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, 20121878 first published online 7 November 2012280 2013 Proc. R. Soc. B

Kevin J. S. Zollman, Carl T. Bergstrom and Simon M. Huttegger

honest communication

Between cheap and costly signals: the evolution of partially

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Research

Cite this article: Zollman KJS, Bergstrom CT,

Huttegger SM. 2013 Between cheap and costly

signals: the evolution of partially honest

communication. Proc R Soc B 280: 20121878.

http://dx.doi.org/10.1098/rspb.2012.1878

Received: 10 August 2012

Accepted: 16 October 2012

Subject Areas:

theoretical biology

Keywords:

handicap theory, costly signalling, hybrid

equilibria, replicator dynamics

Author for correspondence:

Kevin J. S. Zollman

e-mail: kzollman@andrew.cmu.edu

Between cheap and costly signals:

the evolution of partially honest

communication

Kevin J. S. Zollman1, Carl T. Bergstrom2,3 and Simon M. Huttegger4

1

Department of Philosophy, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA

2

Department of Biology, University of Washington, Seattle, WA 98195-1800, USA

3

Sante Fe Institute, Santa Fe, NM 87501, USA

4

Department of Logic and Philosophy of Science, University of California, Irvine, CA 92697, USA

Costly signalling theory has become a common explanation for honest com-

munication when interests conflict. In this paper, we provide an alternative

explanation for partially honest communication that does not require signifi-

cant signal costs. We show that this alternative is at least as plausible as

traditional costly signalling, and we suggest a number of experiments that

might be used to distinguish the two theories.

1. Introduction

Communication is ubiquitous in the biological world. When the interests of signal-

ler and signal receiver are perfectly aligned, the evolutionary benefits of reliable

communication are straightforward. But when interests are not aligned, signallers

might be selected to manipulate signal receivers with misleading signals, and the

signal receivers might evolve to disregard such communications. Why does

communication not break down in situations that involve conflict?

Costly signalling theory provides one explanation. Communication is

framed as a signalling game; with appropriate signal costs, honest communi-

cation is a Nash equilibrium of this game. If signalling is sufficiently costly

so that lying is too costly to be worthwhile, but honest signals are not too

costly to send, all signallers may choose to signal honestly at equilibrium [1– 3].

However, the costly signalling explanation for honest communication is pro-

blematic for several reasons. On the theoretical side, costly signalling can be a

very expensive mode of information exchange, and at equilibrium signals can

be so costly that all involved would be better off simply not communicating [4].

From a dynamical perspective, the necessary level of cost may make it difficult to

evolve signalling at all [5]. On the empirical side, researchers have not always

been able to find substantive signal costs associated with putative costly signal

systems—despite evidence that these systems do convey, at least, some infor-

mation among individuals with conflicting interests [6–14]. What, then, are we

to make of empirical situations in which signals appear to be informative even

without the high costs required by costly signalling models?

One approach to resolving this problem is to recognize that—contra early

work on the problem—the costly signalling mechanism does not require a

cost to the signals that are actually sent in equilibrium. It requires only that

out-of-equilibrium signals, i.e. ‘lies’, be too costly to be worthwhile [15,16]. In

other words, it is not the cost of signalling, but rather the marginal cost of

signalling, that ensures honesty [17].

In this paper, we offer an alternative explanation. We show that the paradig-

matic costly signalling games from the animal behaviour literature allow a

‘hybrid’ signalling equilibrium, in which inexpensive signals are able to support

a partially informative signalling equilibrium. While the existence of hybrid equi-

libria in economic models of signalling has been known for some time [18,19], the

evolutionary significance of these equilibria is only just now being considered.

