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Animal Behaviour 78 (2009) 949–959

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Animal Behaviour

journal homepage: www.elsevier .com/locate/anbehav

Honest and cheating strategies in a simple model of aggressive communication

Ferenc Szalai a,1, Szabolcs Szamado b,*

a Research Institute for Solid State Physics and Optics, Hungarian Academy of Sciencesb Department of Plant Taxonomy and Ecology, HAS Research Group of Ecology and Theoretical Biology, Eotvos Lorand University

a r t i c l e i n f o

Article history:Received 3 August 2007Initial acceptance 12 November 2007Final acceptance 22 June 2009Published online 27 August 2009MS. number: 9479R

Keywords:cheatingcommunicationhonestymixed equilibriumvalue of information

* Correspondence: S. Szamado, Department of PlantResearch Group of Ecology and Theoretical BiologPazmany Peter setany 1/c., H-1117 Budapest, Hungary

E-mail address: szamszab@ludens.elte.hu (S. Szam1 F. Szalal is at the Research Institute for Solid State P

Academy of Sciences, P. O. Box 49, H-1525 Budapest,

0003-3472/$38.00 � 2009 The Association for the Studoi:10.1016/j.anbehav.2009.06.025

The honesty of communication in competitive situations has long been debated. We investigated thecoexistence of a diverse set of strategies in a simple model of aggressive communication by means ofindividual-based computer simulations. The game is an extended Hawk–Dove game in which there aretwo types of individual, weak and strong, and in which individual can communicate by means of cost-free signals before deciding whether to attack. The available strategies can be classified into threecategories: honest, cheaters and those that ignore the signalling system. We found a diverse set ofequilibria, most of them consisting of a mixture of honest and cheating individuals. We found that whenstarting populations consist of all strategies (1) the honest equilibrium can evolve, (2) communication isalmost always present when signals are informative, and (3) strategies that ignore signalling aregenerally rare. Honest individuals need not be the majority in these populations yet communication willbe present and stable in the long run. In contrast, the pure honest equilibrium is unlikely to evolve whenthe starting populations consist of strategies that ignore signals. Strategies that ignore signals are morefrequent in these types of run however, signalling strategies are still present in the most frequentlyevolved equilibria. Even in this simple system two different kinds of use of signals can evolve: the firstwhen signals refer to resource-holding potential and a second where signals are used to create a payoff-irrelevant asymmetry. In general, regardless of the starting conditions, a low resource value favours weakindividuals, both honest and cheaters, and cowards, medium values favour strong individuals that useand listen to signals, and a high resource value favours strong individuals that ignore the signallingsystem and attack under all conditions. Although it is possible to find parameter combinations witha negative value of information, the value of information is positive in the overwhelming majority ofequilibria. Thus one can conclude that for the majority of parameter combinations an equilibriumevolved that might not be honest, not even on average, but communication is present and signals areworth listening to.� 2009 The Association for the Study of Animal Behaviour. Published by Elsevier Ltd. All rights reserved.

Conflict of interest is a major obstacle in the way of honestcommunication. It follows that honest communication in compet-itive situations always demands special explanation, as by defini-tion conflict of interest exists between competitors. Accordingly,some early investigators claimed that honest communication is notpossible under such circumstances (Maynard Smith 1974). Enquist(1985) was able to show with the help of a simple game-theoreticalmodel that honest communication of relevant states can beevolutionarily stable in such a competitive situation provided thatsome conditions are met. It turned out, however, that it is possibleto find a mixed equilibrium in Enquist’s model in which honest and

Taxonomy and Ecology, HASy, Eotvos Lorand University,.ado).hysics and Optics, HunagrianHungary.

dy of Animal Behaviour. Publishe

cheating strategies can coexist and where the frequency of cheaterscan be arbitrarily high (i.e. close to one; Szamado 2000). There isalso a growing literature on the use of honest and cheating signalsin nature. Examples include Batesian mimicry in butterflies (Wiley1983); reproductive strategies of bluegill sunfish, Lepomis macro-chirus (Dominey 1980; Gross & Charnov 1980) and damselflies,Ischnura ramburi (Robertson 1985); aggressive communication instomatopods (Caldwell & Dingle 1975; Adams & Caldwell 1990),fiddler crabs, Uca annulipes (Backwell et al. 2000), snappingshrimps, Alpheus heterochaelis (Hughes 2000), hermit crabs, Pagu-rus bernhardus (Elwood et al. 2006) and American goldfinchesCarduelis tristis (Popp 1987); and finally signalling in cleaner–clientsystems (with the cleaner Labroides dimidiatus; (Bshary & Grutter2002). In parallel with these observations there are a growingnumber of theoretical models that try to explain the existence ofthese signalling systems (Johnstone & Grafen 1993; Adams &Mesterton-Gibbons 1995; Viljugrein 1997; Szamado 2000; Freck-leton & Cote 2003). Although these models established that

d by Elsevier Ltd. All rights reserved.

Table 1The payoff matrix (after Szamado 2000)

SA ScA SF WA WcA WF

SA 0.5V�Css 0.5V�Css V�FA V�Csw V�Csw V�FA

ScA 0.5V�Css�FP V�Css V V�Csw�FP V�Csw VSF �CssþFF 0 0.5V�Css �CswþFF 0 0.5VWA �Cws �Cws V�FA 0.5V�Cww 0.5V�Cww V�FA

WcA �Cws�FP �Cws V 0.5V�Cww�FP 0.5V�Cww VWF �CwsþFF 0 0.5V �CwwþFF 0 0.5V�Cww

S and W denote strong and weak individuals, respectively, and A, cA and F denoteattack, conditional attack and flee behaviours. V denotes the value of the contestedresource, C** denotes the expected cost of a fight between given pairs of individualswhere * can be weak (w) or strong (s). In addition, there is a cost of attackinga fleeing opponent (FA), and of waiting if the opponent attacks unconditionally (FP),and finally, a fleeing player does not have to pay the full cost of the fight, that is, itgets a fixed reduction (FF) of the fighting cost.

F. Szalai, S. Szamado / Animal Behaviour 78 (2009) 949–959950

cheating strategies can coexist with honest strategies under someconditions, they mostly focused on only one kind of cheating,namely when weak individuals pretend to be strong. However,there can be different kinds of cheating strategies, as well asstrategies that ignore the signalling system as opposed to thesimple scenario described above. Although the existence of thesestrategies is known and the stability of the honest equilibriumagainst these strategies has been investigated by several authors(Owens & Hartley 1991; Johnstone & Norris 1993; Hurd 1997;Szamado 2000), the interactions of these strategies and theemerging signalling equilibria have not yet been investigated. Weran a series of computer simulations to investigate the interactionsof eight different strategies. Two of these strategies can beconsidered as honest, two of them as cheaters, and finally four ofthem ignore the signalling system but each behaves in a differentway. These strategies could interact according to a simplesymmetric game of communication introduced by Enquist (1985).We asked the following questions (1) What kinds of equilibriaevolve out of random and honest starting populations as opposedto starting populations that consist of strategies that ignore signals?(2) What is the frequency of the honest equilibrium at theseequilibria? (3) What kinds of equilibria are possible? (4) What is therelation between the value of information (as defined by Lachmann& Bergstrom 2004) and honesty? (5) Is it possible to get equilibriawith a negative value of information?

