can receive the benefit from neighboring coopera-tors. If a defector is connected to k other individuals and j of those are cooperators, then its payoff is bj. Evolutionary dynamics are described by an extremely simple stochastic process: at each time step, a random individual adopts the strategy of one of its neighbors proportional to their fitness.

We note that stochastic evolutionary game dynamics in finite populations are sensitive to the intensity of selection. In general, the reproductive success (fitness) of an individual is given by a constant, denoting the baseline fitness, plus the payoff that arises from the game under consideration. Strong selection means that the payoff is large compared with the baseline fitness; weak selection means the payoff is small compared with the baseline fitness. It turns out that many interesting results can be proven for weak selection, which is an observation also well known in population genetics.

The traditional, well-mixed population of evolutionary game theory is represented by the complete graph, where all vertices are connected, which means that all individuals interact equally often. In this special situation, cooperators are always opposed by natural selection. This is the fundamental intuition of classical evolutionary game theory. But what happens on other graphs?

We need to calculate the probability, pC, that a single cooperator starting in a random position turns the whole population from defectors into cooperators. If selection neither favors nor opposes cooperation, then this probability is 1/N, which is the fixation probability of a neutral mutant. If the fixation probability pC is greater than 1/N, then selection favors the emergence of cooperation. Similarly, we can calculate the fixation probability of defectors, pD. A surprisingly simple rule determines whether selection on graphs favors cooperation. If b/c > k (2.4)

then cooperators have a fixation probability of greater than 1/N and defectors have a fixation probability of less than 1/N. Thus, for graph selection to favor cooperation, the benefit/cost ratio of the altruistic act must exceed the average degree, k, which is given by the average number of links per individual (Ohtsuki et al., 2006). This relationship can be shown with the method of pair-approximation for regular graphs, where all individuals have exactly the same number of neighbors. Regular graphs include cycles, all kinds of spatial lattice, and random regular graphs. Moreover, computer simulations suggest that the rule b/c > k also holds for non-regular graphs such as random graphs and scale-free networks. The rule holds in the limit of weak selection and k << N. For the complete graph, k = N, we always have pD > 1/N > pC. Preliminary studies suggest that eqn 2.4 also tends to hold for strong selection. The basic idea is that natural selection on graphs (in structured populations) can favor unconditional cooperation without any need for strategic complexity, reputation, or kin selection.

Games on graphs grew out of the earlier tradition of spatial evolutionary game theory (Nowak and May, 1992; Herz, 1994; Killingback and Doebeli, 1996; Mitteldorf and Wilson, 2000; Hauert et al., 2002; Le Galliard et al., 2003; Hauert and Doebeli, 2004; Szabo and Vukov, 2004) and investigations of spatial models in ecology (Durrett and Levin, 1994a, 1994b; Hassell et al., 1994; Tilman and Kareiva, 1997; Neuhauser, 2001) and spatial models in population genetics (Wright, 1931; Fisher and Ford, 1950; Maruyama, 1970; Slatkin, 1981; Barton, 1993; Pulliam, 1988; Whitlock, 2003).

The enthusiastic approach of early group selectionists to explain all evolution of cooperation from this one perspective (Wynne-Edwards, 1962) has met with vigorous criticism (Williams, 1966) and even a denial of group selection for decades. Only an embattled minority of scientists continued to study the approach (Eshel, 1972; Levin and Kilmer, 1974; Wilson, 1975; Matessi and Jayakar, 1976; Wade, 1976; Uyenoyama and Feldman, 1980; Slatkin, 1981; Leigh, 1983; Szathmary and Demeter, 1987). Nowadays it seems clear that group selection can be a powerful mechanism to promote cooperation (Sober and Wilson, 1998; Keller, 1999; Michod, 1999; Swenson et al., 2000; Kerr and Godfrey-Smith, 2002; Paulsson, 2002; Boyd and Richerson, 2002; Bowles and Gintis, 2004; Traulsen et al, 2005). We only have to make sure that its basic requirements are fulfilled in a particular situation (Levin and Kilmer, 1974; Maynard Smith, 1976). Exactly what these requirements are can be illustrated with a simple model (Traulsen and Nowak, 2006).

Imagine a population of individuals subdivided into groups. For simplicity, we assume that the number of groups is constant and given by m. Each group contains between 1 and n individuals. The total population size can fluctuate between the bounds m and nm. Again, there are two types of individual, cooperators and defectors. Individuals interact with others in their group and thereby receive a payoff. At each time step a random individual from the entire population is chosen proportional to payoff in order to reproduce. The offspring is added to the same group. If the group size is less than or equal to n then nothing else happens. If the group size, however, exceeds n then with probability q the group splits into two. In this case, a random group is eliminated (in order to maintain a constant number of groups). With probability 1 — q, the group does not divide, but instead a random individual from that group is eliminated (Figure 2.6)*.

