## Abstract

Coexistence of cooperators and defectors is common in nature, yet the evolutionary origin of such social diversification is unclear. Many models have been studied on the basis of the assumption that benefits of cooperative acts only accrue to others. Here, we analyze the continuous snowdrift game, in which cooperative investments are costly but yield benefits to others as well as to the cooperator. Adaptive dynamics of investment levels often result in evolutionary diversification from initially uniform populations to a stable state in which cooperators making large investments coexist with defectors who invest very little. Thus, when individuals benefit from their own actions, large asymmetries in cooperative investments can evolve.

Cooperation is an enduring evolutionary conundrum (*1*–*3*). Its essence is captured by social dilemmas, in which groups of cooperators do better than groups of defectors, yet defecting individuals outcompete cooperative individuals in mixed groups. Considerable efforts have been expended trying to understand the evolution of cooperation on the basis of the prisoner's dilemma (*3*–*9*), which describes situations in which cooperative acts incur costs to the acting individuals, whereas benefits only accrue to others. In this case, cooperators are doomed in the absence of supporting mechanisms (*3*–*9*). In contrast, the social dilemma is relaxed if benefits of costly cooperative acts accrue not only to others but also to the cooperator itself. In classical game theory, this situation is described by the snowdrift game (also called hawk-dove game or chicken game), in which two drivers are trapped on either side of a snowdrift and have the options of staying in their cars or removing the snowdrift. Letting the opponent do all the work is the best option, but if both players refuse to shovel they can't get home. The essential feature of the snowdrift game is that defection is better than cooperation if the opponent cooperates but worse if the opponent defects. This maintains cooperation at a mixed stable state (*10*–*12*).

Cooperative interactions in which benefits accrue to both donor and recipient are common (*3*). They arise when individuals in a population can produce a common resource that is accessible to everybody. For example, yeast produce an enzyme to hydrolyze sucrose, and the enzyme secreted by a single individual can be used by the individual producing it as well as by others (*13*). It is natural to assume that cooperative investments, i.e., the time and effort spent in producing the common good, can vary continuously within a certain range and hence to extend classical games to evolutionary scenarios in which individuals can make continuously varying cooperative investments (*14*–*16*). Here, we define the continuous snowdrift game for the evolution of cooperative investments that incur costs to the donor and accrue benefits to both the donor and the recipient. Each individual has a continuous strategy, or trait, which we take to be a real number, *x*, between 0 and an upper limit, *x*_{max}, and which represents the amount of investment that the individual makes in the cooperative enterprise. For simplicity, we assume that all interactions are pairwise, but the theory presented below immediately extends to interactions in groups of arbitrary size *N* [Supporting Online Material (SOM) Text]. We define the payoff to an *x* strategist interacting with a *y* strategist to be *P*(*x,y*) = *B*(*x + y*) – *C*(*x*), where *B*(*x* + *y*) specifies the benefit that the *x* strategist obtains from the total cooperative investment made by both agents and *C*(*x*) specifies the cost incurred by the *x* strategist due to its own investment. We assume that *B*(*x*) and *C*(*x*) are smooth, strictly increasing functions satisfying *B*(0) = *C*(0) = 0.

In the continuous prisoner's dilemma (*16*), an *x* strategist facing a *y* strategist obtains the payoff *Q*(*x,y*) = *B*(*y*) – *C*(*x*). Because in this case investments incur costs but no benefits to the investing player, investments always evolve to zero (*16*). In contrast, in the continuous snowdrift game one could expect that investments evolve away from zero to some intermediate level provided that benefits outweigh costs [*B*(*x*) > *C*(*x*)] for small *x*. We will see, however, that the continuous snowdrift game displays much richer evolutionary dynamics.

We use adaptive dynamics (*17*–*19*) (SOM Text) to analyze the evolution of the strategy *x*. Consider a monomorphic population in which every individual adopts the same strategy, *x*. It follows from replicator dynamics (*20*) that the growth rate of a rare mutant strategy, *y*, in the resident *x* population is *f*_{x}(*y*) = *P*(*y,x*) – *P*(*x,x*) = *B*(*x + y*) – *C*(*y*) – [*B*(2*x*) – *C*(*x*)]. The evolution of the trait *x* is then governed by the selection gradient *D*(*x*) = ∂*f*_{x}/∂*y*|_{y = x} = *B*′(2*x*) – *C*′(*x*), and the adaptive dynamics of *x* is described by *ẋ* = *D*(*x*) (*17*–*19*) (SOM Text).

