ANIMAL BEHAVIOUR, 2000, 59, 221–230
Article No. anbe.1999.1293, available online at http://www.idealibrary.com on
Cheating as a mixed strategy in a simple model of
aggressive communication
SZABOLCS SZA
u MADO
u
Department of Plant Taxonomy and Ecology, Eötvös Lorand University, Budapest
(Received 30 June 1998; initial acceptance 10 September 1998;
final acceptance 27 September 1999; MS. number: 5926R)
The possibility that frequency-dependent cheating can persist in an evolutionarily stable communication
system has frequently been proposed. Although there is empirical evidence for this idea, however, it has
not been investigated in terms of game theory. In the present paper I show for a simple symmetric game
that cheating can be part of a mixed evolutionarily stable strategy (ESS). Furthermore, despite the
widespread assumption that cheaters must be rare, I show that most of the population can be cheaters,
while the signalling system remains evolutionarily stable. Consequences for signalling theory and
experiments to detect such mixed ESS are discussed.

1980; Gross & Charnov 1980) and damselflies, Ischnura
ramburi (Robertson 1985); and aggressive communication
in stomatopods (Caldwell & Dingle 1975; Adams &
Caldwell 1990).
Cheating has received attention from theoreticians as
well; however, models that have dealt with this problem
(Johnstone & Grafen 1993; Adams & Mesterton-Gibbons
1995; Viljugrein 1997) have all investigated asymmetric
communication games (where the players have different
roles) and where there is no room for a frequencydependent ESS. This follows from Selten’s (1980) theorem
which states that an asymmetric game cannot have a
mixed ESS. Only neutrally stable mixed strategies can
exist in such cases; these can become evolutionarily stable
only if errors of role identification occur (Maynard Smith
1982). Although they have all identified mixed ESS
solutions where cheating was persistent, evolutionarily
stability in these models is not maintained by feedback between the cheater’s fitness and their frequency.
Usually, different types of signallers use different strategies because of different cost–benefit patterns. On the
other hand, a frequency-dependent ESS can easily arise in
symmetric games (where the players can have the same
role at the same time). Here I investigate, with the help of
a simple model of aggressive communication introduced
first by Enquist (1985), whether cheating can be part of
such a mixed ESS, and, if so, what the consequences for
signalling cost may be. Then, I relate the model and the
results to the ‘badge of dominance’ game investigated by
Maynard Smith & Harper (1988). Finally, I investigate
whether the mixed ESS can be interpreted as a case of
‘probing’, as proposed by Dawkins & Guilford (1991).
Honesty is one of the main issues in the field of animal
communication. There is continuing discussion about
whether the information transmitted during communication is reliable, and, if so, what maintains the honesty
of signals. There are two dominant positions in the
literature. One states that animals communicate in order
to manipulate each other; hence there is no reason to
expect them to be honest, except when it is in their own
interests (Krebs & Dawkins 1984). The other, the so-called
‘handicap principle’, first proposed by Zahavi (1975,
1977), states that communication is essentially honest,
and this honesty is maintained by the cost of signals. This
ensures that the investment is acceptable to an honest
signaller but prohibitive to a cheater. However, doubts
have been raised about the handicap principle even after
Grafen’s (1990) game theoretical interpretation. For
instance, many authors have suggested that cheating can
persist in a stable communication system provided that
its incidence is low enough for receivers, on average, to
benefit from the interaction (Krebs & Dawkins 1984;
Grafen 1990; Dawkins & Guilford 1991; Adams &
Mesterton-Gibbons 1995). In other words, if the advantage of cheating is frequency dependent, cheating can be
part of a mixed evolutionarily stable strategy (ESS) played
either at the individual or at the population level. There
are several well-documented examples of such a mixture
of honest and deceptive signals, for example: Batesian
mimicry in butterflies (Whiley 1983); reproductive strategies of bluegill sunfish, Lepomis macrochirus (Dominey
Correspondence: S. Számadó, Department of Plant Taxonomy and
Ecology, Eötvös Lorand University, Budapest, Ludovika tér 2, H-1083,
Hungary (email: [email protected]).
0003–3472/00/010221+10 $35.00/0
2000 The Association for the Study of Animal Behaviour
221

2000 The Association for the Study of Animal Behaviour
222
ANIMAL BEHAVIOUR, 59, 1
Table 1. The expected payoffs of the strategies in a model of aggressive communication
Opponent strength
Strong
Attack
Conditional
attack
Flee
Attack
Conditional
attack
Flee
0.5V−CSS
0.5V−CSS −FP
−CSS
0.5V−CSS
0.5V−CSS
0
V−FA
V
0.5V−CSS
V−CSW
V−CSW −FP
−CSW
V−CSW
V−CSW
0
V−FA
V
0.5V
−CWS
−CWS −FP
−CWS
−CWS
−CWS
0
V−FA
V
0.5V
0.5V−CWW
0.5V−CWW −FP
−CWW
0.5V−CWW
0.5V−CWW
0
V−FA
V
0.5V−CWW
Ego strength
Strong
Attack
Conditional attack
Flee
Weak
Attack
Conditional attack
Flee
Weak
Modified after Hurd (1997). V: Value of the contested resource; CSS, CWW: expected cost of fight between equal
opponents; CSW: cost for strong individual to beat weak one; CWS: cost to weak individual when beaten by strong
one; FA: cost of attacking fleeing opponent; FP: cost of waiting if opponent attacks unconditionally.