Initial analysis focused on the Sir Philip Sidney game [5] and a signalling game

from economics known as the Spence signalling game [20]. In this study, we

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demonstrate that hybrid equilibria in biological settings do not

depend on the particular relatedness structure of the Sir Philip

Sidney game, but rather are of general importance in signalling

games of many types. We show that hybrid equilibria exist

both in differential cost models such as models of mate adver-

tisement [2], and in differential benefit models such as models of

offspring begging [3,21]. Moreover, we demonstrate that

under reasonable evolutionary dynamics, the hybrid equili-

brium has a basin of attraction comparable in size with that

of the classic costly signalling equilibrium. We conclude with

a discussion of the implications of this work for empirical

studies. We argue that most empirical studies taken as evidence

of costly signalling theory fail to discriminate between the clas-

sical costly signalling equilibrium and the hybrid equilibrium

presented here. However, these two different types of signalling

equilibria make different predictions regarding equilibrium be-

haviour. This paves the way for future studies that may be able

to empirically distinguish among these hypotheses.

2. Low-cost signals in costly signalling games

(a) Differential costs and differential benefits

Figure 1 illustrates a canonical two-player signalling game in

extended form. Player 1, the sender is one of two types, T

1

or

T

2

. In the case of signalling between relatives, as exemplified

by Maynard Smith’s Sir Philip Sidney game, the types might

represent the state of need of the receiver. In signalling

between mates or signalling between predators and prey,

these types might represent the quality of the signaller.

Although the sender can condition her behaviour on her

type, player 2, the receiver, cannot directly observe the signal-

ler’s type. Instead, the receiver can observe only whether a

signal, designated Sig, is produced or not. On the basis of

this observation, the receiver can choose one of two actions,

A

1

or A

2

. In the Sir Philip Sidney game, the action A

1

can be

interpreted as transferring a resource and A

2

can be interpreted

as not transferring. In mate selection, A

1

represents mating,

whereas A

2

represents not mating. In predator– prey signal-

ling, A

1

denotes declining pursuit and A

2

denotes pursuit.

This game features four pure (conditional) strategies for the

sender, and four for the receiver. These are listed in table 1.

This game structure reveals that there is a common core to

signalling among relatives, signalling to potential mates or

signalling to predators. The pay-offs of this game result in a

partial conflict of interest between signaller and receiver.

When the signaller is type T

1

(needy, a high-quality mate,

or uncatchable as prey), both the sender and receiver prefer

the same actions: both players prefer that the receiver take

action A

1

. However, when the sender is of type T

2

, the

sender and the receiver have divergent interests. In this

case, the sender still prefers that the receiver take action A

1

,

but a fully informed receiver would prefer to take action A

2

.

This partial conflict of interest introduces the evolutionary

conundrum: a T

1

type individual benefits by advertising its

true quality to the receiver, whereas a T

2

type individual

benefits by deceiving the receiver. The game illustrated in

Table 1. All possible pure strategies in the action–response game pictured

in ﬁgure 1.

label description

S

1

signal if T

1

and do not signal if T

2

S

2

signal always

S

3

never signal

S

4

signal if T

2

and do not signal if T

1

R

1

A

1

if signal is observed, A

2

otherwise

R

2

A

2

always

R

3

A

1

always

R

4

A

2

if signal is observed, A

1

otherwise

1,1 0,0

receiver

receiver

1,0 0,1

1,1 0,0 1,0 0,1

A1

sig.

not

sender

sig.

not

sender

A2

T1T2

A1A2

A1A2A1A2

Figure 1. An action– response game with cost-free signals and a partial conflict of interest. The game begins at the central node (open circle). The first move is a

move by ‘nature’ to determine the type of the signaller; this type is revealed to the signaller but not the receiver. In the second move, the signaller conditions its

behaviour on its type and chooses whether or not to send a signal. As the third move, the receiver must choose between two actions. The receiver can condition on

the signal, but not the type; this uncertainty is represented by the dotted lines.

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figure 1 has no cost associated with signalling. Without signal

costs, this conflict cannot be resolved by honest communi-

cation. There is no stable set of strategies where the type is

revealed by the signal. All the pure-strategy equilibria of

this game are known as pooling equilibria, where both

types send the same signal.

The central insight of costly signalling theory is that signal

costs can stabilize honest communication. For this to work,

different types of signallers must have different incentives

to send signals. Such differing incentives can arise as a

result of differential costs, or differential benefits, or both.

