GROUP FORMATION AND THE EVOLUTION OF SOCIALITY

Authors

  • Thomas Garcia,

    1. École Normale Supérieure, Unité Mixte de Recherche 7625, Écologie et Évolution, 46 rue d'Ulm, 75005 Paris, France
    2. E-mail: t_garcia99@yahoo.fr
    3. Université Pierre et Marie Curie-Paris 6, Unité Mixte de Recherche 7625, Écologie et Évolution, CC 237-7 quai Saint Bernard, 75005 Paris, France
    4. Centre National de la Recherche Scientifique, Unité Mixte de Recherche 7625, Écologie et Évolution, 46 rue d'Ulm, 75005 Paris, France
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  • Silvia De Monte

    1. École Normale Supérieure, Unité Mixte de Recherche 7625, Écologie et Évolution, 46 rue d'Ulm, 75005 Paris, France
    2. Université Pierre et Marie Curie-Paris 6, Unité Mixte de Recherche 7625, Écologie et Évolution, CC 237-7 quai Saint Bernard, 75005 Paris, France
    3. Centre National de la Recherche Scientifique, Unité Mixte de Recherche 7625, Écologie et Évolution, 46 rue d'Ulm, 75005 Paris, France
    Search for more papers by this author

Abstract

In spite of its intrinsic evolutionary instability, altruistic behavior in social groups is widespread in nature, spanning from organisms endowed with complex cognitive abilities to microbial populations. In this study, we show that if social individuals have an enhanced tendency to form groups and fitness increases with group cohesion, sociality can evolve and be maintained in the absence of actively assortative mechanisms such as kin recognition or nepotism toward other carriers of the social gene. When explicitly taken into account in a game-theoretical framework, the process of group formation qualitatively changes the evolutionary dynamics with respect to games played in groups of constant size and equal grouping tendencies. The evolutionary consequences of the rules underpinning the group size distribution are discussed for a simple model of microbial aggregation by differential attachment, indicating a way to the evolution of sociality bereft of peer recognition.

The emergence and persistence of social ventures, where individuals concur to the sustainment of a community at the cost of a personal investment, has been classically addressed in a game-theoretical framework. The evolution of cooperation has been first formalized in the context of dyadic interactions, where the formation of pairs and the accomplishment of the game are concomitant. When individuals play in couple, several mechanisms have been shown to effectively promote cooperation even for a Prisoner’s Dilemma type of interactions, where it is always in one own’s interest to defect in a single round of the game. Cooperators can thrive provided they interact preferentially with other cooperators, for instance via the knowledge of the co-player’s past behavior (Trivers 1971; Axelrod and Hamilton 1981), reputation (Nowak and Sigmund 1998), or mechanisms enhancing individuals’ interactions with kin (Hamilton 1964) such as spatial structure (Nowak and May 1992).

Those results have then been extended to games involving a number N of players, where the public goods game (PGG) plays the same prototypic role as the Prisoner’s Dilemma (Kollock 1998; Doebeli and Hauert 2005). The PGG formalizes the so-called tragedy of the commons (Hardin 1968; Rankin et al. 2007), whereby cheaters who do not contribute to the public goods are always better off, in a one-shot game, than cooperators who pay a cost to sustain the collective entreprise.

Sociality, however, relates not only to the act of helping others, but also affects the context where social games are played, among which the way groups are formed in the first place. In extending the framework from two players to N-players games, the processes that lead to group formation have often been overlooked in holding the group size constant.

This assumption has been recently relaxed in different ways. Group size variations can be externally forced by imposing bottlenecks that periodically increase the variance among groups (Chuang et al. 2009), leading to a “Simpson’s paradox” in which cooperation is disadvantaged locally but a winning strategy on the whole (Wilson 1975). They can also result from demographic fluctuations (Hauert et al. 2006a) or facultative participation to the game (Hauert et al. 2002a,b). The effect of a fixed group size distribution with binomial allocation of individuals within groups has also been investigated in various types of games and can either promote or hinder cooperation (Peña 2011).

