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If animals know their own fighting ability, the evolutionarily stable level of fighting is reduced
ARTICLE IN PRESS
Journal of Theoretical Biology 232 (2005) 1–6 www.elsevier.com/locate/yjtbi
If animals know their own fighting ability, the evolutionarily stable level of fighting is reduced John M. McNamaraa,, Alasdair I. Houstonb a
Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK School of Biological Sciences, Centre For Behavioural Biology, University of Bristol, University Walk, Bristol BS8 1TW, UK
b
Received 23 September 2003; received in revised form 9 July 2004; accepted 16 July 2004 Available online 21 September 2004
Abstract We consider a version of the Hawk–Dove game in which an animal knows its own fighting ability but not the ability of its opponent. For this game at evolutionary stability there is a critical level of ability such that animals with ability greater than the critical level play Hawk and animals with ability below the critical level play Dove. We define the level of fighting to be the probability of a Hawk–Hawk fight when two opponents meet. We show that even if an animal does not know the ability of its opponent, knowing its own ability results in a lower level of fighting at evolutionary stability than is found in the standard Hawk–Dove game in which there are no differences in ability or abilities are not known. r 2004 Elsevier Ltd. All rights reserved. Keywords: Hawk–Dove game; Fighting ability; Evolutionary stable strategy; Threshold
1. Introduction One of the original motivations for applying evolutionary game theory to animal contests was to understand how evolution would shape the level of aggression in a population (Maynard Smith and Price, 1973). The key concept in this analysis is the evolutionarily stable strategy (ESS) (Maynard Smith, 1982; Maynard Smith and Price, 1973). An ESS is defined as follows. If almost all members of a population adopt the ESS, then no other strategy can spread through the population by the action of natural selection. The Hawk–Dove game provides a simple characterization of contests between animals over a resource of value V. In the standard model, all animals are equal in their abilities, but may differ in terms of the behaviour that they adopt in a contest with another animal. An animal that plays Hawk behaves aggressively and continues to escalate Corresponding author. Tel.: +44-117-928-7986; fax: +44-117-928-
7999. E-mail address: [email protected] (J.M. McNamara). 0022-5193/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2004.07.024
until it is injured or the other animal retreats. An animal that plays Dove displays to the other animal and retreats if the other animal behaves aggressively. Thus if a Hawk contests the resource with a Dove the Hawk gets the resource. If two Doves contest the resource each gets it with probability 0.5. If two Hawks contest the resource they fight, each winning the fight with probability 0.5. The winner gets the resource and the loser pays a cost C that represents the loss in fitness as a result of being injured. When V4C then the ESS is always play Hawk. In contrast, when VoC then the ESS is either 1. A polymorphism in which a proportion p ¼ V =C of the population play Hawk and 1 p to play Dove, or 2. A mixed ESS in which each animal plays Hawk with probability p ¼ V =C: This result shows that evolution may not always result in the most aggressive option being adopted by all members of a population.
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J.M. McNamara, A.I. Houston / Journal of Theoretical Biology 232 (2005) 1–6
One way to quantify aggression in the population is to look at the proportion of interactions between individuals that result in a Hawk–Hawk fight. Let p ¼ minð1; V =CÞ denote the probability that a randomly selected population member plays Hawk. When two opponents in this population meet the probability of a Hawk–Hawk fight is ðp Þ2 : Crowley (2000) looks at a modified version of the Hawk–Dove game in which animals may differ in their fighting ability. He establishes the general form of the ESS both when there are discrete ability classes and when there is a continuous distribution of ability, but does not explore the resulting level of fighting in the population. We focus on the consequences for levels of fighting in the Hawk–Dove game when there is a continuous distribution of fighting ability. When animals don’t know their ability or that of their opponent, then including differences in ability does not change the level of fighting from that of the standard Hawk–Dove game. When an animal knows its own ability but does not know the ability of its opponent, the ESS specifies a critical threshold of ability above which the animal plays Hawk and below which it plays Dove. We show that the resulting level of fighting is less than the level of fighting that would be seen in the standard Hawk–Dove game with no differences in (or knowledge of) ability. Thus knowing just one’s own ability can reduce the level of fighting in the population. It seems reasonable that an animal can estimate its fighting ability on the basis of previous experience in aggressive encounters, and in a large population it might be reasonable to assume that the ability of an opponent is not known.
2. The model Our model is based on the standard Hawk–Dove game (e.g. Maynard Smith, 1982) but differs from it in that there is a range of fighting abilities amongst population members. We denote the fighting ability of an animal by a real number x. There is a continuous distribution of fighting abilities in the population specified by the probability density function f(x). We allow x to take any value in the range 1oxo1 although in particular cases the values of x that are found in a population may be limited by the fact that f ðxÞ ¼ 0 for a range of x. We assume that an individual knows its own fighting ability. Opponents are random members of the population and the individual has no cue as to their ability. Thus the ability of opponents is not known; the information about an opponent’s ability is probabilistic, and given by the probability density function f(x). If an individual with ability x meets an opponent with ability y and they fight (i.e. each plays Hawk), then the individual with ability x wins with probability G(x, y).
Since there are no draws in a fight we necessarily have Gðx; yÞ ¼ 1 Gðy; xÞ; and in particular Gðx; xÞ ¼ 0:5: We assume that, for given y, G(x, y) is strictly increasing in x, and that Gðx; yÞ ! 0 as x ! 1 and Gðx; yÞ ! 1 as x ! 1: We refer to the probability that a randomly selected member of a population plays Hawk as the population Hawk probability (PHP). If the PHP ¼ p then the proportion of interactions that result in a Hawk–Hawk fight is p2. We are interested in the proportion of fights at evolutionary stability, i.e. in the value of p2 at the ESS. We refer to this as the evolutionarily stable level of fighting.
