How Altruism Can Prevail Under Natural Selection by Ted Bergstrom and Oded Stark University of Michigan and Harvard University Current version: March 22, 2002 How Altruism Can Prevail Under Natural Selection Ted Bergstrom and Oded Stark Introduction Why have we economists been convinced for so long that our old friend, homo eco- nomicus must be selfish? No doubt we find considerable support for this hypothesis in the behavior of our colleagues. We might also expect that evolutionary pressures tend to produce selfish behavior–with the notable exception of the relation between parents and offspring. But can we expect natural selection to act in favor of altruistic behavior in other relationships? Evolutionary biologists have created a theory that predicts altruistic behavior, not only between parents and children, but also among siblings and other close relatives.1 Richard Dawkins’ expression of this view in The Selfish Gene, is that the replicating agent in evolution is the gene rather than the animal. If a gene carried by one animal is likely to appear in its relatives, then a gene for helping one’s relatives, at least when it is cheap to do so, will prosper relative to genes for totally selfish behavior. This paper presents a series of examples in which natural selection sustains cooperative behavior in single-shot prisoners’ dilemma games. In prisoners’ dilemma, cooperation al- ways gets a lower payoff for oneself and a higher payoff for one’s opponent than defection. Therefore it seems appropriate in this simple case to identify altruism with playing coop- erate in prisoners’ dilemma.2 The reason that cooperative behavior toward siblings can be sustained even where defection is a dominant strategy, is that an individual who has a gene for cooperating with its siblings has a good chance of benefiting from the presence 1 See, for example, William Hamilton (1964a, 1964b), Richard Dawkins (1976), John Maynard Smith (1982), and Robert Trivers (1985). 2 Moresubtlequestionsaboutthenatureofaltruisticpreferencesareleftforotherinvestigations. Each of us has done some work of this kind. See B. Douglas Bernheim and Oded Stark (1988), Oded Stark (1989), and Ted Bergstrom (1988), (1989), (1992). 1 of the same gene in its siblings. Similar reasoning applies to behavior that is imitative rather than genetically inherited if those who share common role models are more likely to interact with each other than with randomly selected members of the population. 1. The Game Creatures Play and the Nature of Equilibrium Individuals will be assumed to play one-period, two-person games of prisoners’ dilemma with their siblings or neighbors. In each game that it plays, an individual can choose one of two strategies, cooperate or defect. The payoffs from this game are listed in the matrix below. If the parameters satisfy the restriction S < P < R < T, then defect will be a dominant strategy for each game. For the game to be called a “prisoners’ dilemma”, it should also satisfy the restriction that S +T < 2R. Prisoners’ Dilemma Player 2 Cooperate Defect Cooperate R,R S,T Player 1 Defect T,S P,P Total payoff to an individual will be the average of its payoffs in the prisoners’ dilemma games that it plays. Where behavior is genetically inherited, we assume that the expected number of surviving offspring that an individual produces will be higher, the higher its total payoff. Where behavior is copied from neighbors, the probability that an individual’s behavior is copied will depend on its payoff. A population can have either a monomorphic equilibrium or a polymorphic equilibrium (or possibly both). In a stable monomorphic equilibrium, only one type of individual is present and if a mutant individual of the other type should arise, it must reproduce less rapidly than normal individuals. In a polymorphic equilibrium, more than one type 2 of individual is present and each type that is present receives the same expected payoff. Stability of polymorphic equilibrium requires that if one type happens to become more common than the equilibrium proportion, it will have a lower expected payoff than the other type. 2. Evolution of Genetically Transmitted Behavior Sincelittleisknownabouttheenvironmentswhichshapedourgeneticinheritance, theevo- lutionary hypothesis may not be very informative about many aspects of our preferences. But the fundamental processes of mating, child-rearing and relations between siblings ap- pear to have changed little over the millennia. Accordingly, we may learn a good deal about the “economics of the family” from a look at the evolutionary theory of relations among kin.3 Altruistic Sororities Without Sex Just to help us understand the logic of inheritance, we begin with a toy model that seems unrealistic for humans–asexual reproduction. Let us assume that any individual will, if she survives to reproductive age, have exactly two children, whose genes are just like her own (except in the case of rare mutations). A surviving individual with a gene for cooperate will have two offspring with genes for cooperate. A surviving individual with a gene for defect will have two offspring like herself. To keep the population size constant, we must assume that only half of the individuals who are born will survive to reproductive age and that the probability that any individual will survive to reproduce will be higher, the greater her payoff in the game that she plays with her sister. We claim that the only equilibrium is a population consisting entirely of cooperators. 