Can natural selection favour indiscriminate spite?

Spiteful behaviours occur when an actor harms its own fitness to inflict harm on the fitness of others. Several papers have predicted that spite can be favoured in sufficiently small populations, even when the harming behaviour is directed indiscriminately at others. However, it is not clear that truly spiteful behaviour could be favoured without the harm being directed at a subset of social partners with relatively low genetic similarity to the actor (kin discrimination). Using mathematical models, we show that: (1) the evolution of spite requires kin discrimination; (2) previous models suggesting indiscriminate spite involve scenarios where the actor gains a direct feedback benefit from harming others, and so the harming is selfish rather than spiteful; (3) extreme selfishness can be favoured in small populations (and in some cases small groups) because this is where the feedback benefit of harming is greatest.


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Spite is the hardest type of social trait to explain. Spiteful behaviour reduces the lifetime fitness of both the recipient and the performer (actor) of that behaviour (Hamilton 1970). In terms of 26 Hamilton's rule, -C + RB > 0, spite represents the case where there is a fitness cost to the actor (positive C), and a fitness cost to the harmed recipient (negative B), which can only be favoured 28 if the genetic relatedness term, R, is negative. Understanding the meaning of negative relatedness is therefore crucial for explaining how and why spite evolves. 30 It has been argued that the evolution of spite requires kin discrimination, allowing the actor to direct harm towards a subset of individuals with whom they share relatively low genetic 32 similarity (Foster &  negative relatedness arises because the actor's genetic similarity to primary recipients is less than its genetic similarity to secondary recipients (Lehmann et al. 2006). In contrast, without kin 38 discrimination, harming behaviours could not be directed at individuals to whom the actor is negatively related, so indiscriminate spite should be impossible. 40 However, a number of theoretical studies have suggested the possibility for indiscriminate spite. Hamilton (1970) originally suggested that if genetic similarity is measured relative to the 42 entire population (including the actor), then there will be a negative relatedness between the actor and all others in the population, especially in small populations. Consequently, several papers 44 have predicted that spiteful harming, directed indiscriminately at others, could be favoured in sufficiently small populations (Hamilton 1970(Hamilton , 1971Grafen 1985;Vickery et al. 2003;Taylor 46 2010; Smead & Forber 2012). As a specific example, Verner (1977) and Knowlton and Parker (1979;Parker & Knowlton 1980) suggested that individuals could be favoured to hold territories 48 that are larger than needed for their own interest ("super-territories"), in order to spitefully exclude others from resources. It is not clear, though, whether such indiscriminate harming traits 50 are truly spiteful.
Here, we resolve this disagreement over indiscriminate spite. Many harming traits will be 52 costly to primary recipients (B < 0) but provide a direct fitness benefit to the actor, because they reduce competition for the actor or its offspring. Consequently, the traits are selfish (-C > 0) 54 rather than spiteful (-C < 0) (Hamilton 1970;Keller et al. 1994;Foster et al. 2001;West & Gardner 2010). We address the possibility that indiscriminate harming traits like territory size 56 have been misclassified as spiteful when they are actually selfish (Colgan 1979;Tullock 1979).
Our specific aims are to: (1) determine generally whether indiscriminate harming evolves as a 58 spiteful or a selfish trait; (2) examine how different modelling approaches can change the meaning of negative relatedness and lead to misclassification of harming traits; (3) re-analyse the 60 Knowlton & Parker (1979) territory-size model to determine whether it predicts spiteful behaviour. 62

Harming traits
We first modelled natural selection acting on a harming trait, following the approach of Lehmann 64 et al. (2006). The trait has a fitness effect on a focal actor (-C) and on two categories of recipients: the harmed primary recipients and the unharmed secondary recipients who benefit 66 from reduced competition (fitness effects B1 and B2, respectively). Crucially, we define an individual's fitness as its number of offspring that survive to adulthood (not simply the number of 68 offspring produced), which is consistent with other definitions used for classifying social traits (Hamilton 1964;Rousset 2004;Lehmann et al. 2006;West et al. 2007). We assume that fitness 70 effects on the actor, primary recipients, and secondary recipients must sum to zero because of competition for finite resources (Rousset & Billiard 2000): 72 implying that any decrease in fitness for one category necessarily means an increase in fitness for 76 another. Our model could apply to any finite population of constant size or to a local "economic neighborhood" (Queller 1994 To predict the direction of natural selection acting on the harming trait, we considered the 82 fate of a mutant harming allele in a population of individuals with a fixed, resident genotype. The success of the mutant allele depends on its "inclusive fitness effect" (Hamilton 1964): the sum of 84 effects from a focal actor's mutant trait on its own fitness and on the total fitness of each recipient category, weighted by their genetic similarity with the actor. Under the usual assumptions of 86 weak selection and additive gene action, the inclusive fitness effect for our model is