Table 2Definition of strategies

Strategies Strong Use signal? Signal Behaviour againstopponent that signals

A B

Honest–strong (Shs) Yes Yes A Fight Conditional attackHonest–weak (Shw) No Yes B Flee FightLiar–strong (Sls) Yes Yes B Flee FightLiar–weak (Slw) No Yes A Flee Conditional attackCoward A (ScA) No No A Flee FleeCoward B (ScB) No No B Flee Flee

THE MODEL

We used Enquist’s (1985) model of aggressive communication.Consider a modified version of the Hawk–Dove game (MaynardSmith 1982), where each player can be weak or strong, witha probability q and 1 � q, respectively, and knows its own level ofstrength but not that of the opponent. The contest consists of twosteps. In the first step, each player can choose between two cost-free signals A or B; in the second step, each animal can give up(flee), attack unconditionally (fight) or attack if the opponent doesnot withdraw (conditional attack). Let V denote the value of thecontested resource, Cww and Css the expected cost of a fightbetween weak and strong individuals, respectively. We assume thatthe expected benefit of a contest between weak individuals isalways greater than zero, 0.5V � Cww > 0. In contrast, we assumethat the expected payoff of a fight between strong individuals issmaller than zero, 0.5V � Css < 0. The reason behind this assump-tion is that weak individuals are expected to settle the contest ina less violent manner (thus probably suffering few or no injuries)than strong individuals. We further assume that a strong animalcan always beat a weak one with a cost Csw, and Cws is the expectedcost that a weak animal should suffer on this occasion. Thefollowing relation holds between these costs: Css > Cws > Cww, andCss > Csw > Cww. In addition, there is a cost of attacking a fleeingopponent (FA), and of waiting if the opponent attacks uncondi-tionally (FP). It is biologically realistic, but not necessary, to assumethat Csw > FA and FP (Hurd 1997). Finally, a fleeing player does nothave to pay the full cost of the fight, that is, it gets a fixed reduction(FF) in the fighting cost. We have chosen the payoffs 0.5V � Css and0.5V � Cww when equal opponents decide to flee instead of theoriginal 0.5V. Then the payoffs for weak and strong contestants canbe written as shown in Table 1.

All-attack A (SaaA) Yes No A Fight FightAll-attack B (SaaB) Yes No B Fight Fight

The table shows the various strategies, whether they use signals or not, and whatbehaviour they select in response to signals A and B. The columns give the name ofthe strategy, its strength, whether the given strategy listens to signals or not, whichsignal the animal should give, and the following behaviour as a function of theopponent’s signal.

Introduction of the Possible Strategies

A strategy is a specification of three things. (1) What kind ofsignal should a player give: A or B? (2) How should it react to signalA: flee, attack or conditional attack? (3) Finally, how should it react

to signal B? Here the options are the same as before. The strategyset (S), which consists of eight strategies (Ns ¼ 8) is as follows.

(1) Honest–strong (Shs): give signal A; if opponent gives signal Athen attack. If it gives signal B then wait until it flees. If it does notflee then attack (that is, after signal B use conditional attack).

(2) Honest–weak (Shw): give signal B; if opponent gives signal Athen flee. If it gives signal B then attack.

(3) Liar–strong (strong) (Sls): give signal B; if opponent givessignal A then flee. If it gives signal B then attack.

(4) Liar–weak (weak) (Slw): give signal A; if opponent givessignal A then flee; if it gives signal B then use conditional attack.

(5) Coward A (weak) (ScA): give signal A; flee regardless of theopponent’s signal.

(6) Coward B (weak) (ScB): give signal B; flee regardless of theopponent’s signal.

(7) All-attack A (strong) (SaaA): give signal A; attack regardlessof the opponent’s signal.

(8) All-attack B (strong) (SaaB): give signal B; attack regardlessof the opponent’s signal.

The strategy set is described in Table 2. Table 3 shows how thestrategies map ðb : S� S/BÞ to behaviours. We call this mappingthe behaviour map.

SIMULATION TECHNIQUE

We investigated the model by means of computer simulationbecause of the complexity of the problem, which makes analyticalsolutions hard to obtain (for a simple three-strategy interaction andinvestigation of cheating as a mixed strategy see Szamado, 2000)The simulation is an individual-based representation of the model.We assume a well-mixed, asexual population of N individuals.Individuals have four genes represented by four bits. These genes

Table 3Behaviour map

Attacker Defender

Shs Shw Sls Slw ScA ScB SaaA SaaB

Shs SA ScA ScA SA SA ScA SA ScAShw WF WA WA WF WF WA WF WASls SF SA SA SF SF SA SF SASlw WF WcA WcA WF WF WcA WF WcAScA WF WF WF WF WF WF WF WFScB WF WF WF WF WF WF WF WFSaaA SA SA SA SA SA SA SA SASaaB SA SA SA SA SA SA SA SA

The table shows how the interaction of various strategies maps to behaviours. Shs:honest–strong; Shw: honest–weak; Sls: liar–strong; Slw: liar–weak; ScA: cowardusing signal A; ScB: coward using signal B; SaaA: all-attack using signal A; SaaB: all-attack using signal B. S and W denote strong and weak individuals, respectively, andA, cA and F denote attack, conditional attack and flee behaviours.

Table 5The parameter space

Parameter Start End Step

V 1 34 3Css 10 35 5Cww 2 32 5Csw 10 35 5Cws 2 32 5FF 0 min (Css, Cww, Csw, Cws) 3

V: the value of the contested resource; Cww: the expected cost of a fight betweenweak individuals; Css: the expected cost of a fight between strong individuals; Csw:the expected cost of a fight for a strong individual versus a weak one; Cws: theexpected cost of fight for a weak individual versus a strong one; FF: the reduction incost of a fight that a fleeing player gets.