THE MODEL
I first use Enquist’s (1985) model to show that cheating as
a mixed strategy can be part of an evolutionarily stable
signalling system. Enquist’s model is as follows. Consider
a modified version of the Hawk–Dove game (Maynard
Smith 1982), where each player can be weak or strong
and knows its own strength but not that of the opponent.
The contest consists of two steps. In the first, each player
can choose between two cost-free signals A or B; in the
second, each animal can give up, attack unconditionally
or attack if the opponent does not withdraw. Enquist
(1985) showed that the following global strategy (Sa) is
evolutionarily stable.
Strong individuals should show A in the first round and
then attack unconditionally if opponent shows A or wait
until opponent flees if it has shown B. Weak individuals
should signal B at the first step and then attack unconditionally if opponent shows B or withdraw if opponent
signals A.
Strategy Sa is a pure strategy in which both strong and
weak animals signal honestly. The corresponding dishonest pure strategy (Sb) can be defined as follows: always
display A in the first round, regardless of strength; then,
in the second round, if strong attack unconditionally if
opponent shows A or wait until opponent flees if it has
shown B; if weak withdraw if opponent signals A or wait
until opponent flees if it has shown B.
Under certain circumstances, as shown below, the following global strategy (SI), which is a mixture of pure
strategies Sa and Sb, can be an ESS. If strong show A in the
first round and then attack unconditionally if opponent
shows A or wait until opponent flees if it has shown B; if
weak show A in the first round with frequency p, then
withdraw if opponent signals A or wait until opponent
flees if opponent signals B; show B in the first round with
frequency 1p, then withdraw if opponent signals A or
attack unconditionally if it has shown B. The mixed
strategy can be played at the level of the population or at
the level of the individual.
Let V denote the value of the contested resource, CWW
and CSS the expected cost of a fight between weak and
strong individuals, respectively. I assume that the
expected utility of a contest between opponents of equal
strength is greater than zero, that is, 0.5VCWW >0,
0.5VCSS >0. I further assume that a strong animal can
always beat a weak one with a cost CSW, and CWS is the
expected cost that a weak animal should suffer on this
occasion. The following relation holds between these
costs: CWS >CSS, CWW >CSW. In addition, there is a cost of
attacking a fleeing opponent (FA), and of waiting if the
opponent attacks unconditionally (FP). It is biologically
realistic, but not necessary, to assume that CSW >FA and FP
(Hurd 1997). The frequencies of weak and strong individuals are denoted by q and 1q, respectively. Then the
payoffs for weak and strong contestants can be written as
shown in Table 1. I have chosen the payoffs 0.5VCSS
and 0.5VCWW when equal opponents decide to flee
instead of the original 0.5V. I will indicate the effect of
this choice.
Using the Bishop & Cannings (1978) theorem, which
states that the fitness of the pure strategies (Sa and Sb)
supporting the mixed evolutionarily stable strategy (SI)
should be equal against the mixed ESS, one can find the
value of p, that is, the proportion of cheaters. From the
equation:
E(Sa,SI)=E(Sb,SI)=E(SI,SI)
the proportion of cheaters can be expressed as:
(1a)
SZA
u MADO
u : CHEATING AS A MIXED STRATEGY
E(SI,SI)>E(SJ,SI)
In order for p* to fall between zero and one the following
inequalities must be fulfilled:
(8)
However, it is possible to show (Appendix 1) that
E(SI,SI)=E(SJ,SI) for any , thus, for SI to be ESS the second
condition must be fulfilled (equation 7):
E(SI,SJ)>E(SJ,SJ)
Merging the two inequalities (3b) and (4b) gives the
range of CWS when cheating can be part of the mixed
strategy:
The right side of the inequality (5) is identical to
Enquist’s result since he addressed the same problem but
from the opposite direction (i.e. what is the condition
when cheaters cannot invade). Changing the payoff
when equal opponents decide to flee changes only the
left-hand side of equation (5). In each case the left-hand
side value is given by the chosen payoff multiplied by the
ratio of q/(1q) (so if we chose a payoff of zero for the
case when equal opponents decide to flee, then the lefthand side is zero; if we chose 0.5V then the left-hand side
is 0.5 q V/(1q)).
The next step is to investigate the evolutionary stability
of the mixed strategy. For SI to be an ESS the following
condition must be fulfilled (Maynard Smith 1982):
E(SI,SI)>E(SM,SI)
(6)
or if E(SI,SI)=E(SM,SI) then
E(SI,SM)>E(SM,SM)
(7)
where SM can be any mutant (pure or mixed) strategy. I
first investigate the invasion of those strategies that
support SI.
(1) Sa, Sb invades as a pure strategy. Since
E(Sa,SI)=E(Sb,SI)=E(SI,SI) then if SM =Sa or SM =Sb then the
second condition (equation 7) must hold. It is easy to see
that if the condition of equation (5) is met then this
condition is fulfilled as well.