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where Q1 and Q2 are probabilities of sharing identical genes between the focal actor and a random individual from the primary and secondary recipients, respectively. We note that the fitness 92 effects in Equation 2 could alternatively be weighted by relatedness coefficients, where genetic similarity is measured with respect to a reference population (e.g., . / = 0 1 20 3 %20 3 , where -3 is the 94 average genetic similarity in the entire population, including the actor; Hamilton 1970). However, doing this would not change any of the results given below. We therefore prefer the simpler 96 approach used in Equation 2 and what follows below.
In the following sections, we examine two different ways of defining the category of 98 secondary recipients and therefore partitioning the fitness effects of harming. Both methods correctly predict the direction of selection (they give the same sum as in Eq. 2). The first 100 partitioning also maintains complete separation of direct and indirect fitness effects (-C and RB, respectively), making it appropriate for classifying harming traits as selfish (-C > 0) or spiteful (-102 C < 0). In contrast, the second partitioning obscures the separation of direct and indirect fitness effects, making it inappropriate for classifying traits in this way. 104

Is indiscriminate harming spiteful or selfish?
We determined the conditions for a harming trait to be classified as spiteful or selfish. For this 106 purpose, we assume that the focal actor, primary recipients, and secondary recipients are mutually exclusive categories. This ensures that the actor is not a recipient of its own behaviour, and so the 6 -C term in the inclusive fitness effect (Eq. 2) captures all effects of the actor's harming behaviour on its own fitness. From Equation 2, we derived the typical two-party version of Hamilton's rule 110 by eliminating the fitness effect on secondary recipients, using B2 = C -B1 (from Eq. 1). After rearrangement, the inclusive fitness effect is positive, and the harming trait is favoured, when 112 which is Hamilton's rule with the relatedness between actor and primary recipients given by 116 0 4 20 5 %20 5 ≡ . % . This is the genetic similarity between the actor and an individual from the potential primary recipients, measured relative to an individual from the potential secondary recipients. 118 Equation 3 implies that indiscriminate spite cannot evolve. This is because negative relatedness (and hence an indirect fitness benefit of harming) will arise only if harm can be 120 directed at primary recipients who are less genetically similar to the actor than secondary recipients are (Q1 < Q2). In contrast, if the actor were harming others indiscriminately-for 122 example, harming a random subset of a population or local economic neighbourhood-then its expected similarity to these primary recipients would be the same as to the set of potential 124 secondary recipients (Q1 = Q2), and relatedness would be zero (R1 = 0). This implies that indiscriminate harming will be favoured when it is a selfish trait with a positive direct fitness 126 benefit (-C > 0).

Why does misclassification occur?
7 Misclassification of harming traits can occur because the fitness effects of social traits can be partitioned in different ways (Frank 1998). An alternative way of partitioning the effects of 130 harming is to include the actor in the set of secondary recipients who may benefit from reduced competition. In fact, it is often implicitly assumed that the set of potential secondary recipients is 132 the entire population (or economic neighbourhood), including the focal actor (Hamilton 1970(Hamilton , 1971; 138 using lower-case letters to indicate that the fitness effects no longer match those from Equation 2. In particular, b2 is now the benefit of reduced competition that may be experienced by all 140 individuals in population (including the actor), and -3 is the probability of genetic identity between the focal actor and a random individual the entire population (including itself). It follows 142 that -c is not a total direct fitness effect because it excludes the secondary benefit of harming that feeds back to the focal actor (increased direct fitness due to reduced competition; Fig. 1 Our key distinction here is that harming behaviours can be either beneficial or costly to 170 the actor (-C > 0 or -C < 0), whereas spiteful behaviours are strictly costly to the actor (-C < 0).
We showed that indiscriminate harming is always favoured because it is beneficial to the actor-172 it has a positive effect on the actor's number of surviving offspring (-C > 0). Moreover, indiscriminate harming can be favoured most in small populations (or small economic 174 neighbourhoods) because this is where the focal actor can benefit most from the reduced competition that results from its harming behaviour. 176