3. Form of the best response Consider a large population in which the probability that an individual of fighting ability y plays Hawk with probability q(y). In this population the PHP is Z 1 p¼ qðvÞf ðvÞ dv: (1) 1
We consider a single individual within this population, and characterize the strategy that maximizes this individual’s fitness. A strategy specifies how an individual following the strategy responds to its fighting ability. In other words, it specifies how the probability of playing Hawk depends on x. Suppose the individual has ability x. If the individual plays Dove, then its payoff is (1p)V/2. Suppose conversely that the individual plays Hawk. Then the probability that the opponent also fights is p. If a fight occurs, the probability that the focal individual wins is the probability that there is a fight and the focal individual wins divided by the probability there is a fight; i.e. Z 1 wðxÞ ¼ Gðx; vÞqðvÞf ðvÞ dv p: (2) 1
The payoff to the focal individual is then ð1 pÞV þ p½wðxÞV ð1 wðxÞÞC : The advantage of playing Hawk rather than Dove is thus ð1 pÞV =2 þ p½ðV þ CÞ wðxÞ C ; which we can write as ðV =2 þ CÞAðxÞ; where AðxÞ ¼ pð1 þ p0 ÞwðxÞ þ p0 p
(3)
and p0 ¼
V : V þ 2C
(4)
If A(x)40 then the unique best action is to play Hawk. Similarly if A(x)o0 the unique best action is to play Dove. Now by Eq. (2), w(x) is a strictly increasing function of x since G(x,y) is a strictly increasing function of x for each y. Thus by Eq. (3) A(x) is a strictly
ARTICLE IN PRESS J.M. McNamara, A.I. Houston / Journal of Theoretical Biology 232 (2005) 1–6
increasing function of x. Our assumptions on G(x,y) also ensure that wðxÞ ! 1 as x ! 1: Thus by Eq. (3) A(x)40 for x sufficiently large. It can hence be seen that there is a critical threshold xc (which may be 1) such that A(x)40 for x4xc and A(x)o0 for xoxc. From the above analysis we see that the strategy that maximizes fitness is the pure contingent strategy: play Dove when fighting ability is below xc, play Hawk when ability exceeds xc. This is the unique optimal strategy (apart from changes of action when the fighting ability is exactly xc, a situation which occurs with probability 0). Thus no mixed strategy does as well. Note that when pop0 we have A(x)40 for all x, so that it is optimal to play Hawk regardless of fighting ability (i.e. xc ¼ 1).
3
population is proportional to Aðx; xc Þ ¼ ð1 þ p0 Þpðxc ÞW ðx; xc Þ þ p0 pðxc Þ:
Let b(xc) satisfy Aðbðxc Þ; xc Þ ¼ 0: If xobðxc Þ then Aðx; xc Þo0 so that it is optimal for the mutant to play Dove. Conversely if x4bðxc Þ then Aðx; xc Þ40 so that it is optimal to play Hawk. In other words the best response of the mutant is to adopt a critical threshold b(xc). By Eq. (7) ð1 þ p0 Þpðxc ÞW ðbðxc Þ; xc Þ þ p0 pðxc Þ ¼ 0:
If almost all population members adopt the same strategy we refer to this strategy as the resident population strategy. For a given resident strategy, the strategy that maximizes the fitness of a single mutant individual in this population is called the best response to the resident strategy. An ESS is a strategy that is a best response to itself. We have established that the best response to any resident population strategy has a threshold form. It follows that an ESS must be of this form. Thus we will from now on restrict attention to strategies of this form in seeking an ESS, i.e. we restrict attention to populations in which the resident strategy is of the form: If x4xc then play Hawk and If xoxc then play Dove In such a population the PHP is Z 1 f ðvÞ dv: pðxc Þ ¼
(5)
xc
We noted that when pðxc Þop0 the population is invadable by the strategy of always playing Hawk. We deduce that at evolutionary stability the PHP p must satisfy p Xp0 : Thus in seeking an ESS we can restrict attention to critical thresholds for which pðxc ÞXp0 : To find the best response to the resident population with critical threshold xc consider a mutant individual with ability x in this population. Let W(x, xc) denote the probability that the individual wins the fight if it plays Hawk and its opponent does likewise. Then Z 1 Gðx; vÞf ðvÞ dv: (6) pðxc ÞW ðx; xc Þ ¼ xc
Following the derivation of Eq. (3) the advantage to an individual of playing Hawk as opposed to Dove in this
(8)
The condition for a threshold level of ability to be an ESS is that it is the best response to itself, i.e. it is a level x such that bðx Þ ¼ x : It follows from this condition that Aðx ; x Þ ¼ 0 and hence ð1 þ p0 Þpðx ÞW ðx ; x Þ þ p0 pðx Þ ¼ 0:
4. Evolutionary stability
(7)
By Eqs. (5) and (6) Z 1 Z ð1 þ p0 Þ Gðx ; vÞf ðvÞ dv ¼ x
(9)
1 x
f ðvÞ dv p0 :
(10)
5. Results We first show that there exists an ESS. Consider the behaviour of the terms in Eq. (10) in the limit as x ! 1: The left-hand side term tends to zero since Gðx ; vÞ tends to 0 for each v. The right-hand side term tends to 1 p0 40: Thus the right-hand side term exceeds the left-hand side term for x sufficiently negative. Now consider the behaviour of the terms in Eq. (10) in the limit as x ! þ1: The left-hand side term tends to 0 while the right-hand side term tends to p0 : Thus the left-hand side term exceed the right-hand side for x sufficiently large and positive. Since all terms are continuous there exists some x such that the left and right-hand side terms are equal. That is Eq. (10) has a solution. We now show that the ESS is unique. To see this we write Eq. (10) as Z 1 1 f ðvÞ dv ¼ p0 ð1 þ p0 Þ 2ð1 p0 Þ x Z 1 ½12 Gðx ; vÞ f ðvÞ dv: ð11Þ x
Consider the term on left-hand side of this equation. Differentiating with respect to x we see that this is this term has derivative 12ð1 p0 Þf ðx Þp0: Thus the lefthand side term decreases as x increases. Using the fact that Gðx ; x Þ ¼ 12 it can be seen right-hand side R 1that the ;vÞ term has derivative ð1 þ p0 Þ x @Gðx @x f ðvÞ dv: This is positive since Gðx ; vÞ is a strictly increasing function of x : Since the left-hand side term decreases and the righthand side term is strictly increasing there exists a unique x at which the two terms are equal.