(2) Sa, Sb invades as a mixed strategy other than SI. If
equation (5) holds, 0<p*<1. Then strategy SI can be
denoted as SI(probability of Sa, probability of Sb), that is,
SI(1p*,p*). Suppose further that there is a mutant strategy SJ playing a different mixture of Sa, Sb, that is,
SJ(1p*, p*+), where is the deviation from p*,
where p*<<1p*. Can this SJ strategy invade our
initial SI strategy? For SI to resist the invasion the first
condition of the ESS definition must be met (equation 6):
(9)
That is, SI should invade the population of any mutant
mixed strategy. It is possible to show (Appendix 2) that
equation (9) is fulfilled for any , so SI is an ESS against
any SJ.
Since there are strategies other then Sa and Sb, we
should investigate the invasion of these strategies in a
pure or in a mixed form as well. Hurd (1997) identified
three significant alternative strategies other than Sa. The
first two are for strong individuals. The ‘coward’ (Sc)
strong individual tries to minimize the cost of fighting,
always signalling B, then fleeing from any opponent. The
‘Trojan horse’ (Sd) signals weakness (B) but then proceeds
with an unconditional attack. The third strategy, bluffing
(Sb), is for weak individuals. Thus we should consider the
invasion of Sc and Sd as pure strategies and any mixture of
Sa, Sb, Sc, Sd other than SJ(Sa,Sb) which have already been
investigated. It can be shown that SI is an ESS against Sc
or Sd as pure strategies (Appendix 3) and against any
combination of strategies (Appendix 4).
In Enquist’s (1985) model the proportion of weak
individuals (q) is fixed. What if q can change? If we allow
for this, then the model can be viewed as the simplest
discrete version of Maynard Smith & Harper’s (1988)
badge of dominance game. They investigated a continuous model where individuals can have different levels of
aggressiveness, more aggressive ones always winning
against less aggressive ones. They were able to show that
if the cost of fighting rises sufficiently steeply with aggressiveness then an evolutionarily stable distribution of
aggressiveness can be found. If q can change, the present
model can be viewed as having only two different levels
of aggressiveness. More aggressive individuals can be
considered strong and less aggressive individuals weak.
From the result of Maynard Smith & Harper (1988),
we can expect that a value of q can be found, provided
that the cost of fighting between strong individuals is
sufficiently high, where the fitness of strong and weak
individuals is equal. Let us denote the strategy playing
strong as SS and weak as SW and the mixture of them as SI.
If SI is an ESS then:
E(SS,SI)=E(SW,SI)=E(SI,SI)
(10)
that is,
From this, the proportion of weak individuals can be
expressed as:
223
ANIMAL BEHAVIOUR, 59, 1
It is easy to see that for q* to be greater than zero the cost
of fighting between strong individuals should be twice as
much as the value of the resource. If this condition does
not holds then the only ESS is to play strong, that is, as
aggressively as possible. So far this is what Maynard Smith
& Harper (1988) found in the continuous model. Moreover, they showed that cheating cannot spread if the
cheater has to pay the full cost of fighting corresponding
to the level of a false signal. If cheating is played as a
mixed strategy and q can change, the following condition
must be fulfilled:
E(SS,SI)=E(SWa,SI)=E(SWb,SI)
(13)
that is:
Proportion of cheaters/weak individuals
224
1
a
0.8
0.6
b
0.4
0.2
0
c
5
10
15
20
Value of the contested resource, V
25
Figure 1. Line a shows the proportion of cheaters (p*) when q is
fixed (at 0.5; q is the frequency of weak individuals). Lines b and c
show the proportions of weak individuals (q*) and cheaters (p*),
respectively, when q can change. CWW =5, CWS =10, CSW =5, CSS =20,
FA =2. See Table 1 for definitions.
where SWa, SWb denote the strategies when a weak individual plays Sa, Sb, respectively. From this the proportion
of cheaters and that of weak individuals can be expressed
as:
It can be shown in a similar way as before that the mixed
strategy playing q*,p* is an ESS against a mixed strategy
playing a different p value, against Sa,Sb,Sc,Sd or any
combination of these strategies and against a mixed
strategy playing a different q value (Appendix 5).
It is worth comparing how the proportion of cheaters
changes in the two versions of the model. If q is fixed
then this proportion increases linearly with the value of
the resource. In an extreme case nearly all of the weak
individuals can be cheaters. For instance, if V=12,
CWW =5, CWS =10, and q=0.9 then p*=0.98 and p*q=0.89.
If, however, q can change then the proportion of cheaters
is a decreasing function of V (an initial increase is possible
depending on the choice of parameters). In this case
the proportion of weak individuals (q*) is a decreasing
function of V as well. Figure 1 depicts the two cases.
