Cheating as a mixed strategy in a simple model of aggressive communication

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1 ANIMAL BEHAVIOUR, 2000, 59, Article No. anbe , available online at on Cheating as a mixed strategy in a simple model of aggressive communication SZABOLCS SZAu MADOu 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 The Association for the Study of Animal Behaviour 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 ( szamszab@ludens.elte.hu). 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) /00/ $35.00/ The Association for the Study of Animal Behaviour

2 222 ANIMAL BEHAVIOUR, 59, 1 Table 1. The expected payoffs of the strategies in a model of aggressive communication Opponent strength Strong Weak Ego strength Attack Conditional attack Flee Attack Conditional attack Flee Strong Attack 0.5V C SS 0.5V C SS V F A V C SW V C SW V F A Conditional attack 0.5V C SS F P 0.5V C SS V V C SW F P V C SW V Flee C SS 0 0.5V C SS C SW 0 0.5V Weak Attack C WS C WS V F A 0.5V C WW 0.5V C WW V F A Conditional attack C WS F P C WS V 0.5V C WW F P 0.5V C WW V Flee C WS 0 0.5V C WW 0 0.5V C WW Modified after Hurd (1997). V: Value of the contested resource; C SS, C WW : expected cost of fight between equal opponents; C SW : cost for strong individual to beat weak one; C WS : cost to weak individual when beaten by strong one; F A : cost of attacking fleeing opponent; F P : 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 (S a )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 S a is a pure strategy in which both strong and weak animals signal honestly. The corresponding dishonest pure strategy (S b ) 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 (S I ), which is a mixture of pure strategies S a and S b, 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 1 p, 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, C WW and C SS 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, 0.5V C WW >0, 0.5V C SS >0. I further assume that a strong animal can always beat a weak one with a cost C SW, and C WS is the expected cost that a weak animal should suffer on this occasion. The following relation holds between these costs: C WS >C SS, C WW >C SW. In addition, there is a cost of attacking a fleeing opponent (F A ), and of waiting if the opponent attacks unconditionally (F P ). It is biologically realistic, but not necessary, to assume that C SW >F A and F P (Hurd 1997). The frequencies of weak and strong individuals are denoted by q and 1 q, respectively. Then the payoffs for weak and strong contestants can be written as shown in Table 1. I have chosen the payoffs 0.5V C SS and 0.5V C WW 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 (S a and S b ) supporting the mixed evolutionarily stable strategy (S I ) should be equal against the mixed ESS, one can find the value of p, that is, the proportion of cheaters. From the equation: E(S a,s I )=E(S b,s I )=E(S I,S I ) the proportion of cheaters can be expressed as: (1a)

3 SZAu MADOu : CHEATING AS A MIXED STRATEGY 223 In order for p* to fall between zero and one the following inequalities must be fulfilled: E(S I,S I )>E(S J,S I ) (8) However, it is possible to show (Appendix 1) that E(S I,S I )=E(S J,S I ) for any ξ, thus, for S I to be ESS the second condition must be fulfilled (equation 7): E(S I,S J )>E(S J,S J ) (9) Merging the two inequalities (3b) and (4b) gives the range of C WS 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/(1 q) (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 qv/(1 q)). The next step is to investigate the evolutionary stability of the mixed strategy. For S I to be an ESS the following condition must be fulfilled (Maynard Smith 1982): or if E(S I,S I )=E(S M,S I ) then E(S I,S I )>E(S M,S I ) (6) E(S I,S M )>E(S M,S M ) (7) where S M can be any mutant (pure or mixed) strategy. I first investigate the invasion of those strategies that support S I. (1) S a, S b invades as a pure strategy. Since E(S a,s I )=E(S b,s I )=E(S I,S I ) then if S M =S a or S M =S b 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) S a, S b invades as a mixed strategy other than S I.If equation (5) holds, 0<p*<1. Then strategy S I can be denoted as S I (probability of S a, probability of S b ), that is, S I (1 p*,p*). Suppose further that there is a mutant strategy S J playing a different mixture of S a, S b, that is, S J (1 p* ξ, p*+ξ), where ξ is the deviation from p*, where p*<ξ<1 p*. Can this S J strategy invade our initial S I strategy? For S I to resist the invasion the first condition of the ESS definition must be met (equation 6): That is, S I should invade the population of any mutant mixed strategy. It is possible to show (Appendix 2) that equation (9) is fulfilled for any ξ, sos I is an ESS against any S J. Since there are strategies other then S a and S b, 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 S a. The first two are for strong individuals. The coward (S c ) strong individual tries to minimize the cost of fighting, always signalling B, then fleeing from any opponent. The Trojan horse (S d ) signals weakness (B) but then proceeds with an unconditional attack. The third strategy, bluffing (S b ), is for weak individuals. Thus we should consider the invasion of S c and S d as pure strategies and any mixture of S a, S b, S c, S d other than S J (S a,s b ) which have already been investigated. It can be shown that S I is an ESS against S c or S d 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 S S and weak as S W and the mixture of them as S I. If S I is an ESS then: that is, E(S S,S I )=E(S W,S I )=E(S I,S I ) (10) From this, the proportion of weak individuals can be expressed as:

4 224 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: that is: E(S S,S I )=E(S Wa,S I )=E(S Wb,S I ) (13) where S Wa, S Wb denote the strategies when a weak individual plays S a, S b, 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 S a,s b,s c,s d 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, C WW =5, C WS =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 Proportion of cheaters/weak individuals Value of the contested resource, V 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. C WW =5, C WS =10, C SW =5, C SS =20, F A =2. See Table 1 for definitions. 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 μ 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: c a b

