Fixation Probabilities for Any Configuration of Two Strategies on Regular Graphs

Population structure and spatial heterogeneity are integral components of evolutionary dynamics, in general, and of evolution of cooperation, in particular. Structure can promote the emergence of cooperation in some populations and suppress it in others. Here, we provide results for weak selection to favor cooperation on regular graphs for any configuration, meaning any arrangement of cooperators and defectors. Our results extend previous work on fixation probabilities of rare mutants. We find that for any configuration cooperation is never favored for birth-death (BD) updating. In contrast, for death-birth (DB) updating, we derive a simple, computationally tractable formula for weak selection to favor cooperation when starting from any configuration containing any number of cooperators. This formula elucidates two important features: (i) the takeover of cooperation can be enhanced by the strategic placement of cooperators and (ii) adding more cooperators to a configuration can sometimes suppress the evolution of cooperation. These findings give a formal account for how selection acts on all transient states that appear in evolutionary trajectories. They also inform the strategic design of initial states in social networks to maximally promote cooperation. We also derive general results that characterize the interaction of any two strategies, not only cooperation and defection.

Mechanisms favoring the emergence of cooperation in social dilemmas have become central focuses of evolutionary game theory in recent years [1][2][3] . The dilemma of cooperation, which is characterized by conflicts of interest between individuals and groups, poses a significant challenge to models of evolution since many of these models predict that cooperation cannot persist in the presence of exploitation by defectors 4,5 . Yet cooperation is widely observed in nature, and the spatial assortment that results from population structure is one element that can promote its emergence. In fact, spatial structure is among the most salient determinants of the evolutionary dynamics of a population  .
In social dilemmas, population structure can allow for the emergence of localized cooperative clusters that would normally be outcompeted by defectors in well-mixed populations 5,29 . However, whether population structure promotes or suppresses cooperation depends on a number of factors such as the update rule, the type of social dilemma, and the spatial details of the structure (which determine the extent of local competition see refs 28 and 30). For example, cooperation need not be favored in prisoner's dilemma interactions under all update rules [31][32][33][34] . As a consequence, population structure should be considered in the context of the game and the underlying update rule.
In the donation game, a cooperator (C) pays a cost, c, to provide the opponent with a benefit, b, and a defector (D) pays no cost and provides no benefit 35 . Provided b > c > 0, this game represents a prisoner's dilemma since then the unique Nash equilibrium is mutual defection, but both players would prefer the payoff from mutual cooperation 36 . In addition to representing one of the most important social dilemmas, the donation game also admits a simple way in which to quantify the efficiency of cooperation: the benefit-to-cost ratio, b/c. As this ratio gets larger, the act of cooperation has a more profound effect on the opponent relative to the cost paid by the cooperator. As we shall see, for any configuration of cooperators and defectors, this ratio is a vital indicator of the evolutionary performance of cooperation.
Evolutionary graph theory is a framework for studying evolution in structured populations 30,32,33,[37][38][39][40][41][42][43][44][45][46][47] . In a graph-structured population, the players reside on the vertices and the edges indicate who is a neighbor of whom. In fact, there are two types of neighborhoods: (i) those that generate payoffs ("interaction neighborhoods") and (ii) those that are relevant for evolutionary updating ("dispersal neighborhoods"). Thus, an evolutionary graph is actually a pair of graphs consisting of an interaction graph and a dispersal graph 46,[48][49][50] . As in many other studies, we assume that the interaction and dispersal graphs are the same. Other extensions of evolutionary graph theory involve dynamic graphs, which allow the population structure to change during evolutionary updating [51][52][53][54] . Our focus is on static, regular graphs of degree k, meaning the population size, N, is fixed and each player has exactly k neighbors.