In differential cost models, one type pays lower signalling

costs than the other. This is the scenario originally described

by Zahavi [1] in the context of signalling between mates;

Grafen [2] treated this case in his landmark 1990 paper. Sig-

nalling from prey to predator also typically involves

differential costs [22,23]. In differential benefit models, one

type reaps a larger reward from the receiver’s response

than the other. Differential benefits drive honest signalling in

Maynard Smith’s Sir Philip Sidney game [3] and in Godfray’s

model of nestling begging [24]. In what follows, we treat

differential cost and differential benefit models in turn.

(b) Hybrid equilibria for differential cost games

Consider the game pictured in figure 2 and table 2. This game

allows for the possibility of stable advertisement of type,

because the signal imposes differential costs on senders of

different types. Take the strategy where T

1

signals and T

2

does not. In such a case, it is obviously best for the receiver

to take action A

1

when the signal is observed and action A

2

if not. So long as the T

1

individuals pay a signal cost of

c

1

,1, it is strictly in their interest to signal in order to

ensure that the receiver takes action A

1

. So long as the T

2

individuals would pay a cost c

2

.1 if they were to signal,

then it is strictly in their interest to refrain from signalling,

because the cost of the signal outweighs the benefit obtained

by inducing the receiver to take action A

1

:

c2.1.c1:ð2:1Þ

This is the familiar separating equilibrium in a differential

costs model.

However, there is another equilibrium that allows for par-

tial transfer of information. Consider the following mixed

strategy pair: the signaller mixes between strategy S

1

with prob-

ability (1 2

a

)andstrategyS

2

with probability

a

.Thereceiver

mixes between strategy R

1

with probability

b

and strategy R

2

with probability (1 2

b

). The resulting behaviour is illustrated

in figure 3. T

1

individuals always send the signal, but T

2

indi-

viduals play a mixed strategy, sending the signal with

probability

a

,1 and refraining from sending the signal with

probability (1 2

a

). If the sender does not observe the signal,

she takes action A

2

with certainty. If she does observe the

signal, she plays a mixed strategy, taking action A

1

with prob-

ability

b

,1 and taking action A

2

with probability (1 2

b

).

1 – c1, 1

A1A2A1A2

0 – c1, 0 1 – c2, 0 0 – c2, 1

receiver

receiver

sig.

not

sender

sig.

not

sender

T1T2

1,1 0,0 1,0 0,1

A1A2A1A2

Figure 2. A differential costs signalling game. Here, the game from figure 1 is modified by adding condition-dependent signal costs to the sender’s pay-offs, where

c

1

represents the cost for senders of type T

1

to send the signal, and c

2

represents the cost to senders of type T

2

.

Table 2. The normal form of the differential costs signalling game in ﬁgure 2. The strategy labels are deﬁned in table 1.

R

1

R

2

R

3

R

4

S

1

(x2xc

1

),1 2xc

1

,(1 2x)(1-xc

1

),x2xc

1

þ(1 2x),0

S

2

(1 2xc

1

2(1 2x)c

2

),x2xc

1

2(1 2x)c

2

,(12x)12xc

1

2(1 2x)c

2

,x2xc

1

2(1 2x)c

2

,(12x)

S

3

0,(1 2x)0,(12x)1,x1,x

S

4

(1 2x)(1 2c

2

),0 (1 2x)c

2

,(12x)12(1 2x)c

2

,xx2(1 2x)c

2

,1

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In plain English, this means that the sender sometimes ‘lies’

and is honest at other times, whereas the receiver only

sometimes chooses the sender’s favoured action.

When the players behave in this way, information transfer

is imperfect, unlike the separating equilibrium; the signal

does not indicate the type of the receiver with certainty. How-

ever, the signaller does transmit some information to the receiver

and the receiver does make some use of that information: because

T

1

signals always and T

2

signals only occasionally, observing

a signal increases the likelihood that the signaller is a T

1

-type.

If the prior probability of T

1

is x, then the probability that a

signalling individual is of type T

1

equals x/(x+

a

(1 2x)).x.