Group size can be directly affected by traits that co-evolve with cooperation. Pfeiffer and Bonhoeffer (2003) illustrated how group clustering (defined in terms of spatial proximity) is selected together with nonexploiting, cooperative behavior if resources are sufficiently concentrated. Avilés et al. (Avilés 2002; van Veelen et al. 2010) showed that grouping tendency and cooperativeness may be favored jointly, resulting in the emergence of viable cooperative groups. This result relies on two features of the model: on the one hand, while cooperation is costly, the ability to join groups is not; thus, as soon as some cooperators are present in the population, individuals are better off in a group than alone, making the lonely lifestyle unprofitable and the “dispersed” population structure unstable. On the other hand, the introduction of a hump-shaped fitness function implies from the start the existence of intermediate “optimal” group sizes, at fixed average level of cooperation within the group. The cheating load is then twofold: “freeloaders” both hamper the benefits retrieved from the group and crowd them uselessly. Powers et al. (2011) similarly evidenced that inheritable aggregative features may evolve jointly with cooperation. They let players have a clear-cut group size preference, whereby groups form by gathering individuals sharing the same preference. In their model, cooperation ends up being tightly linked with small group sizes that support it more easily, even when direct selection pressures for large groups or weaker selection against cooperation is applied.

In line with these studies, we address here the evolution of aggregative traits in a context that is dynamically shaped by the traits themselves. Such traits require an individual investment and produce collective benefits, and can therefore be regarded as a cooperative strategy once individuals have been distributed in groups by the aggregation process. The quest for simple mechanisms allowing grouping to evolve is of particular relevance to understand how sociality can be maintained in microorganisms, where individuals interact in clusters of many individuals, a setting that is recognized as unfavorable to social ventures. We assume that individuals have different tendencies to form cohesive aggregates, and that group cohesion itself is a common good. In our model, individuals are thus endowed with a unique gene that codes for a costly trait (coined hereafter “sociality”). The social trait promotes aggregative cohesion during both the group formation process and the reproductive stage, where the fitness is the individual payoff in a PGG. The outcome of the social interaction is thus considered to hinge upon the physical properties of the groups: more cohesive groups are fitter than groups weakened by looser attachment of their members.

This setting is relevant at least for several microbial organisms usually taken as examples of primitive social behavior (Crespi 2001; West et al. 2006; Smukalla et al. 2008; Nanjudiah and Sathe 2011), where physical stickiness is coupled to cooperative behavior once aggregates are formed. For instance, in Dictyostelium discoideum populations, chimeras composed of aggregative wild type (WT) and nonaggregative mutants produce slugs whose motility increases with the proportion of WT cells (K. Inouye, pers. comm.). In D. discoideum, slug motility allows more efficient chemo- and phototaxis, benefiting all cells equally. In Myxobacteria as well, WT’s social motility multigene system enables cell clumping. The presence of mutants deprived of this social ability impairs swarming and ultimately mar individual fitness in the group.

To evolve, an altruistic trait must ultimately entail some kind of assortment between its bearers (Fletcher and Doebeli 2009). When the gene giving rise to such assortment also codes for cooperative behavior, it is framed under the term of “greenbeards” (Dawkins 1976; Gardner and West 2010). This general definition actually brings together very different mechanisms able to generate assortment, based or not on direct recognition of others’ traits. Here, we will consider a case where no recognition is involved but where assortment emerges spontaneously from blind interactions among individual players. The environment is in this case shaped by the group formation process and changes jointly with the frequency of the social strategy. We present a model showing that the rise of sociality can stem from merely quantitative differences in the probabilities of attachment, so that even mechanisms that do not produce assortment within groups of fixed size can lead to the evolution of sociality if group sizes are distributed.

In “Rooting Payoffs in the Group Formation Process,” we describe the evolutionary consequences of group formation schemes where social and asocial individuals differ quantitatively in their ability to aggregate. Group formation is considered a “black box” generating the group size distributions experienced by players. The average fitness advantage of sociality depends on the distributions of group sizes experienced by players of each strategy. We derive the condition for sociality to outcompete asociality under the assumption that no nepotistic grouping between social individuals generates assortment a priori. In “Group Formation by Differential Attachment,” we apply the results of “Rooting Payoffs in the Group Formation Process” to a toy model based on differential attachment and show that full sociality in a population can be attained, along with sizeable average group sizes, as soon as a threshold frequency of socials is overcome. We also stress the role of lonely individuals, usually neglected when fixed group sizes are considered, in the balance of benefits and costs of the social game. We eventually point out that our mechanism does not condemn large social groups, and may thus be relevant to account for sociality in microorganisms. Different interaction rules leading to nonzero a priori assortment are briefly addressed by numerical simulation. In “Discussion,” we discuss the implications of our results for biological systems and the perspectives in elucidating the mechanistic basis of group formation processes.

