Behavioral Ecology Advance Access originally published online on April 28, 2006
Behavioral Ecology 2006 17(4):633-641; doi:10.1093/beheco/ark009
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Asynchronous snowdrift game with synergistic effect as a model of cooperation
a Department of Plant Taxonomy and Ecology, Eötvös University, Pázmány Péter sétány 1/C, H-1117 Budapest, Hungary, b Collegium Budapest, Institute for Advanced Study, Szentháromság u. 2, H-1014 Budapest, Hungary, and c Department of Plant Taxonomy and Ecology, Research Group of Ecology and Theoretical Biology, Eötvös University, Hungarian Academy of Science, Pázmány Péter sétány 1/C, H-1117 Budapest, Hungary
Address correspondence to Á. Kun. E-mail: kunadam{at}ludens.elte.hu.
Received 2 September 2005; revised 28 March 2006; accepted 3 April 2006.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONAS GAME WITH SYNERGISTIC...AS GAME IN INFINITE,...AS GAME IN FINITE...RESULTSDISCUSSIONAPPENDIXREFERENCES |
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The snowdrift (or chicken) game emerges as a new paradigm in the study of nonkin cooperation in animals. Many situations, for example, cooperative hunting, group foraging, territorial defense, predator watching, or parental care, can be adequately described as a snowdrift game. In this paper, we investigate the asynchronous version of the game in which, contrary to the rather unrealistic assumption of simultaneous moves, one of the players acts first and the other responds by knowing its decision. Players are assigned to be first or second movers randomly and with the same probability. We found that both a synergistic effect of cooperation (i.e., cooperative effort is better than the sum of the individual efforts) and population structure (low dispersal, spatial confinement, or group formation) are crucial for mutual cooperation to emerge. Otherwise, only one of the players will carry the burden of cooperation.
Key words: chicken game, cooperation, game theory, snowdrift game.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONAS GAME WITH SYNERGISTIC...AS GAME IN INFINITE,...AS GAME IN FINITE...RESULTSDISCUSSIONAPPENDIXREFERENCES |
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The evolutionary stability of cooperative behavior is a notoriously hard problem of theoretical biology. Cooperation increases the fitness of cooperating participants, and it seems clear at first sight that cooperative behavior will spread and go to fixation in the population. However, a defecting (or cheating) individual who does not invest in the cooperation gets the benefits from its cooperative partner and has an even higher fitness, whereas the deluded cooperator receives a lower fitness value. Consequently, defectors will spread among cooperators, but cooperators cannot spread among defectors if the fitness of the deluded cooperator is low enough. This situation is described by the classical prisoner's dilemma (PD) game. There is a cooperator (C) and a defector (D) strategy in this game, and the payoff matrix of the game is depicted in Figure 1.
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The game follows the PD if T > R > P > S (see Figure 1). This series of relations guarantees that a defector has a higher fitness than a cooperator (T > R), cooperation is more profitable than defection (R > P), and to be cheated is very costly (P > S). It is clear then that the only evolutionary stable strategy (ESS) is defection in the PD game, and defectors can invade and destroy cooperation in a cooperative population (Trivers 1971
; Axelrod and Hamilton 1981). Whereas the average fitness would be R in a cooperative population, it is only P (P < R) in the evolutionary stable state. However, in contrast to the suggestion of the PD game, intra- and interspecific cooperation is widespread in nature (Brembs 1996; Dugatkin 1997, 2002; Nunn and Lewis 2001). So, elucidation of the mechanisms maintaining the evolutionary stability of cooperation among nonrelated individuals is an important task in evolutionary biology.