When does this pair of mixed strategies constitute an equi-

librium? First, consider the actions of the receiver. Because

only T

2

individuals fail to signal, receiving no signal informs

the receiver that the signaller is a T

2

individual and that the

receiver does best to take action A

2

. If at equilibrium, the recei-

ver randomizes between A

1

and A

2

in response to receiving a

signal, these two actions must yield the same expected pay-off.

Supposing the sender’s strategy conforms to the description

just offered, the expected pay-off to the receiver of playing

A

1

given that a signal is observed is

x

xþ

a

ð1xÞ:

The expected pay-off of playing A

2

given that a signal is

observed is

1x

xþ

a

ð1xÞ:

These expected pay-offs are equal when the signaller’s

mixed strategy uses strategy S

2

with frequency

a

given by

a

=x/(1 2x). This condition can be satisfied only when

T

1

-types are sufficiently rare. (The fact that x,1

2in this

model should not be taken as some deep fact about the possi-

bility of hybrid equilibria. Rather, the threshold of 1

2arises

because we have chosen a symmetric pay-off for the recei-

ver’s success when the sender is type T

1

or type T

2

. Had

we made these pay-offs different, there would be a different

constraint on x.)

Now consider the sender’s strategy. T

1

individuals always

signal. For this to be an equilibrium behaviour, the pay-off

from signalling must be at least as large as that from not sig-

nalling:

b

ð1c1Þþð1

b

Þðc1Þ0, which obtains when

b

c

1

. Individuals of type T

2

mix between signalling and

not signalling. Thus, these two behaviours must yield the

same pay-off:

b

ð1c2Þþð1

b

Þðc2Þ¼0, which reduces

to

b

¼c

2

. Given these requirements,

b

can take on a value

in (0,1) and thus represent a mixed strategy best response

when 1 .c

2

.0 and when c

2

c

1

. Thus, the overall

conditions for the hybrid equilibrium are

a

¼x

1x;ð2:2Þ

1.

b

¼c2.0ð2:3Þ

and c2c1ð2:4Þ

Comparing conditions (2.1) and (2.4), we see that both the

separating equilibrium and the hybrid equilibrium require

c

2

c

1

. Given condition (2.3), the hybrid equilibrium exists

precisely when the cost c

2

is not sufficiently high to maintain

the separating equilibrium, i.e. c

2

,1. Differential cost is still

necessary for the hybrid equilibrium, but the signal can be

considerably cheaper than in the separating equilibrium.

Figure 4aillustrates the regions of parameter space where

the hybrid and separating equilibrium exist.

How do the two players fare at the different equilibria? At

the traditional costly signalling equilibrium, the receiver gets

a pay-off of 1, regardless of the state of the sender. At the

hybrid equilibrium, this pay-off is less, because occasionally

the receiver performs action A

2

when the sender is of type

T

1

and sometimes the receiver performs A

1

when the

sender is of type T

2

. From the perspective of the receiver,

the traditional signalling equilibrium yields a higher pay-off.

Things are more complicated for the sender. Use Hand S

superscripts to the values c

1

and c

2

to refer to costs at the

hybrid or separating equilibrium, respectively. Prior to learn-

ing her type, the sender’s expected fitness in the traditional

costly signalling equilibrium is xð1cS

1Þ. In the hybrid equi-

librium, it is xðcH

2cH

1Þ. Whether it is better for the sender to

be in a signalling or hybrid equilibrium depends on which

signalling and hybrid equilibrium we compare. This much

can be said: for any signalling equilibrium, there exist

hybrid equilibria where the sender fares better than in that

signalling equilibrium.

(c) Hybrid equilibrium for differential benefit games

Differential costs are not essential for a costly signalling equi-

librium. Stable signalling can also arise when the two types of

signallers face the same signal costs, but accrue different

benefits from the receiver’s response. This is the scenario

modelled in Maynard Smith’s Sir Philip Sidney game [3]. A

differential benefits signalling game is illustrated in figure 5

and table 3.

sig. A1

A2

T1

T2not

(1 – a)

(1 – b)

a

Figure 3. An illustration of the hybrid equilibrium. Black lines denote the

strategy of the sender, and grey lines represent the strategy of the receiver.

