## Abstract

In the animal world, collective action to shelter, protect and nourish requires the cooperation of group members. Among humans, many situations require the cooperation of more than two individuals simultaneously. Most of the relevant literature has focused on an extreme case, the *N*-person Prisoner's Dilemma. Here we introduce a model in which a threshold less than the total group is required to produce benefits, with increasing participation leading to increasing productivity. This model constitutes a generalization of the two-person stag hunt game to an *N*-person game. Both finite and infinite population models are studied. In infinite populations this leads to a rich dynamics that admits multiple equilibria. Scenarios of defector dominance, pure coordination or coexistence may arise simultaneously. On the other hand, whenever one takes into account that populations are finite and when their size is of the same order of magnitude as the group size, the evolutionary dynamics is profoundly affected: it may ultimately invert the direction of natural selection, compared with the infinite population limit.

## 1. Introduction

During recent years, evolutionary game theory has been able to provide key insights into the emergence and sustainability of cooperation at different levels of organization (Axelrod & Hamilton 1981; Maynard-Smith 1982; Axelrod 1984; Boyd & Richerson 1985; Hofbauer & Sigmund 1998; Skyrms 2001, 2004; Macy & Flache 2002; Hammerstein 2003; Nowak & Sigmund 2004; Nowak *et al.* 2004; Santos & Pacheco 2005; Nowak 2006; Ohtsuki *et al.* 2006; Santos *et al.* 2006). The most popular and studied game has been the two-person Prisoner's Dilemma (PD). However, other social dilemmas, such as the snowdrift game (Sugden 1986) or the stag hunt (SH) (Skyrms 2004) game also constitute powerful metaphors for many situations routinely encountered in the natural and social sciences (Macy & Flache 2002; Skyrms 2004).

In particular, the SH game constitutes the prototypical example of the social contract, and one can identify instances of SH games in the writings of, for example, Rousseau, Hobbes and Hume (Skyrms 2004). Maynard-Smith & Szathmáry (1995) have discussed the social contracts implicit in some of the major transitions of evolution. After framing most of the discussion in terms of the PD, they remarked that perhaps the SH (their rowing game) is a better model. In a SH there is an equilibrium in which both players cooperate as well as one in which both defect.

Whenever collective action of groups of individuals is at stake, *N*-person games are appropriate. Recent literature has focused on *N*-person Prisoner's Dilemmas (NPDs) in the form of provision of Public Goods games (PGG; Kollock 1998; Hauert *et al.* 2002, 2006, 2007; Brandt *et al.* 2006; Milinski *et al.* 2006, 2008; Rockenbach & Milinski 2006; Santos *et al.* 2008). The prototypical example of a PGG is captured by the so-called NPD. It involves a group of *N* individuals, who can be either cooperators (C) or defectors (D). Cs contribute a cost *c* to the public good, whereas Ds refuse to do so. After all individuals are given the chance to contribute, the accumulated contribution is multiplied by an enhancement factor *F*, and the total amount is equally shared among all individuals of the group. In other words, if there were *k* Cs in a group of *N* individuals, Ds end up with *kFc/N*, whereas Cs only get *kFc/N*−*c*, i.e. in mixed groups Cs are always worse off than Ds. If *F* is smaller than *N*, to cooperate is always disadvantageous against any combination of actions by other group members. In this sense, we have an NPD. Evolutionary game theory directly leads to the tragic outcome in which everybody ends up defecting, hence foregoing the public good. When the group is a mere pair of individuals, this dilemma reduces to the two-person PD.

Consider, however, group hunts of three or four lionesses in Etosha National Park, Namibia (Stander 1992). Two lionesses, the *wings*, attack a group of prey from either side panicking them to run forward. They run right into one or two other lionesses, positioned as *centres*, who are waiting for them. This kind of hunt is highly successful. It is not possible with one or two participants, but it is with three and is even better with four. This is not a generalized PD, but a generalized SH. It is a SH because, unlike the PD, there is a cooperative equilibrium where if others do their part, it is best for you to do yours as well.

Variations on this kind of cooperative hunting have been observed in other species, such as chimpanzees in the Tai forest (Boesch 2002) and African wild dogs (Creel & Creel 1995). In animals, other collective actions, such as lions defending a kill against a pack of hyenas, can also be seen as generalized SH games (Maynard-Smith & Szathmáry 1995).