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F. Szalai, S. Szamado / Animal Behaviour 78 (2009) 949–959 951

code for the strategies. The first gene (Gt) shows the strength of theindividual which can be strong or weak. Individuals with Shs, Slw,SaaA and SaaB strategies are strong, whereas all others are weak.The next gene (Gus) codes whether the individual listens to signalsor not. All individuals use the signalling system except individualswith the strategies Coward (ScA, ScB) and All-attack (SaaA, SaaB).The last two genes (Gss, Gsw) specify the type of signal used by theindividual depending on its strength. These four genes can unam-biguously code for the eight investigated strategies. For example, anindividual with a genome [0, 0, 0, *] is an honest–strong individualand thus behaves accordingly, as specified in Table 2. Table 4summarizes the possible combinations and shows the geneticrepresentations of the different strategies used in the simulation. Atthe start of each simulation N ¼ 1000 individuals were generated.These individuals receive their fitness by fighting a random numberof individuals according to the model described above. Reproduc-tion is proportional to fitness ranking. Four different initial set-upswere studied: (1) all the possible strategies were present in theinitial population with equal probabilities, (2) only Shs (Honest–strong) and Shw (Honest–weak) strategies were in the initialpopulation, also with equal probabilities, (3) only the ScA (CowardA) strategy was present in the population, (4) only the SaaA (All-attack A) strategy was present. The aim of the second type of runwas to study the evolutionary stability of the honest equilibriumand the effect of the initial composition of strategies on the finalresults. The aim of the third and fourth type of run was to study theevolution of signalling from nonsignalling starting positions. Table5 shows the parameter ranges of the simulations; further details ofthe simulation are described in Appendix 1.

Table 4Genetic representation of the strategy set

Gt Gus Gss Gsw

Shs 0 0 0 *Shw 1 0 * 1Sls 0 0 1 *Slw 1 0 * 0ScA 1 1 * 0ScB 1 1 * 1SaaA 0 1 0 *SaaB 0 1 1 *

An asterisk denotes a wildcard; here values were chosen randomly. The first gene(Gt) shows the strength of the individual which can be strong or weak. The next gene(Gus) codes whether the individual listens to signals or not. The last two genes (Gss,Gsw) show the type of signal used by the individuals depending on their strength.Shs: honest–strong; Shw: honest–weak; Sls: liar–strong; Slw: liar–weak; ScA:coward using signal A; ScB: coward using signal B; SaaA: all-attack using signal A;SaaB: all-attack using signal B.

DATA PROCESSING AND RESULTS

The raw result of the simulations consists of the frequencies ofthe different strategies after the iteration I (where I ¼ 1000).Because I was very high and the fluctuation of frequencies (f) waslow (less then 3%) in the last few iterations the Fj ¼ fj(i ¼ I) can beinterpreted as the final equilibrium frequencies of different strat-egies. Figs. 1 and 2 show two examples of two different sets ofparameters (V ¼ 10, Css ¼ 25, Cww ¼ 17, Cws ¼ 20, Csw ¼ 22, FF ¼ 15;and V ¼ 1, Css ¼ 20, Cww ¼ 2, Cws ¼ 10, Csw ¼ 12, FF ¼ 0,

0

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100 200 300 400 500 600 700 800 900 1000Generation

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Figure 1. The frequency of the different strategies as a function of time for the fourstarting positions: example 1. Runs: (a) all strategies, (b) only the honest strategies(Honest–strong and Honest–weak), (c) only the Coward A strategy or (d) only the All-attack A strategy were present in the starting populations respectively. Strategies:green: Honest–strong; red: Honest–weak; blue: Liar–strong; purple: Liar–weak; lightblue: Coward A,B; yellow: All-attack A,B. Parameters: V ¼ 10, Css ¼ 25, Cww ¼ 17,Cws ¼ 20, Csw ¼ 22, FF ¼ 15.

00.10.20.30.40.50.6

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Figure 2. The frequency of the different strategies as a function of time for the fourstarting positions: example 2. Runs: (a) all strategies, (b) only the honest strategies(Honest–strong and Honest–weak), (c) only the Coward A strategy or (d) only the All-attack A strategy were present in the starting populations respectively. Strategies:green: Honest–strong; red: Honest–weak; blue: Liar–strong; purple: Liar–weak; lightblue: Coward A,B; yellow: All-attack A,B. Parameters: V ¼ 1, Css ¼ 20, Cww ¼ 2,Cws ¼ 10, Csw ¼ 12, FF ¼ 0.

F. Szalai, S. Szamado / Animal Behaviour 78 (2009) 949–959952

respectively) for all four types of different starting composition.Fig. 1 gives an example where the same type of equilibrium evolvedregardless of the starting composition of strategies and it remainedstable through time aside from random fluctuations. In contrast,Fig. 2 gives an example where different equilibria evolved in thefirst and in the last two types of run. To decrease the statistical errorof results, R ¼ 100 runs were made with every combination ofparameters using different random number seeds. Our raw data arethe average of these independent runs.

Figure 3 shows the averages of these frequencies as a function ofthe value of the resource (V). It can be seen that the distributions ofF values for a given V value are not uniform.

We introduced a simple code C to represent the differentcombinations of strategies in the final state. The code has valuesbetween 1 and 64. Code number 1 means that only the Shs (Honest–strong) strategy was present in the final state, while 64 means thatall strategies can be found in the final state. Table 6 summarizes thecode values and the corresponding strategy combinations. Astrategy combination distribution P(C) can be calculated using thiscode to represent the frequency of the different final state strategycombinations in the set of all possible final state combinations for allfour kinds of run. See Table 6 for the results.

To measure the value of information at the final stages of thesimulations we had to calculate the expected fitness gain at a given

final stage using signals (A0) and the expected gain at the same finalstage without signals (A1 and A2). In the second case, when playersdo not use signals, we have two assumptions: either they know thefrequency distribution of the strategies at that given final stage, orthey have no information about it at all. Accordingly, we calculatedthe expected gain without signals in two different ways reflectingthe above assumption (A1 and A2, respectively; see Appendix 2 fordetails). A P(G) frequency distribution describes the value ofinformation given our two assumptions for all the four differenttypes of initial conditions. The results are depicted in Fig. 4.

We also calculated the sum of average frequencies of the honeststrategies (Shs (Honest–strong and Shw (Honest–weak)), theliar strategies (Sls (Liar–strong) and Slw (Liar–weak)), and thosestrategies that ignore signals (ScA, ScB (Coward A,B) and SaaA, SaaB(All-attack A,B)) at each kind of equilibrium. Fig. 5 depicts thesesums for all the four types of starting positions; the strategies areordered according to the proportion of honest individuals in thefirst kind of run, and the value of information is also depicted tohelp further evaluation.