Solid lines represent conditionally pure strategies; dashed lines represent

conditionally mixed strategies.

s

h

1c

c2b

1

(a)(b)

p

p

h

s

c1ca

Figure 4. An illustration of the locations of separating, hybrid and pooling

equilibrium for (a) the differential cost game and (b) the differential benefit

game. The regions marked by s and h are regions where the separating and

hybrid equilibria (respectively) exist. Pooling equilibria exist for all parameter

values, but in the regions marked with p, they are the only equilibria.

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In this differential benefits model, the two types pay the

same cost cof signalling, but the benefit they receive from

getting a receiver to take action A

1

varies. The separating

equilibrium requires first that the T

1

individuals do best by

signalling, namely that a.c. Second, it requires that the

T

2

-types do better by not signalling, i.e. that c.b. Thus,

the benefit of A

1

must be higher for the T

1

individuals than

it is for the T

2

individuals, and the cost of signalling must

separate the two:

a.c.b:ð2:5Þ

There is also a hybrid equilibrium. Consider the same

combination illustrated in figure 3: T

1

individuals always

signal, whereas T

2

individuals mix between signalling and

not signalling; receivers mix between A

1

and A

2

in response

to a signal, and always select A

2

in the absence of a signal.

The condition on

a

remains the same as in the differential

costs model:

a

¼x/(1 2x).

The condition on the benefits is as follows:

b

¼c=bc=a.

The conditions are satisfied with

b

taking a value in (0,1)

when ab.c. The overall conditions for a hybrid

equilibrium are thus

a

¼x

1x;ð2:6Þ

ab.cð2:7Þ

and

b

¼c

b:ð2:8Þ

Comparing conditions (2.5) and (2.7), we see that both the

separating and hybrid equilibria require ab, i.e. that

the benefit from A

1

be greater for T

1

individuals than for T

2

individuals. However, the hybrid equilibrium exists precisely

when the signal cost cis too low to sustain the separating

equilibrium.

1

Figure 4billustrates the regions of para-

meter space in which each of these equilibria are found for

differential cost models.

Again, we can compare the pay-offs obtained under the

hybrid and signalling equilibria. As in the differential cost

model, the receiver invariably gets a higher pay-off at the

separating equilibrium. In the differential benefit model, the

expected fitness of the sender is x(a2c

S

) at the separating

equilibrium and xc

H

(a/b21) at the hybrid equilibrium. As

with the differential cost model, for every separating equili-

brium, there exist hybrid equilibria at which the sender fares

better. In both models, separating can be so costly that signal-

lers do better with reduced information transfer. Because in

this game—unlike in the Sir Philip Sidney game—the signal

cost does not decrease the pay-off to the receiver, the receiver

is always better off receiving more information.

Many biologically relevant interactions will feature both

differential cost and differential benefit. Although we have

divided this into two cases for the purposes of clarity, the

same hybrid equilibrium exists in a model that combines

both the differential cost and differential benefit. The con-

ditions for its existence are more complex but essentially

the same conditions as above.

3. Evolutionary implications

The mere existence of Nash equilibria does not guarantee

their evolutionary significance. As is well known, many

Nash equilibria are unlikely evolutionary outcomes in that

they are unstable in one sense or another [25]. In this section,

we will demonstrate the evolutionary significance of the

hybrid equilibria characterized earlier by illustrating that

these equilibria attract open sets of nearby states in the

two-population replicator dynamics.

The two-population replicator dynamics captures the

basic process of evolution by natural selection in asym-

metric games [25]. Let x

i

be the relative frequency of sender

type iand y

j

be the relative frequency of receiver type j,

i,j¼1, ...,4. This dynamics is given by:

_

xi¼xið

p

iðyÞ

p

ðx;yÞÞ;ð3:1aÞ

and

_

yj¼yjð

p

jðxÞ

p

ðy;xÞÞ:ð3:1bÞ

a – c, 1

A1A2A1A2

0 – c, 0 b – c, 0 0 – c, 1

receiver

receiver

sig.

not

sender

sig.

not

sender

T1T2

a, 1 0,0 b, 0 0,1

A1A2A1A2

Figure 5. A differential benefits signalling game. The values aand brepresent the benefit of taking action A

1

to T

1

and T

2

receivers, respectively. Irrespective of the

signaller’s type, the cost of signalling is c.