3 We find ourselves in good company in this heresy. Becker (1976, 1981) and Hirshleifer (1977, 1978), explore genetic explanations for altruism among close family members. Robert Frank (1985) seeks an evolutionaryexplanationforhumanemotions,andArthurRobson(1992a,1992b)exploresanevolutionary explanation of human attitudes toward risk. 3 To see this, consider a population consisting entirely of cooperators. What would happen to a mutant defector that appeared in this population? Since her sister is a cooperator, the mutant gets a payoff of T > R, and so she is more likely to survive than any other member of the population. But her good fortune will not be sustained by her descendants. Her daughters will both inherit her defect gene and will both defect. Sisters in each generation of her descendants will also defect and get P < R, and hence gradually disappear from the population. Similarly, inapopulationofdefectors, amutantcooperatorwouldfaceadefectingsister and would get a payoff of S, while the surrounding defectors would get P > S. Although her survival probability will be lower than the population average, her daughters and their descendants would all be cooperators. Each of them will receive a payoff of R > P and their numbers would grow relative to those of the defectors. Diploid Siblings are Sometimes Altruistic (But Not as Often as Their Parents Would Like) Diploid parents will not be surprised to discover that in our own species, siblings are not always as cooperative as asexual siblings would be. We will show that there is a rich menu of possible equilibria with diploid siblings. Depending on the parameters of the prisoners’ dilemma game, there may be a unique stable monomorphic equilibrium with cooperators only, a unique stable monomorphic equilibrium with defectors only, or there may be two locally stable monomorphic equilibria–one with cooperators only and the other with defectors only. Finally, there are parameter values for which there are no stable monomorphic equilibria, but for these parameters there will be a stable polymorphic equilibrium with some cooperators and some defectors in the population. We consider a large population which reproduces sexually and has diploid genetic struc- ture. Each individual plays a single-shot game of prisoners’ dilemma with each of its sib- lings. To simplify exposition, we will assume that each individual who survives to mate and reproduce has exactly three offspring. The probability that an individual survives to reproduce will be higher, the higher the total payoff that it gets in the games it plays with 4 its two siblings. 4 Individuals are able to distinguish their siblings from other members of the population and may use different strategies in games played with siblings from the strategies used with outsiders. The strategy that any individual uses in play with its siblings is determined by the contents of a single genetic locus. This locus contains two genes, one randomly selected from each of its parents’ two genes. For the present discussion, we assume that mating is monogamous and random with respect to the genes controlling behavior toward sibling. We assume that there are two kinds of genes, a c (cooperate) gene and a d (defect) gene. Then there will be three possible types of individuals, namely type cc homozygotes who carry two c genes, type cd heterozygotes who carry one c gene and one d gene, and type dd homozygotes who carry two d genes. Type cc homozygotes always play cooperate and type dd homozygotes always play defect. If heterozygotes always defect, then the d gene is said to be dominant and the c gene is said to be recessive. If heterozygotes always cooperate, then the c gene is dominant and the d gene is recessive.5 In this paper, we confine our attention to monomorphic equilibria which are stable against invasion by dominant mutant genes. Thus, we will consider whether a population consisting entirely of cc homozygotes could be “invaded” by mutant “dominant” d genes such that cd heterozygotes always play defect. Similarly, we will ask whether a popula- tion consisting entirely of dd homozygotes could be invaded by dominant c genes.6 The possibility of invasion by recessive mutants leads to an interesting, but rather elaborate 4 A richer model would have n siblings play a general n-person game rather than have each individual playtwoseparatetwo-persongameswithitstwosiblings. Suchamodelcouldfocusonquestionsofreturns to scale within families. The assumption that all individuals who survive to mate and reproduce allows us to sidestep the complications that would arise from reconciling the assumption of monogamous random mating with the assumption that the number of children was has depends on ones payoffs in prisoners’ dilemma. What should we do if husband and wife have different expected number of offspring? 