Based on different versions of the PD game, there are numerous solutions to the problem (see reviews in Dugatkin 2002; Sigmund and Hauert 2002; Doebeli and Hauert 2005). For example, cooperation can be stable against the invasion of defecting strategies if the partners interact with each other repeatedly, that is, they play an iterated PD game (Trivers 1971), and cooperation can emerge from a population that contains many different types of strategies initially (Nowak and Sigmund 1992, 1993). Depending on the details of the models, the successful strategies are either those that reward previous cooperation and punish defection in a repeated series of games (Tit for Tat, Generous Tit for Tat) or those that change their previously unsuccessful strategy (Pavlov). It has been shown recently that the invasion and fixation of the Tit for Tat strategy can be very probable in small populations of defecting individuals (Nowak et al. 2004).
Furthermore, cooperators can stably coexist with defectors in populations where mixing of the individuals is weak and partners interact locally (Nowak and May 1992; van Baalen and Rand 1998). Partners can change their level of investment into cooperation depending on their success in the continuous PD game. It has been shown that cooperation typically emerges and goes to fixation in these models, either due to the spatial population structure (Killingback et al. 1999) or because the game is played in an iterated manner (Roberts and Sherratt 1998; Killingback and Doebeli 2002). Models based on the PD game have caused a great intellectual challenge leading to numerous interesting solutions for the problem of evolution of cooperation. However, the PD game is not the only possible model for this problem.
Recently, Hauert and Doebeli (2004) have suggested the so-called "snowdrift" game as an alternative conceptual model of cooperation. The illustrative story for the game is the following: The car of 2 persons sticks in a snowdrift. They can either go out and start to shovel the car out from the snowdrift (cooperation) or remain in the car (defection). The benefit of shoveling is freeing the car from the snow and getting to the warm home, but there is a cost of the work. If the benefit is b and the cost of freeing the car is c, then if both drivers cooperate, the reward is R = b – c/2 because they share the cost. If only one of them cooperates, he gets S = b – c, whereas the defecting partner receives T = b payoff value. They remain in the snowdrift if both defect, that is, P = 0. Assuming that b > c, we receive a game similar to the PD game, except that S > P. This means that to be cheated by the defector is less detrimental than to be a partner in mutual defection in the snowdrift game. Because estimating the payoff values in real biological situations is generally difficult (Milinski 1987; Dugatkin 1991), the S > P relation is at least as probable as the reverse in the PD game (Hauert and Doebeli 2004). Indeed, the PD as a metaphor for cooperation is so widespread in the literature that people are tempted to label any cooperative interaction as resembling the PD game, even if they only show that T > R > P and T > R > S without revealing anything about the relations between P and S. For example, Greig and Travisano (2004) suggest that yeast cells play the PD on a sucrose plate. Here, a fraction of the population is able to secrete an invertase enzyme that transforms sucrose to glucose, which can be stolen by neighboring cells that do not produce the enzyme due to the lack of the gene. In this case, it is evident that without the production of the invertase enzyme none of the cells can grow because they are unable to digest the only available resource. Thus, S > P and the yeast cells play a snowdrift game instead of a PD one.
In the snowdrift game, no pure strategy can be ESS because T > R > S > P, but at equilibrium, cooperators and defectors form a stable polymorphism in a well-mixed, infinitely large population (Maynard Smith 1982; Hauert and Doebeli 2004). Cooperative strategy spreads but cannot go to fixation in an infinite, well-mixed population playing the snowdrift game. In simulations of selection in stochastic cellular automaton models of finite spatially structured populations, cooperation goes to fixation at low, and defection wins at high, c/b ratio (Hauert and Doebeli 2004). If the cost to benefit ratio is intermediate, then coexistence of C and D strategies was observed (Hauert and Doebeli 2004). Consequently, at lower c/b ratio, cooperation is the probable end point of evolution in finite populations; however, this is not an ESS. Defectors can invade the population of cooperators because T > R. A typical evolutionary outcome of continuous snowdrift games (where investments can change continuously) is that cooperating individuals making a large investment coexist with defectors investing low (Doebeli et al. 2004). In the continuous snowdrift game, the evolution of investment generally leads to a cooperator and a defector strategy, which is assumed per se in the simple snowdrift game.