In human affairs, we also find collective action problems that can be viewed as generalized SHs, not only in literal hunts such as the whale hunts discussed in Beding (2008), but also in international relations (Jervis 1978) and macroeconomics (Bryant 1994).

Back to the lionesses in Etosha National Park, two individuals are not enough for a cooperative hunt, three can be successful and four even more so. The average pay-off of an individual depends on the number of participants and may vary according to species and environment. Much empirical evidence supports a U-shaped function for average meat per participant across a number of species, but it is controversial whether this remains true when energetic costs of the hunt are taken into account (Creel & Creel 1995; Packer & Caro 1997).

Here we focus on games where there is a threshold (*M*) for participants below which no public good is produced. We do not make the general assumption that total participation gives each individual the highest pay-off. For instance, we include the possibility of ‘three in a boat, two must row’ (Taylor & Ward 1982; Ward 1990), a generalization of the SH game to three players, where contributions of two out of three players are required for the success of the joint venture. If two others row, there is an incentive to free ride; but if one other rows, there is an incentive to jump in and contribute. There may be an analogue in cooperative hunting by lions in richer environments where prides are larger and the participation of the entire group is not so helpful.

We shall start by investigating the evolutionary dynamics of Cs and Ds in the traditional setting of evolutionary game theory, i.e. infinite well-mixed populations evolving. The fitness of individuals is determined by their pay-off collected when engaging in an *N*-person stag hunt (NSH) dilemma requiring at least *M*<*N* individuals to produce any public good at all. We shall find that the NSH game leads to richer and more interesting evolutionary dynamics scenarios than the corresponding NPD. Subsequently, we investigate the implications of taking into account the fact that populations are finite. Evolutionary dynamics for large finite populations was pioneered in economics by Young (1993) and by Kandori *et al.* (1993). The focus here is on the limiting effect of mutation as it becomes infrequent. Owing to mutation evolutionary dynamics becomes an ergodic Markov chain (Nowak *et al.* 2004). In the classic SH, it is shown that the population spends almost all its time at the non-cooperative equilibrium.

Evolutionary dynamics of a growing (or shrinking) finite population with random deaths is modelled in Schreiber (2001) and by Benaim *et al.* (2004). Either a strategy or the whole population can wander into extinction, but if this does not happen the trajectory of the growing population comes to approximate that of the replicator dynamics.

We shall focus on a (possibly small) well-mixed population of fixed size *Z* without mutation. The dynamics will be a Markov process, with the only possible end states—the absorbing states—being monomorphisms. When the population is large the dynamics approximates the replicator dynamics in the medium run, but it will eventually end up in one of the absorbing states. Thus, it may spend a long time near a stable polymorphic equilibrium of the associated mean-field dynamics before eventually being absorbed by a monomorphism. For small populations where population size is close to group size, there is also the ‘spite’ effect first noted by Hamilton (1970), which works against cooperation.

## 2. Results

### (a) Evolutionary dynamics in infinite populations

Let us assume an infinite, well-mixed population, a fraction *x* which is composed of Cs, the remaining fraction (1−*x*) being Ds, and let us further assume that the groups of *N* individuals are sampled randomly from the population. As shown in appendix A, random sampling of individuals leads to groups whose composition follows a binomial distribution (Hauert *et al.* 2006), which also establishes the average fitness of Cs (*f*_{C}) and Ds (*f*_{D}). In each *N*-individual group with *k* Cs, the fitness of Ds is given by , where the Heaviside step function (*θ*(*x*) satisfies *θ*(*x*<0)=0 and *θ*(*x*≥0)=1. The corresponding fitness of Cs is given by .

The time evolution of the fraction of cooperators *x* in the population is given by the replicator equation,It is straightforward to show that, for the NPD (*M*=0), the right-hand side of the replicator equation will be positive (and hence, the fraction of cooperators will steadily increase) whenever *F*>*N*, since (appendix A). On the other hand, whenever *F*<*N*, for , and cooperators have no evolutionary chance.