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We are concerned with the PHP at the ESS; i.e. with Z 1 f ðvÞ dv: (12) p pðx Þ ¼ x
It can easily be seen from Eq. (10) that p 40: In order to put an upper bound on p we use Eqs. (12) and (4) to rewrite Eq. (11) as Z 1 V V þ1 p ¼ 2 ½12 Gðx ; vÞ f ðvÞ dv: (13) C C x Consider the term Gðx ; vÞ where x ov: By assumption Gðx ; vÞ ¼ 1 Gðv; x Þ: We have also assumed that Gðx; yÞ is strictly increasing in x for fixed y. Thus Gðv; x Þ4Gðx ; x Þ: Since Gðx ; x Þ ¼ 12 we have Gðx ; vÞo12 for x ov: Thus the integral on the righthand side of Eq. (13) is strictly positive. This shows that p o
V : C
(14)
1
(i) 0.8
Best response, b (x c )
4
(ii) 0.6
(iii) 0.4
0.2
0 0
0.2
0.4
0.6
0.8
1
Population threshold, x c
Fig. 2. The critical threshold b(xc) under the best response strategy when the resident strategy has critical threshold xc. (i) V =C ¼ 0:4; (ii) 0.75, (iii) 1.1.
increases and then decreases as xc increases. From Eqs. (10) and (15) it is easy to verify that at evolutionarily stability the PHP is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2r þ 2r2 1 p ¼ ; (16) 1þr
Thus, provided V =Cp1 the PHP is always less than that for the standard Hawk–Dove game. We illustrate the model with the following simple example. The distribution of fighting ability is uniform on (0,1). We consider two forms of the function G. First suppose that the probability that an individual with ability x wins in a fight against an individual of ability y is
where r ¼ V =C: Fig. 3 shows the evolutionarily stable level of fighting ðp Þ2 in the population. As can be seen this is much lower than in the standard Hawk–Dove game, for which
Gðx; yÞ ¼ ð1 þ x yÞ=2:
p ¼ minfr; 1g:
(15)
In this case only when one contestant has the highest possible ability ðx ¼ 1Þ and the other has the lowest ðy ¼ 0Þ is the better animal certain to win. The advantage of playing Hawk as opposed to Dove is illustrated in Fig. 1 for three values of the population threshold xc. In each case the advantage to playing Hawk increases as x increases. Some examples of the best response function are given in Fig. 2. It can be seen that the best response may be non-monotonic; when V/C is low, b(xc) first
0.3
0.2
Advantage, A(x, xc)
The figure also illustrates the evolutionarily stable level of fighting in the case where Gðx; yÞ ¼ 0 for xoy and Gðx; yÞ ¼ 1 for x4y. In this case the individual with the highest level of ability always wins a fight. This function Gðx; yÞ is not strictly increasing in x for fixed y, so that not all of our original assumptions are met. Nevertheless, it is still straightforward to show that the unique ESS is given by Eq. (10). From this equation it can be seen that the PHP is p0. That is p ¼
0.4
(iii)
0.1
(ii)
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.1
(i) -0.2
-0.3
Ability, x
Fig. 1. The advantage A(x, xc) of playing Hawk over that of playing Dove to an individual of ability x when the resident population strategy has critical threshold xc (Eq. (7)). The three cases illustrated are (i) xc ¼ 0:25; (ii) 0.5, (iii) 0.75. V =C ¼ 1 in all cases.
(17)
r : rþ2
(18)
Since p Xp0 whatever the form of the function G, we see that the evolutionarily stable level of fighting is lowest (for given r=V/C) in this case. Comparing the two forms of G illustrated in Fig. 3 we see that the step function results in a lower evolutionarily stable level of fighting than the smooth curve. This illustrates the following general result established in the Appendix: increasing the advantage (in terms of the probability of winning a fight) to the higher ability opponent for a given difference in ability decreases the evolutionarily stable level of fighting. Let the random variable X denote the ability of a randomly selected population member. To investigate the effect of increasing the spread of abilities we can compare this original distribution with the case where
ARTICLE IN PRESS J.M. McNamara, A.I. Houston / Journal of Theoretical Biology 232 (2005) 1–6 1
Level of fighting
0.8
(i)
(ii)
0.6
0.4
(iii) 0.2
0 0
0.5
1
1.5
2
2.5
3
V /C Fig. 3. The evolutionarily stable level of fighting ðp Þ2 as a function of V/C in three cases. (i) The standard Hawk Dove game with no differences in ability. (ii) The game with differences in ability and the probability of winning a fight, G(x,y), given by Eq. (15). (iii) The game with differences in ability and the most able contestant always winning a fight. In (ii) and (iii) ability in the population is uniformly distributed between 0 and 1.
the ability of a randomly selected individual is X~ ¼ kX ; where the constant k satisfies k41. Increasing the spread of abilities in this way while keeping G the same is equivalent to keeping the distribution the same but ~ yÞ ¼ Gðkx; kyÞ: Since G~ is changing G to G~ where Gðx; more step-like than G the result of the transformation is to reduce the evolutionarily stable level of fighting.
