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Journal of Theoretical Biology 242 (2006) 790–797 www.elsevier.com/locate/yjtbi
Why kin and group selection models may not be enough to explain human other-regarding behaviour Matthijs van Veelen CREED, Universiteit van Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands Received 28 February 2006; received in revised form 25 April 2006; accepted 25 April 2006 Available online 9 May 2006
Abstract Models of kin or group selection usually feature only one possible fitness transfer. The phenotypes are either to make this transfer or not to make it and for any given fitness transfer, Hamilton’s rule predicts which of the two phenotypes will spread. In this article we allow for the possibility that different individuals or different generations face similar, but not necessarily identical possibilities for fitness transfers. In this setting, phenotypes are preference relations, which concisely specify behaviour for a range of possible fitness transfers (rather than being a specification for only one particular situation an animal or human can be in). For this more general set-up, we find that only preference relations that are linear in fitnesses can be explained using models of kin selection and that the same applies to a large class of group selection models. This provides a new implication of hierarchical selection models that could in principle falsify them, even if relatedness—or a parameter for assortativeness—is unknown. The empirical evidence for humans suggests that hierarchical selection models alone are not enough to explain their other-regarding or altruistic behaviour. r 2006 Elsevier Ltd. All rights reserved. Keywords: Kin selection; Group selection; Altruism; Hamilton’s rule; Preference relation
1. Introduction In models of selection for altruistic behaviour, all individuals in all generations usually face the possibility of one and the same fixed fitness transfer (see for instance Hamilton, 1964, 1975; Charnov, 1977; Michod and Abugov, 1980; Michod and Hamilton, 1980; Grafen, 1984, Nunney, 1985; Queller, 1985, 1992; Taylor, 1989; Wilson and Dugatkin, 1997). Consequently, the two phenotypes are (1) to make this transfer and (2) not to make it. The fitness transfer is characterized by costs c to the acting individual and benefits b to the receiving individual and the best known prediction from kin selection theory is Hamilton’s rule, that states that phenotype 1 will win if c rbo0, where r is relatedness, and phenotype 2 will win if not. In reality, however, different individuals in different generations may face a variety of possible fitness transfers. If we take as a possibility for a fitness transfer, for instance, a situation Tel.: +31 20 5255293; fax: +31 20 5255283.
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in which a chimpanzee can assist a brother in a fight, then it is clear that not all fights will be the same; the risk of getting hurt yourself varies, as well as the benefit to the brother. Another example is a situation in which a parent can try to save a child from drowning or being eaten by a predator. Such situations will also come with differences in risk of drowning or getting hurt yourself, as well as differences in the odds of actually saving the child. One may conclude that Hamilton’s rule implies that some of these risks will be taken and others will not, and indeed if we treat all of the different fitness transfers separately, Hamilton’s rule does make a prediction as for which phenotype will be selected for that particular fitness transfer. However, it is not very plausible that every such possible fitness transfer has a pair of possible phenotypes of its own and that selection has operated separately on every pair in this whole continuum of pairs of phenotypes. This being a rather unlikely scenario, one at least expects natural selection to exploit the similarities between varying situations more efficiently, in the sense that successive mutations build one coherent system to control behaviour in similar situations. The suggestion made in this paper is