DISCUSSION

The possibility of cheating has been investigated in severalmodels (Johnstone & Grafen 1993; Adams & Mesterton-Gibbons1995; Szamado 2000). The stability of cheating at equilibrium isdue to external factors in the majority of these models, mainlybecause of some qualitative difference between cheaters andhonest individuals (Johnstone & Grafen 1993; Adams & Mesterton-Gibbons 1995). Szamado (2000), however, investigated a situationin which cheating was due to choice of action and thus is main-tained at the equilibrium by endogenous forces. He found thatcheating can be part of a mixed equilibrium, and that contrary toprevious expectations a large proportion of weak individuals cancheat, in the extreme case almost all of them. Here we haveexpanded Szamado’s (2000) investigation by introducing othercheating strategies, as well as strategies that ignore the signallingsystem in one way or the other. Our first main result is that honestcommunication can evolve out of a random starting distribution ofstrategies given the right set of parameters (see Table 6). Thus, inthese cases honest signalling has the properties of both evolu-tionary and convergence stability. Second, the simulations startingfrom honest starting strategies resulted in a very similar distribu-tion of final strategy distributions (see Table 6) compared torandom starting compositions. The number of honest equilibriaincreased but only slightly. This implies that there are only a fewparameter combinations at which honest signalling is evolution-arily stable but not convergence stable; and that the results of thesimulations are fairly robust in respect to the starting distributionof strategies, as long as signalling strategies are present in thestarting population. Third, the pure honest signalling equilibrium isvery sensitive to the composition of the starting population inregard to the absence of signalling strategies, that is, it is unlikely toevolve if the starting population consists only of strategies thatignore signals. Fourth, it follows from the previous points that thepure honest signalling equilibrium is neither evolutionarily norconvergence stable for the majority of the parameter combinations.Fifth, strategies that ignore the signalling system support only a fewof the final states (see Table 6) for the first two types of run (i.e.when signalling strategies were present at the start). When thesimulations start from strategies that ignore signals (i.e. runs star-ted from All-attack A and Coward A) these types are more frequentat the equilibria (see Table 6); however, even in these runs themajority of equilibria consist of a mixture of strategies includingsome of those that use signals. All in all, this supports the analyticalconclusion of Szamado (2000) who said that stable signalling

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Figure 3. The dependence of the average frequencies of the strategies on the value of resource V: (a) when all strategies were represented in the initial states, (b) when only honeststrategies were represented in the initial states, (c) when only the Coward A strategy was present in the initial states, (d) when only the All-attack A strategy was present in theinitial states. Shs (Honest–strong), Shw (Honest–weak), Sls (Liar–strong) and Slw (Liar–weak), ScA, ScB (Coward A,B) and SaaA, SaaB (All-attack A,B).

F. Szalai, S. Szamado / Animal Behaviour 78 (2009) 949–959 953

systems need not be honest, not even on average; however, it begsthe question of what these signals are used for if not to transmitreliable information about strength.

To answer this question it might be worth listing those fiveequilibria that were found in about 10% or more of the simulations,for the first two types of run (note that these equilibria do notcoexist for any given parameter combination, but these are themost frequently observed equilibria within the parameter rangeexplored; the code and frequency of the given strategy combinationin the first two types of run, respectively, are given in parentheses):

(1) Honest–strong and Liar–strong (C ¼ 5; 36%, 32.5%);(2) Honest–strong, Liar–strong and Liar–weak (C ¼ 13; 9.6%,

7.7%);(3) Honest–strong and Honest–weak (C ¼ 3; 7.3%, 10.4%);(4) Honest–strong, Honest–weak and Liar–weak (C ¼ 11; 19.7%,

19.9%);(5) Honest–strong, Honest–weak, Liar–weak and Coward

(C ¼ 27; 7.5%, 9.8%).By Analysing the results of the first type of run we can see the

following.(1) It turns out that the majority of these equilibria consist of

Honest–strong and Liar–strong individuals (C ¼ 5; see Table 6). Thisequilibrium was found in 36% of the parameter combinationsinvestigated. At this equilibrium the signal serves to split thepopulation into two groups, each group fighting only withmembers of the same group and one group yielding to the otherwithout a fight (Liars to Honest ones). Thus, the use of a signalcreates a so-called payoff-irrelevant asymmetry (Maynard Smith &Parker 1976; Hammerstein 1981), which in turn can be used todecide the outcome of the contest (Maynard Smith & Parker 1976;Hammerstein 1981). This way the signal confers information of

value to the receiver, but about the expected choice of behaviouras a function of the signal rather than the level of strength. In thissense both Honest–strong and Liar–strong individuals use thesignal in an honest way at this particular equilibrium. Potentialcheaters would be strong individuals giving signal B and then eitherattacking unconditionally (All-attack A) or fleeing unconditionally(Coward A). However, this kind of equilibrium (Honest–strong,Liar–strong and All-attack A) is not observed if the simulations startfrom the honest equilibrium and it is very rare for random startingstrategies. Also, the equilibrium Honest–strong, Liar–strong, andCoward A is not observed at all. This shows that this particular useof signal is very robust against exploitation.

(2) Yet another frequently found equilibrium is Honest–strong,Liar–strong and Liar–weak (C ¼ 13). It was found in 9.6% of cases.This equilibrium is practically the previous equilibrium with Liar–weak individuals added. The interesting feature of this equilibriumis that the weak individuals give the same signal as Honest–strongones (and of course they have to pay the cost of it). Yet anotherinteresting feature is that the counterpart of this equilibrium, inwhich the weak individuals give the same signal as Liar–strongones (i.e. the Honest–strong, Honest–weak and Liar–strongcombination, C ¼ 6), is 10 times less frequent than this one. Onemight think that this is because Liar–weak individuals try to fleefrom Honest–strong ones, whereas Honest–weak individuals willfight against Liar–strong ones. However, according to the model,both weak types have to pay the full cost of the contest, so theremust be some other factor behind the difference.

(3) The second most frequent type of equilibrium is built aroundthe Honest–strong and Honest–weak strategy pair. The first case ofthis type is the Honest–strong and Honest–weak (C ¼ 3) equilib-rium itself. This is the equilibrium in which the use of signals gives

Table 6Final frequencies of strategy combinations

C Allstrategies

Onlyhoneststrategies

OnlyCoward Astrategy

OnlyAll-attackA strategy

Strategy combination

1 0.001 0.017 0 0 Shs2 0.001 0.001 0 0 Shw3 0.073 0.104 0.0005 0.003 Shs, Shw5 0.363 0.325 0 0 Shs, Sls7 0.018 0.009 8.7e�5 0 Shs, Shw, Sls9 0.0489 0.049 8.7�5 0 Shs, Slw10 0.01 0.015 0.001 0.001 Shw, Slw11 0.197 0.199 0.05 0.048 Shs, Shw, Slw13 0.096 0.077 8.7e�5 0.0001 Shs, Sls, Slw15 0.025 0.025 0.02 0.002 Shs, Shw, Sls, Slw19 0.004 0.006 0.02 0.002 Shs, Shw, Sc (A or B)23 8.7e�05 0 0 0 Shs, Shw, Sls, Sc