The assumption of synchronous games, which is common in the cited works (and in most mathematical models in this field), is usually not appropriate in biology. Imagine, for example, 2 predators hunting for a prey together. They lie in wait for the prey. When the prey is close enough to the predators, one of them starts to chase it (cooperate) and the other either follows it immediately (cooperates) or remains behind (defects). Independent of the hunting strategy, they take equal shares of the prey (which is indeed the case for lions [Packer and Pusey 1997] or chimpanzees [Boesch 1994]). If the success of hunting does not depend on the number of active hunters, this situation can be described by the asynchronous snowdrift (AS) game, where the choice of one of the players is known to the other player. Synchronous games assume that during the game there is no information exchange between the players, which is rarely the case (Dugatkin et al. 1992). Animals can monitor the activities of each other and thus gain information on the behavior of the others, which can influence their behavior.
Moreover, following the above example, 2 hunters are generally more effective than a single one, so if both cooperate, their success is higher than in the case of a single cooperator. Consequently there is a synergistic effect of cooperation. In case of hunting, this is termed cooperative hunting and has been documented, among others, for chimpanzees (Boesch 1994) and African wild dogs (Creel S and Creel NM 1995; Creel 1997) (see more examples in Discussion).
In this paper, we study the evolution of cooperation in the AS game with synergistic effect, in infinitely large, well-mixed populations, and in spatially structured finite populations. The lattice representation in our model can be interpreted in a number of ways. The most straightforward interpretation is that the individuals are constrained in space or their mobility is very low; thus, they interact mostly with the same individuals repeatedly. It can also be interpreted as a nonchanging social network, which allows our study to describe also cases of intragroup cooperation.
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AS GAME WITH SYNERGISTIC EFFECT |
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TOPABSTRACTINTRODUCTIONAS GAME WITH SYNERGISTIC...AS GAME IN INFINITE,...AS GAME IN FINITE...RESULTSDISCUSSIONAPPENDIXREFERENCES |
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In the asynchronous game, a player can adopt different strategies depending on whether it acts first or second. Thus, the overall strategy of an individual is described by a strategy pair, representing the strategies played in the 2 possible roles (first or second). There are 4 possible overall strategies: {C, C}, {C, D}, {D, C}, and {D, D} where the first letter denotes the strategy (cooperation or defection) adopted by an individual, if the individual finds itself in the role of the first-acting player, whereas the second letter marks the strategy adopted, if the given individual acts second.
We describe the snowdrift game with a synergistic effect with the help of 2 parameters: the cost of cooperation in relation to the benefit from cooperation (
) and the level of the synergistic effect (
). More specifically, = c/b is the level of the cost of cooperation in the units of the benefit, and = 
/b is the level of the synergistic effect in the units of the benefit, where is the synergistic benefit present if and only if both individuals cooperate. Naturally, 0 < < 1 and > 0. We note that if > 1, we would arrive at the PD game.
The payoff matrix of the AS game where the cooperation has a synergistic effect is shown in Figure 2.
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AS GAME IN INFINITE, WELL-MIXED POPULATIONS |
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TOPABSTRACTINTRODUCTIONAS GAME WITH SYNERGISTIC...AS GAME IN INFINITE,...AS GAME IN FINITE...RESULTSDISCUSSIONAPPENDIXREFERENCES |
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AS game is described as a simple asymmetric matrix game, so the ESS pairs can be found easily (Hammerstein 1981
). If > /2, then the only ESS is the {C, C} strategy pair. Naturally, C is the only ESS in the synchronous snowdrift game in this situation as well. This situation corresponds to the by-product mutualism.
Let us assume now that < /2. Then, {C, D} and {D, C} are the 2 ESS pairs in the AS game. It can be shown by employing replicator dynamics on this game that {C, D} and {D, C} are the only 2 stable outcomes (see Appendix). Of these 2 strategies, the initially more abundant one will dominate in the population (see Appendix).