6. Discussion Like Crowley (2000), we have shown that when there is a continuous distribution of fighting abilities and an animal knows its own fighting ability but not the ability of its opponent then the ESS is of the form: play Hawk if the animal’s ability is greater than a threshold level and play Dove if its ability is below this threshold level. In contrast to this conclusion, Pagel and Dawkins (1997) consider a strategy that bases its probability of fighting on its estimate of its own ability and argue that such a strategy will not do better than always fighting or never fighting. They thus conclude that a strategy of basing fighting behaviour on one’s own ability cannot be stable. A problem with this conclusion is that it considers a strategy based on a probability of fighting, i.e. it is looking for a mixed strategy. But the ESS cannot be
5
mixed when the distribution of ability is continuous (Crowley, 2000). As we have seen, the ESS is a pure contingent strategy. Our main conclusion is that if an animal knows its own ability but does not know the ability of its opponent, the level of fighting at the ESS is less than the level in the standard Hawk–Dove game in which differences in ability are absent or ignored. Thus even in the absence of knowledge of the ability of an opponent (either by assessment or individual recognition coupled with previous experience) a knowledge of one’s own ability reduces fighting. To see why this reduction occurs compare the standard Hawk–Dove game with the game in which there are differences in ability. In the standard game, if an individual plays Hawk and the opponent also plays Hawk the probability of winning the fight is 0.5. In contrast suppose that an opponent were to play Hawk in our game, then the opponent has ability above the critical threshold and is better than average at winning fights. Thus a randomly selected individual has probability of less than 0.5 of winning a fight if it plays Hawk and the opponents fights back. This deters individuals from playing Hawk unless they are of high ability. Consequently there is less fighting at evolutionary stability. We have assumed that animals ‘‘know’’ both the function G(x,y) that determines the probability that an individual with ability x wins a fight against an individual with ability y and the distribution of abilities in the population. Both of these functions influence the level of fighting at the ESS. We have shown that as the function G(x,y) becomes steeper, the level of fighting goes down. Changing the distribution function can be thought of as rescaling the function G(x,y). As a consequence, increasing the spread of abilities decreases the level of fighting. Crowley (2000) considers the ESS both when an animal knows just its own ability and when it knows both its ability and the ability of its opponent. When both contestants know each other’s ability as well as their own, individuals have well-defined roles and a contest is settled by convention. When the individual with the greater ability always wins (i.e. G is a step function), at evolutionary stability there is no fighting. When G is not so steep, there is still no fighting when VoC, but there may be fights between individuals of similar ability when V4C. There will, however, typically be less fighting than when an individual only knows its own ability. To summarize our conclusions, if an individual only knows its own ability, there is less fighting than when neither animal knows its ability. If both animals know their ability, then the level of fighting will typically be even lower. Which of these cases is likely to hold will depend on a population’s biology. In a small population, it might be possible for an animal to know
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6
both its ability and that of all other members of the population. In a large population in which there are no visual cues that predict ability, it might not be possible to know the ability of other animals. In this case an animal might learn its ability as a result of fighting. We note that there might be a tradeoff between gaining information and avoiding injury (cf. Welton et al., 2003).
Appendix A. The dependence of the PHP on the shape of G
by Eq. (10). Differentiating with respect to a and using the fact that Gðx ; x ; aÞ ¼ 0:5 we obtain Z 1 @G dx 1 f ðvÞ dv 2ð1 p0 Þf ðx Þ þ ð1 þ p0 Þ da x @x Z 1 @Gðx ; v; aÞ f ðvÞ dv: ðA:3Þ ¼ ð1 þ p0 Þ @a x The term multiplying dx =da on the left-hand side of this equation is positive since @G=dx is positive. The right-hand side is also positive by inequality (A.1). Thus dx =da40: Since the threshold increases with a the PHP decreases as a increases.
Suppose that the function G Gðx; y; aÞ depends on the parameter a as well as x and y. We investigate how the shape of G affects the PHP by assuming that @Gðx; y; aÞ o0 for xoy: (A.1) @a Thus as a increases so does the advantage (in terms of the probability of winning a fight) to the higher ability opponent of a given difference in ability. Denoting the ESS critical threshold for given a by x ðaÞ we have Z 1 Gðx ðaÞ; v; aÞf ðvÞ dv ð1 þ p0 Þ x ðaÞ Z 1 f ðvÞ dv p0 ðA:2Þ ¼ x ðaÞ
References Crowley, P.H., 2000. Hawks, doves, and mixed-symmetry games. J. Theor. Biol. 204, 543–563. Maynard Smith, J., 1982. Evolution and the Theory of Games. Cambridge University Press, Cambridge. Maynard Smith, J., Price, G.R., 1973. The logic of animal conflicts. Nature 246, 15–18. Pagel, M., Dawkins, M.S., 1997. Peck orders and group size in laying hens: futures contracts for non-aggression. Behav. Processes 40, 13–25. Welton, N.J., McNamara, J.M., Houston, A.I., 2003. Assessing predation risk: optimal behaviour and rules of thumb. Theor. Popul. Biol. 64, 417–430.