(A or B)24 0 0.0001 0 0 Slw, Sc (A or B)25 8.7e�05 0.0006 0.0003 0.0001 Shs, Slw, Sc (A or B)26 0.006 0.008 0.015 0.015 Shw, Slw, Sc (A or B)27 0.075 0.098 0.119 0.122 Shs, Shw, Slw, Sc

(A or B)29 0.032 0.03 0.0003 0.0003 Shs, Sls, Slw, Sc

(A or B)31 0.008 0.006 0.017 0.0156 Shs, Shw, Sls,

Slw, Sc (A or B)32 0.021 0.005 0 0 Saa (A or B)33 0.007 0.009 0.157 0.156 Shs, Saa (A or B)35 0 0 0.034 0.034 Shs, Shw, Saa (A or B)37 0 0.006 0.176 0.174 Shs, Sls, Saa (A or B)39 0 0 0.039 0.0396 Shs, Shw, Sls, Saa

(A or B)41 8.7e�05 0.001 0.033 0.033 Shs, Slw, Saa (A or B)43 0 0 0.155 0.157 Shs, Shw, Slw, Saa

(A or B)45 0 0 0.054 0.055 Shs, Sls, Slw, Saa

(A or B)47 0 0 0.048 0.048 Shs, Shw, Sls, Slw,

Saa (A or B)48 0 0.0006 0 0 Sc (A or B), Saa

(A or B)51 0 0 0.001 0.001 Shs, Shw, Sc

(A or B), Saa (A or B)53 0 0 8.7e�5 0.0001 Shs, Sls, Sc (A or B),

Saa (A or B)55 0 0 0.002 0.002 Shs, Shw, Sls, Sc

(A or B), Saa (A or B)56 0.0002 0.001 0 0 Slw, Sc (A or B), Saa

(A or B)57 0.0003 0.003 0.005 0.005 Shs, Slw, Sc

(A or B), Saa (A or B)59 0 0 0.028 0.026 Shs, Shw, Slw, Sc

(A or B), Saa (A or B)61 0 0 0.01 0.01 Shs, Sls, Slw, Sc

(A or B), Saa (A or B)63 0 0 0.04 0.042 Shw, Sls, Slw, Sc

(A or B), Saa (A or B)

Summary of the distribution P(C) of strategy combinations for the four differentinitial conditions. In the case of missing C values both of the initial conditionsresulted in zero values. Shs: honest–strong; Shw: honest–weak; Sls: liar–strong;Slw: liar–weak; ScA: coward using signal A; ScB: coward using signal B; SaaA: all-attack using signal A; SaaB: all-attack using signal B.

0

0.05

0.1

0.15

0.2

0.25

−2 0 2 4 6 8 10 12 14

P(G

)

G

Gf1Gf2Gh1Gh2Gc1Gc2Ga1Ga2

Figure 4. The value of information. The P(G) distribution depicts the frequency of finalstrategy combinations as a function of the benefit resulting from the use of signals. TheX axis denotes the fitness gain G resulting from the use of signals, and the Y axisdenotes the proportion of strategy combinations that gets the given amount of gain.The Gf, Gh, Gc and Ga values denote those distributions in which random, only honest,only Coward A and only All-attack A strategies were present in the initial state,respectively.

F. Szalai, S. Szamado / Animal Behaviour 78 (2009) 949–959954

reliable information about the strength of the signaller. This equi-librium was found in 7.3% of cases starting from a random distri-bution of strategies. This is very far from the majority, but it isimpressive enough if we consider that this means that honestevolutionarily stable communication can evolve out of a randomdistribution of strategies. This type of equilibrium is less robustagainst exploitation as the next equilibrium shows.

(4) The fourth equilibrium that was found often is an extensionof the previous one: Honest–strong, Honest–weak and Liar–weak(C ¼ 11). This is the classical ‘cheating’ scenario. Weak individuals

give the signal used by strong ones to deceive other weak indi-viduals. This type of equilibrium was found in 19.7% of cases. Thus,after the Honest–strong, Liar–strong (C ¼ 5) equilibrium this is thesecond most frequent final state. The proportion of cheaters canvary considerably at this equilibrium.

(5) The last of the frequently found equilibria (7.5%) consists ofthe same three strategies as in the previous one plus the cowardstrategy: Honest–strong, Honest–weak, Liar–weak and Coward(C ¼ 27). This is the only equilibrium out of the five most frequentones that includes a strategy that ignores the signalling system.Notably, this strategy is the Coward strategy, which always fleesregardless of the signal, and not the all-attack strategy (whichwould always attack).

By analysing the most frequent equilibria, one can concludethat even in this simple system two different kinds of use ofsignals are possible. In the first use, signals are used to transmitinformation about strength (resource-holding potential, RHP,Parker 1974), whereas in the second signals are used to createa payoff-irrelevant asymmetry. Accordingly, the majority ofequilibria are built either around the Honest–strong, Honest–weak signalling pair (signalling RHP, i.e. the ‘traditional’ use) oraround the Honest–strong, Liar–strong signalling pair (i.e. payoff-irrelevant asymmetry). However, while the second pair is themost frequent equilibrium, the second most frequent is not theHonest–strong, Honest–weak pair, but the ‘traditional’ cheatingscenario. Thus, the second kind of use (payoff-irrelevant asym-metry) seems to be robust against cheaters, while the ‘traditional’use is much less so.

The simulations that started with honest strategies gave verysimilar results (see Table 6). The five most frequent strategies werethe same five as above. There are, however, some differences.Combinations consisting of Honest–strong and Liar–strong indi-viduals (Honest–strong and Liar–strong, C ¼ 5; and Honest–strong,Liar–strong, Liar–weak, C ¼ 13) both decreased in frequency (from36.3% to 32.5% and from 9.6% to 7.7%, respectively), whereas theHonest–strong, Honest–weak, C ¼ 3 equilibrium was found moreoften (from 7.3% to 10.4%, see Table 6. This shows that the results of

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2 3 1 7 19 5 11 37 15 9 13 27 31 10 26 29 33 57 56 32 8 63 62 61 60 6 59 58 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 4 39 38 36 35 34 30 28 25 24 23 22 21 20 18 17 16 14 12 0

Freq

uen

cyIn

form

atio

n v

alu

e

Code

(a)

0

0.2

0.4

0.6

0.8

1(b)

0

0.2

0.4

0.6

0.8

1

(c)

0

0.2

0.4

0.6

0.8

1(d)

0

0.2

0.4

0.6

0.8

1

10 20 30 40 50 60 70

(e)