Rooting Payoffs in the Group Formation Process

In this work, we want to address the emergence and maintenance of social behavior in organisms whose life cycle consists of a phase of aggregation, a phase of differential reproduction that modifies the frequencies of each type in the following generation, and a dispersal phase (see Fig. 1). The whole population is re-shuffled at each generation, unlike models involving lasting groups (Fletcher and Zwick 2004; Killingback et al. 2006; Traulsen and Nowak 2006). This requires resolving a process, group formation, that happens on a time scale much faster than evolutionary changes. Our working assumption is that sociality consists primarily in a quantitative difference in the ability to aggregate, that affects both the group formation phase and the competitive success of aggregates. Once groups are formed, their cohesion constitutes a public good, so that fitter groups are those comprising a larger fraction of cohesive individuals.

Figure 1.

Scheme of the model resolving the process of group formation in an evolving population characterized by a fraction x of social players. At each generation, we distinguish three processes: aggregation, trait frequency evolution, and dispersal (arrows). Initially, scattered individuals undergo a group formation process giving rise to groups of different sizes. Differences in attachment ability between the two strategies result in distinct distributions ds and da for the size of the group an individual player belongs to. A social game takes place within each group in the form of a linear public goods games where social individuals contribute to group welfare. The performance of the two strategies is computed in terms of the difference in average payoff. The frequency of the social type x is updated according to this payoff difference and groups are dispersed. If the aggregation phase occurs on a fast time scale with respect to the change in frequency of the two strategies, such evolutionary dynamis can be described by a continuous-time replicator equation.

We assume that individuals are either social or asocial, these two strategies being genetically encoded. A social individual pays a cost c for being more likely to aggregate. Asocial individuals do not pay this cost and have a lower probability of aggregating. After group formation has occurred, both social and asocial players belong to a group or can remain alone. For the sake of generality, we do not specify the grouping process, but characterize it by its outcome: the distributions ds(n) and da(n) (inline image) of group sizes in which social (resp. asocial) individuals are found. During group formation, assortment may be generated within groups. For instance, processes leading to positive within-group assortment of social individuals may rely on preferential interactions (Wilson and Dugatkin 1997), or on a probability to join a group proportional to the number of social players it contains (Avilés 2002). In this section, we point out that whereas some kind of assortment is necessary for sociality to evolve, no preferentially assortative feature needs to be assumed a priori as soon as the group size is not fixed; as such, it is compatible with the scenario where groups form by random and blind interaction processes.

Once groups are assembled, social players contribute b to a linear PGG, whereas asocial players do not contribute. In a group of size n with m social players, all individuals thus gain b m/n irrespective of their strategy. Different choices of the gain function (notably accounting for discount or synergy, as in Hauert et al. 2006b and Archetti and Scheuring 2010) are possible, but we opt here for the standard linear formulation, so as to focus only on the nonlinearities generated by the aggregation process. We thus assume that no group size is inherently beneficial to group members, so that the payoff only depends on the proportion and not on the absolute number of social players in a group. This is a conservative hypothesis because any payoff function increasing with group size would be further favored whenever sociality is associated with larger groups.

Let us compute the average payoff of each (social or asocial) strategy in a population where a fraction x of individuals is social and a fraction (1 −x) is asocial. After the aggregation process, social and asocial players belong to groups of variable sizes. A PGG is played within each group, and the resulting average payoffs for both types of conditions the evolution of their frequencies at the next generation.

We first consider groups of size n (n≥ 2). Following Fletcher and Doebeli (2009), we split the payoff of each player in a part due to self and an other due to the interaction environment, that depends only on the composition of the group. The payoff due to self is b/nc for a social player, who pays a cost c for sociality and gets a share b/n of its own contribution to the common goods; for an asocial player, who does not contribute to the PGG, it is 0.

For a linear PGG, the payoff due to the interaction environment is proportional to the average number es(n) (resp. ea(n)) of social individuals among the n− 1 co-players of a social (resp. asocial) player, so that the average payoffs of social individuals in a group of size n is:

image(1)

and for asocial ones:

image(2)

The “interaction neighborhoods”es(n) and ea(n) are in general different, for example, if assortative mechanisms such as peer or group recognition are involved in the process of group formation. In these cases, the local environment of a social player is enriched in social players compared to that of an asocial one (es(n) > ea(n)). For instance, total segregation between social and asocial players would yield es(n) =n− 1 and ea(n) = 0.

