analysis when analyzing the prospects for information transfer in strategic
1 Although cheap talk is thought to be uninformative when interests oppose, economists and
biologists agree that communication can be kept honest in such situations through costly
signalling (Spence [1973]; Zahavi [1975]; Grafen [1990]).
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interactions. Lewis ([1969]) proposed a refinement of Nash equilibria that he
called a proper coordination equilibrium. Economists frequently identify
equilibria in signalling games by using something called the intuitive criterion
(Cho and Kreps [1987]). And biologists generally turn to Maynard Smith
and Prices ([1973]) concept of an evolutionarily stable strategy. What all of
these approaches to analyzing strategic communication share in common is
that they apply refinements of Nash equilibria. But static equilibrium analysis
leaves out part of the story: actors have to find their ways to equilibria.
Human players may reach an equilibrium through learning and evolutionary
systems may reach one through natural selection,2 but even in the simplest of
games, not all adaptive systems reach an equilibrium.
In this article, I investigate information transfer in a signalling game in
which interests are as opposed as possible. In other words, the game is
zero-sum: any gain by one player is a loss to the other. To model biological
evolution or social learning, it is assumed that the system evolves according
to the replicator dynamic. Although the signals in this game are meaningless
when the system is at an equilibrium, the system never reaches one. Instead,
it exhibits a very complicated form of out-of-equilibrium behaviour:
Hamiltonian chaos. This is the first observation of chaotic behaviour in a
Lewis signalling game.3 Since the system doesnt reach an equilibrium, infor-
mation transfer is sustained indefinitely. And thus, adaptive dynamics makecommunication possible in a zero-sum signalling game.
Section 2 describes the framework of Lewis signalling games and the math-
ematical machinery necessary to quantify the informational content of a mes-
sage in such a game. In Section 3, I describe the dynamics of Lewis signalling
games in which the communicators have aligned preferences. Section 4 ex-
tends this framework to signalling games in which the interests of sender and
receiver totally oppose. The game is zero-sum and contains best-response
cycles similar to those found in rockpaperscissors. The dynamics of this
game, including the deterministic chaos, is described in detail in Section 5.
Lyapunov exponents are used to present strong numerical evidence that the
dynamics are indeed chaotic. Section 6 spells out the consequences of such
chaos for information transfer. It is found that information transfer in this
system is partial and that the meaning of the signals fluctuates as the dynamics
unwind. Section 7 concludes.
2 Skyrms ([2002]) explores the ways in which preplay cheap talk can influence the sizes of the
basins of attraction of various equilibria in signalling games when the interests of the commu-
nicators are not perfectly aligned.3 Mitchener and Nowak ([2004]) have identified chaos in a different sort of language game. The
chaos in their setup is due to mutation, not conflicting payoffs.
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2 Lewis Signalling Games and Information Transfer
In this article, I examine a Lewis signalling game with modified payoffs. Lewis
([1969]) introduced signalling games to argue that language and semantic
meaning could be the product of a self-sustaining convention. The standard
Lewis signalling game involves two players: a sender and a receiver. Nature
flips a coin to determine the state of the world. The sender witnesses the state
of the world and then sends a message to the receiver. The receiver does
not observe the state, but does observe the message sent by the sender.
After observing the message, the receiver takes some action. Lewis assumed
that each action is correct for exactly one state of the world, so that if the
receiver performs the correct action for the state that obtained, then both
players receive a payoff of one. Otherwise they both receive a payoff of
zero. The senders pure strategies in such a game are functions that map
states of nature into messages. Likewise, the receivers strategies are functions
that map messages into actions. An extensive form representation of this game
(with two states, two messages, and two actions) is shown in Figure 1. Table 1
shows a state-act payoff matrix for such a game.
Every Lewis signalling game has several Nash equilibria. There are poolingequilibria in which the sender sends the same message regardless of the state
and the receiver performs the same action regardless of message. Such strategy
profiles are Nash equilibria because neither player can gain by unilaterally
deviating. And there are also separating equilibria in which the message pre-
cisely identifies the state and the receiver always performs the proper action
in the state that obtains. Lewis ([1969]) noted that at such separating equilibria
(he called these states signalling systems) it appears that the messages
have semantic meaning. For instance, if the players are using the separating
strategies shown in Figure 2, it looks as though m1 means something like s1
has occured or take action a1. This sort of rudimentary semantic meaning
has been called a pushmi-pullyu representation by Millikan ([1984]) and
primitive content by Harms ([2004]). Since there are two signalling system
s2s1 N
m2
m1
1m2
m1
1
2
2
a2
0, 0
a1
1, 1
a2
1, 1
a1
0, 0
a2
0, 0
a1
1, 1
a2
1, 1
a1
0, 0
Figure 1. An extensive form representation of the standard Lewis signalling game
with two states, two messages, and two actions.
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equilibria that are both equally effective at coordinating action yet use differ-
ent signals for each state, Lewis argued that meaning here is conventional.