Finally, I investigate the possibility of a mixed receiver
strategy. Probing, proposed by Dawkins & Guilford
(1991), is an example of this. They argued that receiving
as well as giving a signal can be costly; moreover, the cost
that the receiver should pay is probably proportional to
the signal cost. In this case, however, it may pay the
receiver to settle for a less costly, less reliable signal, and
only occasionally ask for the reliable, costly one (probing). This may allow cheating to persist at a low frequency in the population. There are two main aspects of
this proposal. The first is about signal cost. I discuss this
question, that is, whether signal A or signal B is more
costly, in detail later. The second aspect, which I consider
next, is that the receiver should play in a probabilistic
manner (it should occasionally ask for the costly signal)
that is, it should play a mixed strategy. Can a mixed
receiver strategy be an ESS for strong or for weak individuals? For strong individuals it means that they should
occasionally accept signal A without questioning the
underlying quality. This could certainly increase the frequency of cheating, but what would it mean in terms of
behaviour? Should strong individuals flee or should they
wait for a conditional attack? If the opponent signals A
then unconditional attack is the ESS for strong individuals, so any other behaviour would result in a loss of
fitness. Thus, for strong individuals there is no room for a
mixed receiver strategy. For weak individuals it means
that they should not accept signal A on every occasion;
instead they should sometimes attack unconditionally
(let denote the probability of doing so). Were this
strategy to be successful it would decrease, not increase,
the frequency of cheating. For this mixed receiver strategy to be an ESS the following condition must be fulfilled:
Rearranging gives:
SZA
u MADO
u : CHEATING AS A MIXED STRATEGY
However, equation (5) reveals that this condition cannot
be met. Thus, there is no room for a mixed receiver
strategy for either strong or weak individuals.
DISCUSSION
With the aid of a simple game theoretical model, I have
confirmed the frequent proposal that a mixture of honest
and deceptive signals can be evolutionarily stable in a
symmetric conflict situation. If the proportion of weak
individuals (q) is fixed then the stability of such a mixed
strategy depends only upon the value of the contested
resource and the expected cost of fighting for weak
individuals against strong and weak animals (CWW, CWS).
If, however, the proportion of weak individuals (q) is an
evolutionary variable, then the cost of fighting between
strong individuals (CSS) and the cost of chasing a fleeing
opponent (FA) should be taken into account as well.
The oft-proposed assumptions about probing are unnecessary to explain the persistence of cheating (Dawkins &
Guilford 1991; Semple & McComb 1996). In fact, a mixed
receiver strategy is not an ESS for strong or for weak
individuals. Furthermore, my results suggest that if the
proportion of weak individuals is fixed (q), theoretically
and in contrast to the assumption that cheating must be
rare in an evolutionarily stable signalling system (Krebs &
Dawkins 1984; Grafen 1990; Dawkins & Guilford 1991;
Adams & Mesterton-Gibbons 1995), many weak individuals can cheat in an ESS (p*q). In an extreme case this can
mean nearly all of the weak animals (since p* can vary
from zero to one). In this case p* is an increasing function
of the value of the resource (V). If, however, q can change
then both the proportion of weak individuals and that of
cheaters decreases with V.
Relation to Previous Models
The present model differs from previous ones in several
respects. Johnstone & Grafen (1993) investigated an
asymmetric situation with two types of signaller. Each
type plays a pure strategy and cheating is possible because
the cost-benefit patterns are different for the different
types of signallers. The conclusions of Adams &
Mesterton-Gibbons (1995) are very similar to that of
Johnstone & Grafen (1993). They have three types of
individuals (weak, intermediate, strong) in an asymmetric
communication game, each type plays a pure strategy,
and cheating is favoured for weak individuals because different types have different cost-benefit patterns.
Viljugrein (1997) investigated a simple model of mate
choice which is also about an asymmetric situation, with
an equilibrium at which both males and females play a
mixed strategy. However, this equilibrium is evolutionarily unstable since there is no negative feedback between
the frequencies and the fitness of the different strategies.
Example: the Badge of Dominance Game
An example of a symmetric conflict investigated in the
present model is the badge of dominance game observed
in many bird species (Rohwer & Rohwer 1978; Fugle et al.
1984; Jarvi & Bakken 1985; Rohwer 1985; Watt 1986;
Møller 1987). Indeed the focal question of several studies
has been why cheating does not occur in such systems
(Rohwer & Rohwer 1978; Møller 1987; Maynard Smith &
Harper 1988). Authors have suspected that cheating can
persist in a frequency-dependent manner (Krebs &
Dawkins 1984; Dawkins & Guilford 1991) and my model
supports this claim. However, this kind of frequencydependent process has not been observed in any avian
signalling system (Rohwer 1985). Why cannot such
mixed strategies be observed in the wild? There are two
competing hypotheses to explain honesty in the case of
status signalling. The first assumes that submissive individuals must make the ‘best of a bad job’, that is, they are
intrinsically worse fighters than dominants (Rohwer &
Rohwer 1978; Jarvi & Bakken 1984). In contrast, according to the second hypothesis, there need not be intrinsic
differences between subordinates and dominants because
these are genetically controlled alternative strategies with
equal fitness, that is, the solution of the status-signalling
game itself is a mixed strategy (Maynard Smith 1982; Jarvi
& Bakken 1984; Studd & Robertson 1985). In other words,
the first hypothesis relates to the model where q is fixed,
and the second where q is a variable. What is common in
both hypotheses is that each assumes that there is a cost
of being a dominant. Several possibilities have been
proposed.