Considering all possible group sizes, the payoff for social and asocial individuals is obtained as an averaged sum of these payoffs, weighted by the group size distributions ds(n) and da(n). In doing this, one has to consider separately the contribution of lonely individuals, who do not engage in a PGG, and whose payoffs are c for socials and 0 for asocials.

Since evolutionary consequences are measured in terms of relative advantages, we only display here the difference in the average payoff of social and asocial individuals:

image(3)

This formula is composed of three terms: the cost to the individual for its investment in a social action, which is paid also when the social player remains alone; the marginal gain for being social, averaged over groups of all sizes; and a third term combining the effect of within-group assortment to that of differential allocation in groups. Although the second term necessarily declines when groups of larger size form in the population, the third term allows for different repartitions between groups (ds and da) to compensate for unfavorable average interaction environment within groups of a given size. This compensatory effect may in principle even overcome negative within-group assortment (i.e., ea(n) > es(n)).

When only one group size is present in the population, from equation (3) one immediately retrieves the condition for the evolution of sociality found in Fletcher and Doebeli (2009). In case group formation is governed by an extreme recognition process leading socials to form groups only with their kind (es(n) =n− 1 and ea(n) = 0 for all n), the condition ΔP(x) > 0 reduces to b/c > 1/(1 −ds(1)). Were no individual left alone, b > c suffices in this case for sociality to evolve.

When groups of different size are present, sociality can however thrive even in the absence of such within-group assortment, that is, when es(n) =ea(n) for all n≥ 2. In this case of random within-group repartition, the interaction neighborhoods are equal:

image(4)

with p(s|n) the fraction of social players within groups of size n. Given the distributions ds(n) and da(n), this fraction is:

image(5)

In this case, equations (3), (4), and (5) thus yield:

image(6)

We have highlighted the dependence of the payoff difference upon the fraction x of social players to remind that the population composition and the aggregation rules, which together determine the distributions ds and da, are held fixed during group formation.

The fraction of social players will increase in the next generation whenever ΔP(x) is positive, and the evolutionary equilibria xeq of the system are those such that ΔP(xeq) = 0. A condition for sociality to be favored when initially absent in the population writes ΔP(x= 0) > 0, that is, inline image. It is also the requirement for sociality to evolve when grouping tendencies are equal for both strategies (ds(n) =da(n) ∀ n) and can be interpreted as the condition for sociality to pertain to directly beneficial cooperation (see Section S2 of the Supporting Information). As exemplified by a toy model in the next section, sociality may be favored even when it is altruistic as soon as the distributions da(n) and ds(n) seen by the two strategies differ sufficiently. In the following, we will consider that selection is weak enough to guarantee a small change in frequencies from one generation to the next. The evolutionary dynamics is in this case approximated by a continuous-time replicator equation (Taylor and Jonker 1978; Schuster and Sigmund 1983; Hofbauer and Sigmund 1998):

image(7)

where the aggregation phase occurs infinitely fast with respect to evolutionary changes. If the time scales of aggregation and evolution were not separated, the evolutionary dynamics would be more correctly described by a discrete-time replicator equation that displays a potentially much more complex behavior (Villone et al. 2011).

In the next section, we illustrate our conclusions through a toy model with an explicit mechanism of aggregation underpinning different group size distributions for the two strategies. This mechanism is chosen such that it creates no assortment a priori. We can thus apply the equations derived in this section to study the evolutionary dynamics of the social strategy along with that of the group size distributions.

Group Formation by Differential Attachment

In this section, we apply the results of “Rooting Payoffs in the Group Formation Process” to an illustrative model where group formation is based on simple hypotheses regarding individual interactions. We explain how social behavior characterized by an increase in individual “stickiness” might evolve, and clarify the mechanism giving rise to assortment at the population level even in the absence of peer recognition. A social individual produces a costly “glue” that increases its chances to attach to any individual it comes in contact with. At the same time, he enhances overall group cohesion to a higher extent than an asocial individual more loosely glued to its group. This is consistent with our assumption that sociality entails differences both in the process of group formation, and in the contribution to group welfare. The following scheme for group formation is deliberately crude so as to remain analytically tractable and make the conditions for evolution of social attachment explicit. It is nonetheless an example of the evolution of social behavior via a biologically plausible mechanism of “blind” interactions among unrelated individuals, where assortment is an emergent property of the group formation process.