Assuming that an individual can be the first or the second player with the same probability, individuals get 1 – /2 payoff value on average in the ESS. However, mutually cooperating ({C, C}) individuals would get an even higher payoff, that is, 1 – /2 + ! The situation is similar to the PD game, where the average payoff in the ESS is smaller than it would be in the cooperative state (P < R).
If we consider the synchronous synergistic snowdrift game with < /2 and use a replicator dynamics, the equilibrium frequency of cooperators is (1 – )/(1 – /2–) (Maynard Smith 1982; Hofbauer and Sigmund 1998). When the cost of cooperation increases relative to the benefit ( 
1), the frequency of cooperators decreases to zero and will be close to one if cooperation is very beneficial relative to the cost ( << 1). However, in the AS game, individuals behave cooperatively in 50% of the cases, independent of the cost–benefit ratio. This means that the frequency of the cooperative acts is lower in the synchronous snowdrift game if < 3/2 – 1 and > 2/3 and higher otherwise.
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AS GAME IN FINITE POPULATIONS |
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TOPABSTRACTINTRODUCTIONAS GAME WITH SYNERGISTIC...AS GAME IN INFINITE,...AS GAME IN FINITE...RESULTSDISCUSSIONAPPENDIXREFERENCES |
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In the AS game with finite populations, we considered 2 cases: 1) the dispersal rate of the individuals was low or 2) it was high. High dispersal rate resulted in a well-mixed population, in which the interaction partners changed constantly. On the other hand, individuals interacted repeatedly with the same individual for a long time in the low-dispersal case. The finite population was modeled as a population on a finite lattice, where each individual interacted only with a finite set of other individuals.
The model is similar to the spatial, asynchronously updated stochastic game employed by Nowak et al. (1994a, 1994b), except that in our case there was an asymmetry regarding the player that acted first. Note that asynchrony or synchrony in this case refers to the method of updating the cellular automaton, not to the possibility or lack of information shared among players, which is one of the key issues of this study.
Individual players were confined to sites on regular 100 x 100 lattices with periodic boundary condition and interacted with their 4 orthogonal neighbors.
An elementary step implies updating the strategy of a randomly chosen individual. The fitness of the focal individual is calculated as the average of payoffs collected against each of its neighbors (naturally, the fitness of its neighbors are computed in a similar way). In each interaction, one of the players is randomly selected to be the first-acting player, and the other player assumes the role of the second-acting player. There are alternative deterministic and stochastic updating rules in the literature (see, e.g., Hauert and Doebeli 2004). Because deterministic rules have less biological relevance than stochastic ones, they are not considered here.
The generally used stochastic update rule is the "proportional rule" (Nowak et al. 1994b; Hauert 2002; Hauert and Doebeli 2004). The probability of the focal player to adopt the strategy i is 
where i means the focal individual's or its neighbors' strategy and wi is the average fitness of player i. 
is the sum of the average payoffs of the neighbors and the focal individual.
This update rule mimics the following biological situation: the number of the offspring (or their survival probability) is proportional to the success of a local competition. Offspring are dispersed evenly among the neighboring sites and the focal site to create a local "offspring pool." After the focal individual dies, one offspring is selected randomly from the offspring pool to become the new adult individual.