We study two prominent update rules: birth-death (BD) and death-birth (DB). In both processes, players are arranged on a graph and accumulate payoffs by interacting with all of their neighbors. This payoff, π, is then converted to fitness, f, via f = 1 + wπ, where w ≥ 0 is the intensity of selection 5 . For BD updating 5,55 , a player is chosen with probability proportional to fitness for reproduction; the offspring of this player then replaces a random neighbor (who dies). For DB updating 32 , a player is chosen uniformly at random for death; a neighbor of this player then reproduces (with probability proportional to fitness) and the offspring fills the vacancy. For each of these processes, we assume that w is small, which means selection is weak. Weak selection is often a biologically meaningful assumption since an individual might possess many traits (strategies), and each trait makes only a small contribution to fitness 5,[56][57][58][59][60][61] .
The effects of selection on fixation probability have been studied chiefly for states with just a single cooperator since, if the mutation rate is small, the process will reach a monomorphic state prior to the appearance of another cooperator through mutation 62,63 . Although small mutation rates are often reasonable from a biological standpoint [64][65][66][67] , there are several reasons to study arbitrary cooperator configurations. Even when starting from a state with a single cooperator, an evolutionary process typically transitions subsequently through states with many cooperators. From a mathematical standpoint, it is therefore natural to ask how selection affects the fixation probability of cooperators from each possible transient state that might arise in an evolutionary trajectory. Furthermore, many-mutant states could arise through migration [68][69][70][71] or environmental mutagenic agents 72,73 , which, even when rare, might result in several cooperators entering the population at once. In the case of social networks, cooperators could arise through design rather than mutation or exploration; if cooperators can be strategically planted within the population, then one can ask how to do so in order to maximize the chances that cooperators take over. Therefore, the effects of selection on arbitrary numbers and configurations of cooperators and defectors play an important role in the evolutionary dynamics of cooperation.
When starting from a configuration with n cooperators and N − n defectors, weak selection is said to favor the evolution of cooperation (on a regular graph) if the probability that cooperators fixate exceeds n/N if w is sufficiently small but positive. This comparison is based on the fact that the fixation probability of n cooperators for neutral drift (w = 0) is n/N. Ohtsuki et al. 32 show that, on large regular graphs of degree k, selection favors the fixation of a single, randomly-placed cooperator under DB updating as long as Taylor et al. 46 show that for finite bi-transitive graphs of size N and degree k, the condition for selection to favor the fixation of a single cooperator is Bi-transitive graphs constitute a subset of regular graphs. In another refinement of the 'b/c > k' result, Chen 39 shows that, for any n with 0 < n < N, selection favors cooperation when starting from a random configuration of n cooperators and N − n defectors on a regular graph of size N and degree k if and only if Eq. (2) holds. This ratio, which characterizes when selection increases the fixation probability of cooperators, is independent of the location of the mutants, despite the fact that the probability of fixation itself depends on the location 74 . As the population size, N, gets large, the critical benefit-to-cost ratio of Eq. (2) approaches k, which recovers the result of Ohtsuki et al. 32 . Our goal here is to move beyond Eq. (2) and give an explicit, computationally feasible critical benefit-to-cost ratio for any configuration of cooperators and defectors on any regular graph.
Given the profusion of possible ways to structure a population of a fixed size, it quickly becomes difficult to determine when a population structure favors the evolution of cooperation. Here, we provide a solution to this problem for BD and DB updating on regular graphs. We show that, for any configuration of cooperators and defectors, (i) cooperation is never favored for BD updating, and (ii) for DB updating, there exists a simple, explicit critical benefit-to-cost ratio that characterizes when selection favors the emergence of cooperation. Moreover, if N is the population size and k is the degree of the graph, then the complexity of calculating this ratio is O(k 2 N), and, in particular, linear in N. Thus, while the calculations of fixation probabilities in structured populations are famously intractable [75][76][77] , the determination of whether or not selection increases the probability of fixation, for weak selection, is markedly simpler.