This model reflects some features of social microbes that are able to produce adhesive proteins at their surface. Although in some cases, adhesion proteins are strain-specific and allow to recognize other bearers by direct matching, we can imagine that, in the early stages of social evolution, cells might have been endowed with generic adhesion-enhancing properties. In this case, one can regard “stickiness” as an a priori property of a subpopulation of cells, that is energetically costly and entrains higher group-level productivity (e.g., in the search for prey, protection against predators, dispersal efficiency) because aggregates composed of a higher proportion of adhesive cells are more cohesive.

This model mirrors the properties of, for instance, social amoebas and bacteria. These microbes are thought to possess inheritable social strategies, whereby cells have different propensities to sacrifice for others, participating to the construction of the stalk of a fruiting body rather than becoming spores. The success of the genes that are passed on to the following generation is hence determined by the composition of the spore pool in all the groups (fruiting bodies) that are formed within the population at the aggregation stage of the life cycle. Enhanced probabilities to end up in the stalk are moreover often found associated to a higher stickiness (Strassmann and Queller 2011).

A TOY MODEL FOR DIFFERENTIAL ATTACHMENT

We consider an infinite population composed of a fraction x of social and a fraction (1 −x) of asocial individuals that differ in their attachment abilities. At each generation, aggregates form from sets of T individuals that are randomly drawn from the population pool. Group formation in each set is nucleated by one individual, named “recruiter,” that is chosen at random within the set. The remaining (T− 1) individuals are sequentially given one possibility to attach to the recruiter and hence to join the group. This one-shot adhesion step leaves some players outside the groups. Such lonely individuals are commonly observed in microorganisms (see, for instance, Bonabeau et al. 1999; Smukalla et al. 2008) and will play an important role in the emergence of sociality in our model. Attachment probabilities are fixed for any couple of strategies: between two social individuals (resp. a social individual and an asocial individual; two asocial individuals), it is denoted by πss (resp. πas; πaa).

Social individuals attach more efficiently, so that πss≥πas≥πaa. Moreover, we choose these probabilities such that no preferential interactions favor assortment between social players. This hypothesis reflects the requirement that interactions are not assortative a priori, unlike when social individuals recognize and select groups that are composed of a larger fraction of their kind. For a given composition of the population, it means that the expected proportion of socials among those a focal individual attaches to does not depend on its type (social or asocial) in dyadic interactions:

image(8)

that is verified for every x when:

image(9)

When inline image, the expected proportion of social co-players is higher for social than for asocial individuals, that is, positive assortment among socials would occur if the interactions were only pairwise. On the other hand, inline image would denote prior negative assortment in a dyadic context. In our analytical calculation, we will choose attachment probabilities satisfying equation (9), but we will relax this assumption at the end of the section and consider rules of attachment generating nonnull prior assortment.

GROUP SIZE DISTRIBUTIONS AND PAYOFF DIFFERENCE

In the Supporting Information, we analytically derive the size distributions for social and asocial types given the previously described mechanism for group formation. These distributions are illustrated in Figure S1. They are the superposition of a component in n= 1 (players remaining alone) and of two binomial distributions of respective averages T[xπss+ (1 −xas] and T[xπas+ (1 −xaa], corresponding to group nucleated by social (resp. asocial) recruiters. Their relative weights depend on the social type: social individuals are less often alone than asocials. Increasing the fraction of social individuals, the two nonsingular distributions shift toward higher group sizes.

Knowing the group size distributions emerging from the aggregation process, the payoff difference ΔP(x) can be computed for a given composition of the population. Figure 2 shows such payoff difference obtained by substituting equations (1) and (2) in the Supporting Information in equation (6). ΔP(x) is displayed for different values of the game parameters b and c and the aggregation parameters πss, πaa, and T. The advantage of social over asocial players increases monotonically with x, and is zero at most at one (unstable) equilibrium x*.

Figure 2.

Average payoff difference between social and asocial individuals as a function of the frequency x of socials as one parameter is changed. (a) The benefit-to-cost ratio b/c, with πss= 0.6, πaa= 0.1, T= 100; (b) the social-to-social attachment probability πss, with πaa= 0.1, b/c= 20, T= 100; (c) the upper bound for group size T, with πss= 0.6, πa= 0.1, b/c= 20. In each case inline image. Sociality evolves and invades the population as soon as it reaches a threshold frequency x*. Invasion by the social strategy is facilitated by either a large benefit-to-cost ratio, a high adhesiveness, or smaller maximal group sizes. However, there exists a threshold inline image above which sociality evolves for any maximal group size (panel c).