Journal of Theoretical Biology 232 (2005) 1–6 www.elsevier.com/locate/yjtbi
If animals know their own fighting ability, the evolutionarily stable level of fighting is reduced John M. McNamaraa,, Alasdair I. Houstonb a
Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK School of Biological Sciences, Centre For Behavioural Biology, University of Bristol, University Walk, Bristol BS8 1TW, UK
b
Received 23 September 2003; received in revised form 9 July 2004; accepted 16 July 2004 Available online 21 September 2004
Abstract We consider a version of the Hawk–Dove game in which an animal knows its own fighting ability but not the ability of its opponent. For this game at evolutionary stability there is a critical level of ability such that animals with ability greater than the critical level play Hawk and animals with ability below the critical level play Dove. We define the level of fighting to be the probability of a Hawk–Hawk fight when two opponents meet. We show that even if an animal does not know the ability of its opponent, knowing its own ability results in a lower level of fighting at evolutionary stability than is found in the standard Hawk–Dove game in which there are no differences in ability or abilities are not known. r 2004 Elsevier Ltd. All rights reserved. Keywords: Hawk–Dove game; Fighting ability; Evolutionary stable strategy; Threshold
1. Introduction One of the original motivations for applying evolutionary game theory to animal contests was to understand how evolution would shape the level of aggression in a population (Maynard Smith and Price, 1973). The key concept in this analysis is the evolutionarily stable strategy (ESS) (Maynard Smith, 1982; Maynard Smith and Price, 1973). An ESS is defined as follows. If almost all members of a population adopt the ESS, then no other strategy can spread through the population by the action of natural selection. The Hawk–Dove game provides a simple characterization of contests between animals over a resource of value V. In the standard model, all animals are equal in their abilities, but may differ in terms of the behaviour that they adopt in a contest with another animal. An animal that plays Hawk behaves aggressively and continues to escalate Corresponding author. Tel.: +44-117-928-7986; fax: +44-117-928-
7999. E-mail address: [email protected] (J.M. McNamara). 0022-5193/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2004.07.024
until it is injured or the other animal retreats. An animal that plays Dove displays to the other animal and retreats if the other animal behaves aggressively. Thus if a Hawk contests the resource with a Dove the Hawk gets the resource. If two Doves contest the resource each gets it with probability 0.5. If two Hawks contest the resource they fight, each winning the fight with probability 0.5. The winner gets the resource and the loser pays a cost C that represents the loss in fitness as a result of being injured. When V4C then the ESS is always play Hawk. In contrast, when VoC then the ESS is either 1. A polymorphism in which a proportion p ¼ V =C of the population play Hawk and 1 p to play Dove, or 2. A mixed ESS in which each animal plays Hawk with probability p ¼ V =C: This result shows that evolution may not always result in the most aggressive option being adopted by all members of a population.
ARTICLE IN PRESS 2
J.M. McNamara, A.I. Houston / Journal of Theoretical Biology 232 (2005) 1–6
One way to quantify aggression in the population is to look at the proportion of interactions between individuals that result in a Hawk–Hawk fight. Let p ¼ minð1; V =CÞ denote the probability that a randomly selected population member plays Hawk. When two opponents in this population meet the probability of a Hawk–Hawk fight is ðp Þ2 : Crowley (2000) looks at a modified version of the Hawk–Dove game in which animals may differ in their fighting ability. He establishes the general form of the ESS both when there are discrete ability classes and when there is a continuous distribution of ability, but does not explore the resulting level of fighting in the population. We focus on the consequences for levels of fighting in the Hawk–Dove game when there is a continuous distribution of fighting ability. When animals don’t know their ability or that of their opponent, then including differences in ability does not change the level of fighting from that of the standard Hawk–Dove game. When an animal knows its own ability but does not know the ability of its opponent, the ESS specifies a critical threshold of ability above which the animal plays Hawk and below which it plays Dove. We show that the resulting level of fighting is less than the level of fighting that would be seen in the standard Hawk–Dove game with no differences in (or knowledge of) ability. Thus knowing just one’s own ability can reduce the level of fighting in the population. It seems reasonable that an animal can estimate its fighting ability on the basis of previous experience in aggressive encounters, and in a large population it might be reasonable to assume that the ability of an opponent is not known.
2. The model Our model is based on the standard Hawk–Dove game (e.g. Maynard Smith, 1982) but differs from it in that there is a range of fighting abilities amongst population members. We denote the fighting ability of an animal by a real number x. There is a continuous distribution of fighting abilities in the population specified by the probability density function f(x). We allow x to take any value in the range 1oxo1 although in particular cases the values of x that are found in a population may be limited by the fact that f ðxÞ ¼ 0 for a range of x. We assume that an individual knows its own fighting ability. Opponents are random members of the population and the individual has no cue as to their ability. Thus the ability of opponents is not known; the information about an opponent’s ability is probabilistic, and given by the probability density function f(x). If an individual with ability x meets an opponent with ability y and they fight (i.e. each plays Hawk), then the individual with ability x wins with probability G(x, y).
Since there are no draws in a fight we necessarily have Gðx; yÞ ¼ 1 Gðy; xÞ; and in particular Gðx; xÞ ¼ 0:5: We assume that, for given y, G(x, y) is strictly increasing in x, and that Gðx; yÞ ! 0 as x ! 1 and Gðx; yÞ ! 1 as x ! 1: We refer to the probability that a randomly selected member of a population plays Hawk as the population Hawk probability (PHP). If the PHP ¼ p then the proportion of interactions that result in a Hawk–Hawk fight is p2. We are interested in the proportion of fights at evolutionary stability, i.e. in the value of p2 at the ESS. We refer to this as the evolutionarily stable level of fighting.