Skyrms ([2010]) proposed a more technical notion of information content
designed to make discussion of the evolution of semantic meaning more
precise. A messages information content is just how the message affects prob-
abilities.4 Since a signal may impact the probabilities of as many states as exist
in whatever model is under consideration, informational content must there-
fore be a vector with components for each state of the world. For example,
the information m1 contains about the state in a three-state, three-message,
three-action signalling game is the vector:
Im1 logPrs1jm1
Prs1
!; log
Prs2jm1
Prs2
!; log
Prs3jm1
Prs3
!( )1
If the logarithms here are given in base two, then the informational content
is yielded in bits. As an example, suppose that nature chooses between three
equiprobable states and that the sender only sends message m1 in state s2. Then
the informational content of message m1 is simply the vector:
Im1 1; 1:58; 1h i 2
The 1 components indicate that the states have probability zero given that
message m1 is sent.5 So from the informational content vector it is possible to
read off the meaning of the signal. The negative infinity components make iteasy to see that the signal rules out states s1 and s3. Therefore, following Lewis
Table 1. A stateact payoff matrix for a standard three-state, three-message,
and three-action Lewis signalling game
a1 a2 a3
s1 1, 1 0, 0 0, 0s2 0, 0 1, 1 0, 0
s3 0, 0 0, 0 1, 1
s1 m1s2 m2
m1 a1m2 a2
Figure 2. An example of a signalling system strategy profile. These two strategies
constitute a strict Nash equilibrium in the standard Lewis signalling game.
4 The probabilities here can be either the probability that a certain state occurred (Skyrms calls
this information about the state of nature) or the probability that the receiver will perform a
certain action (information about the act). For brevity in this article, I focus on information
about the state, but all of the findings discussed below also extend to information about the act.5 The 1 is an artifact of taking the logarithm and not a reason to worry.
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we can take the signal to mean something like s2 has occurred, which is
appropriate given that this signal is only sent in state s2.
The informational content of a signal is a vector, so in order to compute an
overall measure of information in a message we can take a weighted average
over the components in the vector. In other words, the overall quantity of
information in signal m1 is equal to
KLm1 X
i
Prsijm1 logPrsijm1
Prsi
!3
This quantity is often called the KullbackLeibler divergence or distance
(Kullback and Leibler [1951]). Since receiving a signal is just like looking at
the outcome of an experiment, this quantity is called the information pro-
vided by an experiment by Lindley ([1956]).
The above information theoretic account of the meaning of a signal is ne-cessary in order to discuss the partial information transfer that emerges in the
models below. At separating equilibria it is easy to talk loosely about how the
signals seem to have gained meaning. In fact, at separating equilibria it is as
though the signals have propositional content; i.e. each signal can be under-
stood as identifying a particular world from a set of possible worlds.6 But it is
not always the case that messages in a signalling game have propositional
content. For example, if the sender randomizes between several different stra-
tegies, the signals will only carry partial information about the state of nature.To make this precise, consider the two sending strategies in Figure 3. These
are strategies for a signalling game with two messages and two states. If the
sender flips a biased coin to decide which of these two strategies to use, then
she will not perfectly communicate her private information to the receiver.
This is because sometimes the sender will employ the first signalling system
that associates s1 with m1 and s2 with m2, and other times the sender will
employ the second signalling system that associates a different message with
each state. Suppose that the coin is biased so that the sender uses the first
strategy with probability 0.7 and the second strategy with probability 0.3.
s1 m1s2 m2
s1 m1s2 m2
Figure 3. Two sending strategies that transmit partial information about the stateof the world when the sender mixes between them.
6 Skyrms ([2010]) argues that a considerable advantage of this information theoretic account of
meaning is that it subsumes propositional content as a special case of the information content
vector.
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Information transfer here is not perfect, but since, for example, m1 is more
likely to be sent when s1 is the case, this mixture of sending strategies does
communicate some information. The information content of the messages is
I(m1) < 0.485, 0.737> and I(m2) . To put an English
gloss on the signals, m1 can be taken to mean something like s1 is probably the
case and m2 conveys something like s2 is probably the case. Neither signal
rules out either state, but each signal is more likely in one of the states
than in the other. Therefore, the signals carry some information. Solving
for the quantity of information in both messages confirms this fact.
KL(m1) KL(m2) 0.119. Since the logarithms are taken to base two, this
quantity is in bits. The amount of information in these messages is obviously
greater than 0 bits, which is the quantity transmitted by the messages in an
equilibrium in which messages do not have meaning. And it is also less than 1bit, which is the quantity transferred by messages in a signalling system. Thus,
the mixed strategy here is partially communicative. That is, the messages
reveal some information, but do not completely identify the state of the world.
The sort of situation described above is not special or unique. Almost all
strategies in the senders entire mixed strategy space transmit partial informa-
tion about the state. The illusion that messages in a signalling game either
transmit information or do not is an artifact of a tendency to focus on pure
strategies and strategies that form part of a Nash equilibrium. In mixed pro-
files and many out-of-equilibrium strategy profiles, messages carry partial
information about the state. The dynamical systems investigated in Sections
4, 5, and 6 never reach pure strategy states and never reach equilibrium.
Therefore, the information conveyed by the signals is always partial. The
information-theoretic account of content described above is indispensable
for investigating the emergence of partial communication in such systems.
3 Evolution and Lewis Signalling Games
Following Lewis and Skyrms, we see that when players in a signalling game
adopt certain strategy profilesnamely, signalling systemsthe messages
convey information about the state of the world and it looks as though
theyve gained some semantic meaning. In the standard Lewis signalling
game, such strategy profiles are Nash equilibria. Lewis argued that signalling
systems are the unique rational solution to signalling games. To advance this
point, he developed an equilibrium refinement that he called a proper coord-
ination equilibrium. But this equilibrium concept requires a lot from the
actors in the game, for example common knowledge that every player expects
every other player to conform to the equilibrium. What about simpler agents?