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therefore to take preference relations for phenotypes, because they allow for the possibility that individual behaviour in a variety of dilemmas can be characterized with only a few parameters. Preference relations also allow for a description of behaviour that more closely matches with how other-regarding behaviour actually seems to be implemented in, for instance, humans. With selection operating on preference relations, we can prove a result that has two different implications. The first one is positive: when phenotypes are taken to be preference relations, we still find that the end result is that individuals behave as if selection would have operated on every possible fitness transfer separately. That is good news, because it solves the unrecognized problem when going from Hamilton’s rule—as a prediction in a setting with one fixed fitness transfer—to a prediction of behaviour in a world where individuals may face differing situations. The other implication is less reassuring. It is relatively straightforward to show that all preference relations that result from a process of kin selection must be linear in fitnesses. The same applies to standard group selection models. This new implication of most known hierarchical selection models can therefore in principle also rule them out as the sole explanations of other-regarding, altruistic behaviour. Observed human altruistic preference relations, for instance, are rather hard to interpret as being linear in fitnesses, and therefore we either have to find reasons why payoffs in monetary or food terms translate to fitnesses in rather peculiar ways, or accept that there is more to the evolution of other-regarding behaviour than the standard hierarchical selection models only. 2. What is a preference relation? In mathematical economics, or microeconomics, the concept of a preference relation is used as a general way to handle human behaviour when faced with choices. There is however nothing uniquely human to being faced with choices and therefore there is also no reason to restrict the use of preference relations to economics. Because not all readers will be familiar with microeconomics, a few central definitions will be repeated (see Mas-Colell et al., 1995). Assume a set of alternatives X . Elements x and y from this set X are to be compared to each other, and therefore we have a binary relation on X, denoted by h. This binary relation is called a preference relation, and we read xhy as ‘‘x is weakly preferred to y’’. Given any preference relation h, we can derive two other important relations on X: 1. The strict preference relation, , defined by x y 3 xhy but not yhx. 2. The indifference relation, , defined by xy 3 xhy and yhx.
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In the examples in this section, X will be the set of twodimensional vectors in R2þ , where x ¼ xself ; xother 2 X is a combination of fitnesses of the individual itself and the other. If x y that means that when the status quo is y, an individual with this preference relation does make the fitness transfer that consists of incurring a fitness loss c ¼ yself xself himself, while the other gains b ¼ xother yother . Usually we restrict attention to preference relations that have some consistent structure. That can be done by looking at preference relations that can be represented by (continuous) utility functions. A function u : X ! R is a utility function representing preference relation h if, for all x and y in X, the following holds: xhy3uðxÞXuðyÞ. It is important to note that all utility functions represent some preference relation, but not all preference relations have a utility function that represents them. Utility functions also have the practical advantage that they can describe preference relations in a concise and insightful manner, but there is also a conceptual reason why they are useful here. If we can represent choice behaviour by a utility function with one or a few parameters, it seems likely that there are also shortcuts that in one go code for behaviour in all the different dilemmas, rather than a preference relation being merely a list of which alternative would be chosen for each possible fitness transfer separately. The latter case would bring us back in the situation where a preference relation is nothing but a combination of phenotypes for every possible fitness transfer as described in the Introduction. A utility function on the other hand captures the idea that a simple structure can lead to choice behaviour for a variety of dilemmas, and that the parameters of this choice behaviour can evolve. In the remainder of this section, a few examples of utility functions are given. They stem from the literature in experimental economics, in which the elements of X represent money combinations rather than combinations of fitnesses. The examples will nonetheless help to illustrate the possible shapes that preference relations can have. They will also provide material for the final section, where we will discuss whether money, food or risks can plausibly be linked to fitness such that the preferences we observe can be explained with hierarchical selection models. To visualize the different preference relations, pictures with iso-utility curves will be drawn. An iso-utility curve is a set of points that represent alternatives that yield the same value of the utility function. 2.1. Altruism only (or spite) The first family of preference relations has members that are determined by a parameter a and are defined by utility functions ua ðxself ; xother Þ ¼ xself þ axother . These utility functions are linear in both xself and xother , where xself has coefficient 1 and xother has coefficient a. An intuitive interpretation of a as the weight that one
