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Evolution of cooperation by phenotypic similarity

Communicated by Simon A. Levin, Princeton University, Princeton, NJ, March 10, 2009 (received for review June 15, 2008)
Abstract
The emergence of cooperation in populations of selfish individuals is a fascinating topic that has inspired much work in theoretical biology. Here, we study the evolution of cooperation in a model where individuals are characterized by phenotypic properties that are visible to others. The population is well mixed in the sense that everyone is equally likely to interact with everyone else, but the behavioral strategies can depend on distance in phenotype space. We study the interaction of cooperators and defectors. In our model, cooperators cooperate with those who are similar and defect otherwise. Defectors always defect. Individuals mutate to nearby phenotypes, which generates a random walk of the population in phenotype space. Our analysis brings together ideas from coalescence theory and evolutionary game dynamics. We obtain a precise condition for natural selection to favor cooperators over defectors. Cooperation is favored when the phenotypic mutation rate is large and the strategy mutation rate is small. In the optimal case for cooperators, in a onedimensional phenotype space and for large population size, the critical benefittocost ratio is given by
 coalescent theory
 evolutionary dynamics
 evolutionary game theory
 mathematical biology
 stochastic process
Evolutionary game theory is the study of frequencydependent selection (1–8). Fitness values depend on the relative abundance, or frequency, of various strategies in the population, for example, the frequency of cooperators and defectors. Evolutionary game theory has been applied to understand the evolution of cooperative interactions in viruses, bacteria, plants, animals, and humans (9–13). The classical approach to evolutionary game dynamics assumes wellmixed populations, where every individual is equally likely to interact with every other individual (4). Recent advances include the extension to populations that are structured by geography or other factors (14–25).
The term “greenbeard effect” was coined in sociobiology to describe the result of the following thought experiment (26, 27). What evolutionary dynamics will occur if a single gene is responsible for both a phenotypic signal (“a green beard”) and a behavioral response (for example, altruistic behavior toward individuals with like phenotypes)? Later, the term “armpit effect” was introduced (28) to refer to a selfreferent phenotype that is used in identifying kin (29–31).
Both of these concepts are now seen as cases of “tagbased cooperation,” in which a generic system of phenotypic tags is used to indicate similarity or difference, and the evolutionary dynamics of cooperation are studied in the context of these tags. A first approach, based on computer simulations, assumed a wellmixed population, a continuum of tags, and an evolving threshold distance for cooperation (32). More recent models use numerical and analytic methods and often combine tags with viscous population structure (33–37). A general finding of these articles is that it is difficult to obtain cooperation in tagbased models for wellmixed populations, indicating that some spatial structure is needed (14).
Inspired by work on tagbased cooperation (32–34, 38) and building on a previous approach (39), we study evolutionary game dynamics in a model where the behavior depends on phenotypic distance (40, 41). As a particular example we explore the evolution of cooperation (42, 43). Studies of different organisms, including humans, support the idea that cooperation is more likely among similar individuals (31, 44–49). Our model applies to situations where individuals tend to like those who have similar attitudes and beliefs. We introduce a natural model in which individuals mutate to adjacent phenotypes in a possibly multidimensional phenotype space. We study one and infinitely many dimensions in detail. We develop a theory for general evolutionary games, not just the evolution of cooperation. Spatial structure is not needed for cooperation to be favored in our model. Moreover, in contrast to previous work (39), we develop an analytic machinery for describing heterogeneous populations in phenotype space.
Consider a population of asexual haploid individuals, with a population size N that is constant over time. Each individual is characterized by a phenotype, given by an integer i that can take any value from minus to plus infinity. Thus, this phenotype space is a onedimensional and unbounded lattice. Individuals inherit the phenotype of their parent subject to some small variation. If the parent's phenotype is i, then the offspring has phenotype i − 1, i, or i + 1 with probabilities v, 1 − 2v, and v, respectively. The parameter v can vary between 0 and 1/2.
Let us consider a Wright–Fisher process. In each generation, all individuals produce the same large number of offspring. The next generation of N individuals is sampled from this pool of offspring. To introduce some fundamental concepts and quantities, we first study the model without any selection. No evolutionary game is yet being played, and there is only neutral drift in phenotype space. The entire population performs a random walk with a diffusion coefficient v, and by this process will tend to disperse over the lattice. In opposition to this, all of the individuals in the population will be, to some degree, related due to reproduction in a finite population. Thus, while occasionally the population may break up into two or more clusters, typically there is only a single cluster (50, 51). The standard deviation of the distribution in phenotype space, which is a measure for the width of the cluster, is
Next, we superimpose the neutral drift of two types: the strategies A and B (Fig. 1). Still for the moment, assuming no fitness differences, we have reproduction subject to mutation between A and B. Specifically, with probability u the offspring adopts a random strategy. The mutation–reproduction process defines a stationary distribution (52). If u is very small relative to N, the population tends to be either allA or allB. If u is large, the population tends toward onehalf A and onehalf B. Fig. 2 illustrates the random walk in phenotype space of the population composed of the two types A and B.