In addition to providing a computationally feasible way of determining whether selection favors cooperation on a particular graph, our results highlight the importance of the initial configuration for the emergence of cooperation. Depending on the graph, adding additional cooperators to the initial condition can either suppress or promote the evolution of cooperation. A careful choice of configuration of cooperators and defectors can Scientific RepoRts | 6:39181 | DOI: 10.1038/srep39181 minimize the critical benefit-to-cost ratio for selection to favor cooperation. If cooperation is not favored by selection in such a strategically chosen initial state, then it cannot be favored under any other initial configuration. In this sense, there exists a configuration that is most conducive to the evolution of cooperation, which is not apparent from looking at single-cooperator configurations or random configurations with n cooperators since these initial configurations need not minimize the critical benefit-to-cost ratio.

Results
Critical benefit-to-cost ratios. Let ξ be a configuration of cooperators and defectors on a fixed regular graph of size N and degree k, and let C denote the configuration consisting solely of cooperators. For the donation game, the probability that cooperators take over the population when starting from state ξ may be viewed as a function of the selection intensity, ρ ξ,C (w). We consider here the following question: when does weak selection increase the probability that cooperators fixate? In other words, when is ρ ξ,C (w) > ρ ξ,C (0) for sufficiently small w > 0? Note that if there are n cooperators in state ξ, then ρ ξ,C (0) = n/N, so this condition is equivalent to ρ ξ,C (w) > n/N for small w > 0.
To answer this question, we first need to introduce some notation. If x is a vertex of the graph and ξ is a configuration, then let f 1 (x, ξ) and f 0 (x, ξ) be the frequencies of cooperators and defectors, respectively, among the neighbors of the player at vertex x. Similarly, let f 10 (x, ξ) be the fraction of paths of length two, starting at x, that consist of a cooperator followed by a defector. From these quantities, let x V 10 10 x V 1 0 1 0 which are obtained by averaging these 'local frequencies' over all of the players in the population. From these local frequencies, which are straightforward to calculate (see Fig. 1), we obtain our main result: for small w > 0, ρ ξ,C (w) > ρ ξ,C (0) if and only if the benefit-to-cost ratio exceeds the critical value  2 N), it follows that the complexity of finding the critical benefit-to-cost ratio is O(k 2 N), so it is feasible to calculate even when the population is large. Note also that if ξ is the state obtained by swapping cooperators and defectors in ξ, then both ξ and ξ have the same critical benefit-to-cost ratio. We discuss these 'conjugate' states further in our treatment of structure coefficients.  , ξ), where ξ is the configuration consisting of a defector at vertex y and cooperators elsewhere. Among the three neighbors of the player at vertex x, two are cooperators (u and v) and one is a defector (y); thus, f 1 (x, ξ) = 2/3 and f 0 (x, ξ) = 1/3. Furthermore, of the nine paths of length two that begin at vertex x, only two (x → u → y and x → v → y) consist of a cooperator followed by a defector, and it follows that f 10 (x, ξ) = 2/9. When ξ has just a single cooperator, the ratio of Eq. (7) reduces to that of Eq. (2), which, in particular, does not depend on the location of the cooperator. This property is notable because the fixation probability itself usually does depend on the location of the cooperator, even on regular graphs 74 . We show in Methods that one recovers from Eq. (7) the result of Chen 39 that Eq. (2) gives the critical benefit-to-cost ratio for a randomly-chosen configuration with a fixed number of cooperators.
For fixed k ≥ 2, the critical benefit-to-cost ratio in Eq. (7) converges uniformly to k as N → ∞ (see Methods). Therefore, on sufficiently large graphs, the critical ratio is approximated by k for any configuration, regardless of the number of cooperators. As a result, on large graphs there is less of a distinction between the various transient (non-monomorphic) states in terms of whether or not selection favors the fixation of cooperators. On smaller graphs, these transient states can behave quite differently from one another. This effect is particularly pronounced on very small social networks in which cooperators can be strategically planted in the population to ensure that cooperators are favored by selection.