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individual attaches to the interest of another is a correct one. (Interesting detail: Edgeworth (1881) already called a a coefficient of efficient sympathy.) Because a preference relation is much more general than just a choice between only two possible alternatives, it makes sense to define altruism as a property of a preference relation as a whole, rather than as a property of one choice or act. Some preference relations will imply a stronger inclination than others to decide in favour of the other individual, even if that comes at a cost to the focal individual itself. Within this family of preference relations, the parameter a is a good measure of this inclination. Altruism therefore is also a quantitative notion; preference relations with high a are more altruistic than preference relations with low a. A natural restriction on a could be to demand that it lies within the interval ½0; 1, but in principle there can be preferences with a negative altruism parameter, representing spite, or with a’s that exceed 1, which means that an individual with such a preference relation attaches a greater weight to the other person than it does to itself. An altruism parameter that is equal to 0 can be seen as the selfish benchmark, which is a utility function that disregards whatever gains or losses the other would incur; uðxself ; xother Þ ¼ xself . Fig. 1 depicts two such preference relations, one with a ¼ 1, one with a ¼ 0:5. Note that the sharper ascend of the iso-utility curves in the second picture implies that it takes a larger gain of the other individual to make an individual with this preference relation go for the altruistic alternative. Within our definition, the roles of the different alternatives are symmetric; no information is given as to whether either one can be described as a status quo, making the other the result of an action to be taken. If we
would, then this could make a difference in interpretation. For instance, if a ¼ 0:5, then ð3; 1Þ ð2; 2Þ. If ð3; 1Þ is the status quo and this preference relation describes the behaviour of the individual, it means that it does not give one unit to the other. However, if ð2; 2Þ is the status quo, it means that the individual does take one unit from the other. When looking at isolated acts, the act of not giving in the first case may at worst only not get the label ‘‘altruistic’’, while taking in the second case sometimes is described as ‘‘selfish’’ (see Engels, 1983). In the more general setting, however, we do not qualify acts as altruistic or selfish. Instead, we look at preference relations, and in this case we can even quantify altruism, as it is measured by the altruism parameter a. 2.2. Altruism and inequality aversion A more general family of preference relations has parameters a and r and is represented by utility functions ua;r ðxself ; xother Þ ¼ ðxrself þ axrother Þ1=r for a 2 ½0; 1 and r 2 ð1; 1. This type is known in economics as a constant elasticity of substitution (CES) utility function. If we choose r ¼ 1, Xother
= 1, = 1 Xother
Xself
Xother
Xself u (xself , xother) = xself + xother Xother
= 1, → 0
Xself
Xother
Xself u (xself , xother) = xself + 12– xother Fig. 1. Different Degrees of altruism.
= 1, → −∞
Xself
Fig. 2. Different degrees of inequality aversion.
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then we have ua;1 ðxself ; xother Þ ¼ xself þ axother , so the family of altruistic preference relations discussed above is included in this one for a 2 ½0; 1. For ra1, however, we get utility functions that are no longer linear. For a ¼ 1, Fig. 2 shows iso-utility curves for r ¼ 1; r ! 0 and r ! 