Let us now consider the NSH, where 1<*M*≤*N*. Let us assume that the return from the public good increases linearly with the number *k* of Cs, inasmuch as *k*≥*M.* In view of the previous definitions, whenever *k*<*M* no public good is produced, and hence Ds have a pay-off of zero whereas Cs have a pay-off of −*c*. The evolutionary dynamics of Cs and Ds in the NSH game with a minimum threshold *M* can again be studied by analysing the sign of *f*_{C}−*f*_{D}. We may write (see the electronic supplementary material)where the polynomial *R(x)* and its properties have been defined in appendix A, whereas the details are provided in the electronic supplementary material. In a nutshell, the properties of *Q*(*x*) lead to very interesting dynamics of the replicator equation, with possibly two interior fixed points (*x*_{L} and *x*_{R}, with *x*_{L}≤*x*_{R}), as illustrated in figure 1, for *N*=20, different values of 1<*M*≤20 and variable *F*. Note, in particular, the fact that and (electronic supplementary material) allows us to classify immediately *x*_{L} as an unstable fixed point whereas *x*_{R}, if it exists, corresponds to a stable fixed point, as also illustrated in figure 1. Moreover, when (*F*/*N*)=*R*(*M*/*N*), *M/N* is the unique interior and unstable fixed point.

Between these two limiting values of *F*, and given the nature of the interior fixed points *x*_{L} and *x*_{R}, one can easily conclude that below *x*_{L} all individuals will ultimately forego the public good. Conversely, for all *x*>*x*_{L}, the population will evolve towards a mixed equilibrium defined by *x*_{R}, corresponding to a stable fixed point of the associated replicator equation (even if, initially, *x*>*x*_{R}). ‘Three in a boat’ provides the simplest possible case of this scenario. Similar to the NPD, whenever *F*/*N*<*R*(*M*/*N*), for all *x*, which means that all individuals will end up foregoing the public good.

### (b) Evolutionary dynamics in finite populations

Let us focus on a well-mixed population of size *Z* in the absence of mutations. Sampling of individuals is no longer binomial, following a hypergeometric distribution (see appendix B). The fraction of cooperators is no longer a continuous variable, varying in steps of 1/*Z*. We adopt a stochastic birth–death process (Karlin & Taylor 1975) combined with the pairwise comparison rule (Traulsen *et al.* 2006, 2007*a*,*b*) in order to describe the evolutionary dynamics of Cs (and Ds) in a finite population. Under pairwise comparison, two individuals from the population, A and B are randomly selected for update (only the selection of mixed pairs can change the composition of the population). The strategy of A will replace that of B with a probability given by the Fermi function (from statistical physics),The reverse will happen with probability 1−*p*. The quantity *β*, which in physics corresponds to an inverse temperature, controls the intensity of selection: for *β*≪1, selection is weak, and one recovers the replicator equation in the limit *Z*→∞ (Traulsen *et al.* 2006, 2007*a*,*b*). The pairwise comparison rule is similar to the so-called logit rule (Sandholm in press), according to which an individual A is selected with a probability proportional to ; here the noise parameter *η* plays the role of the temperature above; in fact, both processes share the same fixation probabilities, despite leading to different evolutionary dynamics equations.

For arbitrary *β*, the quantity corresponding to the right-hand side of the replicator equation, specifying the ‘gradient of selection’, is given in finite populations by (Traulsen *et al.* 2006, 2007*a*,*b*)(2.1)The right-hand side of *g*(*k*) is similar to the replicator equation, only that the pairwise comparison leads to the appearance of the hyperbolic tangent of the fitness difference, instead of the fitness difference. This has implications in the characteristic evolutionary times, which now depend on *β* (Traulsen *et al.* 2006, 2007*a*,*b*), but not in what concerns the roots of *g*(*k*). Importantly, the evolutionary dynamics in finite populations will only stop whenever the population reaches a monomorphic state (*k/Z=*0 or *k/Z=*1). Hence, the sign of *g*(*k*), which indicates the direction of selection, is important in that it may strongly influence the evolutionary time required to reach any of the absorbing states.

Whenever *M*=0 (NPD) we may write (see appendix B)(2.2)*which is independent of k* being, however, *population and group size dependent*. This means *frequency independent* selection. In particular, whenever the size of the group equals the population size, *N*=*Z*, we have and cooperators have no chance irrespective of the value of the enhancement factor. This contrasts with the result in infinite, well-mixed populations (*Z*→∞), where to play C would be the best option whenever *F*>*N*. For finite populations, the possibility that group size equals population size leads to the demise of cooperation.

Given the independence of *f*_{C}−*f*_{D} on *k* in finite populations, for a given population size there is a critical value of *F* for which selection is neutral, and above which cooperators will win the evolutionary race. From the two equations above, this critical value readsIn figure 2, we show the *Z*-dependence of *g*(*k*) for fixed group size *N*=10 and fixed *F*=12 leading to a critical population size *Z*=55.