Players that are only boundedly rational? Or agents that learn through some
sort of nave imitation? Or organisms that evolve their strategies through a
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process of frequency dependent selection? Since Lewis wrote Convention,
there have been methods developed to address these questions. This section
describes two such methodsevolutionarily stable strategies and the replica-
tor dynamicand their applications to Lewis signalling games.
Maynard Smith and Price ([1973]) proposed a refinement, called an
evolutionarily stable strategy or ESS, of Nash equilibria inspired by biological
explanations of the limited wars seen in animal conflicts. An ESS is a strategy
such that if the entire population played it, a small number of mutants would
always do worse against the population than the dominant type. This poor
performance would drive the mutants extinct. More precisely, a strategy S
is evolutionarily stable if for any other strategy M either:
(1) u(S, S) > u(M, S), or
(2) u(S, S) u(M, S) and u(S, M) > u(M, M)
where u(a, b) is the payoff received by the player of strategy a when matched
against a player of strategy b. When the game is asymmetric, this notion of an
ESS is equivalent to that of a strict Nash equilibrium (Weibull [1997],
Proposition 5.1).
Warneryd ([1993]) and Skyrms ([1996]) noted that the only evolutionarily
stable states of Lewis signalling games are the separating equilibria. This was
the first triumph of evolutionary game theory as applied to Lewis signalling
games. If we buy Maynard Smith and Prices supposition that biological sys-
tems will be found in equilibrium at an ESS, then we see that meaning will
evolve in Lewis signalling games.
The second triumph of evolutionary game theory applied to Lewis signal-
ling games originated from theories of adaptive dynamics. The replicator dy-
namic, which was introduced by Taylor and Jonker ([1978]), is a simple model
of an asexually reproducing population. The story behind it is as follows.
There is a large population of individuals and each individual uses the same
pure strategy throughout her lifetime. Additionally, each individual producesoffspring which faithfully inherit their parents strategy, so that the fluctu-
ations in each strategys frequency within the population is just given by the
rates at which the users of each strategy reproduce. Since this is a game dy-
namic, the simplest assumption is that the fitness of each strategy type is just
that types expected payoff when matched against a randomly chosen member
of the population. In other words, the fitness of type i is just (Ax)i where A is
the payoff matrix of the game and x is a vector in which the j-th component
gives the frequency of type j in the population. If we assume that in time
teach individual spawns (Ax)it additional individuals, then (as t is taken to
zero) the continuous time dynamic equation becomes
_xi xi Axi x Ax
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This dynamic has a tight connection to Maynard Smith and Prices evolution-
arily stable states: any ESS is an attractor of the one-population replicator
dynamic. The derivation of the replicator dynamic can be extended to
asymmetric games by increasing the number of populations (with one popu-
lation for each player role). The two population replicator dynamic, which is
used extensively throughout the sections below, is given by the differential
equations
_xi xi Ayi x Ay
_yj yj Bxj y Bxh i
In the multi-population replicator dynamic, a state is asymptotically stable if
and only if it is a strict Nash equilibrium (Weibull [1997], Propostion 5.13).
In addition to its popularity as a simple model of biological evolution,
the replicator dynamic is also often employed to model social learning and
cultural evolution. In fact, many different models of social learning have been
shown to yield the replicator dynamic (see, e.g. Binmore et al. [1995];
Bjornerstedt and Weibull [1996]; Schlag [1998]). One such model works as
follows. As before, suppose there are one or more large populations of
individuals. As time passes, individuals are randomly offered opportunities
to adjust their strategies. These individuals revise by picking a player at
random and then imitating this players strategy only if this players expectedpayoff is higher than her own and carrying out this imitation with probability
proportional to the payoff difference. This imitation protocol, called pairwise
proportional imitation by Schlag ([1998]), generates the replicator dynamic as
its aggregate behaviour. Although the replicator dynamic may not be the
whole story on either biological or cultural evolution, it surely provides a
natural starting point for investigation.
Skyrms ([1996]) observed that every computer simulation of a population
playing the standard Lewis signalling game with two states and evolving
according to the discrete-time replicator dynamic converges to a separating
equilibrium. In other words, meaning and perfect information transfer is
guaranteed to spontaneously emerge under the discrete-time replicator dy-
namic. Figure 4 illustrates this creation of information. The replicator dynam-
ic carries the system to a separating equilibrium, and along the way the
messages gain informational content. Huttegger ([2007]) provided an analytic
proof of the same fact with respect to the replicator dynamic: almost all initial
population states evolve to separating equilibria.7
7 This brief survey of the dynamics of signalling games necessarily obscures many interesting
complications. For example, if the states are not equiprobable then there is a nonnegligible
chance that the system will evolve to a pooling equilibrium in which there is no information
transfer. Additionally, if there are more than two states, there is a non-negligible chance that the
system will evolve to a pooling equilibrium in which some information is conveyed by the
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So things look pretty good for the evolution or emergence of meaning
in standard Lewis signalling games. The only ESSs are separating equilibria.
And in two-state games with equiprobable states, the replicator dynamic guar-
antees convergence to these separating equilibria. We see how it is that terms
can naturally acquire semantic meaning through a mindless process of bio-logical evolution or cognitively nave social learning, at least when the two
parties share an interest in communicating.