Strategic placement of cooperators in (small) social networks. Among the more interesting consequences of Eq. (7) are its implications for the success of cooperators as a function of the initial configuration. Recall that Eq. (2) gives the critical benefit-to-cost ratio for both (i) configurations with a single cooperator and (ii) random configurations with a fixed number of cooperators. When cooperators and defectors are configured randomly, this critical ratio is independent of the number of cooperators, which suggests that the effects of selection cannot be improved by increasing the initial abundance of cooperators. Equation (7), on the other hand, shows that the initial configuration of cooperators, including their abundance, does affect how selection acts on the population. First of all, there are graphs for which the critical benefit-to-cost ratio is infinite for configurations with a single cooperator but finite for some configurations with multiple cooperators (see Fig. 2(a)). In contrast, there are graphs for which this ratio is finite for configurations with a single cooperator but infinite for some states with multiple cooperators (see Fig. 2(b)). Therefore, despite the fact that the critical ratio for a single mutant is the same as the critical ratio for a random configuration with any fixed number of mutants, the critical ratio does (in general) depend on the number of mutants present in the configuration.
We say that a configuration has isolated cooperators (resp. defectors) if the minimum distance between any two cooperators (resp. defectors) is at least three steps. Let N 0 denote the maximum number of isolated strategies that a configuration can carry. (Examples of configurations with isolated cooperators on a graph with N 0 = 3 are given in Fig. 3). If a strategy (cooperate or defect) appears only once in a configuration, then that strategy is clearly isolated, so N 0 ≥ 1. We show in Methods that if N > 2k, then cooperation can be favored for a mixed initial condition with n cooperators whenever 1 ≤ n ≤ N 0 + 1 or 1 ≤ N − n ≤ N 0 + 1, and, moreover, these bounds on n are sharp. Stated differently, under these conditions any configuration with n cooperators has a finite critical benefit-to-cost ratio. Furthermore, if N 0 ≥ 2, then, for any n with 2 ≤ n ≤ N 0 , there exists a configuration with n cooperators whose critical ratio is strictly less than the ratio for a single cooperator (Eq. (2)). Such a configuration necessarily has no isolated strategies since the minimum critical ratio among configurations with an isolated strategy is attained by any state with just a single cooperator.
The strategic placement of cooperators and defectors can therefore produce a critical benefit-to-cost ratio that is less than the ratio for a single cooperator among defectors. In fact, starting from a configuration with just one

Figure 2. Two graphs showing configurations of cooperators (blue) and defectors (red). (a) Cooperation
can be favored for the initial condition that is shown since the critical benefit-to-cost ratio is 42 and, in particular, finite. However, the fixation of cooperation cannot be favored for any initial configuration with a single cooperator on this graph. (b) Cooperation cannot be favored for the initial condition that is shown since the critical benefit-to-cost ratio is infinite. However, any initial configuration with a single cooperator has a critical benefit-to-cost ratio of 28. Therefore, the addition of cooperators to the initial configuration can either favor cooperation, (a), or suppress it, (b). The critical benefit-to-cost ratio can also be expressed in terms of a well-known quantity known as a "structure coefficient", σ, which satisfies (b/c) * = (σ + 1)/(σ − 1).
cooperator, one can reduce this critical ratio by placing a second cooperator adjacent to the first cooperator (see Methods). If b/c lies below Eq. (2) and above Eq. (7), then a strategically chosen configuration can ensure that the fixation of cooperation is favored by selection even if it is disfavored for any single-cooperator state. This behavior is particularly pronounced on small social networks, where the critical ratios take on a significant range of values (see Fig. 3), and less apparent on large networks, where the critical ratios are much closer to the degree of the graph, k. Fortunately, on small networks it is easier to directly search for configurations that have small critical benefit-to-cost ratios via Eq. (7).
. The configuration of (e) is 'optimal' for cooperation in the sense that if selection increases the fixation probability of cooperators in some state, then it does so in state (e) as well. Relative to all possible initial states, selection can increase the fixation probability of cooperators in (e) under the smallest b/c ratio. Panels (g-i) show that when cooperators are added in a different order (starting with just a single cooperator), the critical benefit-to-cost ratio can actually be increased. Each of these three configurations has isolated cooperators, and (i) gives the global maximum of ξ ⁎ b c ( / ) , which is achieved by just (i) and its conjugate. Since N 0 = 3, (i) is a maximal isolated configuration. The initial state in (i) is least conducive to cooperation in the sense that, relative to all other initial configurations, (i) requires the largest b/c ratio for selection to increase the fixation probability of cooperators. If selection increases this fixation probability when starting from state (i), then it does so when starting from any other mixed initial configuration.