Let us now discuss the NSH with 1<*M*<*N*≤*Z*. Whenever *N*=*Z*, the result is easily inferred from the NPD above—all individuals in the population will ultimately forego the public good. This will happen, in finite populations, irrespective of the existence (or not) of a threshold *M*. Whenever *N*<*Z* the threshold brings about a strong disruption of the finite population dynamics, which we illustrate numerically, given the unappealing look of the analytical equations (see appendix B).

Let us start with the case in which *F*>*N*, that is, the regime for which we obtain a pure coordination game with a single (unstable) fixed point in the replicator dynamics equation (cf. figure 1). The possible scenarios are depicted in figure 3*a*.

Clearly, for small population sizes, cooperators are always disadvantageous. With increasing *Z*, however, one approaches the replicator dynamics scenario (coordination game), despite the fact that, e.g. for *Z*=20, convergence towards the absorbing state at 100 per cent Cs is hindered because Cs become disadvantageous for large *k*. Indeed, for this population size, Cs are advantageous only in a small neighbourhood of *k*/*Z*=0.5, being disadvantageous both for smaller and larger values of *k*/*Z*. In other words, and despite the fact that evolution will stop only at *k*=0 or *k*=*Z*, the time it takes to reach an absorbing state will depend sensitively on the population size, given the occurrence (or not) of interior roots of *g*(*k*).

Whenever *F*<*N*, yet above the critical limit below which Cs become disadvantageous for all *x* in figure 1, we observe that for small population sizes Cs are always disadvantageous, and the two interior fixed points of the replicator dynamics equation only manifest themselves above a critical population size *Z*_{CRIT}, as illustrated in figure 3*b*.

## 3. Discussion

In this paper, we extend the range of PGG to systems where a minimum of coordinated collective action is required to achieve a public good. By doing so, we generalized the two-person SH game to *N*-person games. In infinite, well-mixed populations, the existence of a threshold opens the possibility for the appearance of two interior fixed points in the replicator equation. The one at the lower frequency of cooperators is always an unstable fixed point, which determines a threshold for cooperative collective action. The other, at a higher frequency of cooperators, is a stable fixed point, and hence determines the final frequency of cooperators in the population, assuming that the coordination threshold is overcome. Besides this most interesting regime, there are also the possible outcomes of no cooperation or of a pure coordination game with a threshold that depends sensitively on the minimum number of cooperators *M* in a group of *N* individuals required to produce any public good.

Once the simplifying assumption of an infinite population size is abandoned, the evolutionary dynamics of the NSH game is profoundly affected, mostly when the population size is comparable with the group size (see table 1 for a summary). In this regime, one observes an overlap of the different scenarios obtained in infinite populations. Hence, for *Z=N*, cooperators are always disadvantageous, irrespective of the existence or not of a threshold. For *Z*>*N*, the direction of selection in a finite population is strongly size dependent. For fixed *F*>*N*, there is a critical value of *Z*_{1}, above which the interior roots of *g*(*k*) emerge, which constitute the finite-population analogues of *x*_{L} and *x*_{R} in infinite populations (cf. figure 1). Above a second critical value *Z*_{2}, *x*_{R} disappears, and one ends up with a coordination game. For *M*<*F*<*N* and a small population size, i.e. *F*<*N* but yet above the critical value defined in appendix A and the electronic supplementary material, cooperators are always disadvantageous; however, above a critical population size (*Z*_{C}) the interior roots of *g*(*k*) emerge simultaneously and the evolutionary dynamics approach that observed in infinite populations. Finally, for *F*<*M* cooperators have no chance irrespective of the population size. Such strong size dependence, with an impact that is stronger for smaller population sizes, can be directly traced back to the fact that, for smaller populations, the hypergeometric sampling of individuals into groups significantly deviates from binomial sampling. This, in turn, reflects the intuition that, in small populations, choices are reduced, and this must influence the overall evolutionary dynamics.

In this work we have always assumed that the benefit returned by the PGG scales linearly with the amount contributed. This need not be the case, and it is possible to find examples in which a nonlinear return paradigm would be more appropriate. Hence, it will prove interesting to understand in which way deviations from a linear return will affect evolutionary game dynamics, mostly in finite populations. Work along these lines is in progress.

## Acknowledgments

This work was supported by FCT Portugal (J.M.P.), FNRS Belgium (F.C.S.) and FAPERJ Brazil (M.O.S.).

## Footnotes

- Received June 6, 2008.
- Accepted September 2, 2008.

- © 2008 The Royal Society