4 Signalling Games with Opposing Interests
The standard Lewis signalling thats been under consideration thus far pre-
sumes a very strong common interest. This strong common interest is readily
evident in the state-act payoffs shown in Table 1. Both players receive identical
payoffs in each stateact combination (these stateact combinations are the
leaves of the extensive-form game tree). But there is no reason to suppose that
real-life communication interactions are like this. In fact, there is reason to
suspect that senders and receivers rarely have identical interests. Think of
bacteria sending signals that cause their neighbours to produce and secrete
an extracellular enzyme that digests protein so that the bacteria can consume
the digested nutrients. A bacterium sending the signal has an opportunity to
freeride by inducing his neighbour to pay the metabolic cost for creating the
enzyme but then reaping the reward of absorbing the nutrients (Keller and
10 20 30 40 50t
0.5
1.0
1.5
KL m1
Figure 4. The creation of information by the replicator dynamic in a three-state,
three-message, and three-action Lewis signalling game.
messages, but that two or more states are pooled together. For explorations of these and other
issues, see (Huttegger [2007]; Pawlowitsch [2008]; Huttegger et al. [2010]; Barrett and Zollman
[2009]; Wagner [2009]; Skyrms [2010]).
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Surette [2006]). Or think of stomatopods settling conflicts over nesting areas
by displaying colored spots on the undersides of their raptorial appendages
(Dingle [1969]). Signalers may gain by signalling an exaggerated fighting abil-
ity.8 Or think of Harris sparrows that signal their position in the dominance
hierarchy by the size of their black chest markings. Since more dominant
sparrows have increased access to food and mates, senders may gain by dis-
playing a larger bib than they deserve (Rohwer [1975]). Or think of used car
salesmen.
In any case, there is no a priori reason to assume the interests of potential
communicators must be aligned. In this article, I investigate an extreme form
of opposed interests. The setup is the same as a three-state, three-message, and
three-act Lewis signalling game, but the stateact payoffs are altered so thatthe game is zero-sum. Any gain by the sender is a loss to the receiver and vice
versa. Interests here are as opposed as possible. The stateact payoff is shown
in Table 2. Notice that this stateact payoff has a parameter , which can
range from 0 to 1. This parameter determines the payoffs in the stateact
outcomes that are intermediate between a win and a loss for both players.
If 0, then neither player does better than the other in these outcomes.
But when > 0, then the sender reaps some reward at the receivers expense.
In the next section, I explore the dynamics of this game as this parameter is
varied.
Since this is a three-state, three-message, and three-act signalling game, the
players strategy spaces are fairly complex. Since pure strategies in a signalling
game are functions mapping states to messages and messages to actions and
there are 33 27 such functions, each players strategy space is the
26-dimensional simplex 27. But despite the high dimensionality of the state
space, some features of the game are easy to see. For one, separating strategies
(i.e. Lewiss signalling systems) are not Nash equilibria. Heres why: Imagine
that the sender and receiver have adopted a perfectly communicative
Table 2. A stateact payoff matrix for a modified version of a Lewis signalling
game with totally opposed payoffs
a1 a2 a3
s1 1, 1 , 1, 1s2 1, 1 1, 1 ,
s3 , 1, 1 1, 1
These stateact payoffs yield a zero-sum signalling game.
8 In fact, stomatopods continue to perform meral spread displays even after a molt, when their
exoskeleton is still hardening and their fighting ability is dramatically diminished. For this
reason, the display is sometimes considered a paradigm example of a deceptive signal (Searcy
and Nowicki [2005], ch. 4).
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signalling system strategy profile. Suppose this strategy profile stipulates send-
ing message m1 when s1 occurs and performing action a1 upon receipt of
message m1. This situation is great from the receivers point of view (she
earns her highest payoff when performing a1 in s1), but the sender would
prefer that the receiver perform action a3 in state s1. Consequently, the
sender has an incentive to deviate from this strategy profile so that in s1 she
sends whichever message causes the receiver to perform a3. Since the sender
has an incentive to deviate from any separating profile, no separating strategy
profile can be a Nash equilibrium.
It is also easy to see that this game has many Nash equilibria. For example
heres one. The sender always sends messages m1 regardless of the state that
obtains. And the receiver always chooses action a1 regardless of the message
received. This is a Nash equilibrium because neither player has an incentive todeviate. The sender doesnt have such an incentive because her messages are
ignored. And the receiver has no incentive because the messages dont carry
any information about the state of the world.
Another fact is that the Nash equilibria of this game consist of strategy
profiles in which the messages do not carry useful information. The reason is
easy to see. If the messages were informative, then the receiver would be able
to use that information to increase her odds of choosing the action that she
most prefers for that state. But since this game is zero-sum, any increase in
expected payoff for the receiver is a decrease to the expected payoff for the
sender. Therefore, messages sent in equilibrium must not be informative.
It is for this reason that researchers interested in the theoretical foundations
of strategic signalling have thought that communication is impossible in
zero-sum games. Separating profiles are not Nash equilibria and at Nash
equilibria signals do not transmit information. Consequently, when interests
slightly diverge, researchers must hypothesize mechanisms (e.g. signal cost or
reputation in repeated interactions) that make deception too costly to pay
off. And when interests are totally opposed (as in this zero-sum signallinginteraction), conventional wisdom says that communication is impossible.
But this judgement is too quick. As is shown below, once we look beyond
static equilibrium analysis we see that adaptive dynamics can allow persistent
information transfer.