Some recent papers propose an alternative update rule (Hauert 2002, 2005; Hauert and Doebeli 2004; Traulsen et al. 2005). Here, whenever a randomly chosen site i is updated, the fitness of the individual living in this focal site is compared with the fitness of its one randomly selected neighbor j. The site i switches to the neighbor's strategy with probability Pi
j = (1 – 
) + [(wj – wi)/
Pmax], where Pmax is the maximum possible "payoff difference" (Pmax = 1 in our case) and 0 < 
1 is the strength of the selection. The rationality of this payoff difference update rule is that it leads to the replicator dynamics in the well-mixed infinite population size limit (Traulsen et al. 2005). The weakness of this update rule is that it has no so clear biological interpretation like the proportional update rule has. On the other hand, the proportional update generates an adjusted replicator dynamics in the infinite, well-mixed population size limit, whose dynamics has the same fixed points as the replicator dynamics (Maynard Smith 1982; Traulsen et al. 2005). The 2 dynamical equations behave in a qualitatively different manner only if the game they describe is asymmetrical. However, in this case, the adjusted replicator dynamics is the correct description (Maynard Smith 1982). Consequently, we focus on results obtained by using the proportional update rule, but results are mentioned for the payoff difference rule too.
In order to compare our results with the usually assumed synchronous game, we have made simulations without asynchrony (i.e., individuals acted simultaneously), but otherwise, the method was the same. This also allows us to compare the results from simulations where = 0 to the results of Hauert and Doebeli (2004), which can serve as a test for the proposed method.
Simulations proceeded either until only one of the strategies remained in the lattice or until the 5 x 109th update, whichever occurred first. In order to map the parameter space, we used different values and varied in order to know at which value (with 0.01 precision) the outcome changes. These values were recorded and plotted against , and a line was fitted to these points (Figures 3 and 4).
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RESULTS |
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TOPABSTRACTINTRODUCTIONAS GAME WITH SYNERGISTIC...AS GAME IN INFINITE,...AS GAME IN FINITE...RESULTSDISCUSSIONAPPENDIXREFERENCES |
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The simulations indicate that {C, C} strategies win the competition in the structured population (i.e., when dispersal is low) if the synergistic effect (
) is strong enough compared with the cost to benefit ratio of cooperation (), namely, when > 0.353 (r2 = 0.996) (Figure 3). This relation can be explained in a relatively simple way.
The cooperative {C, C} strategy cannot be an ESS in a strict sense because a single {D, D}, {C, D}, or {D, C} strategy has a higher fitness than their neighboring {C, C} strategies, independent of the number of neighbors. The same holds true for 2 and 3 neighboring individuals with noncooperative strategy as well. However, if the synergistic effect is high enough, then individuals with {C, D} (or {D, C}) strategy in blocks of 4 have a lower fitness than one of their neighboring {C, C} strategies. For example, if there are 4 neighbors, then the average fitness of {C, D} in the block is [2(1–/2) + 2(1–/4 + /2)]/4), whereas the average fitness of the adjacent {C, C} strategies is [1 – 3/4 + /2 + 3(1 – /2 + )]/4, assuming that individuals have done the first and the second decision with the same frequency. Comparison of these values shows that the {C, C} strategy has a higher fitness than {C, D} in the rectangular block if 3/8 < . (For the 8-neighbor model, 4 players with {C, D} strategy, which live in a rectangular block, are annihilated if 3/10 < .) Thus, all the {C, D} players in the rectangular block can be eliminated in a single generation, and this will happen sooner or later due to the stochasticity and finiteness of the system. Similarly, the fitness of {D, D} individuals living in a rectangular block will be smaller than that of the neighbors playing {C, C}, if 5/6 < + 2/3. This relation is valid if 3/8 < ; thus, the 4-member {D, D} blocks can be eliminated at a lower synergistic effect than the {C, D} blocks. (The situation is the same in the 8-neighbor model.) Snapshots from the simulations (Figure 5) indicate that the {C, C}, {C, D}, and {D, C} strategies form clusters and the {D, D} strategy disappears from the population quickly.
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Consequently, if the synergistic effect exceeds a critical level, the {C, C} strategy will be the winner of the competition in spatially structured finite populations. This result is robust with respect to whether we employ synchronous or asynchronous updating. On the other hand, with the payoff difference update rule we found no parameter combination resulting in {C, C} being the only surviving strategy. However, there was a very small region of the parameter space where {C, C} and {D, D} coexisted, and {C, C} had a higher density than {D, D}. This indicated that the frequency of cooperative acts was higher than in the region where {C, C} or {C, D} won the competition.