1. The first picture is discussed above. The second utility function is limr!0 u1;r ðxself ; xother Þ ¼ xself xother and the third utility function is limr!1 u1;r ðxself ; xother Þ ¼ minfxself ; xother g. (See Mas-Colell et al. (1995, Exercise 3.C.6) for technical details.) Within this family we can disentangle altruism and inequality aversion and a and r are the parameters that indicate the extent to which they play a role in an individual’s preferences. If we take a constant r and vary a, we see that for increasing a the preference relation becomes more altruistic in the sense that the set of dilemma’s in which the utility function decides in favour of the altruistic option increases. If we on the other hand fix a, then r can be seen as a parameter that indicates the aversion to inequality; for r ¼ 1 inequality does not play a role at all and iso-utility curves are straight lines, while for r ! 1 inequality ends up being the only determinant of choices. The lower the r, the more the iso-utility curves are actually curved, which means that it matters more how equal or unequal the alternatives are. 2.3. Mixed emotions: altruism, inequity aversion and spite For all preference relations in Section 2.2, we have that whatever ðxself ; xother Þ is, if xother increases, then utility ua;r increases too. The same holds for utility functions from Section 2.1 with a40, while for ao0 the converse is true: if xother increases, then utility ua decreases, whatever ðxself ; xother Þ. It is, however, conceivable that preferences are not so unambiguous regarding increases in xother . One example from Fehr and Schmidt (1999), reduced to the two-player case, is the following utility function: ua;b ðxself ; xother Þ ( xself aðxother xself Þ if xother Xxself ; ¼ xself bðxself xother Þ if xself 4xother ; where it is assumed that bpa and 0pbo1. Note that ua;b increases in xother if xself 4xother , while it decreases in xother if xother 4xself . A picture for a ¼ 2=3; b ¼ 1=3 is Fig. 3. Please note that these utility functions are called inequity averse, as opposed to inequality averse in Section 2.2. Of the three examples given above, the first family contains linear utility functions, the second also nonlinear ones and the third even utility functions that are not everywhere increasing in both xself and xother . 3. Hierarchical selection models, linear preferences The question we would like to answer is which preference relation, or which utility function, would be favoured by kin selection. The answer is given by a theorem
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Xother
u, (xself , xother)
Xself
Fig. 3. Altruism as well as spite.
proved in the Appendix and a corollary. We begin with assuming that in a model with only one possible fitness transfer, inclusive fitness drives the population dynamics, and phenotypes with highest inclusive fitness will spread. This may not always be justified (see Rice, 2004), but this theorem only concerns the step from binary decisions in a model with only one possible fitness transfer to more general behaviour in a variety of dilemmas. Therefore, we change the possibilities for fitness transfers by assuming there is a range of them, over which a probability measure is defined that determines the odds with which the different possibilities of fitness transfers occur. Phenotypes will be preference relations. The theorem states that behaviour that follows an altruistic utility function in which all the relatives have weights according to their measures of relatedness can never be outcompeted by another preference relation. P This means that if we take the utility function ur ðxÞ ¼ ni¼1 ri xi , where ri is the measure of relatedness of individual i to individual 1, then this preference relation cannot be invaded successfully by another preference relation. The corollary in addition states that as long as there are possibilities for fitness transfers for which a preference relation describes behaviour different from what utility function ur would give, and they occur with positive probability, then ur has a selective advantage. Although this may not seem to add much to the idea of inclusive fitness maximization, it is important to realize that what it shows is that inclusive fitness can be maximized by decisions in a variety of possibilities for fitness transfers, while all that evolves is a parameter in a utility function. Suppose for instance that all interactions take place between duos with relatedness r and that the population consists of a resident with utility function ua ðxself ; xother Þ ¼ xself þ axother with aor. Then any mutant with a linear utility function with parameter b; aobor, will do at least as good, because 1. no good decisions by ua are reversed by ub , that is, if ua ðxÞ4ua ðyÞ and ur ðxÞ4ur ðyÞ, then also ub ðxÞ4ub ðyÞ and if ua ðxÞoua ðyÞ and ur ðxÞour ðyÞ, then also ub ðxÞoub ðyÞ, and 2. there are wrong decisions by ua that are reversed by ub , that is, there are x and y for which ua ðxÞ4ua ðyÞ but ur ðxÞour ðyÞ for which ub ðxÞoub ðyÞ.