(1) The social control hypothesis: the cost of fighting
among dominants may be high, for two reasons.
(a) There are frequent fights among dominants, who
have similar levels of aggression. Some authors have
indeed found that dominants fight more frequently with
each other (Jarvi & Bakken 1984; Møller 1987), but others
have not (Keys & Rothstein 1991; Slotow et al. 1993).
(b) Fighting among dominants is not more frequent than
expected on a random basis but it is particularly costly.
Keys & Rothstein (1991) found that this holds for whitecrowned sparrows, Zonotrichia leucophrys; however, they
found equally costly fights among immatures as well.
Thus, they concluded that costly fights cannot explain
honest signals. However, what really matters is not the
cost of a fight between dominants but the cost a cheater
suffers if it is beaten by a strong individual (CWS). This
should exceed a critical value (the inverse of equation 3b:
Enquist’s result). However, this can be investigated only
by experimental manipulation. Indeed, in those experiments where ‘cheaters’ were introduced into a natural
heterogeneous flock, the true dominants have shown
considerably increased aggression towards them (Rohwer
1977).
(2) The signal may confer a cost unrelated to aggressive
communication.
(a) Dominants have a substantially increased resting
metabolic rate. This was found for great tits, Parus major
(Jarvi & Bakken 1984).
(b) The avoidance of dominant individuals by subordinates can also be a cost to dominants. Senar & Caminero
(1997) have shown that siskins, Carduelis spinus, are able
to recognize dominant individuals by the size of the black
bib and they actively avoid them, preferring to feed in the
225
226
ANIMAL BEHAVIOUR, 59, 1
company of subordinates. This can be a considerable
cost for dominants, especially during winter, because it
makes foraging more difficult, and increases the risk of
predation.
(c) Dominants may suffer higher predation risk because
of their more conspicuous colours (Keys & Rothstein
1991; Slotow et al. 1993).
In addition to these costs, two other factors can prevent
cheating being part of an evolutionarily stable mixed
strategy.
(3) An asymmetry, such as ownership, may be
observed unambiguously (Maynard Smith 1982).
(4) The opponents may identify each other and
remember the outcome of previous encounters. This is a
problem only if the mixed strategy is played at an individual level (that is, when each weak individual cheats
with probability p). However, if the mixed strategy is
played at a population level (that is, a proportion p of the
weak individuals cheats), this is not an obstacle.
Relation to Signalling Theory
One may ask how the findings of the present model can
be related to signalling theory, especially to Grafen’s
(1990) results, since the explicit formulation of Zahavi’s
‘handicap principle’ is widely accepted as a mechanism
that maintains reliable and stable communication.
Grafen’s result that communication is honest has been
replaced by the view that signalling systems need not be
entirely honest to be evolutionarily stable (Zahavi 1987;
Grafen 1990; Dawkins & Guilford 1991; Johnstone &
Grafen 1993; Adams & Mesterton-Gibbons 1995). Signals
should be honest ‘on average’ (Grafen 1990; Johnstone &
Grafen 1993), otherwise the receivers would not use them
(Maynard Smith & Harper 1995), so cheaters should be
rare (Grafen 1990; Dawkins & Guilford 1991; Adams &
Mesterton-Gibbons 1995; Viljugrein 1997). However, my
results show that the frequency of cheaters can be quite
high. Since the value of p can vary from zero to one,
nearly every weak individual can be a cheater, while the
signalling system remains stable. If we take into account
strong individuals, the proportion of dishonest signallers
can still exceed half of the population, that is, the signal
‘on average’ is dishonest, yet the signalling system is still
stable.
Grafen’s second and third results are that signals are
costly, and costlier for low-quality individuals. I would
argue that one cannot ask whether these results are valid
in the context of the present model. If there is a conflict
of interest, only physical correlation between signal and
trait or strategic cost can guarantee the stability of the
communication (Grafen 1990; Maynard Smith & Harper
1995). There are two ways in which the strategic cost of
signalling can be paid (Dawkins 1993): (1) as a production
cost, paid on every occasion; or (2) as a consequence of
giving a signal, but not necessarily on every occasion.
Dawkins argued that this difference opens up the way to
different kinds of selection pressures. In the former case,
the signal is designed primarily by strategic considerations: it should reflect the quality being signalled.
However, in the latter case, efficacy considerations may
dominate the signal design, since the cost is unrelated to
the form of the signal.
Based on this distinction, Hurd (1997) constructed a
classification of strategic signalling of fighting abilities:
(1) handicaps, which are a function of the signal and the
signaller’s state; (2) vulnerability handicaps or risks,
which are the function of signal, signaller’s state and the
receiver’s reply; and (3) conventional signals, which are
maintained by cost defined only by the signaller’s state
and the reply. Crucial to this classification is the assumption that the consequence can be accounted as a signalling cost. This makes it possible to state only that a signal
is costly because it is risky to make (Adams & MestertonGibbons 1995; Hurd 1997). Only in this sense can Grafen
(1990) claim that Enquist’s model is in agreement with
his results since signalling A or B has costly consequences.