Scientific RepoRts | 6:39181 | DOI: 10.1038/srep39181 Structure coefficients. Consider now a generic 2 × 2 game whose payoff matrix is The donation game is a special case of this game with A indicating a cooperator and B indicating a defector. If A denotes the monomorphic state consisting of only A-players and if ξ is a configuration of A-and B-players, then a natural generalization of the question we asked for the donation game is the following: when is ρ ξ,A (w) > ρ ξ,A (0) for sufficiently small w > 0? That is, when does (weak) selection favor the fixation of A when starting from state ξ? For technical reasons, this question is more difficult to answer when the payoff matrix is Eq. (8) instead of that of the donation game. There is, however, an alternative way of generalizing the critical benefit-to-cost ratio to Eq. (8).
When considering the evolutionary success of strategy A based on configurations with only one mutant, another standard measure is whether the fixation probability of a single A-mutant in a B-population exceeds that of a single B-mutant in an A-population [see ref. 34,Eq. 2]. That is, one compares the fixation probability of A to the fixation probability of B after swapping A and B in the initial state. This interchange of strategies may be defined for any initial state: formally, if ξ is a configuration of A-players and B-players, the conjugate of ξ, written ξ, is the state obtained by swapping A and B in ξ. In other words, the A-players in ξ are the B-players in ξ.
A natural generalization of this criterion to arbitrary configurations involves comparing the fixation probability of A in ξ to the fixation probability of B in ξ. Let A and B be the monomorphic states consisting of all A-players and all B-players, respectively. In this context, our main result is that for all sufficiently small w > 0 if and only if where, for DB updating, In Methods, we give an explicit formula for the structure coefficient, σ ξ , for BD updating as well. Just as it is for the critical benefit-to-cost ratio of Eq. (7), the complexity of calculating σ ξ is O(k 2 N). In fact, the relationship between ξ ⁎ b c ( / ) and σ ξ is remarkably straightforward: which, for DB updating, generalizes a result of Tarnita et al. 34 to arbitrary configurations. Note that the critical benefit-to-cost ratio increases as σ ξ decreases. Moreover, unlike the critical benefit-to-cost ratio, σ ξ is always finite. Interestingly, both ξ ⁎ b c ( / ) and σ ξ are invariant under conjugation, meaning they are the same for ξ as they are for ξ. For the donation game, Eq. (9) . Of course, Eq. (9) applies to a broader class of games as well and represents a simple way to compare the success of a strategy (A) relative to its alternative (B) when selection is weak. In this sense, Eq. (9) may be thought of as a generalization of the critical benefit-to-cost rule to arbitrary 2 × 2 games.

Discussion
Selection always opposes the emergence of cooperation for BD updating, regardless of the configuration of cooperators and defectors (see Methods). This result is consistent with previous studies showing that cooperation cannot be favored under random configurations 32,33,78,79 , and it specifies further that cooperation cannot be favored under any configuration. For general 2 × 2 games given by Eq. (8), we show in Methods that one can also find a simple formula for σ ξ in the selection condition of Eq. (9) that can be easily calculated for a given graph.
Remarkably, for DB updating, both the critical benefit-to-cost ratio and σ ξ depend on only local properties of the configuration, which makes these quantities straightforward to calculate. Furthermore, the complexity of calculating both of these quantities is O(k 2 N), where N is the size of the population and k is the degree of the graph, so they are computationally feasible even on large graphs. Therefore, our results provide a tractable way of determining whether or not selection favors cooperation for any configuration.