5 Dynamics of Zero-Sum Signalling Games
As reviewed in Section 3 above, the replicator dynamic always carries popu-
lations playing a standard Lewis signalling game to an equilibrium. But stand-
ard Lewis signalling games are games of common interest. What happens to
the dynamics when the players are playing the game of completely conflicting
interests described above in Section 4? Immediately, we know that the
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replicator dynamic cannot converge to a stable state in which the messages
convey information. This is because the replicator dynamic only converges to
Nash equilibria (Weibull [1997], Proposition 3.5), and we know from Section 4
that Nash equilibria here are states in which the messages are necessarily
meaningless. But what about out-of-equilibrium information transfer? This
section will explore the out-of-equilibrium behaviour of this signalling game.
Since both the sender and receiver choose from 3
3
27 strategies, the rep-licator dynamics live in the 52-dimensional space 27 27. Unfortunately it
is difficult to analyze this high-dimensional space directly. So to get a handle
on the dynamics, lets start by looking at the behaviour on some lower dimen-
sional faces of the entire phase space. Faces are forward-invariant under the
replicator dynamic, so the behaviour of these smaller systems remains the
same as it is in the larger system. Additionally, because the dynamic is
smooth, the interior of phase space is a combination of the behaviours on
the faces. So by analyzing the dynamics in these smaller faces we can gain
insight into the behaviour of the entire 52-dimensional system.
Lets start by considering the 4-dimensional space (3 3) composed of
the sending and receiving strategies shown in Figure 5.9 Each of these strate-
gies, which are labeled S1, S2, S3 and R1, R2, R3 for convenience, are half of a
fully communicative signalling system. If the sender and receiver use a signal-
ling system strategy profile, then action ai is always performed in state si.
Remember from the payoffs in the signalling game with totally opposed inter-
ests (shown in Table 2), that this guarantees the receivers preferred payoff.
But since this is a zero-sum game, any gain by the receiver is a loss to the
S1 :
s1 m1s2 m2s3 m3
S2 :
s1 m1s2 m2s3 m3
S3 :
s1 m1s2 m2s3 m3
R1 :
m1 a1m2 a2m3 a3
R2 :
m1 a1m2 a2m3 a3
R3 :
m1 a1m2 a2m3 a3
Figure 5. The three sending and three receiving strategies that yield rockpaper
scissors payoffs in the signalling game with the stateact payoffs shown in Table 2.
9 This space is 4-dimensional because it contains two populations of three types each. The
frequencies of each type in a population must sum to one, so each population lives on a
2-dimensional simplex. Thus, the whole system is 4-dimensional.
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sender. The sender would prefer that action ai1 be performed in state si.
Therefore, if the receiver is using one of the strategies Ri in Figure 5, then
the sender would prefer to use the sending strategy Si1. Such a strategy profilewill guarantee the sender her most preferred outcome.
The above description is a little dense, but the upshot is that these six
strategies lead to best-response cycles like the following: Suppose the sender
plays S1. Then the receivers best response is to play R1. But if the receiver
plays R1, the senders best response is to play S3. But if the sender plays S3, the
receivers best response is to play R3. And so on. Such best response cycles are
the hallmark of rockpaperscissors. Indeed, if we assume that the three states
of nature are equiprobable, then the normal form game yielded by the ex-pected payoffs of the extensive form signalling game is exactly rockpaper
scissors. This resulting normal form game is shown in Table 3. This game has a
single Nash equilibrium. At this equilibrium both players mix uniformly over
their three strategies.
Conveniently, the behaviour of the two population replicator dynamic in
this exact rockpaperscissors game has been studied by Sato et al. ([2002]).
These authors found that the resulting dynamical system is incredibly com-
plex. In fact, it is so complex that for certain parameter values the system
exhibits Hamiltonian chaos.10 There is no universally accepted definition of
dynamical chaos, but in a fairly representative quote, Strogatz ([1994]) defines
chaos as aperiodic long-term behaviour in a deterministic system that exhibits
sensitive dependence on initial conditions. This system (described in detail
below) fulfills all three criteria and is additionally Hamiltonian (Hofbauer
[1996]). Hamiltonian systems have no attractors, and thus any particular
orbit can be either chaotic or quasi-periodic.
Since this system is 4-dimensional it is difficult to visualize. To get a feel for
the dynamics, I will numerically integrate
11
some initial conditions for various
Table 3. The normal form game that results from taking the expected payoffs
to the strategies shown in Figure 5 when the three states of nature are
equiprobable
R1 R2 R3S1 1, 1 1, 1 ,
S2 , 1, 1 1, 1
S3 1, 1 , 1, 1
These are the payoffs of a two player asymmetric rockpaperscissors game.
10 To my knowledge, Sato et al. ([2002]) were the first to note Hamiltonian chaos in the replicator
dynamic. For strange attractors in the one population replicator dynamic, see (Skyrms [1992]).11 All numerical integrations are performed using Mathematicas fourth-order symplectic parti-
tioned Runge Kutta method.
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values of. One way to visualize these solutions is to look at time-series data.Figure 6 shows the evolution of the population frequencies of one initial
condition when 0.0. This initial condition leads what looks to be a
quasi-period trajectory. Another way to visualize the system follows from
the fact that it is composed of two populations, each with three strategies.