Re-visiting "super-territories"
We next re-examined the territory size model from Knowlton & Parker (1979;Parker & 178 Knowlton 1980). We first analysed the model to fully separate direct and indirect fitness effects (applying Eq. 2), asking whether the model predicts selfish behaviour, as expected. We then used 180 the alternative approach (applying Eq. 4) to illustrate why previous studies have interpreted territory size as a spiteful trait. 182 We considered a finite, deme-structured population ("island model"; Wright 1943) with d demes (assuming d > 1) and n individuals competing for territory in each deme (total population 184 size is N = dn). Individuals that secure a territory have offspring and then die before a fraction m of their offspring disperse independently to a random deme in the entire population. All 186 individuals have a genetically-determined strategy for the size of territory that they try to obtain (a continuous trait). Taking over a larger territory has three key effects: (1) it incurs a fecundity 188 cost for the actor (we assume a linear cost with increasing trait size, with slope -a and a Î [0,1]); (2) it harms the actor's deme mates by taking resources away and reducing their fecundity; (3) it 190 reduces the competition faced by all remaining offspring in the population to secure a territory in the next generation. 192 We first assumed that the actor, primary recipients, and secondary recipients are mutually exclusive categories (as in Eq. 2). In the Appendix, we derive an expression for the fitness, W, of 194 a focal actor as a function of its own territory-size strategy, x; the average strategy of its deme mates (primary recipients), y; and the average strategy of individuals in other demes (secondary 196 recipients), z. We used this "neighbour-modulated" fitness function to derive the inclusive fitness 202 where all partial derivatives are evaluated in a monomorphic population (x = y = z). We derive expressions for Q1 and Q2 in the Appendix, and with these we determined the equilibrium of the 204 model (D, where directional selection stops) by solving ΔWIF = 0. We also checked that the equilibrium is a convergence-stable strategy, denoted z*, meaning that if the population is 206 perturbed from the equilibrium then natural selection will push it back ( We found that the equilibrium of our model, z* = 1/(aN), is identical to that originally 208 predicted by Parker & Knowlton (1980); however, our analysis shows that the optimal territory size strategy is selfish rather than spiteful. Territory size cannot be spiteful in this model because 210 the actor's genetic similarity to individuals in other demes is always equal to or less than the similarity to deme mates (Q1 ≥ Q2). Accordingly, the relatedness to primary recipients (measured 212 relative to secondary recipients) is never negative (R1 ≥ 0), and so there is no indirect benefit of larger territory size. Moreover, when offspring dispersal is limited (m < 1) and deme mates are 214 positively related (R1 > 0), there is no indirect benefit of smaller territory size (as a form of helping). This is because limited dispersal increases competition among offspring within the 216 deme, which promotes harming and exactly cancels the effect of positive relatedness (as in Taylor 1992). Territory size therefore evolves for its direct benefit only, with larger territories promoted 218 by a smaller fecundity cost to the actor (smaller a) and smaller population size (smaller N).
Specifically, the direct fitness effect at equilibrium (z = z*) is 220 which is either positive (when m < 1) or zero (when m = 1). In the case of full offspring dispersal 224 (m = 1), the equilibrium is the point where the fecundity cost to the actor is exactly balanced by the feedback benefit experienced by its offspring (reduced competition for space in the next 226 generation). As the population approaches this equilibrium, however, direct fitness is always positive (-C > 0), confirming that territory size evolves as a selfish trait (Fig. 2). 228 We next assumed that the set of secondary recipients is the entire population, including the focal actor (as in Eq. 4). In this case, the inclusive fitness effect is gives the same answer as before, z* = 1/(aN). 236 This version of the model shows, however, why territory size could be misclassified as  , aN/(N -1). The partitioning in Equation 9 therefore splits the 248 total direct fitness effect of territory size into two separate terms, -c + (-1/[N-1])b1 or -c + (1/N)b2, which could be misinterpreted as a direct fitness cost (-C < 0) and an indirect fitness 250 benefit (RB > 0).

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We examined a general model of harming traits and a specific model where larger territory size is an indiscriminate harming trait. In both models we found that: (1) the evolution of spite requires 254 kin discrimination, where the actor harms only a subset of other individuals (those with relatively low genetic similarity); (2) without kin discrimination, harming can be favoured but only when 256 there is a sufficient direct, feedback benefit to the actor (reduced competition for the actor or its offspring); (3) indiscriminate harming can be favoured most in small populations (or small 258 economic neighbourhoods), where the feedback benefit to the actor is greatest; (4) previous studies have misclassified indiscriminate harming as spite, partly because they misinterpret the 260 feedback benefit as an indirect (kin-selected) benefit (RB > 0). Overall, our analyses illustrate why indiscriminate harming traits are selfish rather than spiteful. 262