EVOLUTIONARY DYNAMICS AND EFFECT OF THE PARAMETERS

The internal equilibrium x* exists in a large region of the parameters space. In this region, the evolutionary dynamics ruled by the replicator (eq. 7) is bistable, with two additional stable monomorphic equilibria of full asociality x= 0 and full sociality x= 1. Sociality invades as soon as x is larger than the threshold value x*. Once established, full sociality is stable against the invasion by asocials. This scenario is qualitatively different from the case of one single group size, where the evolutionary dynamics can only lead to full asociality for linear PGG.

Figure 3 recapitulates the evolutionary dynamics by displaying the threshold frequency x* for sociality to invade. Figure 3a confirms that sociality evolves more easily the bigger the difference between social and asocial individuals’ attachment probabilities. Figure 3b shows that the threshold frequency x* decreases, as one would expect, as b/c increases. The region where sociality is evolutionary stable is larger than the region where the social behavior implies direct benefits, that is, when the marginal gain of a social individual is larger than c (Wilson 1979; Pepper 2000). We refer to Section S2 of the Supporting Information for the definition and calculation of the condition that delimits such region where sociality trivially evolves. Figure 3 shows that sociality evolves and is maintained in the population for a wide range of nontrivial parameters, that is, even when it is an essentially altruistic act.

Figure 3.

Threshold frequency x* of social individuals required to trigger the evolution of sociality through the replicator equation (7). (a) x* as a function of adhesiveness πss and πaa (with inline image, b/c= 20, T= 100); (b) x* as a function of b/c and T (πss= 0.6, πaa= 0.2, inline image). For values of b/c below the gray line, the evolution of cooperative sociality is trivial, corresponding to direct benefits, whereas above the line it is altruistic.

The threshold x* increases with T, consistently with the common claim that the evolution of altruism is easier in small groups (Olson 1971). However, when inline image, it converges to a value x* < 1 (Fig. 2c), meaning that there exist a critical initial frequency of social individuals such that sociality will invade whatever is the maximal group size.

The fact that x* is always positive means that, in general, an infinitesimal initial load of social players, as is generated by extremely rare random mutations, is not sufficient for sociality to evolve in the first place. However, when the threshold is low, numerous mechanisms can lead the frequency of the social strategy over the threshold, for example, random fluctuations due to finite-size effects, noninfinitesimal mutation rates or incomplete reshuffling from one generation to the next. Numerical simulations show that in finite populations subjected to a small mutation rate, the evolution of sociality is indeed easier than analytically expected.

Figure 4 displays the coupled dynamics of the social strategy and of the group size distributions in a numerical simulation of a large population (see Section S3 of the Supporting Information for details on the algorithm). Initially, only asocial individuals are present in the population, and the threshold is reached because of random mutations. As the frequency of social individuals increases, groups of progressively larger size form and concomitantly the fraction of lonely individuals decreases. The difference between the distributions for social and asocial individuals is enhanced by the fact that the balance between the solo and group components of these distributions is affected in opposite directions by the evolutionary dynamics.

Figure 4.

One run of the evolutionary algorithm (Section S3 of the Supporting Information), starting from a population of asocial individuals with random mutations on newborns, with rate u= 0.01. (a) Time evolution of the frequency x of the social type. (b–d) Snapshots at generation 120 (b), 260 (c), and 400 (d), indicated by the arrows in panel (a), of the group size distributions experienced by social (black) and asocial (light gray) individuals. The social trait rises toward fixation in the population while distributions displace toward larger group sizes. Parameters: πss= 0.8, πaa= 0.3, inline image, b/c= 20, T= 100, K= 106, x(0) = 0, inline image, inline image.

When all players are social, a fraction 1 −πss of individuals remains alone, whereas the others belong to groups binomially distributed around an average size Tπss. Notice that the group size at the social equilibrium is not influenced by the parameters b and c defining the PGG, but only by parameters determining the group formation process. In particular, the average group size arising in fully social populations linearly depends on the maximal possible group size T. Although higher cooperation levels are believed to occur more easily in small groups, group formation by differential attachment thus does not impose an a priori burden on large groups. This suggests that unsophisticated interaction rules may be relevant in explaining how sociality is maintained in the microbial world, where social aggregates are commonly composed of a large number of cells, for example, thousands of them in flocculating yeast (Smukalla et al. 2008), or up to 105 in D. discoideum (Tang et al. 2002).