Equilibrium points of the adaptive dynamics are called singular strategies and are solutions of *D*(*x**) = *B*′(2*x**) – *C*(*x**) = 0. If there is no such solution, the trait *x* montonically increases or decreases over evolutionary time, depending on the sign of *D*(*x*). If *x** exists, it is convergent stable and, hence an attractor for the adaptive dynamics, if *dD*/*dx*|_{x = x*} = 2*B*″(2*x**) – *C*″(*x**) < 0 (*17*–*19*) (SOM Text). If this inequality is reversed, *x** is a repeller.

Initially, the population will converge to an attracting singular point *x**, but its subsequent evolutionary fate depends on whether *x** is a maximum or minimum of the invasion fitness *f*_{x}(*y*). If *x** is a maximum, i.e., if ∂^{2}*f*_{x*}/∂*y*^{2}|_{y = x*} = *B*″(2*x**) – *C*″(*x**) < 0, then *x** is an evolutionary stable strategy (ESS), representing an evolutionary end state in which all individuals make intermediate cooperative investments. If, however, *B*″(2*x**) – *C*″(*x**) > 0, then a population of *x** strategists can be invaded by mutant strategies on either side of *x**. In this case, the population undergoes evolutionary branching (*18*, *19*) and splits into two distinct and diverging phenotypic clusters.

For example, such dynamics can be observed for quadratic cost and benefit functions *B*(*x*) = *b*_{2}*x*^{2} + *b*_{1}*x* and *C*(*x*) = *c*_{2}*x*^{2} + *c*_{1}*x*. In this case, the singular point is unique if it exists and is given by ; it is an evolutionary branching point if 2*b*_{2} < *c*_{2} < *b*_{2} < 0 (SOM Text). If evolutionary branching occurs, the two phenotypic clusters diverge and evolve to the borders of the trait interval (Fig. 1A) (SOM Text). The population emerging from this evolutionary process contains both defectors that do not make any investment at all and cooperators making the maximal investment (Fig. 1A). In this state, the continuous snowdrift game transforms into its traditional form with two coexisting strategies (SOM Text). Therefore, adaptive dynamics of continuous strategies yields a natural explanation of the evolutionary emergence of the pure cooperator and defector strategies of the traditional snowdrift game (*21*).

A complete classification of the adaptive dynamics for quadratic cost and benefit functions is provided in Fig. 1: evolutionary branching (Fig. 1A), convergent stable ESS (Fig. 1B), evolutionary repellor leading to bistable evolutionary dynamics (Fig. 1C), and, in the absence of a singular strategy, either uniform selection for defectors (Fig. 1D) or cooperators (Fig. 1E). We note that maximal mean payoffs in monomorphic populations cannot be used as a predictor of the evolutionary outcome (Fig. 1 and SOM Text). This is obvious in cases of evolutionary branching, but it is also generally true if the system exhibits an attracting ESS or unidirectional evolutionary dynamics (Fig. 1).

We illustrate the connection between the adaptive dynamics of the continuous snowdrift game and traditional game theory by considering local games between two strategies with similar investment levels (Fig. 2). For traditional two-player games, there are three basic types of outcomes (*22*): (i) one strategy dominates and reaches fixation, (ii) a globally stable mixed equilibrium occurs (as, for example, in the classical snowdrift game), or (iii) the game exhibits bistability, with one or the other strategy going to fixation depending on initial conditions.

Away from the vicinity of a singular point, the adaptive dynamics is directional, and the local game between a resident, *x*, and a nearby mutant, *x* + ϵ (ϵ small, > 0), is of the first type, so that one of the two investment strategies is dominant. If an attracting singular point exists, successive local games exhibit dominance and induce evolutionary convergence to the vicinity of the singular strategy (Fig. 2A). Near the singular point, local games can be of any type. If the singular strategy is an ESS, local games between the singular strategy and neighboring strategies exhibit either domination by the ESS or bistability, and hence neighboring strategies cannot invade. In contrast, if the singular strategy is an evolutionary branching point, local games between the singular strategy and neighboring strategies always exhibit a mixed stable state; in particular, neighboring strategies can invade (Fig. 2A, insets). On the other hand, evolutionary repellors are characterized by the fact that local games on either side of the repellor again exhibit dominance but now induce directional evolution away from the singular point (Fig. 2B). This results in bistability of the global evolutionary dynamics.