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The vectors x¼ðx1;...;x4Þand y¼ðy1;...;y4Þare the

vectors of strategy frequencies for signaller and receiver,

respectively. The expression

p

iðyÞgives the pay-off of i

against yand

p

jðxÞgives the pay-off of jagainst x. The aver-

age pay-off in the sender population is given by

p

ðx;yÞand

the average pay-off in the receiver population by

p

ðy;xÞ.

The two-population replicator dynamics is thus a simple

formalization of the idea that, in each population, strategies

with above-average fitness thrive, whereas strategies with

below-average fitness decline.

We can now consider the evolutionary properties of both

the signalling and hybrid equilibria. In both the differential

cost game and the differential benefits game, the signalling

equilibrium is a strict Nash equilibrium (when the listed strict

inequalities are satisfied). Therefore, the signalling equilibrium

is necessarily an attracting state in the two-population replica-

tor dynamics [25]. Although this does not guarantee that it is

a global attractor—meaning that we cannot be assured that

the signalling equilibrium be reached from any ancestral

state—we can conclude that a population at a signalling equili-

brium will remain there if subject to only small perturbations.

The hybrid equilibria in both games are slightly more

complex. On the plane defined by the four strategies (two

sender and two receiver) that make up the hybrid equilibria,

populations cycle around the hybrid equilibrium indefinitely.

States near, but not on this plane, converge to the plane. So,

the hybrid equilibrium characterizes a set of states that are

also evolutionarily significant. In §3a,b, we demonstrate this

for both games.

(a) Differential cost games

There are two major claims that must be established. First, we

will characterize trajectories on the plane composed by strat-

egies S

1

,S

2

,R

1

and R

2

. Second, we will show that population

states near the plane will converge to it.

We will begin by considering the game comprised by

the four strategies listed earlier. This game produces the

following two pay-off matrices:

S¼xð1c1Þxc1

xð1c1Þþð1xÞð1c2Þxc1ð1xÞc2

R¼1x

ð1xÞð1xÞ

:

By subtracting a (different) constant from each column,

one obtains the following equivalent game:

2

S0¼0ð1xÞc2

ð1xÞð1c2Þ0

R0¼02x1

x0

:

The hybrid equilibrium corresponds exactly to the interior

Nash equilibrium of this game (whenever it exists). The

interior Nash equilibrium is either unstable for the replicator

dynamics restricted to the plane, or there are closed orbits

around it. Hofbauer & Sigmund [25] show that there are

three necessary conditions that are required for the two-

population replicator dynamics to produce the second case,

i.e. cyclic behaviour. First, the non-zero entries in S

0

must

be of the same sign. Second, the non-zero entries in R

0

must

also be of the same sign. Finally, the non-zero entries in S

0

and R

0

must be of different signs from one another. The

first condition holds on assumption that c

2

,1, which is

required for the existence of the hybrid equilibrium. The

second condition holds on condition that x,1/2, again a

requirement for the hybrid equilibrium. From these two

requirements, the final condition follows. It is shown in

Hofbauer & Sigmund [25] that when these conditions are sat-

isfied, there exists a constant of motion. It follows that the

replicator dynamics cycle around the hybrid equilibrium

in closed orbits on the plane comprising the four strategies

S

1

,S

2

,R

1

and R

2

.

What about states off this plane? Consider perturbations

from the hybrid equilibrium into the interior of the space.