OTHER RULES FOR GROUP FORMATION

So far, we have assumed that players undergo blind interactions whereby individuals attach, according to their strategies, with probabilities πaa, πas, and πss in geometric progression, ensuring no a priori assortment. However, other formulations of the adhesion probability, reflecting different settings of pairwise interactions, can be contemplated, and will in general result in nonrandom assortment within groups of fixed size. In these cases, equation (6) does not hold any longer, but the payoff difference can anyway be numerically computed by repeatedly simulating the aggregation process.

Figure 5 displays the difference in payoff between social and asocial players for the two cases where the attachment probabilities take up their extreme values: πasss and πasaa. In these model configurations, it is the social (resp. asocial) co-player that takes the lead in deciding the outcome of binary interactions. The first rule, where the asocial–social attachment probability is maximal, reduces the threshold for sociality to spread in the population. At the same time, the fully social equilibrium is destabilized: when chances to encounter a social individual are high, sociality becomes a “wasted investment” and asociality is favored again. The resulting evolutionarily stable equilibrium is polymorphic: the social and asocial strategies coexist. In contrast, when the social–asocial attachment probability is minimal, the invasion barrier x* is more difficult to reach compared with null a priori assortment. However, the fact that asocials are more efficiently segregated when the population is largely social, makes the fully social equilibrium even more stable with respect to larger attachment probabilities. Any other choice for πas such that πaa≤πas≤πss leads to thresholds x* between those two extremal values. Therefore, the potential for social behavior to become stable in the population is not challenged by the amount of a priori assortment generated by the attachment rules.

Figure 5.

Payoff differences between social and asocial individuals obtained by simulation of the group formation process, for different rules of attachment: inline image, corresponding to null within-group assortment (full line); πas= min (πss, πaa) =πaa, corresponding to positive prior assortment (dashed line); πas= max (πss, πaa) =πss, corresponding to negative prior assortment (dotted line). Parameters values: πss= 0.6, πaa= 0.2, b/c= 20, T= 100. For the third rule of interaction, the threshold x* to trigger the evolution of sociality decreases, but sociality does not invade fully and is only profitable up to a frequency xpoly where the population attains a polymorphic equilibrium.

Discussion

SOCIAL GROUPS FORMATION AND EVOLUTION

In this work, we address the coupling between the process of group formation and the evolutionary dynamics of individual “social” traits that affect both aggregation propensity and group cohesion, for example, cell-to-cell adhesiveness. In addition to entailing a different contribution to group welfare, such traits underpin a difference in expected group size distributions. This difference ultimately generates assortment at the population level even in the absence of preferentially directed interactions based on peer recognition. Rather, we evidence that, whenever the size of groups is not fixed, simple nonassortative rules can still generate average local environments that favor the evolution of sociality even when it is not associated to direct benefits. We have illustrated our claim via a toy model where groups form by blind interactions among individuals with different attachment abilities, stemming for instance from signaling or due to the production of a costly glue. In this deliberately simple setting, we showed that even when attachment rules are indiscriminate toward the strategies of partners (and groups of any size are randomly assorted), socials individuals fare better than asocials because of the distinct allocations of the two types in groups of various sizes and in particular to different chances of ending up alone. The emergent population structure gives rise to a Simpson’s paradox where one strategy’s advantage is reversed when one goes from the group to the population level. This has already been related to the evolution of cooperation when group size changes in time (Hauert et al. 2006a; Chuang et al. 2009). It is noteworthy that in our toy model, there is no intrinsic limitation to the size of the evolutionarily viable groups, contrary to most previous models of N-players games (e.g., Matessi and Jayakar 1976; Powers et al. 2011). This suggest that sociality in large groups, such as in microbial communities, can be sustained with unsophisticated mechanisms that do not require information transfer between partners.