More complicated evolutionary scenarios than those shown in Fig. 1 are possible with other cost and benefit functions. For example, the equation *B*′(2*x**) – *C*″(*x**) = 0 may have more than one solution, in which case there are several singular points of the adaptive dynamics. Figure 3 shows two cases of an evolutionary branching point co-occurring with a repellor and illustrates that evolution does not always drive the phenotypic clusters emerging through evolutionary branching to the boundaries of the strategy space. Our numerical simulations (SOM Text) confirm that evolutionary branching is a generic and robust outcome for many different cost and benefit functions, *C*(*x*) and *B*(*x*), and always occurs under suitable conditions, i.e., whenever a singular point *x** satisfies 2*B*″(2*x**) < *C*″(*x**) < *B*″(2*x**) < 0. These conditions require that both cost and benefit functions are saturating, i.e., have negative curvature, near the singular point. Saturating benefits are clearly realistic, whereas costs could often be expected to accelerate. However, diminishing additional costs of larger cooperative investments are reasonable whenever the initiation of cooperative acts, such as turning on enzyme production machinery, is more costly than subsequent increases in cooperative investments. Given an instance of evolutionary branching, it is in principle possible that further branching events occur that would lead to the establishment of more than two phenotypic clusters in the population (*18*, *19*). In the case of quadratic cost and benefit functions, such secondary branching does not occur (SOM Text).

The paradox of altruism, of which the tragedy of the commons (*23*) is a celebrated avatar, is that although populations of altruists outperform populations of nonaltruists, selection will act to eliminate altruism altogether. Here, we have unveiled a different paradox of cooperation, which could be termed the “tragedy of the commune”: In a cooperative system, in which every individual contributes to a common good and benefits from its own investment, selection does not always generate the evolution of uniform and intermediate investment levels but may instead lead to an asymmetric stable state, in which some individuals make high levels of cooperative investment and others invest little or nothing.

In practice, it is often difficult to determine the payoffs in social interactions and hence to distinguish prisoner's dilemma and snowdrift interactions [a phage system marks a rare exception, but interestingly, selection turns the prisoner's dilemma into a snowdrift game (*24*)]. Nevertheless, the mere existence of high- and low-investing individuals has often been taken as prima facie evidence that the interaction is governed by a prisoner's dilemma, with some additional mechanism, such as reciprocity, responsible for the co-existence of altruists and nonaltruists. The tragedy of the commune, however, provides a quite different and, in many ways, simpler explanation for the coexistence of high- and low-investing individuals, which potentially applies to a wide range of cooperative and communal enterprises in biological systems.

For example, in the aforementioned case of enzyme production in yeast, the tragedy of the commune suggests evolution toward a state in which some cells produce the enzyme whereas others do not and instead simply exploit the common resource. Exactly this situation has been found experimentally (*13*). The same mechanism may drive the evolution of defective interference in viruses (*25*). When viruses coinfect a cell, the replication enzymes they produce are a common resource. Selection could then favor diversification into two coexisting viral types: one producing the replicase and the defective interfering particles in which replicase production is down-regulated (*25*). Similar remarks apply to social dilemmas in RNA phages due to coinfection of host cells (*24*). In all these examples, differences between cooperators and defectors have their basis in complex regulatory processes. Therefore, these differences likely involve multiple genetic changes, allowing for gradual evolution of different cooperative investment levels. We also note that one can draw interesting parallels between the tragedy of the commune and models for the evolution of anisogamy (i.e., for the emergence of asymmetric investments into gamete size (*26*, *27*)), which represents a common good in sexual species.

In behavioral ecology, classical examples of cooperation include collective hunting and territory defense in lions (*28*), predator inspection in sticklebacks (*29*), and alarm calls in meerkats (*30*). In theoretical discussions of these examples, the existence of cooperators providing a common good and defectors exploiting it has been assumed a priori. The tragedy of the commune, however, suggests an evolutionary mechanism for the emergence of distinct behavioral patterns with differing degrees of provisions to the common good. This mechanism may also apply to cultural evolution in human societies, in which large differences in cooperative contributions to communal enterprises could give rise to conflicts on the basis of accepted notions of fairness.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/306/5697/859/DC1

SOM Text

References and Notes