Finding an optimal configuration, which is one that minimizes the critical benefit-to-cost ratio, seems to be a difficult nonlinear optimization problem. The critical ratio is easily computed for any given configuration, but a graph of size N has 2 N possible configurations, which makes a brute-force search unfeasible for all but small N. Our results qualitatively show that both the abundance and the configuration of cooperators can strongly influence the effects of selection. We leave as an open problem whether it is possible to find a polynomial-time algorithm that produces an optimal configuration on any regular graph. However, since Eq. (7) is extremely easy to compute for a given configuration, and since small graphs generally exhibit broader variations of critical ratios than do larger graphs (since → ξ ⁎ b c k ( / ) uniformly as N → ∞ ), it is typically feasible to find a state that is more conducive to cooperation than a random configuration.
Our analysis of arbitrary configurations uncovers two important features of the process with DB updating: (i) there exist graphs that suppress the spread of cooperation when starting from a single mutant but promote the spread of cooperation when starting from configurations with multiple mutants (Fig. 2(a)), and (ii) there exist graphs that promote the spread of cooperation when starting from a single mutant but suppress the spread of cooperation when starting from configurations with many mutants (Fig. 2(b)). The proper initial configuration is thus a crucial determinant of the evolutionary dynamics, and our results help to engineer initial conditions that promote the emergence of cooperation on social networks. More importantly, these results provide deeper mathematical insights into the complicated problem of how selection affects the outcome of an evolutionary process at each point along an evolutionary trajectory.

Methods
Notation and general setup. In what follows, the population structure is given by a simple, connected, k-regular graph, G = (V, E), where V denotes the vertex set of G and E denotes the edge set. For x, y ∈ V, we write x~y to indicate that x and y are neighbors, i.e. (x, y) ∈ E. Throughout the paper, we assume that #V = N is finite and k ≥ 2.
The payoff matrix for a generic game with strategies A and B is A configuration on G, denoted ξ, is a function from V to {0, 1}. If ξ(x) = 1, then the player at vertex x is using A; otherwise, this player is using B. A special case of Eq. (12) is the donation game, When we are considering the donation game, ξ(x) = 1 indicates a cooperator at vertex x and ξ(x) = 0 indicates a defector at vertex x. For any such configuration, ξ, the conjugate configuration, ξ, is defined as For any configuration, ξ, on a k-regular graph, G, and for x ∈ V and i, j ∈ {0, 1}, let For any function, f(x, ξ), let x V be the arithmetic average of f with respect to the vertices of G. (Fig. 1 in the main text gives an example of how these quantities are calculated). The arithmetic averages of the functions formed from these local frequencies admit simple probabilistic interpretations: If a random walk is performed on the graph at a starting point chosen uniformly-at-random, then ) is the probability that the player at the first step is a cooperator (resp. a defector), and ξ f ( ) 10 is the probability that the player at the first step is a cooperator and the player at the second step is a defector. If two independent random walks are performed at the same starting point, is the probability of finding a cooperator at step one in the first random walk and a defector at step one in the second random walk. If one chooses an enumeration of the vertices and represents G by an adjacency matrix, Γ , and ξ as a column vector, then Scientific RepoRts | 6:39181 | DOI: 10.1038/srep39181 1 0 2 2 which gives a simple, alternative way to calculate each of ξ f ( ) 10 and ξ f f ( ) 1 0 . Let w ≥ 0 be a sufficiently small selection intensity. The effective payoff of an i-player at vertex x in configuration ξ, denoted ξ e x ( , ) i w , for the game whose payoffs are given by the generic matrix of Eq. (12), is defined via w 0 1 0 The basic measure we use here to define the evolutionary success of a strategy is fixation probability. If X is a strategy (either in {A, B} or in {C, D}), let X denote the monomorphic configuration in which every player uses X. For any configuration, ξ, and a fixed game, we write ρ ξ,X (w) to denote the probability that strategy X fixates in the population given an initial configuration, ξ, and selection intensity, w.
In the following sections, we consider DB and BD updating under weak selection (w ≪ 1).

DB updating.