Figure 5. The value of information and the frequency of different strategies supporting the given strategy combination as a function of the four starting set-ups. The codes of thestrategy combinations are given above Fig. 5a. The strategies are ordered according to the average proportion of honest–strong individuals in the first type of run. (a) The value ofinformation. the -: All the possible strategies were present in the starting population; :: only honest strategies were present; >: only cowards were present; B: only the all-attack strategy was present. In b–e the frequency of different strategies supporting the given strategy combination is given on the Y axis. -: Sum of the frequency of all the honeststrategies (Shs (Honest–strong) and Shw (Honest–weak)); 7: sum of the frequencies of all the liar strategies (Sls (Liar–strong) and Slw (Liar–weak)); >: sum of the frequencies ofall the strategies that ignore signals (ScA, ScB (Coward A,B) and SaaA, SaaB (All-attack A,B)). Runs: (b) all the possible strategies were present in the initial population, (c) only honeststrategies were present, (d) only the coward strategy was present, (e) only the all-attack strategy was present.

F. Szalai, S. Szamado / Animal Behaviour 78 (2009) 949–959 955

the simulations are fairly robust in respect to the distribution ofstarting strategies, and that most of the honest equilibria are bothevolutionarily and convergence stable as long as the honest strat-egies are present in the starting populations.

The simulations that started from the ScA (Coward A) strategyand from the SaaA (All-attack A) strategy gave a different picture.The most frequent strategies (found in more than 5% of thesimulations) are as follows (frequency of these strategies forScA (Coward A) and SaaA (All-attack A) starting strategy,respectively):

(1) Honest–strong, Honest–weak and Liar–weak (C ¼ 11; 5% and4.89%);

(2) Honest–strong, Honest–weak, Liar–weak and Coward(C ¼ 27; 11.9% and 12.2%);

(3) Honest–strong and All-attack (C ¼ 33; 15.7% and 15.5%);(4) Honest–strong, Liar–strong, and All-attack (C ¼ 37; 17.6%

and 17.4%);(5) Honest–strong, Honest–weak, Liar–weak and All-attack

(C ¼ 43; 15.5% and 15.7%);(6) Honest–strong, Liar–strong, Liar–weak and All-attack

(C ¼ 45; 5.4% and 5.5%).By comparing these results with the first two types of run one

can see that only the traditional ‘cheating scenario’, that is, C ¼ 11(Honest–strong, Honest–weak and Liar–weak) and the

F. Szalai, S. Szamado / Animal Behaviour 78 (2009) 949–959956

combination of this with the Coward strategy (i.e. C ¼ 27)frequently emerged in all types of run (see Table 6). However,comparing the rest of the strategies shows a clear pattern,namely, that the most frequent equilibria in these types of runare the same as the other three equilibria in the first two types ofrun combined with the All-attack A strategy. Thus while theHonest–strong and Liar–strong (C ¼ 5) equilibrium is rare, theHonest–strong, Liar–strong and All-attack A (C ¼ 37) one isfrequent. The Honest–strong, Liar–strong and Liar–weak (C ¼ 13)equilibrium is rare the Honest–strong, Liar–strong, Liar–weak andAll-attack A (C ¼ 45) one is frequent. Finally, the Honest–strong,Honest–weak and Liar–weak (C ¼ 11) equilibrium is less frequentwhile the Honest–strong, Honest–weak, Liar–weak and All-attackA one (C ¼ 43) is a lot more frequent. The only exception is thepure honest equilibrium (Honest–strong and Honest–weak,C ¼ 3). While reasonably frequent in the first two types of run ithas no ‘counterpart’ in the simulations that started from theCoward A and All-attack A strategies. The strategy Honest–strongand All-attack (C ¼ 33) is the closest to it; however, the Honest–weak strategy is ‘missing’. This shows that the traditional honestsignalling equilibrium is the most sensitive to the composition ofthe starting population in respect to signalling strategies. If thestarting population consists to signalling strategies then the purehonest signalling equilibrium can evolve; if, however, the startingpopulation consists of a type that ignores signalling (i.e. CowardA or All-attack) then the pure honest signalling equilibrium haslittle chance of appearing.

This makes sense but it prompts the obvious question of howhonest signalling can evolve, as most would regard the non-signalling equilibrium as the obvious starting position in nature.First, signalling does evolve even from the starting position ofCoward A or All-attack populations, and not just from the purehonest signalling of RHP. Both the ‘cheating scenario’ (with orwithout Coward A or All-attack, i.e. C ¼ 11, C ¼ 27, C ¼ 43) and thesignalling of a payoff-irrelevant asymmetry (i.e. equilibria C ¼ 37and C ¼ 45) readily evolve. Second, this strongly suggests that forthe traditional honest signalling of RHP to emerge there has to besome cues already present in the population. Cues are morpho-logical features or behaviours that evolved for reasons other thansignalling yet can reveal some relevant piece of information aboutthe animal. For example, one of the most obvious cues is the size ofthe animal, as it can be correlated with strength and condition.Such cues, if present, can be worth heeding and in turn may evolveinto signals (i.e. may be enlarged or emphasized in some otherway). Actually, this is exactly what we can see in the case of threatdisplays. Here we see the first steps of species-specific fightingtechniques (which usually involve the presentation of weapons)‘frozen’ into threat displays (Walther 1984; see Szamado 2003) fora more detailed discussion). Obviously, for one to fight, weaponshave to be prepared. This is a cue, because the intention is not toinform the other; however, it can be used by one opponent to readits intention. This is turn can select for the first step to be presentedfor longer than is necessary for strict fighting efficiency simply togive more time for the opponent to read intentions. Of course, thiswill happen only if it is in the interest of the first player to informthe other player of its intentions (and intentions in the case of thehonest equilibrium are correlated with RHP). In terms of the modelit means that the situation is in the range of the parameter spacethat allows the evolution of honest signalling. If so, then at the endof this process cues presented for reasons of fighting efficiency willbe frozen into signals that can transfer information about inten-tions and RHP.

Investigating the effect of the value of the resource on thefrequency of the various strategies at equilibrium (Fig. 3) revealsfour robust results.

(1) The frequency of weak individuals decreases monotonicallyas the value of the resource increases; conversely, the frequency ofstrong individuals increases monotonically.

(2) Within strong strategies both the Honest–strong and theLiar–strong have a maximum.

(3) Conversely, the frequency of individuals playing the All-attack strategy increases monotonically as a function of the value ofthe resource.

(4) The frequency of Cowards decreases monotonically as thevalue of the resource increases.