However, the interpretation of this assumption in the
present model (which is basically Enquist’s original
model) is not without problems. One cannot specify the
cost of signal A or B without specifying the action
(unconditional attack, attack, flee) following the signal.
The cost in the present model is a function of the
signaller’s state, the signal, the following action and the
receiver’s reply. However, if we specify all these things we
specify a game and not a signal. Thus the cost is better
viewed as a cost inherent in the game and not as a cost
related to any particular signal. This means, in turn, that
it is pointless to ask whether Grafen’s definitions are valid
since they are about the cost of signals and not about
costs inherent in signalling games. Moreover, the cost of
a signal is usually interpreted as a difference between
giving a signal and doing something else (for instance,
doing nothing; Bergstrom & Lachmann 1998). Here again
there is conflict with the present model. What is the
difference between giving signal A or B for a strong
individual? We cannot answer this question without
specifying which behaviour follows these signals. However, that Grafen’s definitions are not valid in the present
context does not imply that they are invalid in the right
context. Moreover, the results of the present model are in
full agreement with those of Getty (1998a, b), that in an
evolutionarily stable signalling system high-quality individuals should convert advertising into fitness more
efficiently than low-quality individuals do.
Detecting a Mixed Strategy
Finally, I discuss the problem of searching for such
mixed strategies. A simple observational technique may
not suffice. To an observer, a cheater is a strong individual
that loses against some other strong individuals (honest
strong ones) and wins against others (other cheaters). If
cheating is rare then it loses against every other strong
individual. Thus, for any observer a cheater is a ‘weak
strong individual’ or ‘the weakest of the strong individuals’. However, there may be differences even between
honest strong individuals, either genetic or conditional,
owing to nutrition, parasite infections, etc. Thus, even
amongst honest strong individuals one can expect some
difference in fighting success, which in turn implies that
SZA
u MADO
u : CHEATING AS A MIXED STRATEGY
it is impossible to decide solely on the basis of observation whether we face a less strong honest individual
or a weak cheater signalling it is strong. One way to
overcome such difficulties would be experimental
manipulation. Generally, subordinates are painted as
dominants (see for exceptions: Rohwer 1977; Senar &
Camerino 1998). In our case, however, one should paint
dominant individuals a submissive colour. If it is a real
strong individual it should win most of its battles against
weak animals. Indeed this was found in Rohwer’s (1977)
experiment. On the other hand, if it is a weak cheater
then it has only a 50% chance, on average, of winning
against any honest weak individual. That is, the efficiency
of strong individuals should not decrease (considerably);
however, the efficiency of cheaters depends upon the
false signal.
Acknowledgments
I thank Eörs Szathmáry and István Scheuring for helpful
comments and discussion, and Magnus Enquist and an
anonymous referee for helpful comments. This work was
supported by the Pál Juhász-Nagy junior-fellowship of
Collegium Budapest.
References
Adams, E. S. & Caldwell, R. L. 1990. Deceptive communication in
asymmetric fights of the stomatopod crustacean Gonodactylus
bredini. Animal Behaviour, 39, 706–716.
Adams, E. S. & Mesterton-Gibbons, M. 1995. The cost of threat
displays and the stability of deceptive communication. Journal of
Theoretical Biology, 175, 405–421.
Bergstrom, C. T. & Lachmann, M. 1998. Signaling among relatives.
III. Talk is cheap. Proceedings of the National Academy of Sciences
U.S.A., 95, 5100–5105.
Bishop, D. T. & Cannings, C. 1978. A generalized war of attrition.
Journal of Theoretical Biology, 70, 85–124.
Caldwell, R. L. & Dingle, H. 1975. The ecology and evolution of
agonistic behavior in stomatopods. Naturwissenschaften, 62, 214–
220.
Dawkins, M. S. 1993. Are there general principles of signal design?
Philosophical Transactions of the Royal Society of London, Series B,
340, 251–255.
Dawkins, M. S. & Guilford, T. 1991. The corruption of honest
signalling. Animal Behaviour, 41, 865–873.
Dominey, W. J. 1980. Female mimicry in male bluegill sunfish: a
genetic polymorphism? Nature, 284, 546–548.
Enquist, M. 1985. Communication during aggressive interactions
with particular reference to variation in choice of behaviour.
Animal Behaviour, 33, 1152–1161.
Fugle, G. N., Rothstein, S. I., Osenberg, C. W. & McGinley, M. A.
1984. Signals of status in wintering white-crowned sparrows,
Zonotrichia leucophrys gambelii. Animal Behaviour, 32, 86–93.
Getty, T. 1998a. Handicap signalling: when fecundity and viability
do not add up. Animal Behaviour, 56, 127–130.
Getty, T. 1998b. Reliable signalling need not be a handicap. Animal
Behaviour, 56, 253–255.
Grafen, A. 1990. Biological signals as handicaps. Journal of Theoretical Biology, 144, 517–546.