3. Form of the best response Consider a large population in which the probability that an individual of fighting ability y plays Hawk with probability q(y). In this population the PHP is Z 1 p¼ qðvÞf ðvÞ dv: (1) 1
We consider a single individual within this population, and characterize the strategy that maximizes this individual’s fitness. A strategy specifies how an individual following the strategy responds to its fighting ability. In other words, it specifies how the probability of playing Hawk depends on x. Suppose the individual has ability x. If the individual plays Dove, then its payoff is (1p)V/2. Suppose conversely that the individual plays Hawk. Then the probability that the opponent also fights is p. If a fight occurs, the probability that the focal individual wins is the probability that there is a fight and the focal individual wins divided by the probability there is a fight; i.e. Z 1 wðxÞ ¼ Gðx; vÞqðvÞf ðvÞ dv p: (2) 1
The payoff to the focal individual is then ð1 pÞV þ p½wðxÞV ð1 wðxÞÞC : The advantage of playing Hawk rather than Dove is thus ð1 pÞV =2 þ p½ðV þ CÞ wðxÞ C ; which we can write as ðV =2 þ CÞAðxÞ; where AðxÞ ¼ pð1 þ p0 ÞwðxÞ þ p0 p
(3)
and p0 ¼
V : V þ 2C
(4)
If A(x)40 then the unique best action is to play Hawk. Similarly if A(x)o0 the unique best action is to play Dove. Now by Eq. (2), w(x) is a strictly increasing function of x since G(x,y) is a strictly increasing function of x for each y. Thus by Eq. (3) A(x) is a strictly
ARTICLE IN PRESS J.M. McNamara, A.I. Houston / Journal of Theoretical Biology 232 (2005) 1–6
increasing function of x. Our assumptions on G(x,y) also ensure that wðxÞ ! 1 as x ! 1: Thus by Eq. (3) A(x)40 for x sufficiently large. It can hence be seen that there is a critical threshold xc (which may be 1) such that A(x)40 for x4xc and A(x)o0 for xoxc. From the above analysis we see that the strategy that maximizes fitness is the pure contingent strategy: play Dove when fighting ability is below xc, play Hawk when ability exceeds xc. This is the unique optimal strategy (apart from changes of action when the fighting ability is exactly xc, a situation which occurs with probability 0). Thus no mixed strategy does as well. Note that when pop0 we have A(x)40 for all x, so that it is optimal to play Hawk regardless of fighting ability (i.e. xc ¼ 1).
3
population is proportional to Aðx; xc Þ ¼ ð1 þ p0 Þpðxc ÞW ðx; xc Þ þ p0 pðxc Þ:
Let b(xc) satisfy Aðbðxc Þ; xc Þ ¼ 0: If xobðxc Þ then Aðx; xc Þo0 so that it is optimal for the mutant to play Dove. Conversely if x4bðxc Þ then Aðx; xc Þ40 so that it is optimal to play Hawk. In other words the best response of the mutant is to adopt a critical threshold b(xc). By Eq. (7) ð1 þ p0 Þpðxc ÞW ðbðxc Þ; xc Þ þ p0 pðxc Þ ¼ 0:
If almost all population members adopt the same strategy we refer to this strategy as the resident population strategy. For a given resident strategy, the strategy that maximizes the fitness of a single mutant individual in this population is called the best response to the resident strategy. An ESS is a strategy that is a best response to itself. We have established that the best response to any resident population strategy has a threshold form. It follows that an ESS must be of this form. Thus we will from now on restrict attention to strategies of this form in seeking an ESS, i.e. we restrict attention to populations in which the resident strategy is of the form: If x4xc then play Hawk and If xoxc then play Dove In such a population the PHP is Z 1 f ðvÞ dv: pðxc Þ ¼
(5)
xc
We noted that when pðxc Þop0 the population is invadable by the strategy of always playing Hawk. We deduce that at evolutionary stability the PHP p must satisfy p Xp0 : Thus in seeking an ESS we can restrict attention to critical thresholds for which pðxc ÞXp0 : To find the best response to the resident population with critical threshold xc consider a mutant individual with ability x in this population. Let W(x, xc) denote the probability that the individual wins the fight if it plays Hawk and its opponent does likewise. Then Z 1 Gðx; vÞf ðvÞ dv: (6) pðxc ÞW ðx; xc Þ ¼ xc
Following the derivation of Eq. (3) the advantage to an individual of playing Hawk as opposed to Dove in this
(8)
The condition for a threshold level of ability to be an ESS is that it is the best response to itself, i.e. it is a level x such that bðx Þ ¼ x : It follows from this condition that Aðx ; x Þ ¼ 0 and hence ð1 þ p0 Þpðx ÞW ðx ; x Þ þ p0 pðx Þ ¼ 0:
4. Evolutionary stability
(7)
By Eqs. (5) and (6) Z 1 Z ð1 þ p0 Þ Gðx ; vÞf ðvÞ dv ¼ x
(9)
1 x
f ðvÞ dv p0 :
(10)
5. Results We first show that there exists an ESS. Consider the behaviour of the terms in Eq. (10) in the limit as x ! 1: The left-hand side term tends to zero since Gðx ; vÞ tends to 0 for each v. The right-hand side term tends to 1 p0 40: Thus the right-hand side term exceeds the left-hand side term for x sufficiently negative. Now consider the behaviour of the terms in Eq. (10) in the limit as x ! þ1: The left-hand side term tends to 0 while the right-hand side term tends to p0 : Thus the left-hand side term exceed the right-hand side for x sufficiently large and positive. Since all terms are continuous there exists some x such that the left and right-hand side terms are equal. That is Eq. (10) has a solution. We now show that the ESS is unique. To see this we write Eq. (10) as Z 1 1 f ðvÞ dv ¼ p0 ð1 þ p0 Þ 2ð1 p0 Þ x Z 1 ½12 Gðx ; vÞ f ðvÞ dv: ð11Þ x
Consider the term on left-hand side of this equation. Differentiating with respect to x we see that this is this term has derivative 12ð1 p0 Þf ðx Þp0: Thus the lefthand side term decreases as x increases. Using the fact that Gðx ; x Þ ¼ 12 it can be seen right-hand side R 1that the ;vÞ term has derivative ð1 þ p0 Þ x @Gðx @x f ðvÞ dv: This is positive since Gðx ; vÞ is a strictly increasing function of x : Since the left-hand side term decreases and the righthand side term is strictly increasing there exists a unique x at which the two terms are equal.