These trends are present in all types of run. A low resource valuefavours weak individuals, both honest and cheaters, and Cowards,medium values favour strong individuals that use and listen tosignals (i.e. Honest–strong and Liar–strong), while weak individ-uals are still present, and finally, a high resource value favoursstrong individuals that ignore the signalling system and attackunder all conditions (i.e. All-attack). These results support previousmodels that predicted that high values of the resource compared tothe cost of fighting do not favour communication (Maynard Smith &Harper 1988; Enquist & Leimar 1990). The results also support theconjuncture that honest communication about RHP is most likelywhen the value of the resource is smaller than the expected cost offighting.

It is also worth investigating the relation between honesty andthe value of information. One way to investigate the potentialbenefits of using signals is to calculate the value of information(Lachmann & Bergstrom 2004). This value shows the expectedbenefit that the receiver can gain by listening to the signalscompared to a situation when no signals are used. Analytical resultspredict that this expected benefit of listening to signals mustalways be greater than zero at separating signalling equilibria(Lachmann & Bergstrom 2004) but the value of signals might benegative at pooling equilibria (Lachmann & Bergstrom 2004). Atseparating equilibrium each signaller type gives a signal of differentintensity, so receivers can reliably infer the quality of the signallarfrom the signal (Bergstrom et al. 2002), whereas at a poolingequilibrium different types of signaller might give the same signal(i.e. form ‘pools’, Lachmann & Bergstrom 1998). Thus any equilib-rium at which cheaters or those strategies that ignore the signallingsystem can be found is a pooling equilibrium. It is thus interestingto ask whether the value of information is negative or positive atthese equilibria. We have calculated the value of information in twodifferent ways, according to whether we assume the members ofthe population have expectations about the frequency distributionof strategies of the given population or not (see Appendix 2 fordetails). The results are depicted in Fig. 5, which shows that in theoverwhelming majority of the final states it is worth using signals,that is, the value of signals is higher than zero (G > 0). Fig. 5 alsoshows that the value of information on average is higher at thoseequilibria that evolved out of starting populations that consisted ofsignalling strategies. It is clear that the value of information is alsohigher when we assume that the individuals do not have expec-tations about the frequencies of the different strategies in thepopulation, which is reasonable as they gain more with the use ofsignals. Overall, for the first two types of run, there are 563parameter combinations for which the value of information at thegiven final state is smaller than zero. This is around 0.05% of all theparameter combinations investigated . Most of these final statesconsist of either the All-attack strategy (C ¼ 32), or the Honest–strong, Liar–weak mixed strategy (C ¼ 9). Thus, one can concludethat even at those equilibria supported by cheating strategies orstrategies that ignore the signalling system the value of informationis greater than zero.

Finally, one can ask what the relation of the results is to theobserved frequencies of cheaters in natural populations. Here we

F. Szalai, S. Szamado / Animal Behaviour 78 (2009) 949–959 957

discuss only those examples in which cheating was observed in thecontext of aggressive communication and in which the proportionof cheaters is documented. In American goldfinches Popp (1987)found that subordinates occasionally (10.9% and 9.3% in two groups,respectively) tried to bluff against dominants by initiatingencounters with the highest intensity threat. Importantly, thesesubordinates lost the majority of these encounters. In thestomatopod Gonodactylus bredini, ‘meral spread’, which serves asa threat display, is given by some of the newly moulted individualsthat are unable to back up the threat with force (Adams & Caldwell1990). The proportion of these bluffing individuals among thenewly moulted ones depends on the size asymmetry between thecontestants. Larger individuals bluff more, and the frequency ofbluffing varies between 30 and 60% of individuals (Adams & Cald-well 1990) and can be as high as 88% (Steger & Caldwell 1983).Because moulting occurs continuously during the year there isroughly a constant proportion of freshly moulted individuals in anygiven population. Steger & Caldwell (1983) estimated that up to 20%of the individuals using the meral spread display could havemoulted in the past 5 days and hence would be bluffing. In thefiddler crab Uca annulipes, males can lose their claw as a result ofpredator attacks. The regenerated claw is very similar to the orig-inal both in size and in shape and apparently serves equally well interritorial displays. However, because of its lower mass it is muchless efficient in fighting (Backwell et al. 2000). The frequency of‘cheaters’ can be very high in some populations, ranging from 16 to44% (Backwell et al. 2000). While there are other documentedexamples of cheating in an aggressive context (Hughes 2000;Elwood et al. 2006), in snapping shrimps bluffing seems to occurduring the assessment stage and not as a threat display (Hughes2000), and in the study on hermit crabs the proportion of dishonestsignals was not given (Elwood et al. 2006). There is one commonthing, however, namely that in both cases the weaker, smalleropponent seemed to be more predisposed towards bluffing thanthe larger one. Although in the fiddler crab example the frequencyof cheaters is maintained by an exogenous force (i.e. by thefrequency of predation attempts), in the other situations theadvantage of bluffing seems to be frequency dependent and hencethe probability of bluffing seems to be maintained by this effect(goldfinches; stomatopods). Given the previous theoretical result(Szamado 2000) and the results of this study, similar futureobservations cannot be excluded. One of the major challenges forfuture empirical studies is to measure and document this frequencydependence.

Conclusions

The results of our computer simulations reinforce theprevious analytical result (Szamado 2000) that there is noa priori reason to assume that an evolutionarily stable commu-nication system should be honest, even on average. The simu-lations have shown that a large variety of equilibria is possiblewith varying degrees of honesty. The honest equilibria canevolve from random combinations of strategies but are unlikelyto evolve when the starting population consists of strategies thatignore signalling. In general, two kinds of honest use of signalsare possible: in the first case signals refer to strength (RHP),whereas in the second case signals create a payoff-irrelevantasymmetry. The first type where signals refer to RHP seems to bemore sensitive both to the composition of starting strategies andto cheating. A low resource value compared to the cost offighting favours weak individuals and signalling, although itfavours cheaters as well. A high resource value favours strongindividuals that attack and ignore signals. Intermediate valuesfavour strong individuals that use and listen to signals and weak

types can also be present. In almost all of the equilibria the valueof information is above zero, that is, signals are worth heeding,even when the given equilibrium is supported by cheatingstrategies and/or by strategies that ignore signals. All in all, ourresults show that even in this simple model various equilibriacan evolve and can be evolutionarily stable. Signalling is notblack or white where the majority of the population have to behonest signallers. Various strategies can coexist and this coex-istence cannot be excluded in nature. When researchers seek outthis variety, a more diverse picture of signalling will emerge.

Acknowledgments

We thank Istvan Scheuring for his helpful comments. This workwas supported by the European Union Framework 6 (Grant IST-FET1940), the National Office for Research and Technology (GrantNAP2005/CKCHA005) and the Hungarian Research Foundation(OTKA) under grant no. T049692.