At the hybrid equilibrium, the average pay-off for the

sender is x(c

2

2c

1

) and (1 2x) for the receiver. The pay-

off of S

3

and S

4

against the hybrid equilibrium is 0 in both

cases; as a result, the motion in the interior close to the

hybrid equilibrium will be in the direction of the hybrid equi-

librium. The pay-off for strategies R

3

and R

4

against the

hybrid equilibrium is x. Because the hybrid equilibrium

exists only when x,1

2, it follows that whenever the hybrid

equilibrium exists, the motion of the system will be towards

the hybrid equilibrium from the interior when the system is

close to the hybrid equilibrium. These considerations show

more formally that the so-called transversal eigenvalues of

the hybrid equilibrium [25] are negative. Together with the

results regarding the plane comprised of S

1

,S

2

,R

1

and R

2

,

it follows that interior trajectories approach the hybrid

equilibrium in a spiralling manner. The hybrid equilibrium

is ‘strongly stable’ relative to outside states and ‘weakly

stable’ towards states on the plane.

3

That stable cycles are observed on the face of the simplex is

structurally unstable—small changes in the underlying

dynamics (as might occur with the introduction of mutation

or other stochastic effects) are likely to alter this prediction of

our model. A number of different outcomes can occur depend-

ing on what changes are made to the underlying dynamical

system [25]. We leave the details of these changes for future

research. On the other hand, the stability of the plane itself

against invasion from types off the plane is unlikely to be

affected by small changes in the dynamics. While we are

unsure whether cycles should be expected in natural

Table 3. The normal form of the differential beneﬁts signalling game in ﬁgure 5. The strategy labels are deﬁned in table 1.

R

1

R

2

R

3

R

4

S

1

x(a2c),1 2xc,(1 2x)x(a2c)þ(1 2x)b,x2cx þ(1 2x)b,0

S

2

xa þ(1 2x)b2c,x2c,(1 2x)xa þ(1 2x)b2c,x2c,(1 2x)

S

3

0,(1 2x) 0,(1 2x)xa þ(1 2x)b,xxaþ(1 2x)b,x

S

4

(1 2x)(b2c),0 2(1 2x)c,(1 2x)xa þ(1 2x)(b2c),xxa2(1 2x)c,1

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populations, the stability of populations on the hybrid

equilibrium plane to invasion by other types suggests that

one should expect to find natural populations on or near

this plane.

(b) Differential benefit games

We will now complete the same analysis for the differential

benefit version of the signalling game. R

0

remains as before,

but S

0

is different:

S0¼0ð1xÞc

ð1xÞðbcÞ0

:

Again, the three conditions are satisfied when the hybrid

equilibrium exists, so we can conclude that the motion on the

plane is cyclic. In the differential benefit game, the average

pay-off for sender in the hybrid equilibrium is xc(a/b21),

which is positive so long as a.b. The pay-off for S

3

and S

4

are both 0, and so we can conclude that the motion in these

directions is towards the hybrid equilibrium. The analysis

for the receiver is identical as above.

(c) Basins of attraction

Thus far, we have shown that a non-zero fraction of the state

space will converge to the hybrid equilibrium when it exists.

A similar result holds for signalling equilibria, provided that

they exist. Figure 6 illustrates simulation results that capture

the relative sizes of the basins of attraction for both signalling

and hybrid equilibria. This graph reveals that in both the

differential cost and differential benefit models, the hybrid

equilibrium is as significant an evolutionary outcome as the

signalling equilibrium.

In these graphs, the basin of attraction of the hybrid equi-

librium is modelled by measuring how many initial states

converge to the surface which only consists of strategies S

1

,

S

2

,R

1

and R

2

. We have already illustrated that once on this

face, populations will enter stable cycles around the hybrid

equilibrium. This represents populations that are hetero-

geneous with respect to their willingness to signal honestly

and respond to the signal.

Notethattheanalysisinthissectiondealswiththesituation

in which mixed strategies are manifested by population-level

polymorphism of pure strategists rather than a monomorphic

population of pure strategists [26]. There is no straightforward

way to characterize the dynamic stability of a hybrid equilibrium

manifested as a monomorphic population of mixed strategists.

The hybrid equilibrium is not a strict Nash equilibrium because

mixed Nash equilibria are never strict. Nor is it an evolutionarily

stable strategy (ESS), because by Selten’s theorem [27], there are

no ESSs in mixed strategies in role-asymmetric games.