5 Alternatively,onecouldassumethatheterozygotesplayamixedstrategywithsomefixedprobabilities of cooperation and defection. 6 Notice that we do not take the view that either c genes or d genes must be intrinsically dominant. Instead we ask whether in a monomorphic population, if a dominant mutant of the opposite type should arise, the mutant strain would increase as a proportion of the population or would ultimately disappear. 5 analysis which will not be pursued here.7 First let us ask when a population consisting only of cooperators would be resistant to invasion by mutants with a (dominant) gene for defection. Suppose that the entire pop- ulation consists of type cc homozygotes, all of whom cooperate. Now let some individual experience a mutation which changes one of its c genes to a d gene which is dominant over the type c gene. The mutant will therefore be a cd heterozygote and will play defect. In the games it plays with its siblings, this mutant will get a higher payoff than normal members of the population, since its normal siblings cooperate while it defects. Therefore the mutant receives T in each game while ordinary members of the population receive R < T. But in order to find out whether the mutant type will invade the population in the long run, we must follow the fortunes of its offspring who inherit the mutant gene. When the mutant cd type is rare, it will almost certainly mate with a normal type cc. The mutant’s offspring will therefore be of type cd with probability 1/2 and of type cc with probability 1/2. An offspring of the mutant who carries the mutant gene will be of type cd and will play defect. With probability 1/2, a randomly chosen sibling of this individual will be a type cc homozygote and with probability 1/2, that sibling will be another type cd. Therefore with probability 1/2, this individual can exploit a cooperative sibling and receive a payoff of T, but with probability 1/2, the sibling will also defect. It then follows that the expected payoff to each heterozygote offspring of the mutant is (T +P)/2. The offspring of normal cc types will receive a payoff of R in the games they play with their siblings. It follows that while the mutant gene is rare, carriers of the mutant gene will reproduce more rapidly than normal individuals if T +P > 2R and less rapidly if T +P < 2R. Now let us ask when a population consisting entirely of type dd individuals could be invaded by a mutant c gene where the mutant gene is dominant over the normal genes. A single mutating gene would first appear in a cd heterozygote. Assuming the c gene is 7 A heterozygote with a recessive mutant gene will act just like the normal population and so there will be no selection either for or against recessive genes until “genetic drift” produces enough mutant heterozygotes so that they occasionally mate, thereby producing homozygotes who act differently from the remaining population. Equilibria which are resistant to invasion both by dominant and by recessive mutants are studied by Bergstrom (1992). 6 dominant, the mutant individual would cooperate. The mutant individual would receive the low payoff S, since it plays its siblings who play defect. But on average, its offspring will do better than S and perhaps will do better than the normal population of defectors, all of whom receive P. When the mutant type is rare, a mutant will almost certainly mate with a normal dd type. Half of the mutant’s offspring will be cd heterozygotes, who cooperate, and half of them will be dd homozygotes, who defect. An offspring of the mutant who carries the mutant gene will be of type cd and will play cooperate. With probability 1/2, a randomly chosen sibling of this individual will be a type dd homozygote who defects and with probability 1/2, that sibling will be another type cd who cooperates. Therefore with probability 1/2, an offspring that carries of the mutant gene will be exploited by its sibling and get a payoff of S, but with probability 1/2, its sibling will also cooperate and each of them will receive a payoff of R. The expected payoff to a type cd offspring of the mutant is therefore (S +R)/2. This payoff will be smaller than the payoff to normal dd types if 2P −S −R > 0 and larger if the inequality is reversed. As we have shown, there will be a stable monomorphic equilibrium with all cooperators if T +P −2R > 0 and there will be no such equilibrium if the inequality is reversed. There will be a stable monomorphic equilibrium with all defectors if 2P −S−R > 0 and no such equilibrium if the inequality is reversed. It turns that there are prisoners’ dilemma games where each of these inequalities takes either sign. The possibilities are illustrated in Figure 1. In this figure we have normalized the game to set S = 0 and T = 1.8 With this normalization, there will be a stable monomorphic equilibrium with type cc only if R > (P+1)/2 > 0 and there will be a stable monomorphic equilibrium with type dd only if R < 2P. For parameter values in Region C of Figure 1 there is a stable monomorphic equilibrium with cooperators only and no stable equilibrium with defectors only. For parameter values in Region D of Figure 1, there is a stable monomorphic equilibrium with defectors only and no stable equilibrium with cooperators 8 This can be done without loss of generality, since the population dynamics discussed in this paper are invariant to affine transformations of the payoff matrix. 