5 SZAu MADOu : CHEATING AS A MIXED STRATEGY 225 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 (C WW, C WS ). If, however, the proportion of weak individuals (q) is an evolutionary variable, then the cost of fighting between strong individuals (C SS ) and the cost of chasing a fleeing opponent (F A ) 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 (C WS ). 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

6 226 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 & Mesterton- Gibbons 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

7 SZAu MADOu : CHEATING AS A MIXED STRATEGY 227 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. 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8 228 ANIMAL BEHAVIOUR, 59, 1 Appendix 1 Here I investigate whether the condition E(S I,S I )>E(S J,S I ) holds for any ξ. Since from equation (1) we know that E(S I,S I )=E(S a,s I ) equation (6) can be written as: E(S a,s I )>E(S J,S I ) (A1) The fitness of S J has two parts: (1) when it plays S a with probability 1 p ξ; (2) when it plays S b with probability p+ξ. Let denote E(S a,s I ) with E ai and E(S b,s I ) with E bi, then equation (A1) has the form: E ai >(1 p) E ai +pe bi ξe ai +ξe bi (A2) Since E ai =E bi the right side equals the left side for any ξ. Thus, the first condition of the definition of ESS cannot hold. Appendix 2 Here I investigate whether the condition E(S I,S J )>E(S J,S J ) holds for any ξ. Let us denote E(S a,s J ) with E aj and E(S b,s J ) with E bj. Then equation (9) can be written as follows: (1 p) E aj +pe bj >(1 p ξ) E aj +(p+ξ) E bj (A3) after rearrangement: 0>ξE bj ξe aj (A4) The payoff for E aj is: which can be rewritten with E ai as: The payoff for E bj is: Substituting E bi results: Since E ai =E bi : E aj =E bj +ξq C WW (A7) Thus equation (A4) holds, the condition of E(S I,S J )>E(S J,S J ) is fulfilled for any ξ, S I is an ESS. Appendix 3 Here I investigate the invasion of S c and S d as pure strategies against S I. First, I show that S a and S b alone are both ESS against S c or S d. Then it possible to show that S I is an ESS as well. (1) S a and S b is an ESS against S c,thatis: E aa >E ca and E bb >E cb (A8) Hurd (1997) has shown that the honest strategy is an ESS against the coward provided: that is:

9 SZAu MADOu : CHEATING AS A MIXED STRATEGY 229 S b 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 (0.5V C SS )>0 and C SS >F A. (2) S a and S b is an ESS against S d,thatis: E aa >E da and E bb >E db (A13) The honest strategy is an ESS against the Trojan horse provided: S b is also an ESS against the Trojan horse provided: This should hold since C SW >F A >0. It can be shown that strategies S a and S b have equal payoffs playing against S a or S b for strong individuals. From Table 1 we can see that: (3) S I against S c or S d. Equation (9) in this case can be written as: (1 p) E aa +pe ab >(1 p) 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 S I is an ESS against S c or S d. Appendix 4 Here I show that S I is an ESS against the invasion of any mixed strategy consisting of any combination of S a, S b, S c or S d. The payoff for the mutant mixed strategy is always a linear combination of the payoffs that the supporting pure strategies get against S a and S b, 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 u i denotes the probability of playing the ith strategy. For the invasion of S a and S b it was shown in Appendix 2 that S I is an ESS. Thus here I consider only those cases where at least one strategy supporting the mutant is S c or S d. Since (1) the fitness of the mutant mixed strategy is always a linear combination of the payoffs that the supporting pure strategies have against S a and S b ; (2) both the coward and the Trojan horse are worse against S a or S b than S a or S b against themselves;

10 230 ANIMAL BEHAVIOUR, 59, 1 (3) E ab =E bb it follows (equations A8, A13 and A17) that the fitness of S I is always greater than any mutant mixed strategy that consists of S c or S d. Thus S I is an ESS against any mutant mixed strategy. Appendix 5 Here I show that the mixed strategy playing q*,p*(s I ) is an ESS against any mixed strategy playing a different p value and against S a, S b, S c, S d 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 S I is an ESS against S c. In order for q to be greater than zero, the inequality C SS >0.5V should hold (equation 12), thus it is difficult to judge from equations (A10) and (A12) whether S I is an ESS against S c. However, it can be proved the other way round. From equation (13) it follows that: that is: E(S Wa,S I )>E(S c,s I ) (A20) Assuming that 0.5V C WW >0 and that C SW >0 this should hold. In the remaining part of the Appendix, I show that S I is an ESS against any mixed strategy playing a different q value. That is, I investigate whether the condition E(S I,S J )>E(S J,S J ) holds for any ξ, where S I (play weak, play strong) is S I (q*,1 q*) and S J (q*+ξ,1 q* ξ). Let us denote E(S Wa,S J ) with E WaJ, E(S Wb,S J ) with E WbJ and E(S S,S J ) with E SJ. Then equation (9) can be written as follows: (1 q) E SJ +q(1 p) E WaJ +qp E WbJ >(1 q ξ) E SJ +(q+ξ) (1 p) E WaJ +(q+ξ) pe WbJ after rearrangement: 0>ξ(1 p) E WaJ +ξp E WbJ ξe SJ The payoff for E SJ can be written with E SI as: (A22) (A23) The payoff for E WaJ can be written with E WaI as: The payoff for E WbJ can be written with E WbI as: Substituting equations (A24), (A25), (A26) into equation (A23) gives: Since E SI =E WaI =E WbI, thus: Rearrangement gives: Suppose that p.1 then equation (A28a) simplifies to: Suppose that p.0 then it simplifies to: C SS >p(c WS +F A ) C WW (1+2p(1+p)) C SS >C WS +F A 3 C WW (A28a) C SS > C WW (A30) Since both C SS and C WW is greater than zero, equation (A30) should hold. Thus, if equation (A29) holds then S I is an ESS against any strategy playing a different q other than q*. (A29)