References

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Bergstrom, C. T., Szamado, S. & Lachmann, M. 2002. Separaing equilibria incontinuous signalling games. Philosophical Transactions of the Royal Society B,357, 1595–1606.

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APPENDIX 1. STEPS IN THE SIMULATION

In the present simulation the number of iterations I was 1000.This number is large enough to get stable nonoscillatory equilib-rium solutions in a wide range of parameter space. To decrease thestatistical error of results R ¼ 100 runs were made with everycombination of parameters using different random number seeds.The result of the simulations consists of the frequencies of thedifferent strategies after the iteration I (where I ¼ 1000). Thesefrequencies can be defined with the following formula:

fjðiÞ ¼1R

XR

r¼1

Nrj ðiÞN

; i ¼ 1.I (A1)

where Nrj ðiÞ is the number of individuals with the Sj strategies in the

ith iteration and in the r repetition of simulation.Each iteration consists of two steps: fight and reproduction.

During a fight there are N number of turns; in each turn we selectone individual randomly, which then fights Fn fights, where Fn isa random number between 0 and 50. Reproduction comes after thefighting stage.

The fitness of an individual is based on its success competing fora resource with a value V. The opponent is chosen randomly foreach fight from the set of all other individuals. The two chosenindividuals then play the game as described in the model section.The fitness gain of a fight is calculated from the payoff matrix(Table 1).

The second stage of each iteration, after the fighting stage, isreproduction. First, a rank order is created based on the fitness ofthe individuals. This rank order will determine which individualscan reproduce and which ones have to die. The number of repro-ducing individuals depends on the Ps ‘survival’ probability. With thehelp of Ps we divide the population into three parts: the firstN(1 � Ps), the middle N(2Ps � 1) and the last N(1 � Ps) individuals.The first NPs individuals can reproduce. Out of these individuals thefirst N(1 � Ps) individuals produce two offspring, while the nextN(2Ps � 1) individuals produce one new individual. The remaining(N(1 � Ps)) individuals die without leaving their genes to the nextgeneration. We used Ps ¼ 0.9 during the simulations. For furtherexplanation see below.

The production of new individuals is asexual. There is a Pm

mutation probability with which genes can flip from one state tothe other. This probability Pm ¼ 0.001 was fixed during the simu-lations for all genes. Test simulations showed that provided themutation rate is sufficiently small the value of mutation onlymodifies the length of the initial transient behaviour but it will notmodify the finial states.

The Parameter Space

The simulation has a huge parameter space with a decentnumber of different model- and simulation technique-specificparameters. The relevant parameters are collected in thissection.

The payoff matrix is influenced by eight parameters (V, Css, Cww,Csw, Cws, FF, FA, FP). Two of these parameters (FA, FP) were keptconstant during the simulations (because it is biologically reason-able to keep these parameters at a low value).

Table 5 shows the ranges of parameters with the appropriatestep widths. The value of the fixed parameters was chosen to be 2.0.According to the theory, they should be greater than zero but stillsmall so their value was chosen to be the lowest boundary of thepayoff parameters.

The value of the remaining simulation parameters, Fn and Ps,were chosen to represent possible realistic values. However, as wefound in a detailed study of the sensitivity of the model to theseparameters extreme values of these parameters can greatly modifythe final states of the simulations. There are two main phases in themodel: one with normal a nonoscillatory phase and anotherstrange one with high fluctuations. In general, this strange phaseappears near to the minima of the Ps ¼ 0.5. It is easy to see why,because in this case half of the population dies at the same time,which can cause huge fluctuations in the composition of thepopulation.

To run the total number of parameter combinations with fourdifferent types of initial condition required huge computationalpower for which we used a grid system.

APPENDIX 2. THE VALUE OF INFORMATION

The value of information is calculated using the followingequation:

Gi ¼ A0 � Ai; i ¼ 1;2 (A2)

The A0 value is the expected fitness at a given final state whensignals were used and listened to (except, of course, those strategiesthat do not listen to signals). This is a weighted average of all thepayoff values for all the possible behaviour combinations. This A0

value is calculated as follows:

A0 ¼1

Ns � 1

XNs�1

i¼0;j¼0

FjPoff�b�Si; Sj

�; b�Sj; Si

��(A3)

where Sj and Fj denote the jth strategy and its frequency in the finalstate, respectively; Poff is the payoff matrix defined in Table 1, andb denotes the behaviour map defined in Table 3.

The A1 value is the expected fitness gain when players do not usesignals but have a (correct) expectation about the frequency of thedifferent strategies at the given final state.

In this case we assume that individuals will select the possiblebehaviours that are defined in their behaviour map b according tothe expected frequency of different strategies, that is, the fitness isa weighted average of all the payoff values for all the possiblebehaviour combinations.

To calculate this value we first introduce the subset of behav-iours associated with a given strategy:

BjS ¼�

B���cS0˛S

�B ¼ b

�S; S0

��(A4)

Table A1Modified behaviour map

Attacker Defender

Shs Shw Sls Slw ScA ScB SaaA SaaB

Shs SA SA SA SA SA SA SA SAShw WF WF WF WF WF WF WF WFSls SF SF SF SF SF SF SF SFSlw WF WF WF WF WF WF WF WFScA WF WF WF WF WF WF WF WFScB WF WF WF WF WF WF WF WFSaaA SA SA SA SA SA SA SA SASaaB SA SA SA SA SA SA SA SA

Modified behaviour map when the individuals do not use signals and have noinformation about the frequency distribution of the strategies at the given equi-librium. Shs: honest–strong; Shw: honest–weak; Sls: liar–strong; Slw: liar–weak;ScA: coward using signal A; ScB: coward using signal B; SaaA: all-attack using signalA; SaaB: all-attack using signal B. S and W denote strong and weak individuals,respectively, and A, cA and F denote attack, conditional attack and flee behaviours.

F. Szalai, S. Szamado / Animal Behaviour 78 (2009) 949–959 959

Using this notion the A1 average can be defined as follows:

A1 ¼1

Ns � 1

XNs�1

i¼0;j¼0

Fj1

NBjSiNBjSj

X

b1˛BjSi

X

b2˛BjSj

Poff ðb1; b2Þ (A5)

where NBjSiis the number of elements in the BjSi subset of

behaviours associated with the Sith strategy.The A2 value is the expected fitness gain when players do not use

signals and have no expectation about the frequency of thedifferent strategies at the given final state.

In this case we assume that players cannot decide between thepossible opponents and thus pick the same reply against all theopponents. Thus the A2 value is calculated as follows:

A2 ¼1

Ns � 1

XNs�1

i¼0;j¼0

FjPoff�b1�Si; Sj

�;b1�Sj; Si

��(A6)

where b1 denotes the behaviour map defined in Table A1.