A three-strategy population lives on the 2-dimensional simplex 3. So it is
possible to chart the movement of the population states on two 2-dimensional
simplexes (one for each population). These charts, which make the quasi-
periodic structure of the orbit quite conspicuous, are shown in Figure 7.
But by increasing this orbits structure appears to change. Figure 8 shows
the systems behaviour starting from the same initial condition but with
0.5. The time series demonstrate that the fluctuations in population fre-
quencies appear aperiodic and unpredictable. And the charts showing the
Figure 7. The evolution of the two populations beginning from the starting state
(x1, x2, x3, y1, y2, y3) (0.5, 0.01, 0.49, 0.5, 0.25, 0.25) when 0.0. This orbit is
quasi-periodic.
Figure 6. Two time series illustrating the evolution of the initial condition (x1, x2,
x3, y1, y2, y3) (0.5, 0.01, 0.49, 0.5, 0.25, 0.25) when 0.0. This trajectory
is quasi-periodic.
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evolution of the two populations no longer exhibit regular quasi-periodic
structure. Instead, the population frequencies look like they meander random-
ly over the entire simplexes. These features suggest that this same initial con-
dition leads to a chaotic trajectory with 0.5.
Poincare sections allow us to get another look at the dynamics of this
system. Following Sato et al. ([2002]), Figure 9 shows three Poincare sections
for the values 0.00, 0.25, 0.5, and the initial conditionsx1; x2; x3;y1;y2;y3 0:5; 0:01k; 0:5 0:01k; 0:5; 0:25; 0:25
with k 1, 2, . . . , 25. These images show the points where the trajectories
originating from these twenty-five initial conditions intersect the hyperplane
x2 x1 +y2 y1 0.12 When 0, these numerical integrations indicate that
the system is not chaotic. The trajectories appear to be quasi-periodic.
However, when > 0 it is easy to see the creation of chaotic orbits. As is
varied from 0 to 0.5, these Poincare sections show that some quasi-period
trajectories collapse and become chaotic. As is shown in Figure 9 below,these chaotic trajectories cover a larger region of strategy space than their
Figure 8. The evolution of the system beginning from the initial condition (x1, x2,
x3, y1, y2, y3) (0.5, 0.01, 0.49, 0.5, 0.25, 0.25) when 0.5. This orbit is chaotic.
12 This particular hyperplane was chosen due to the fact that all of these twenty-five orbits intersect
it, but it is not unique in this respect.
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Figure 9. Poincare sections at x2 x1 +y2 y1 0 for the 4-dimensional face con-sisting of the six strategies shown in Figure 5. Moving from the top downwards are
the maps for the system with parameter 0.0, 0.25, 0.5. These maps show that as
is increased quasi-period orbits become chaotic.
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quasi-periodic counterparts. And, as is expected by systems exhibiting
Hamiltonian chaos, quasi-periodic and chaotic orbits are finely interwoven,
meaning that a quasi-periodic orbit can be found arbitrarily close to anychaotic orbit (Lichtenberg and Lieberman [1983]).
To numerically demonstrate that these orbits are in fact chaotic, it is pos-
sible to compute their Lyapunov exponents.13 These exponents can be thought
of as generalizations of the eigenvalues of the Jacobian matrix of a system that
remain well defined for chaotic dynamics. A system exhibits sensitive depend-
ence on initial conditions if the distance between the trajectories originating
from one point and another infinitesimally close to it increase exponentially
with time. Lyapunov exponents quantify this sensitivity. A positive Lyapunov
exponent indicates a direction of local exponential expansion. A negative
Lyapunov exponent indicates a direction of local exponential contraction.
An orbit has as many Lyapunov exponents as the dynamical system has di-
mensions. A positive Lyapunov exponent is one of the hallmarks of a chaotic
orbit (see Strogatz [1994] for an introduction to Lyapunov exponents).
Lyapunov exponents for five initial conditions are shown in Table 4 for
each of 0.0, 0.25, 0.5. The largest Lyapunov exponent is clearly positive
for some of the orbits when 0.25 and 0.5. This presents very strong
numerical evidence that the orbits are indeed chaotic. An indication ofthe accuracy of these numerical computations can be obtained in two ways.
Table 4. Lyapunov exponents () for the initial conditions (x1, x2, x3, y1, y2, y3)
(0.5, 0.01k, 0.5 0.01k, 0.5, 0.25, 0.25) with k 1, 2, 3, 4, 5
k 1 2 3 4 5
0.0 1 +1.1 +1.4 +0.4 +0.4 +0.42 +0.2 +0.4 +0.3 +0.3 +0.3
3 0.5 0.4 0.3 0.3 0.3
4 0.8 1.2 0.4 0.4 0.4
0.25 1 +49.1 +35.2 +16.5 +0.4 +0.4
2 +0.3 +0.3 +0.4 +0.2 +0.3
3 0.2 0.1 0.4 0.2 0.3
4 49.1 35.4 16.4 0.4 0.4
0.50 1 +61.5 +34.9 +28.0 +12.0 +0.2
2 +0.6 +0.3 +0.1 +0.0 +0.2
3 0.5 0.4 0.2 0.1 0.24 61.4 35.7 27.9 12.1 0.3
The Lyapunov exponents in this chart have been multiplied by 103. The positiveLyapunov exponents indicate chaotic trajectories. These Lyapunov exponents are shownin boldface.
13 All Lyapunov exponents were computed in Mathematica using an algorithm adapted from
Sandri ([1996]). Integration was performed to t 10,000 with an accuracy of 1011.