Classifying harming traits
For the purposes of classifying harming traits, we found that it is easiest to treat the actor, 264 primary recipients, and secondary recipients as separate categories. This makes it straightforward to separate the total direct and indirect fitness effects of harming (-C and RB, respectively) and 266 ensures that non-zero relatedness will always be associated with an indirect fitness effect. For example, spiteful harming (-C < 0, B < 0) requires that harm is directed at primary recipients to 268 whom the actor is negatively related (with respect to secondary recipients; Q1 < Q2 and R1 < 0), resulting in a positive indirect fitness effect (R1B > 0) (Lehmann et al. 2006). In contrast, when harming is indiscriminate, the actor has zero relatedness to primary recipients (with respect to secondary recipients; Q1 = Q2 and R1 = 0), and so harming can be favoured as a selfish trait only 272 We showed that misclassification of indiscriminate harming is due to an implicit 274 assumption that the focal actor is a secondary recipient of its own behaviour (Hamilton 1970(Hamilton , 1971; Others have suggested that harming traits should be classified based on their primary 282 effects only, rather than their total fitness effects (Krupp 2013). This means that indiscriminate harming traits like larger territory size, which may be associated with a survival or fecundity cost 284 (-c < 0 in the terms of our model), would be classified as spiteful, despite the feedback benefit to the focal actor. We argue, however, that a classification based on total fitness effects (-C and RB) 286 is more useful (Hamilton 1964;West et al. 2007). This is because it emphasises the fundamental distinction between spiteful harming, which is favoured by indirect fitness benefits and requires 288 kin discrimination, versus selfish harming, which is favoured by direct fitness benefits and does not require kin discrimination (West & Gardner 2010). Similar arguments have been made for 290 maintaining the distinction between altruistic helping (-C < 0, B > 0) and mutually-beneficial helping (-C > 0, B > 0) ). 292

Indiscriminate harming in nature
We found that selfish indiscriminate harming can be favoured most in small populations or small 294 economic neighbourhoods (e.g., small groups with relatively local competition). This is because harming primary recipients leads to reduced competition for all individuals in the population or 296 group, and a focal actor receives a larger fraction of this secondary benefit when it makes up a larger fraction of the population or group. Indiscriminate harming can therefore be thought of as 298 producing a type of public good for secondary recipients (Tullock 1979), analogous to indiscriminate helping, which is often thought of as a public good for primary recipients. A key 300 difference is that indiscriminate helping is inhibited by local competition (Taylor 1992;Griffin et al. 2004); in contrast, indiscriminate harming requires local competition so that the focal actor 302 can actually benefit the reduced competition that results from its harming (Gardner & West 2004b). Here, we derive an expression for the fitness of a focal actor with a mutant territory size strategy, 390 based on the models of Knowlton and Parker (1979;Parker and Knowlton 1980). We consider a population that is structured into d demes of n individuals competing for territories, where each 392 deme has A units of available territory. The focal actor's strategy, x, represents a continuous number of territory units that it attempts to gain (x > 0). The average strategy of the actor's deme mates is y, and the average strategy in all other demes is z.
We first calculate the expected offspring production (expected fecundity, F) for the focal 396 actor, an individual in the actor's deme, and an individual in another deme. These expected values depend on: (1) the probability of an individual acquiring a territory (assuming that 398 available spaces are acquired completely randomly); (2) the cost associated with the individual's strategy (assuming fecundity declines linearly with increasing territory size strategy; f (x) = 1 -400 ax, where 0 < a < 1). For the focal actor, there are A/y spaces available in the deme, and we use the simplifying assumption that a mutant individual has priority to claim the territory units 402 denoted by its strategy (Knowlton and Parker 1979). Therefore, the focal actor has a 1/n probability of acquiring a territory, and its expected fecundity is 404 (A1) 406 The space available for others in the patch depends on whether or not the focal actor claims a 408 territory. The actor gains access to the patch with probability A/ny, and in this case (Ax)/y spaces remain; otherwise, A/y spaces are available. The expected fecundity for one of the n -1 410 deme mates of the focal actor is therefore 412 Finally, for an individual in another deme in the population, there are A/z spaces available, and so the expected fecundity for one of these individuals is 416 We next calculate the focal actor's fitness, *(W, Z, D), which is the number of its offspring 420 that survive to compete for a territory in the next generation. This can be partitioned into two terms, the first term accounting for offspring that compete on the focal actor's natal deme (those 422 that did not disperse, with probability 1-m, and those that dispersed but landed on the natal deme, with probability m/d) and the second term accounting for offspring that disperse with probability 424 m to compete in the d -1 non-natal demes: These are the probabilities of genetic identity used in Equations 7 and 9 of the main text. 458 Figure 1. Partitioning the fitness effects of a harming trait. When a focal actor harms a primary recipient, this reduces competition and may therefore benefit the unharmed secondary recipients and the actor itself ("feedback benefit"). Some modelling approaches include the actor in the set of secondary recipients of the harming trait. However, the total direct fitness effect (-C in Hamilton's rule) includes the fecundity cost of expressing the harming trait plus the feedback benefit. . Territory size and direct fitness. Larger territory size is promoted by smaller population size (smaller dn) and reduced offspring migration from the deme (smaller m), both of which increase the direct benefit to an actor for harming its deme mates. However, reduced migration also increases the relatedness among deme mates, which inhibits larger territory size. Ultimately, the optimal territory size strategy (z*, dashed line) is independent of migration rate and evolves as if the population were fully mixed (m = 1). Other parameters used: d = 5, a = 0.05.