By using coalescence theory (53, 54) many interesting and relevant properties of the distributions of both the strategies and phenotypic tags can be calculated. For example, the probability that two randomly chosen individuals have the same phenotype is
We can now use these insights to study game dynamics. We investigate the competition of cooperators, C, and defectors, D. Cooperators play a conditional strategy: they cooperate with all individuals who are close enough in phenotype space and defect otherwise. The notion of being close enough is modeled by a lattice structure. In particular, a cooperator with phenotype i cooperates only with other individuals of phenotype i. Defectors, in contrast, play an unconditional strategy: they always defect. Cooperation means paying a cost, c, for the other individual to receive a benefit b. The larger the total payoff of an individual, interacting equally with every member of the population, the larger the number of offspring it will produce on average. We want to calculate the critical benefittocost ratio, b/c, that allows the game in phenotype space to favor the evolution of cooperation.
A configuration of the population is specified by m_{i} and n_{i}, which are the number of cooperators with phenotype i and the total number of individuals with phenotype i, respectively. The total payoff of all cooperators is
When the population size is large, the averages in inequality (Eq. 1) are proportional to the probabilities g, z, and h respectively, which we introduced earlier. Consequently, inequality (Eq. 1) can be written as bg − cz > (b − c)h. Using the values of z,g, and h given above we obtain which is approximately 2.16. If the benefittocost ratio exceeds this number, then cooperators are more abundant than defectors in the mutationselection process. The success of cooperators results from the balance of movement and clustering in phenotype space. Inequality (Eq. 2) represents the condition for cooperators to be more abundant than defectors in a large population when the strategy mutation rate u is small (Nu ≪ 1) and the phenotypic mutation rate v is large (Nv ≫ 1). In SI Appendix, we derive conditions for any population size and mutation rates. Fig. 3 shows the excellent agreement between numerical simulations and analytical calculations. In general, we find that both lowering strategy mutations and increasing phenotypic mutations favor cooperators.
We can expand our analysis to study any 2 × 2 game, not only the interaction between cooperators and defectors. Consider two strategies A and B and the general payoff matrix The payoffs for A versus A, A versus B, B versus A, and B versus B are given by R, S, T, P, respectively. A players use strategy A against other individuals with the same phenotype, otherwise they use B. B players always use strategy B. For the game in a onedimensional phenotype space and large population size we find that A is more abundant than B if For the derivation see Section 5.2 in the SI Appendix. This formula can be used for evaluating any twostrategy symmetric game in a onedimensional phenotype space. In the SI Appendix, we discuss the snowdrift game and the staghunt game as particular examples.
We can also study higherdimensional phenotype spaces. In general, for higher dimensions, it is easier for cooperators to overcome defectors. The intuitive reason is that in higher dimensions phenotypic identity also implies strategic identity. In Section 5.3 of the SI Appendix, we show that, in the limit of infinitely many dimensions, and under the same assumptions that produced conditions 2 and 4, the crucial benefittocost ratio in the Prisoner's Dilemma converges to b/c > 1. For general games, the equivalent result of condition 4 becomes R > P, which means the evolutionary process always chooses the strategy with the higher payoff against itself. Our basic approach can also be adapted to continuous, rather than discrete, phenotype spaces. In this case, no two individuals have exactly the same phenotype, but the conditional behavioral strategy is triggered by sufficient phenotypic similarity.
In summary, we have developed a model for the evolution of cooperation based on phenotypic similarity. Our approach builds on previous ideas of tagbased cooperation, but in contrast to earlier work (33–37), we do not need spatial population dynamics to obtain an advantage for cooperators. We derive a completely analytic theory that provides general insights. We find that the abundance of cooperators in the mutationselection equilibrium is an increasing function of the phenotypic mutation rate and a decreasing function of the strategic mutation rate. These observations agree with the basic intuition that higher phenotypic mutation rates reduce the interactions between cooperators and defectors, whereas higher strategic mutation rates destabilize clusters of cooperators by allowing frequent invasion of newly mutated defectors. Therefore, cooperation is more likely to evolve if the strategy mutation rate is small and if the phenotypic mutation rate is large. In a genetic model this assumption may be fulfilled if the strategy is encoded by one or a few genes, whereas the phenotype is encoded by many genes. Also in a cultural model, it can be the case that the phenotypic mutation rates are higher than the strategic mutation rates; for example, people might find it easier to modify their superficial appearance than their fundamental behaviors. Furthermore, we show how the correlations between strategies and phenotypes can be obtained from neutral coalescence theory under the assumption that selection is weak (54, 56). Our theory can be applied to study any evolutionary game in the context of conditional behavior that is based on phenotypic similarity or difference.
Acknowledgments
This work was supported by the John Templeton Foundation, the National Science Foundation/National Institutes of Health (R01GM078986) joint program in mathematical biology, the Bill and Melinda Gates Foundation (Grand Challenges Grant 37874), the Japan Society for the Promotion of Science, and J. Epstein.
Footnotes
 ^{1}To whom correspondence should be addressed. Email: martin_nowak{at}harvard.edu

Author contributions: T.A., H.O., J.W., P.D.T., and M.A.N. performed research; and T.A., H.O., J.W., P.D.T., and M.A.N. wrote the paper.

The authors declare no conflict of interest.

See Commentary on page 8405.

This article contains supporting information online at www.pnas.org/cgi/content/full/0902528106/DCSupplemental.
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