ARTICLE IN PRESS J.M. McNamara, A.I. Houston / Journal of Theoretical Biology 232 (2005) 1–6
We are concerned with the PHP at the ESS; i.e. with Z 1 f ðvÞ dv: (12) p pðx Þ ¼ x
It can easily be seen from Eq. (10) that p 40: In order to put an upper bound on p we use Eqs. (12) and (4) to rewrite Eq. (11) as Z 1 V V þ1 p ¼ 2 ½12 Gðx ; vÞ f ðvÞ dv: (13) C C x Consider the term Gðx ; vÞ where x ov: By assumption Gðx ; vÞ ¼ 1 Gðv; x Þ: We have also assumed that Gðx; yÞ is strictly increasing in x for fixed y. Thus Gðv; x Þ4Gðx ; x Þ: Since Gðx ; x Þ ¼ 12 we have Gðx ; vÞo12 for x ov: Thus the integral on the righthand side of Eq. (13) is strictly positive. This shows that p o
V : C
(14)
1
(i) 0.8
Best response, b (x c )
4
(ii) 0.6
(iii) 0.4
0.2
0 0
0.2
0.4
0.6
0.8
1
Population threshold, x c
Fig. 2. The critical threshold b(xc) under the best response strategy when the resident strategy has critical threshold xc. (i) V =C ¼ 0:4; (ii) 0.75, (iii) 1.1.
increases and then decreases as xc increases. From Eqs. (10) and (15) it is easy to verify that at evolutionarily stability the PHP is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2r þ 2r2 1 p ¼ ; (16) 1þr
Thus, provided V =Cp1 the PHP is always less than that for the standard Hawk–Dove game. We illustrate the model with the following simple example. The distribution of fighting ability is uniform on (0,1). We consider two forms of the function G. First suppose that the probability that an individual with ability x wins in a fight against an individual of ability y is
where r ¼ V =C: Fig. 3 shows the evolutionarily stable level of fighting ðp Þ2 in the population. As can be seen this is much lower than in the standard Hawk–Dove game, for which
Gðx; yÞ ¼ ð1 þ x yÞ=2:
p ¼ minfr; 1g:
(15)
In this case only when one contestant has the highest possible ability ðx ¼ 1Þ and the other has the lowest ðy ¼ 0Þ is the better animal certain to win. The advantage of playing Hawk as opposed to Dove is illustrated in Fig. 1 for three values of the population threshold xc. In each case the advantage to playing Hawk increases as x increases. Some examples of the best response function are given in Fig. 2. It can be seen that the best response may be non-monotonic; when V/C is low, b(xc) first
0.3
0.2
Advantage, A(x, xc)
The figure also illustrates the evolutionarily stable level of fighting in the case where Gðx; yÞ ¼ 0 for xoy and Gðx; yÞ ¼ 1 for x4y. In this case the individual with the highest level of ability always wins a fight. This function Gðx; yÞ is not strictly increasing in x for fixed y, so that not all of our original assumptions are met. Nevertheless, it is still straightforward to show that the unique ESS is given by Eq. (10). From this equation it can be seen that the PHP is p0. That is p ¼
0.4
(iii)
0.1
(ii)
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.1
(i) -0.2
-0.3
Ability, x
Fig. 1. The advantage A(x, xc) of playing Hawk over that of playing Dove to an individual of ability x when the resident population strategy has critical threshold xc (Eq. (7)). The three cases illustrated are (i) xc ¼ 0:25; (ii) 0.5, (iii) 0.75. V =C ¼ 1 in all cases.
(17)
r : rþ2
(18)
Since p Xp0 whatever the form of the function G, we see that the evolutionarily stable level of fighting is lowest (for given r=V/C) in this case. Comparing the two forms of G illustrated in Fig. 3 we see that the step function results in a lower evolutionarily stable level of fighting than the smooth curve. This illustrates the following general result established in the Appendix: increasing the advantage (in terms of the probability of winning a fight) to the higher ability opponent for a given difference in ability decreases the evolutionarily stable level of fighting. Let the random variable X denote the ability of a randomly selected population member. To investigate the effect of increasing the spread of abilities we can compare this original distribution with the case where
ARTICLE IN PRESS J.M. McNamara, A.I. Houston / Journal of Theoretical Biology 232 (2005) 1–6 1
Level of fighting
0.8
(i)
(ii)
0.6
0.4
(iii) 0.2
0 0
0.5
1
1.5
2
2.5
3
V /C Fig. 3. The evolutionarily stable level of fighting ðp Þ2 as a function of V/C in three cases. (i) The standard Hawk Dove game with no differences in ability. (ii) The game with differences in ability and the probability of winning a fight, G(x,y), given by Eq. (15). (iii) The game with differences in ability and the most able contestant always winning a fight. In (ii) and (iii) ability in the population is uniformly distributed between 0 and 1.
the ability of a randomly selected individual is X~ ¼ kX ; where the constant k satisfies k41. Increasing the spread of abilities in this way while keeping G the same is equivalent to keeping the distribution the same but ~ yÞ ¼ Gðkx; kyÞ: Since G~ is changing G to G~ where Gðx; more step-like than G the result of the transformation is to reduce the evolutionarily stable level of fighting.
