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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.

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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]).

Elliott O. Wagner548



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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

Documents

Br J Philos Sci 2012 Wagner 547 75