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If possible fitness transfers in the second category occur with positive probability, ub will even do strictly better than ua . We therefore expect to see mutations until the altruism parameter equals r. Similarly, if we have a family of utility functions that allows for altruism as well as inequality aversion (Section 2.2), then the optimal parameters will be a ¼ r and r ¼ 1, that is, altruism to the degree of relatedness and no inequality aversion. The flip side of the theorem is that it contains a hitherto overlooked implication of hierarchical selection models. For Hamilton’s rule applied to preference relations, we found that on the domain of alternatives that occur often enough, they must be linear in xi for all i, in the sense that they can P be represented by a utility function ur1 ;...;rn ðxÞ ¼ ni¼1 ri xi . This linearity intuitively makes sense, for it is clear that if c rb40, then also ac rab40 for all a40, and it implies that kin selection can only explain preference relations that are linear in fitnesses. Linearity is in fact inevitable for all hierarchical selection models that have a structure similar to that of kin selection models. With kin selection models, we start with different genotypes that cause different fitness transfers between individuals. (In many examples, one of the genotypes causes fitness transfer 0, that is, no fitness transfer.) Then we figure out where costs and benefits go and calculate expected or average fitness changes for different genotypes, which tells us which alleles will increase and which alleles will decrease in frequency. The set-up of most group selection models is similar. We start with different types of individuals that behave differently when faced with the possibility of a (fixed) fitness transfer: some do and some do not make the transfer. For a given way of grouping the population, we figure out where costs and benefits go in order to find out whether the frequencies of the different types will increase or decrease. Within and between groups selection then work in opposite directions, and the assortativeness of the grouping determines which fitness transfers will and which will not be selected. This is the setting for which Price (1972) meant the group selection application of his equation to work (see also Van Veelen, 2005 for a critical evaluation) and that is discussed in Hamilton (1975). Assortativeness takes over the role of relatedness and the result takes the same form as Hamilton’s rule with a parameter for assortativity replacing r. For all models with this structure, the result must be linear utility functions. That is, assume that we want to find out whether a fitness transfer will be selected for in a model. This fitness transfer is given by a vector y x if x is the vector of fitnesses in the status quo and y the vector of fitnesses for mutant behaviour. If the model outcome is that y is favoured by selection over x, then the same will be true for x þ aðy xÞ for all a40. In other words, if fitness transfers are multiplied by a constant, nothing changes as for the direction of selection. The reason is that all we do in the model is bookkeeping in order to find out whether
frequency is going up or down in the (aggregate) population, and all numbers in the books reflecting changes will just shrink or be inflated by a. Please note that not all models where group formation plays a role have this structure. One exception in Sober and Wilson (1998, p. 27–29) is their model for the Dicrocoelium dendriticum (brain worm), the working of which crucially depends on assumptions that ensure that these brain worms are only present in ants in multiples of 10. This model also does not imply a version of Hamilton’s rule, but predicts a stable polymorphism. Other models with group formation but a different structure are for instance Aviles (1999, 2002) and Aviles et al. (2002). The argument that standard hierarchical selection models must lead to linear preferences does assume that selection can indeed reach the optimum. There may be reasons—for instance, related to the neural architecture— that inhibit selection arriving at the optimal utility function, in which case the implication does not hold. The richness of observed human other-regarding behaviour combined with the relatively simple structure of the optimum, when expressed in fitnesses, suggests that at least for humans the optimum is not infeasible. 3.1. Do we observe linear preference relations? Given that we know that hierarchical selection models imply linear preference relations, it would be worthwhile to find out what shapes they actually have in real life examples. Some efforts are made to recover other-regarding preference relations in humans (see for instance Fehr and Schmidt, 1999; Bolton and Ockenfels, 2000; Andreoni and Miller, 2002). Unfortunately, however, fitness is not an easily measurable unit and in experimental economics money is much more obvious a choice. For the question whether or not it is plausible that preferences are linear in fitnesses, we would have to find ways of going from one to the other. Although it will be rather hard to recover the current relation between wealth and fitness, and close to impossible to find it for pre-historic humans, we can at least try to come up with some reasonable assumptions. One such assumption could be that the marginal contribution of money to fitness is decreasing in money. That would be the case if money only determines survival probabilities, which may increase a lot per euro at a low budget, but less so on a high income. Formally, if xi is i’s monetary payoff and f : Rþ ! Rþ maps money to fitness, then we can assume that f 0 40 and f 00 o0. Such a function would turn linear utility functions into quasi-concave ones, which means that uðax þ ð1 aÞyÞX minfuðxÞ; uðyÞg (see MasColell et al., 1995). All utility functions mentioned in Section 2 are quasi-concave. Any explanation for utility functions that are decreasing in xother on a subset of R2þ would require a reason why more money for the other would imply a lower fitness of the self. For males such a reason could be that money also determines status, and status determines reproductive