7 only. For parameter values in Region B, there will be stable monomorphic equilibria of both types, and for parameter values in Region A, there will not be a stable monomorphic equilibrium of either type. In order for the game to be a prisoners’ dilemma, it must also be that R > P and that R > .5. The region in Figure 1 above the two dotted lines satisfies these conditions. We note that each of the regions A, B, C, and D can occur with parameters suitable for prisoners’ dilemma. Figure 1 justifies the claims made in the title of this section. For prisoners’ dilemma games with parameter values in Region C, diploid siblings will cooperate, even though it is to their selfish advantage to defect. For prisoners’ dilemma games with parameter values in Region D, diploid siblings will both defect, although parents who wish them to maximize their joint payoff would prefer them both to cooperate. It is interesting to consider Regions A and B. For parameter values in Region B, there are two stable equilibria–one with a monomorphic equilibrium of each kind. For parameter values in Region A, there are no stable monomorphic equilibrium. In order to understand thesecases, itisnecessarytoworkoutthedetailedlawsofmotionforthedynamicalsystem that results from this model. This is done by Bergstrom and Bergstrom (1992), where it is found that for parameter values in Region A, there exists exactly one stable polymorphic equilibrium and for parameter values in Region B, there is one unstable polymorphic equilibrium and no stable polymorphic equilibria. 3. When Children Imitate their Parents or Teachers Here we study a model in which behavior is acquired by imitation, rather than geneti- cally. The model discussed here is a variant of models of cultural transmission which were developed by Cavalli-Sforza and Feldman (1980), and Boyd and Richardson (1985). Weassumethateachindividualhastwosiblingsandplaysagameofprisoners’dilemma with each of them. We will also assume that the probability that any individual survives to mate and reproduce is proportional to the average payoff that it receives in the games it plays with its siblings. 8 Assumethatwithprobabilityv,achildadoptsthestrategythatwasusedbyarandomly chosen one of its two parents and with probability 1−v it adopts the strategy used by a nonparent, randomly selected from the entire population. We assume that marriage is monogamous, so that all siblings share the same mother and father. Parent-couples can be one of three possible types; two-cooperator couples, “mixed couples” with one cooperator and one defector, and two-defector couples. Mating is said to be assortative if adults always mate with individuals of their own type.9 Let x be the fraction of the adult population who are cooperators. If marriage is purely random, the fraction of all marriages which are mixed couples will be 2x(1 − x). We define a parameter m where 0 ≤ m ≤ 1 in such a way as to allow mating patterns that lie between the polar cases of purely random (m = 0) and purely assortative (m = 1) mating. In the population at large, the proportion of mixed couples is 2(1−m)x(1−x), the proportion of two-cooperator couples is x2+mx(1−x), and the proportion of two-defector couples is (1−x)2 +mx(1−x).10 Given the proportions of couples of each type, we can determine the proportions of all sibling pairs consisting, respectively, of two cooperators, one cooperator and one defector, and two defectors. This enables us to determine not only the proportion of offspring of each type, but also the expected payoffs to offspring of each type, since we will know the probability that a randomly chosen sibling of an individual of each type will be a cooperator or a defector. With this information, we are be able to determine the relative growth rates of the population of cooperators and of defectors. The details of this process are worked out in the Appendix of this paper. This model turns out to have a remarkably convenient mathematical structure. The rate of change of the number of surviving individuals of each type in any generation turns 9 If mating requires mutual consent, if types are costlessly recognizable and search costs are negigible, this would be a natural outcome, since cooperators can expect more offspring if they mate with other cooperators than if they mate with defectors so long as 2R>S+P. 10 Cavelli-Sforza and Feldman attribute this parameterization of assortative mating to Sewall Wright (1921).These proportions would be achieved if couples were first randomly matched and then the fraction m of the mixed couples were broken up and the freed individuals paired with persons of their own type. 9