Under DB updating, a player is first selected for death uniformly-at-random from the population. The neighbors of this player then compete to reproduce, with probability proportional to fitness (effective payoff), and the offspring of the reproducing player fills the vacancy. We assume that the strategy of the offspring is inherited from the parent. Therefore, if the player at vertex x dies when the state of the population is ξ, then the probability that this vacancy is filled by an i-player is Critical benefit-to-cost ratios. Recall that our goal is to determine when, for any configuration, ξ, ρ ξ,C (w) > ρ ξ,C (0) for all sufficiently small w > 0. We first need some technical lemmas: Lemma 1. For any configuration, ξ, we have the following first-order expansion as w → 0 + : By the definition of π i w , Eq. (21), we have  In order to compute the voter-model integrals in Eq. (22), we now turn to coalescing random walks on graphs. Suppose that {B x } x∈ V is a system of rate-1 coalescing random walks on G, where, for each x ∈ V, B x starts at x. These interacting random walks move independently of one another until they meet, and thereafter they move together. The duality between the voter model and these random walks is given by for each S ⊆ V, t > 0, and strategy configuration, ξ. For more information on this duality, including a proof of Eq. (28) and its graphical representation, see §III.4 and §III.6 in ref. 80.
Consider now two discrete-time random walks on G, ⩾ X ( ) n n 0 and ⩾ Y ( ) n n 0 , that start at the same vertex and are independent of {B x } x∈V . If the common starting point is x ∈ V, then we write  x to denote the expectation with respect to this starting point. If the starting point is chosen with respect to the uniform distribution, π, then this expectation is denoted by  π . The random-walk probabilities,  x and  π , are understood in the same way.
x V N y V y x k 1 : 1 0 1 , for example, we will use these random walks to save notation when we compute the local frequencies of strategy configurations.
□ We are now in a position to prove the first of our main results: Theorem 3. In the donation game, for any configuration, ξ, we have the following expansion as w → 0 + : Scientific RepoRts | 6:39181 | DOI: 10.1038/srep39181 Proof. By Lemma 1, it suffices to obtain the coefficient of w, i.e. the first order term, on the right-hand side of Eq. (22). Since the game under consideration is the donation game, a simple calculation gives Therefore, Eq. (42) follows from the calculations of Lemma 2, which completes the proof. □ From Theorem 3, we see that, for small w > 0, which gives the critical benefit-to-cost ratio of Eq. (7).
Structure coefficients. We now turn to a generalization of the critical benefit-to-cost ratio for arbitrary 2 × 2 games in which the payoff matrix is given by Eq. (12). Our main result is the following: for all sufficiently small w > 0 if and only if Proof. By the neutrality of the voter model, we have ρ ρ By the first-order expansion of Eq. (23), it follows that, for small w > 0, , 0 0 0 0 By Eqs (22) and (23) and the neutrality of the voter model, we have  and form the averages ξ f ( ) ijl via Eq. (16). For any configuration, ξ, we have the following first-order expansion as w → 0 + :  We then obtain Eq. (57) by applying this calculation to the first-order expansion of Eq. (23). □ Our main result for BD updating is the following: In the donation game, for any configuration, ξ, we have the following expansion as w → 0 + : Proof. For the donation game, the function D of Eq. (58) simplifies to 10 1 0 10 Therefore, by the calculations of Lemma 2, we see that for each mixed state, ξ, it follows that ρ ξ,C (w) < ρ ξ,C (0) for all sufficiently small w > 0 whenever ξ is not an absorbing state, so cooperation is always suppressed by weak selection under BD updating.
Structure coefficients. Although cooperation is never favored by weak selection under BD updating, we can still write down a condition for selection to favor strategy A in an arbitrary 2 × 2 game whose payoff matrix is given by Eq. (12): for all sufficiently small w > 0 if and only if Proof. The same argument given in the proof of Theorem 4 shows that Eq. (62) is equivalent to for all sufficiently small w > 0 if and only if where σ ξ is the structure coefficient given by Strategic placement of cooperators for DB updating. We turn now to the consequences of Theorem Proposition 8. Let k ≥ 2 be fixed. In the limit of large population size, N → ∞ , the critical benefit-to-cost ratio converges uniformly to k over all k-regular graphs, G, of size N and all configurations, ξ, on G.