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opportunities, of which there may only be a fixed amount. Formally, we then should be able to define a function f : R2þ ! R2þ that maps combinations of monetary payoffs to combinations of fitnesses, for which there are values ðx1 ; x2 Þ such that qf 1 =qx2 o0. Andreoni and Miller (2002) report an experiment that focusses on recovering preferences as purely as possible, without strategic interaction. They find that individual behaviour is such that quasi-concave utility functions could have generated it. Furthermore, they find that preference relations are quite heterogeneous, ranging from a ¼ 0 to 1 and from r ¼ 1 to r ! 1 in the family of preference relations from Section 2.2 (They do not use the same family of utility functions in their analysis of the data, but they find Selfish (a ¼ 0), Leontief (a ¼ 1; r ! 1) and Perfect Substitutes (a ¼ 1; r ¼ 1) utility functions.) Jealous preferences are not found in this setting, where jealous means that there is a part of the domain where utility is decreasing in xother , implying that subjects would be willing to pay for a reduction of the payoff of the other. Fehr and Schmidt (1999) look at preferences in a setting with more strategic interaction. Their results are consistent with the claim that a part of the subjects are purely selfish and part of them are inequity averse, as described in Section 2.3. This implies that some subjects do have jealous preferences (see also Bolton and Ockenfels, 2000). While all of these utility functions in monetary terms are not a priori at odds with utility functions that are linear in fitnesses, it would take a sound link between money and fitness to really find out. But even if we would establish such a link, it seems quite a challenge to explain why there is so much heterogeneity. The rather wide range of observed preference relations suggests that they can at least not all be (close to) linear in fitnesses. Again, if we observe nonlinearity and the real reason behind it is that the optimum preference relation is infeasible, then it would be erroneous to conclude that some selective force other than kin or group selection must have been involved. Here, however, we find a range of preference relations, some of which will be closer to linearity than others. It is the large diversity that suggests that even if some of those preference relations are (almost) linear in fitnesses, it is impossible that they all are. The results from Section 3 and from the Appendix imply that kin selection would have favoured the more linear preference relations over the less linear ones, and that the same holds for standard group selection models. 3.2. Possibilities for further testing: not all explanations can be right While money and food do not translate into fitness without complications, this may be slightly less problematic for the examples involving risks, as discussed in the Introduction. In the examples there are two options for behaviour: to help or not to help. Suppose that the latter option can be characterized as follows: the deciding
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individual survives for sure, while the other for sure does not. The first option—to help—might then be a situation where the helping individual has a probability p of surviving and the helped individual will survive with probability q. Suppose that we also observe that individuals choose to help given those two alternatives. The result derived in the Appendix implies that if this helping behaviour has evolved through kin selection, or is explained by a group selection model with a similar structure, then we should expect the same individuals to also come to the other’s aid if the chances of surviving in case of help are ap þ 1 a and aq for the helping and helped individual, respectively. This can be computed as follows: if 1 þ r 0op þ r q, then also að1 þ r 0Þoaðp þ r qÞ, which with a little rearranging leads to 1 þ r 0oap þ 1 a þ r aq. Note that it is not necessary to know relatedness to test for violations of linearity, which is a big advantage of this implication. Although laboratory experiments are obviously no option here, it might be that there are some natural experiments that could be informative. Hierarchical selection models are, for instance, mentioned as an explanation for alarm calls in birds (Sober and Wilson, 1998; Frank, 1998) but so is sexual selection (Zahavi, 1997; Miller, 2001; see also Grafen, 1990a,b). Empirical evidence on the linearity of such behaviour would, together with our result, offer a way to possibly falsify hierarchical selection models and help determine which selective model applies where. Acknowledgements I would like to thank Maus Sabelis for useful comments and advice and The Netherlands’ Organisation for Scientific Research (NWO) for financial support. Appendix A A.1. Definitions r ¼ ðr1 ; . . . ; rn Þ is a vector containing measures of relatedness r1 ; . . . ; rn of n individuals to individual 1, respectively. Obviously r1 ¼ 1. S is a set of convex, compact subsets of Rnþ . That means that a set S 2 S represents a choice situation an individual can be in, where the choice made in that situation influences the fitness of the focal individual and those of n 1 of its relatives. m is a probability measure on S, representing the odds with which different choice situations occur; mðAÞ is, for all A that are subsets of S, the probability that the choice situation will be in the set A. U is the set of all continuous utility functions, which makes u : Rnþ ! R an element of U if u is continuous. Different elements of this set may represent the same preference relation, but different preference relations cannot be represented by the same utility function.