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First, by comparison with the results of Sato et al. ([2002]). Sato et al. com-
puted these same Lyapunov exponents and their results match my own at least
to a factor of 104. Second, it is possible to compare the computed values with
some general facts about the Lyapunov exponents of Hamiltonian systems.
Since volume is conserved in Hamiltonian systems and the Lyapunov expo-
nents measure the expansion and contraction of an orbit, these Lyapunov
exponents should sum to zero. This is true of these calculated exponents up
to 104. Additionally, the second and third exponents should sum to zero.
This is also true up to 104.
So, the dynamics of this 4-dimensional system are incredibly complex. And
these three sending and three receiving strategies represent only half of the
signalling system strategies available to the players in the full signalling game.
When paired against each other these other six separating strategies formbest-response cycles just like those weve been investigating thus far. And
taking the expected payoffs to these other six strategies yields a normal
form representation that is identical to the one shown in Table 3. Since the
payoff matrices are identical, the dynamics on this other four dimension face
will also be identical. Consequently, the entire phase space has two disjoint
4-dimensional faces that display chaotic behaviour. Since movement inside the
interior of the system is determined by movement on the faces, the dynamics of
the entire 52-dimensional system must be very complex indeed!
Since the entire system is very high dimensional, it is obviously difficult
to visualize. But, as before, we can numerically integrate the evolution of indi-
vidual initial conditions and can compute Lyapunov exponents. Figure 10
shows the behaviour of an initial condition that is inside the interior of
phase space but is near its boundary14 (the three signalling system strategies
from above dominate the population). The four frequencies shown in these
time series are the first four signalling system strategies from Figure 5. Its
clear from the time series that these orbits are quasi-periodic. This intuition is
confirmed by the computation of the orbits Lyapunov exponents. The highestexponent is .0014 (a spurious zero).
On the other hand, Figure 11 shows the evolution of the same initial con-
dition when is increased from 0.0 to 0.5. The trajectorys aperiodic behaviour
is demonstrated by the seemingly random jumps in frequency. And, as before,
sensitive dependence on initial conditions is demonstrated by running the
numerical integration and then taking a second point that is very close to
the current location of the first trajectory (the first sending strategy is increased
in frequency by 10
6
and the other twenty-six sending strategies are decreased14 The three sending strategies from Figure 5 have frequencies 0.1k, .49 .01k, and .5. The other 24
sending strategies are each initialized with frequency 1=2400. The three receiving strategies from
Figure 5 have frequencies 0.25, 0.24, and 0.5. And the other twenty-four receiving strategies are
each initialized with frequency 1=2400.
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0 50 100 150 200t
0.2
0.4
0.6
0.8
1.0
x6
0 50 100 150 200t
0.2
0.4
0.6
0.8
1.0
y6
0 50 100 150 200t
0.2
0.4
0.6
0.8
1.0
x16
0 50 100 150 200t
0.2
0.4
0.6
0.8
1.0
y16
Figure 10. Charts showing the evolution of the initial condition described in
footnote 14 when 0.0 and k 1. This trajectory is quasi-periodic. The strategies
shown are the first two sending strategies and first two receiving strategies from
Figure 5.
Figure 11. Charts showing the evolution of the initial condition described in
footnote 14 when 0.5 and k 1. The strategies shown are the first two sending
strategies and first two receiving strategies from Figure 5.
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uniformly to compensate). As demonstrated in Figure 12. The two initial
conditions diverge very quickly, indicating local exponential expansion.
And again, the intuition that this orbit is chaotic is confirmed through the
numerical calculation of its Lyapunov exponents. The largest exponent is
0.0688, which is clearly positive and an indication of chaos.
Lyapunov exponents are more systematically calculated for three initial
conditions as is varied in Table 5. Just as was found on the systems
4-dimensional rockpaperscisssors faces, several of these initial conditions
lead to chaotic trajectories (as indicated by their positive Lyapunov expo-
nents) as is increased from 0.0 to 0.5. In summary, the dynamics of the
signalling game with totally opposed interests never brings the populations
to equilibrium. Instead, the populations remain out-of-equilibrium in either
quasi-periodic or chaotic orbits.
6 Deterministic Chaos and Information Transfer
The previous section showed that the dynamics of the signalling game with
opposed interests can be chaotic. This section explores this facts consequences
for the possibility of communication in zero-sum games.
Figure 12. These charts illustrate the sensitive dependence on initial conditions
when 0.5. The solid line shows the evolution of the initial condition described
above. The dashed line shows the evolution of an alternative initial condition
taken by slightly perturbing the original orbit at t 150. The strategies shown
are the first sending strategy and first receiving strategy from Figure 5.
Table 5. The maximal Lyapunov exponents (max) for the initial conditions
described in footnote 14 with k 1, 2, 3
k 1 k 2 k 3
0.0 max 1.44 1.98 1.85
0.25 max 30.4 46.7 15.3
0.50 max 69.7 38.9 11.8
This Lyapunov exponents in this chart have been multiplied by 103
. The positiveLyapunov exponents indicate chaotic trajectories. These exponents are shown in boldface.
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Recall that the equilibria of this game are uncommunicative. That is, mes-
sages sent by players at an equilibrium necessarily do not convey information.
But the dynamics here do not bring the system to equilibrium. The system has
no attractors. Instead of approaching a rest point, orbits traverse phase space
indefinitely. Therefore, the fact that messages in equilibrium do not transmit
information is irrelevant. Instead we must look for information transfer along
these non-convergent trajectories.