This minimalist model of multilevel selection has some interesting features. Note that the evolutionary dynamics are entirely driven by individual fitness. Only individuals are assigned payoff values. Only individuals reproduce. Groups can stay together or split (divide) when reaching a certain size. Groups that contain fitter individuals reach the critical size faster and therefore split more often. This concept leads to selection among groups, although only individuals reproduce. The higher level selection emerges from lower level reproduction. Remarkably, the two levels of selection can oppose each other.

As before, we can compute the fixation probabilities, pC and pD, of cooperators and defectors to check whether selection favors one or the other. If we add a single cooperator to a population of defectors, then this cooperator must first take over a group. Subsequently the group of cooperators must take over the entire population. The first step is opposed by selection, the second step is favored by selection. Hence, we need to find out if the overall fixation probability is greater to or less than

Figure 2.6 A simple model of group selection. A population consists of m groups of maximum size n. Individuals interact with others in their group in the context of an evolutionary game. Here we consider the game between cooperators, C, and defectors, D. For reproduction, individuals are chosen from the entire population with a probability proportional to their payoff. The offspring is added to the same group. If a group reaches the maximum size, n, then it either splits in two or a random individual from that group is eliminated. If a group splits, then a random group dies, in order to keep the total population size constant. This metapopulation structure leads to the emergence of two levels of selection, although only individuals reproduce.

Figure 2.6 A simple model of group selection. A population consists of m groups of maximum size n. Individuals interact with others in their group in the context of an evolutionary game. Here we consider the game between cooperators, C, and defectors, D. For reproduction, individuals are chosen from the entire population with a probability proportional to their payoff. The offspring is added to the same group. If a group reaches the maximum size, n, then it either splits in two or a random individual from that group is eliminated. If a group splits, then a random group dies, in order to keep the total population size constant. This metapopulation structure leads to the emergence of two levels of selection, although only individuals reproduce.

what we would obtain for a neutral mutant. An analytic calculation is possible in the interesting limit q << 1, where individuals reproduce much more rapidly than groups divide. In this case, most of the groups are at their maximum size and hence the total population size is almost constant and given by N = nm. We find that selection favors cooperators and opposes defectors, pC > 1/N > pD, if b/c > 1 + n/(m — 2) (2.5a)

This result holds for weak selection. Smaller group sizes and larger numbers of competing groups favor cooperation. We also notice that the number of groups, m, must exceed 2. There is an intuitive reason for this threshold. Consider the case of m = 2 groups with n = 2 individuals. In a mixed group, the cooperator has payoff — c and the defector has payoff b; the defector/cooperator difference is b + c. In a homogeneous group, two cooperators have payoff b — c, while two defectors

* This is not the same q as in section 2.3; we have run out of convenient letters.

have a payoff of 0. Thus the disadvantage for cooperators in mixed groups cannot be compensated for by the advantage they have in homogeneous groups. Interestingly, however, for larger splitting probabilities, q, we find that cooperators can be favored even for m — 2 groups. The reason is the following: for very small q, the initial cooperator must reach fixation in a mixed group; but for larger q, a homogeneous cooperator group can also emerge if a mixed group splits, giving rise to a daughter group that has only cooperators. Thus, larger splitting probabilities make it easier for cooperation to emerge.

Let us also consider the effect of migration between groups. The average number of migrants accepted by a group during its lifetime is denoted by z. We find that selection favors cooperation provided that b/c > 1 + z + n/m (2.5b)

In order to derive this condition we have assumed weak selection and q << 1, as before, but also that both the numbers of groups, m, and the maximum group size, n, are much larger than 1. For more information, see Traulsen and Nowak, 2006.

Group selection (or multilevel selection) is a powerful mechanism for the evolution of cooperation if there is a large number of relatively small groups and migration between groups is not too frequent.

We end by listing the five rules that we mentioned in the beginning. These rules represent laws of nature governing the natural selection of cooperation.

1. Kin selection leads to cooperation if b/c > 1/r, where r is the coefficient of genetic relatedness between donor and recipient.

2. Direct reciprocity leads to cooperation if b/c > 1/w, where w is the probability of playing another round in the repeated Prisoner's Dilemma.

3. Indirect reciprocity leads to cooperation if b/ c > 1/ q, where q is the probability of knowing the reputation of a recipient.

4. Graph selection (or network reciprocity) leads to cooperation if b/c > k, where k is the degree of the graph; that is, the average number of neighbors.

5. Group selection leads to cooperation if b/c > 1 + z + n/m, where z is the number of migrants accepted by a group during its lifetime, n is the group size, and m is the number of groups.

In all five theories, b is the benefit for the recipient and c the cost for the donor of an altruistic act.

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