4. Discussion

We have shown that hybrid equilibria are general features of

the costly signalling games studied in evolutionary biology.

They arise and facilitate low-cost, low-fidelity communication

not only in differential benefit games used to model nestling

begging, but also in the differential cost games used to model

sexual signalling and prey-to-predator communication. Our

findings suggest an important empirical avenue for investi-

gation. Experiments that purportedly demonstrate costly

signalling in animal populations have traditionally compared

two hypotheses: the null hypothesis—that no communication

is taking place—against an alternative hypothesis—that some

communication is taking place. While these studies have con-

clusively rejected the hypothesis that no signalling is taking

place, they were not designed to distinguish between the tra-

ditional signalling hypothesis and the hybrid equilibrium

hypothesis presented here.

Empirical studies could distinguish between the hybrid

equilibrium and the separating equilibrium. First, the hybrid

equilibrium hypothesis predicts that signalling will be honest

some, but not all of the time. Moreover, the deviation from

honesty should take on a particular direction that distinguishes

the hybrid equilibrium from a separating equilibrium mud-

died by noise or mistakes. At the hybrid equilibrium, ‘low’

type individuals (T

2

in our model) should sometimes deviate

and send the costly signal, whereas ‘high’ type individuals

(T

1

) should never deviate by failing to signal. Similarly, recei-

vers should sometimes deviate and fail to respond positively

C2

C1

0

0.5

1.0

1.5

2.0

b

0

0.5

1.0

1.5

2.0

0

0.2

0.4

0.6

0.8

1.0

0

0.2

0.4

0.6

0.8

1.0

0.51.0

(a)(b)

1.52.0

a

0.51.01.52.0

00.5

S

1.01.52.000.51.01.52.0

H

H

S

Figure 6. Simulation results establishing the basins of attraction for signalling and hybrid equilibria. In both cases, x(the probability of being T

1

) is equal to 0.25. In (b),

c¼1. Each point in the graph represents one setting of the variables, and the colour represents the estimated size of the basin of attraction for the replicator dynamics.

(a) Varying cost signalling and hybrid; (b) varying benefit signalling and hybrid.

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to a signal, but they should never deviate by responding

positively in the absence of a signal.

Second, our model predicts that the population would be

heterogeneous with respect to both the sending and receiving

strategies. Third, our model suggests that it is unlikely that

the population would be in equilibrium, but rather would

be observed cycling around the equilibrium. This last predic-

tion is perhaps the most tenuous, because the cycling

behaviour observed here will not be stable to small pertur-

bations in the underlying dynamics.

Beyond the empirical significance of our results, we have

demonstrated an alternative evolutionary explanation for (par-

tially) honest communication in situations of conflict of interest,

which can range from parent–offspring interactions to mating

advertisement to predator– prey interactions. Although this

equilibrium had been previously observed [5,20], it was not

known if this phenomena was an artefact of the particular

games or if it represented a general phenomenon common to

many different signalling interactions. In this paper, we show

that this type of signalling is at least as evolutionarily plausible

as that offered by the traditional costly signalling models and

may fit better with the observed data on signal costs. In this

respect, it may represent a superior theory to traditional

handicap theory.

The authors thank the anonymous reviewers for helpful comments

on earlier drafts of this paper. This material is based upon

work supported by the National Science Foundation under grant

no. EF-1038456. Any opinions, findings and conclusions or rec-

ommendations expressed in this material are those of the

authors and do not necessarily reflect the views of the National

Science Foundation.

Endnotes

1

Huttegger & Zollman [5] prove the existence of this equilibrium in

the Sir Philip Sidney game, a model of signalling among relatives

that involves relatedness between signaller and receiver, requiring

additional parameters and a more byzantine form than the game con-

sidered here.

2

The game is equivalent in the sense of having the same equilibria.

3

See Huttegger & Zollman [5] for technical details in the case of the

Sir Philip Sidney game; they easily carry over to this case. Further

technical details are provided in Hofbauer & Sigmund [25].

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