Fortunately, to investigate out-of-equilibrium information transfer here, we
can simply apply the information-theoretic account of meaning developed by
Skyrms ([2010]) (as outlined in Section 2 above). At the initial conditions
specified above in footnote 14 with k 1, the information content vector of
signal m1 is approximately . None of the components of
this vector are 1, so none of the states are ruled out by the signal. But just
the same, the signal does convey information about the state. The first entry is
negative. This is because state s1 is unlikely to be the case given that the signal
m1 is sent. The value of Pr(s1Wm1) is approximately 0.013. On the other hand,
the second and third entries are positive. This means that states s2 and s3 are
likely when m1 is sent. And indeed they are. The values are Pr(s2Wm1)&0.503
and Pr(s3Wm1)& 0.483. To put an English gloss on the information content one
might say that signal m1 indicates something along the lines of probably not
s1. But the English gloss is not the important point here. What is important isthat although the system is not in equilibrium and although the messages may
not perfectly communicate the state, the messages do indeed transfer
information.
Furthermore, as the dynamics unwind, this partial communication is not
eliminated. Recall that the trajectory beginning at this initial condition is
chaotic. Consequently, the systems state wanders unpredictably through-
out phase space. And as the state evolves the information content of the
messages changes. For example, at t 10 the information content vector of
signal m1 is . The English gloss for this information
content vector would be something like probably not s3. But once again
the gloss is not too important. What is important is that the signal still
conveys information and, as the system evolves, the meanings of the sig-
nals change. This information fluctuation of signal m1 is shown in
Figure 13. But because the state never reaches any of the equilibria, the
messages never lose all meaning. At some times the messages may be more
informative than at other times, but the messages never cease to transmit
information.
Trajectories in this system clearly do not converge to equilibria. One branch
of literature (see Fudenberg and Levine [1998]) downplays the importance of
non-convergent dynamics by arguing that learning or evolutionary models
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that fail to converge are not plausible models of natural behaviour. For
example:
Our argument here is that the learning models that have been studiedso far do not do full justice to the ability of people to recognize patterns
of behaviour by others. Consequently, when learning models fail to
converge, the behaviour of the models individuals is typically quite
naive; for example, the players may ignore the fact that the model is
locked in to a persistent cycle. We suspect that if the cycles persist long
enough, the agents will eventually use more sophisticated inference rules
that detect them; for this reason we are not convinced that models of
cycles in learning are useful descriptions of actual behaviour. (Fudenberg
and Levine [1998], p. 3)
The thought is that actors with even crude abilities to learn will figure out
when their opponents are choosing their strategies according to a pattern.
Once the pattern is learned, the agent will be able to exploit it. This exploit-
ation will break the cycles and (arguably) drive the system to equilibrium.
The importance of the chaotic trajectories in this system is that it makes
such pattern learning and exploitation impossible. Along orbits with a positive
Lyapunov exponent, local expansion is exponential. This makes prediction
impossible because any slight error in estimation of the states current position
will be magnified exponentially. In particular, since this is a signalling game,
it is impossible to predict the future meaning or information content of the
messages. The sensitive dependence of meaning on initial conditions and the
impossibility of the prediction of the future informational content of a signal
20 40 60 80 100 120 140t
0.2
0.4
0.6
0.8
KL m1
Figure 13. The quantity of information conveyed by signal m1 (as measured by the
KullbackLeibler divergence) as the system evolves from t 0 to 150. Logarithms
are taken to base 2 so information transmission is measured in bits. 0.5 and the
initial conditions are given in footnote 14 above.
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is illustrated in Figure 14. If prediction of the future state and message
meaning is impossible, then it is also impossible to use historical information
about past play to exploit the receivers behaviour.15 Therefore, the out-of-
equilibrium play and the partial information transmitted by the messages are
both sustained indefinitely.
One could call this sort of partial communication when interests conflict an
example of deception. Searcy and Nowicki ([2005]) say deception occurs when
(1) A receiver registers something Y from a signaler;
(2) The receiver responds in such a way that
(a) Benefits the signaler and
(b) Is appropriate ifY means X; and
(3) It is not true that X is the case.
Many of the signals sent in the system described here fit this description of
deception.16 Suppose the system is in the state described above in which the
information content vector of signal m1 is . The English
gloss for this vector would be something like s1 is probably not the case.
200 220 240 260 280 300 320t
0.2
0.4
0.6
0.8
1.0
KL m1
Figure 14. An illustration of the sensitivity of meaning on initial conditions. This
chart shows the quantity of information conveyed by signal m1 along two nearby
trajectories. The solid line shows information from the orbit beginning at the initial
conditions above. The dashed line shows information from an orbit with the initial
conditions given by slightly perturbing the location of the first orbit in phase space
at t 180. 0.5.
15 This argument is made with respect to chaotic dynamics in the rockpaperscissors game by
Sato et al. ([2002]).16 In fact, these signals qualify as deception according to all information-based accounts of de-
ception. See (Skyrms [2010]).
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But all strategy types are present in this population, so there exists a small
quantity of senders who always send signal m1 regardless of the true state of
the world. Consider such a sender matched with a receiver who performs a3
upon receipt ofm1. Suppose Nature flips its fair coin and the state of the world
is s1. This sender then sends message m1
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