Gross, M. R. & Charnov, E. L. 1980. Alternative male life histories in
bluegill sunfish. Proceedings of the National Academy of Sciences
U.S.A., 77, 6937–6940.
Hurd, P. L. 1997. Is signalling of fighting ability costlier for weaker
individuals? Journal of Theoretical Biology, 184, 83–88.
Jarvi, T. & Bakken, M. 1984. The function of the variation in the
breast stripe of the great tit (Parus major). Animal Behaviour, 32,
590–596.
Johnstone, R. A. & Grafen, A. 1993. Dishonesty and the handicap
principle. Animal Behaviour, 46, 759–764.
Keys, G. C. & Rothstein, S. I. 1991. Benefits and costs of dominance
and subordinance in white-crowned sparrows and the paradox of
status signalling. Animal Behaviour, 42, 899–912.
Krebs, J. R. & Dawkins, R. 1984. Animal signals: mindreading and
manipulation. In: Behavioural Ecology. An Evolutionary Approach
(Ed. by J. R. Krebs & N. B. Davies), pp. 380–402. Oxford: Blackwell
Scientific.
Maynard Smith, J. 1982. Evolution and the Theory of Games.
Cambridge: Cambridge University Press.
Maynard Smith, J. & Harper, D. G. C. 1988. The evolution of
aggression: can selection generate variability? Philosophical Transactions of the Royal Society of London, Series B, 319, 557–570.
Maynard Smith, J. & Harper, D. G. C. 1995. Animal signals: models
and terminology. Journal of Theoretical Biology, 177, 305–311.
Møller, A. P. 1987. Variation in badge size in male house sparrows
(Passer domesticus): evidence for status signalling. Animal Behaviour, 35, 1637–1644.
Robertson, H. M. 1985. Female dimorphism and mating behaviour
in a damselfly, Ischnura ramburi: females mimicking males. Animal
Behaviour, 33, 805–809.
Rohwer, S. 1977. Status signaling in Harris sparrows: some experiments in deception. Behaviour, 61, 107–129.
Rohwer, S. 1985. Dyed birds achieve higher social status than
controls in Harris sparrows. Animal Behaviour, 33, 1325–1331.
Rohwer, S. & Rohwer, F. C. 1978. Status signalling in Harris
sparrows: experimental deceptions achieved. Animal Behaviour,
26, 1012–1022.
Selten, R. 1980. A note on evolutionarily stable strategies in asymmetric animal conflicts. Journal of Theoretical Biology, 84, 93–101.
Semple, S. & McComb, K. 1996. Behavioural deception. Trends in
Ecology and Evolution, 11, 434–437.
Senar, J. C. & Camerino, M. 1998. Status signalling and the ability
to recognize dominants: an experiment with siskins (Carduelus
spinus). Proceedings of the Royal Society of London, Series B, 265,
1515–1520.
Slotow, R., Alcock, J. & Rothstein, S. I. 1993. Social status signalling in white-crowned sparrows: an experimental test of the social
control hypothesis. Animal Behaviour, 46, 977–989.
Studd, M. V. & Robertson, R. J. 1985. Evidence for reliable badges
of status in territorial yellow warblers (Dendroica petechia). Animal
Behaviour, 33, 1102–1113.
Viljugrein, H. 1997. The cost of dishonesty. Proceedings of the Royal
Society of London, Series B, 264, 815–821.
Zahavi, A. 1975. Mate selection: a selection for handicap. Journal of
Theoretical Biology, 53, 205–214.
Zahavi, A. 1977. The cost of honesty. Journal of Theoretical Biology,
67, 603–605.
Zahavi, A. 1987. The theory of signal selection and some of its
implications. In: International Symposium of Biological Evolution (Ed.
by V. P. Defino), pp. 305–327. Bari: Adriatica Editrice.
Watt, D. J. 1986. A comparative study of status signalling in
sparrows (genus Zonotrichia). Animal Behaviour, 34, 1–15.
Whiley, R. H. 1983. The evolution of communication: information
and manipulation. In: Animal Behaviour. Vol. 2. Communication (Ed
by T. R. Halliday & P. J. B. Slater), pp. 156–189. Oxford: Blackwell
Scientific.
227
228
ANIMAL BEHAVIOUR, 59, 1
Appendix 1
Here I investigate whether the condition E(SI,SI)>E(SJ,SI) holds for any . Since from equation (1) we know that
E(SI,SI)=E(Sa,SI) equation (6) can be written as:
E(Sa,SI)>E(SJ,SI)
(A1)
The fitness of SJ has two parts: (1) when it plays Sa with probability 1p; (2) when it plays Sb with probability p+.
Let denote E(Sa,SI) with EaI and E(Sb,SI) with EbI, then equation (A1) has the form:
EaI >(1p) EaI +pEbI EaI +EbI
(A2)
Since EaI =EbI the right side equals the left side for any . Thus, the first condition of the definition of ESS cannot
hold.