AGGREGATIVE SOCIALITY IN MICROORGANISMS

In the microbial world, the formation of biofilms and their cohesion are reckoned to be beneficial to cells in many respects (Velicer 2003). In several microorganims, the same costly individual traits that support the stability of groups may enhance the probabilities for cells to be part of them in the first place. Velicer and Yu featured costly “stickiness” as an adaptive prerequisite in swarming microorganisms (Velicer and Yu 2003). In D. discoideum, the production of cell adhesion molecules required for the aggregation cycle is thought to reduce the chances to become a spore: more adhesive strains are primarily found in the dead tissues of the fruiting body (Ponte et al. 1998; Strassmann and Queller 2011). Myxobacteria form multicellular aggregates as well that enhance survival by decreasing predation and favoring dispersal (Shimkets 1986a,b). Both agglutination and social cohesion are mediated in these bacteria by the production of a costly extracellular matrix of fibrils, increasing at the same time cell adherence and enabling collective gliding (Velicer and Yu 2003). Mutations that affect a gene located at a single-locus impair fibril binding and result in both lower cell–cell adhesion and cohesion of aggregates (Shimkets 1986b). In Saccharomyces cerevisiae, an adhesion protein expressed by a social gene (FLO1) prompts individuals to form flocs that provide them with enhanced resistance to chemical stresses (Smukalla et al. 2008). When this strain is mixed with nonflocculating variants, heterogeneous aggregates still contain a majority of FLO1+ cells, whereas individuals outside groups are more often FLO1, thus denoting assortment emerging from mere different adhesive abilities.

Although the processes involved in group formation become more complex as the cognitive abilities of players increase, our general conclusions might be also of interest for higher organisms that interact via mechanisms parallel to physical adhesiveness. For instance, Dunbar interpreted grooming in monkeys as a behavior likely to provide higher grouping opportunities as well as cement social bonding once the group is formed (Dunbar 1993), and further extended the argument to humans, based on the presumed genetic foundations of language (Pinker and Bloom 1990). Even if we have focused here on an aggregation mechanism that is more promptly related to social microorganisms, our conclusions hold in general for any inheritable trait, not necessarily involving physical adhesion, that plays a role both in group formation and group cohesion.

NONNEPOTISTIC GREENBEARDS?

In our model, assortment is generated among carriers of the social gene alone, and not on the whole genome. Therefore, sociality here pertains to greenbeard mechanisms as termed by the recent classification of Gardner and West (West et al. 2007; Gardner and West 2010). In their review, the authors stressed that such genes need not code for conspicuous traits as was posited in the original formulations (Hamilton 1964; Dawkins 1976). We argue that assortment at a single locus does not require nepotistic behavior of the gene toward other carriers neither, at least not in the usual sense imposed by dyadic or fixed-N frameworks. Indeed, assortment may mechanistically occur even when social individuals interact with each type in the same proportions as asocial individuals, provided they do it more often. A blind increase in the propensity to interact can thus have the same effect as preferentially directed interactions with peer discrimination, that may be more demanding on the cognitive level. This might be of interest for the interpretation of social behavior in organisms where the existence of recognition mechanisms is not straightforward. More in general, it might be useful to disentangle more explicitly greenbeard mechanisms that rely on active sorting of interaction partners from passive, indiscriminate mechanisms generating assortment with weaker requirements (Eshel and Cavalli-Sforza 1982). Such differentiation would echo and complement that of “obligate” versus “facultative” greenbeards formulated in the case of dyadic interactions (Gardner and West 2010).

TOWARD A RE-EVALUATION OF THE GROUP FORMATION STEP

We have stressed that the process of group formation can play an essential role in the unfolding of the evolutionary dynamics of social traits. A complete account of the evolution of cooperative groups requires to trace back the entire process leading to their formation. The toy model used here is a useful tool for illustrating our conclusion in a simple and extreme setting. It is however missing many features of actual biological systems. One could instead wish to predict, based on individual properties of physical attachment, the group size dynamics and the degree and nature of assortment between the social and asocial types in a more realistic aggregation model. This requires to further specify the mechanism of group formation, and notably explicit the individual rules of interaction and the topology structuring individual encounters. For organisms moving on a plane, such as cells gliding on a surface, grouping patterns, and the resulting group size distributions have been mimicked by models based on simple rules (e.g., Okubo 1986; Vicsek et al. 1995; Bonabeau et al. 1999). Recently, social games have been implemented in explicit schemes of aggregation for self-propelled particles interacting locally with their neighbors (Chen et al. 2011). The way aggregative traits themselves can be sustained in a landscape shaped by a realistic group formation process however is still to be explored. Yet, the propensity to seek interactions, before that of behaving altruistically once the interaction is established, may be the very first, and a prerequisite, of all social actions.

Associate Editor: M. Doebeli

ACKNOWLEDGMENTS

TG and SDM are extremely grateful to M. van Baalen, J.-B. André, associate editor M. Doebeli, and two anonymous referees for their insightful comments and suggestions.

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