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h; i isP used to denote inner products, that is: ha; bi ¼ ni¼1 ai bi .
as long as the expected inclusive fitness is not at the maximum value.
A.2. Results
Corollary 2. Successful mutations of the preference relation represented by utility function u are possible as long as Em r; arg max uðsÞ oEm max hr; si.
The following theorem states that if we choose ur ðxÞ ¼ P hr; xi ¼ ni¼1 ri xi for a utility function, expected inclusive fitness is maximized by the behaviour it represents, whatever the odds are for different choice situations to occur. Theorem 1. For all probability measures m, the following holds: ur ðxÞ ¼
n X
ri xi maximizes Em
i¼1
n X
ri arg max uðsÞi s2S
i¼1
over u 2 U.
s2S
s2S
Together, Theorem 1 and Corollary 2 imply that if we take preference relations as phenotypes, we can expect to see a sequence of successful mutations of the preference relation with ever increasing expected inclusive fitnesses, until we are at, or close to, a preference relation for which expected inclusive fitness is at the maximum value. Reformulation with the set U containing all possible complete preference relations—including intransitive ones—is straightforward.
Or, more concisely, ur ðxÞ ¼ hr; xi maximizes Em hr; arg max uðsÞi s2S
References
over u 2 U. Proof. Since maximizing u for a given u 2 U represents just one way of choosing an element s from S, we have that Em r; arg max uðsÞ pEm max hr; si s2S
s2S
for all u 2 U and all m. If we choose ur , however, then obviously, Em r; arg max ur ðsÞ ¼ Em max hr; si s2S
s2S
holds too, whatever m, because Em r; arg maxs2S ur ðsÞ ur ðsÞi ¼ Em r; arg maxs2S hr; si ¼ Em maxs2S hr; si. Therefore, for all m; ur ðxÞ ¼ hr; xi maximizes Em hr; arg maxs2S uðsÞi over u 2 U. & For a given m, the preference relation represented by ur may not be the only one that maximizes expected inclusive fitness. If we take for instance m to be a probability measure that is degenerate in S—which means that mðAÞ ¼ 1 if S 2 A and mðAÞ ¼ 0 if SeA—then we are back in the situation with only one possible fitness transfer. For this case, a utility function or a preference relation is actually a needlessly complex thing, because there is only one set of alternatives S from which the individual has to choose an element. In other words, all choice situations are identical and therefore any utility function u 2 U for which maxs2S hr; uðsÞi ¼ maxs2S hr; si for this particular S will do to achieve the maximum, whatever the definition of u on all kinds of alternatives that do not change the outcome of the maximization on this one S. However, more interesting probability measures m impose more restrictions on the preference relation that maximizes expected inclusive fitness. One can summarize that informally by saying that kin selection will force preference relations to be in line with ur on sets that occur often enough. More formally, successful mutations of the preference relation are possible
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