Appendix 2
Here I investigate whether the condition E(SI,SJ)>E(SJ,SJ) holds for any . Let us denote E(Sa,SJ) with EaJ and E(Sb,SJ) with
EbJ. Then equation (9) can be written as follows:
(1p) EaJ +pEbJ >(1p) EaJ +(p+) EbJ
(A3)
0>EbJ EaJ
(A4)
EaJ =EbJ +q CWW
(A7)
after rearrangement:
The payoff for EaJ is:
which can be rewritten with EaI as:
The payoff for EbJ is:
Substituting EbI results:
Since EaI =EbI:
Thus equation (A4) holds, the condition of E(SI,SJ)>E(SJ,SJ) is fulfilled for any , SI is an ESS.
Appendix 3
Here I investigate the invasion of Sc and Sd as pure strategies against SI. First, I show that Sa and Sb alone are both ESS
against Sc or Sd. Then it possible to show that SI is an ESS as well.
(1) Sa and Sb is an ESS against Sc, that is:
Eaa >Eca and Ebb >Ecb
Hurd (1997) has shown that the honest strategy is an ESS against the coward provided:
that is:
(A8)
SZA
u MADO
u : CHEATING AS A MIXED STRATEGY
Sb is an ESS against the coward provided:
that is:
It is easy to see that equations (A10) and (A12) should hold since we assume that (0.5VCSS)>0 and CSS >FA.
(2) Sa and Sb is an ESS against Sd, that is:
Eaa >Eda and Ebb >Edb
(A13)
The honest strategy is an ESS against the Trojan horse provided:
Sb is also an ESS against the Trojan horse provided:
This should hold since CSW >FA >0.
It can be shown that strategies Sa and Sb have equal payoffs playing against Sa or Sb for strong individuals. From
Table 1 we can see that:
(3) SI against Sc or Sd. Equation (9) in this case can be written as:
(1p) Eaa +pEab >(1p) E*a +pE*b
(A18)
where the asterisk denotes either c or d. From equations (A8), (A13) and (A17) it follows that this condition holds for
any p, thus SI is an ESS against Sc or Sd.
Appendix 4
Here I show that SI is an ESS against the invasion of any mixed strategy consisting of any combination of Sa, Sb, Sc or
Sd. The payoff for the mutant mixed strategy is always a linear combination of the payoffs that the supporting pure
strategies get against Sa and Sb, thus equation (9) can be written as:
where i can range from 2 to 4 depending upon how much pure strategy supports the invading mutant and ui denotes
the probability of playing the ith strategy. For the invasion of Sa and Sb it was shown in Appendix 2 that SI is an ESS.
Thus here I consider only those cases where at least one strategy supporting the mutant is Sc or Sd. Since (1) the fitness
of the mutant mixed strategy is always a linear combination of the payoffs that the supporting pure strategies have
against Sa and Sb; (2) both the coward and the Trojan horse are worse against Sa or Sb than Sa or Sb against themselves;
229
230
ANIMAL BEHAVIOUR, 59, 1
(3) Eab =Ebb it follows (equations A8, A13 and A17) that the fitness of SI is always greater than any mutant mixed strategy
that consists of Sc or Sd. Thus SI is an ESS against any mutant mixed strategy.
Appendix 5
Here I show that the mixed strategy playing q*,p* (SI) is an ESS against any mixed strategy playing a different p value and
against Sa, Sb, Sc, Sd or against any combination of these strategies. It can be shown in a similar way as before
(Appendices 2, 3, 4). The only difference is to show that SI is an ESS against Sc. In order for q to be greater than zero, the
inequality CSS >0.5V should hold (equation 12), thus it is difficult to judge from equations (A10) and (A12) whether SI
is an ESS against Sc. However, it can be proved the other way round. From equation (13) it follows that:
E(SWa,SI)>E(Sc,SI)
(A20)
that is:
Assuming that 0.5VCWW >0 and that CSW >0 this should hold.
In the remaining part of the Appendix, I show that SI is an ESS against any mixed strategy playing a different q value.
That is, I investigate whether the condition E(SI,SJ)>E(SJ,SJ) holds for any , where SI (play weak, play strong) is
SI(q*,1q*) and SJ(q*+,1q*). Let us denote E(SWa,SJ) with EWaJ, E(SWb,SJ) with EWbJ and E(SS,SJ) with ESJ. Then
equation (9) can be written as follows:
(1q) ESJ +q(1p) EWaJ +qp EWbJ >(1q) ESJ +(q+) (1p) EWaJ +(q+) p EWbJ
(A22)
after rearrangement:
0>(1p) EWaJ +p EWbJ ESJ
(A23)
The payoff for ESJ can be written with ESI as:
The payoff for EWaJ can be written with EWaI as:
The payoff for EWbJ can be written with EWbI as:
Substituting equations (A24), (A25), (A26) into equation (A23) gives:
Since ESI =EWaI =EWbI, thus:
Rearrangement gives:
CSS >p(CWS +FA)CWW(1+2p(1+p))
(A28a)
Suppose that p.1 then equation (A28a) simplifies to:
CSS >CWS +FA 3 CWW
(A29)
CSS > CWW
(A30)
Suppose that p.0 then it simplifies to:
Since both CSS and CWW is greater than zero, equation (A30) should hold. Thus, if equation (A29) holds then SI is an ESS
against any strategy playing a different q other than q*.