6. Discussion Like Crowley (2000), we have shown that when there is a continuous distribution of fighting abilities and an animal knows its own fighting ability but not the ability of its opponent then the ESS is of the form: play Hawk if the animal’s ability is greater than a threshold level and play Dove if its ability is below this threshold level. In contrast to this conclusion, Pagel and Dawkins (1997) consider a strategy that bases its probability of fighting on its estimate of its own ability and argue that such a strategy will not do better than always fighting or never fighting. They thus conclude that a strategy of basing fighting behaviour on one’s own ability cannot be stable. A problem with this conclusion is that it considers a strategy based on a probability of fighting, i.e. it is looking for a mixed strategy. But the ESS cannot be
5
mixed when the distribution of ability is continuous (Crowley, 2000). As we have seen, the ESS is a pure contingent strategy. Our main conclusion is that if an animal knows its own ability but does not know the ability of its opponent, the level of fighting at the ESS is less than the level in the standard Hawk–Dove game in which differences in ability are absent or ignored. Thus even in the absence of knowledge of the ability of an opponent (either by assessment or individual recognition coupled with previous experience) a knowledge of one’s own ability reduces fighting. To see why this reduction occurs compare the standard Hawk–Dove game with the game in which there are differences in ability. In the standard game, if an individual plays Hawk and the opponent also plays Hawk the probability of winning the fight is 0.5. In contrast suppose that an opponent were to play Hawk in our game, then the opponent has ability above the critical threshold and is better than average at winning fights. Thus a randomly selected individual has probability of less than 0.5 of winning a fight if it plays Hawk and the opponents fights back. This deters individuals from playing Hawk unless they are of high ability. Consequently there is less fighting at evolutionary stability. We have assumed that animals ‘‘know’’ both the function G(x,y) that determines the probability that an individual with ability x wins a fight against an individual with ability y and the distribution of abilities in the population. Both of these functions influence the level of fighting at the ESS. We have shown that as the function G(x,y) becomes steeper, the level of fighting goes down. Changing the distribution function can be thought of as rescaling the function G(x,y). As a consequence, increasing the spread of abilities decreases the level of fighting. Crowley (2000) considers the ESS both when an animal knows just its own ability and when it knows both its ability and the ability of its opponent. When both contestants know each other’s ability as well as their own, individuals have well-defined roles and a contest is settled by convention. When the individual with the greater ability always wins (i.e. G is a step function), at evolutionary stability there is no fighting. When G is not so steep, there is still no fighting when VoC, but there may be fights between individuals of similar ability when V4C. There will, however, typically be less fighting than when an individual only knows its own ability. To summarize our conclusions, if an individual only knows its own ability, there is less fighting than when neither animal knows its ability. If both animals know their ability, then the level of fighting will typically be even lower. Which of these cases is likely to hold will depend on a population’s biology. In a small population, it might be possible for an animal to know
ARTICLE IN PRESS J.M. McNamara, A.I. Houston / Journal of Theoretical Biology 232 (2005) 1–6
6
both its ability and that of all other members of the population. In a large population in which there are no visual cues that predict ability, it might not be possible to know the ability of other animals. In this case an animal might learn its ability as a result of fighting. We note that there might be a tradeoff between gaining information and avoiding injury (cf. Welton et al., 2003).
Appendix A. The dependence of the PHP on the shape of G
by Eq. (10). Differentiating with respect to a and using the fact that Gðx ; x ; aÞ ¼ 0:5 we obtain Z 1 @G dx 1 f ðvÞ dv 2ð1 p0 Þf ðx Þ þ ð1 þ p0 Þ da x @x Z 1 @Gðx ; v; aÞ f ðvÞ dv: ðA:3Þ ¼ ð1 þ p0 Þ @a x The term multiplying dx =da on the left-hand side of this equation is positive since @G=dx is positive. The right-hand side is also positive by inequality (A.1). Thus dx =da40: Since the threshold increases with a the PHP decreases as a increases.
Suppose that the function G Gðx; y; aÞ depends on the parameter a as well as x and y. We investigate how the shape of G affects the PHP by assuming that @Gðx; y; aÞ o0 for xoy: (A.1) @a Thus as a increases so does the advantage (in terms of the probability of winning a fight) to the higher ability opponent of a given difference in ability. Denoting the ESS critical threshold for given a by x ðaÞ we have Z 1 Gðx ðaÞ; v; aÞf ðvÞ dv ð1 þ p0 Þ x ðaÞ Z 1 f ðvÞ dv p0 ðA:2Þ ¼ x ðaÞ
References Crowley, P.H., 2000. Hawks, doves, and mixed-symmetry games. J. Theor. Biol. 204, 543–563. Maynard Smith, J., 1982. Evolution and the Theory of Games. Cambridge University Press, Cambridge. Maynard Smith, J., Price, G.R., 1973. The logic of animal conflicts. Nature 246, 15–18. Pagel, M., Dawkins, M.S., 1997. Peck orders and group size in laying hens: futures contracts for non-aggression. Behav. Processes 40, 13–25. Welton, N.J., McNamara, J.M., Houston, A.I., 2003. Assessing predation risk: optimal behaviour and rules of thumb. Theor. Popul. Biol. 64, 417–430.