Proposition 8 follows immediately from the following technical result: Lemma 9. For fixed k ≥ 2 and for N > 4k 2 + 1 such that there exists a k-regular graph with N vertices, where G ranges over all k-regular graphs on N vertices, and, for each G, ξ ranges over all mixed configurations.
Proof. By the Cauchy-Schwarz inequality and the reversibility of the random walk, both ξ f ( ) 10 and 1/2 . Therefore, for any such G and any mixed ξ, it follows from Eq. (7) that Proof. Fix a k-regular graph with N vertices and, for 0 < n < N, let u n denote the uniform distribution on the set of configurations, ξ, with exactly n cooperators. Since u n is independent of the graph geometry, n whenever x ≠ y. Therefore, by the definitions of f i and f ij in Eqs (14) and (15), It follows from Eq. (7) in the main text that the critical benefit-to-cost ratio for u n is  (69) and (70) to see that the coefficient of w on the right-hand side of Eq. (42) under the random placement u n is equal to which is consistent with Theorem 1 in ref. 39.   73), the benefit-to-cost ratio for any n-random configuration is independent of n, so random configurations with more cooperators are neither more nor less conducive to cooperation than those with fewer. For a fixed graph, G, let N 0 be the maximum number of vertices that can be chosen in such a way that no two of these vertices are within two steps of one another. We say that a subset of vertices with this property is isolated. If the defectors in a configuration lie on isolated vertices, then we say that defectors are isolated. Proposition 11. If N > 2k, then cooperation can be favored for a mixed configuration with n cooperators whenever either 1 ≤ n ≤ N 0 + 1 or 1 ≤ N − n ≤ N 0 + 1.
Proof. For any configuration, ξ, with n cooperators, we have the inequalities In Eqs (74) and (75), equality is obtained by a configuration with n isolated cooperators. Indeed, ξ f ( ) 10 and ξ f f ( ) 1 0 depend on the number of cooperator-defector paths and the number of cooperator-anything-defector paths in ξ, respectively, and each such path is defined by either an edge or two incident edges. On the other hand, at least one of these inequalities is strict whenever ξ does not have isolated cooperators: If two cooperators are  In panels (a-i), cooperators are added sequentially, with each new cooperator neighboring a cooperator in the previous configuration. These panels clearly demonstrate that a configuration and its conjugate have the same critical ratio and structure coefficient. Panels (j-l) show that when cooperators are added in a different order, the critical ratios can increase rather than decrease. The configurations of (j-l) each have isolated cooperators. adjacent to one another, then Eq. (74) is strict; if two cooperators are adjacent to the same defector, then Eq. (75) is strict. In order to establish the proposition, we need to show that  . Suppose, on the other hand, that N − N 0 − 2k < 0. By the definition of N 0 , we can then find a configuration with N − 2k + 1 isolated cooperators. Since each of these cooperators has k neighboring defectors, and since none of these defectors have more than one cooperator as a neighbor, we have which contradicts the assumption that N > 2k, as desired. □

Remark 3.
The proof of Proposition 11 shows that whenever N > 2k, in fact − ⩾ N k N 2 max G 0 holds, where G ranges over all k-regular graphs on N vertices. This lower bound, max G N 0 , is sharp, which can be seen from the graph in Fig. 2(b) since this graph has size 9, is 4-regular, and satisfies N 0 = 1.
Scientific RepoRts | 6:39181 | DOI: 10.1038/srep39181 It then follows from the definition of the critical benefit-to-cost ratio that which is smaller than the ratio for random placement, Eq. (2), because N > 2k, which gives part 2. □

Examples. In Figs 4 and 5, we
give examples of the relationship between the configuration and the critical benefit-to-cost ratio on three small graphs.