However, if the mobility of the individuals is high, then {C, D} or {D, C} is the winning strategy irrespective of the magnitude of synergy because the mixing of individuals prohibits the emergence of blocks of players. This result is the same as that derived for the infinite, well-mixed population. Our simulations of the finite, well-mixed population can be considered as numerical approximation to the replicator dynamics, thus giving further credence to the derived results.
In the synchronous snowdrift game with synergy, there are 3 discernible regions in the parameter space (Figure 4): full defection, coexistence of cooperators and defectors, and full cooperation. Full defection is the ESS when the cost of cooperation is very high (close to = 1) and the synergistic effect is very low. The = 0.77 – 0.59 (r2 = 0.96) line separates this region from another in which cooperators and defectors coexist. In this parameter range, an intermediate frequency of cooperative acts can be observed. The frequency of cooperative acts increases with the simultaneous decrease in the cost of cooperation (lower ) and increase in the synergistic effect (higher ). The = 0.64 – 0.25 (r2 = 0.999) line separates this region from that region where full cooperation is the winning strategy. For = 0, we have got the same result as Hauert and Doebeli (2004, see their Figure 4 in the supplementary information).
In the symmetric case, simulations with the payoff difference rule yield the qualitatively same result, that is, there are 3 distinct regions: 1) defectors win, 2) coexistence of cooperators and defectors, and 3) cooperators win. Without noise ( = 0), the lines separating the regions are = 1.97 – 1.55 (r2 = 0.996) and = 0.76 – 0.23 (r2 = 0.996). With moderate noise ( = 0.5), the lines separating the regions are = 1.68 – 1.17 (r2 = 0.998) and = 1.33 – 0.78 (r2 = 0.996). In both cases, the regions (1) and (3) are larger than those observed using the proportional update rule.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONAS GAME WITH SYNERGISTIC...AS GAME IN INFINITE,...AS GAME IN FINITE...RESULTSDISCUSSIONAPPENDIXREFERENCES |
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In our study of the evolution and maintenance of cooperation in a snowdrift game, we consider the effect of 3 key features: 1) asynchrony in the moves of the players, 2) synergistic effect of cooperation, and 3) structured population. We found that if the synergistic effect is high enough (
> /2) then cheating is not favored and cooperative behavior is an ESS and can readily evolve. In this case, the payoff matrix conforms to the so-called assurance game or by-product mutualism.
On the other hand, if the synergistic effect of cooperation is not so strong, then in the AS game, cooperative behavior ({C, C}) can be stabilized in a structured population. Otherwise, the strategy {C, D} or {D, C} is an ESS even in a well-mixed population. Here, the degree of cooperation is 50% independent of the value of . This can be regarded as reciprocal altruism.
Asynchrony has been studied in the context of the alternating PD (Frean 1994; Nowak and Sigmund 1994), where players act in a strictly alternating manner (but see Nowak and Sigmund 1994) and they remember the previous move or moves (e.g., Hauert and Schuster 1998; Neill 2001) of the other player. They found that the average level of cooperation can be high in such a system (Frean 1994; Nowak and Sigmund 1994). The inclusion of memory results in higher average cooperation even in the simultaneous PD game (Nowak and Sigmund 1992, 1993), and although the inclusion of asynchrony modifies the best strategy, the general message remains the same. Moreover, although the PD paradigm is the workhorse metaphor of social dilemmas (Kollock 1998; Dawes and Messick 2000) and fits many situations human society faces, its utility in the study of cooperation of nonkin individuals has been questioned (Noë 1990; Dugatkin et al. 1992). As an alternative, the snowdrift game (also called the chicken game or the hawk–dove game if the cost of defection outweighs the cost of cooperation) was suggested (Hauert and Doebeli 2004), which might not be uncommon in nature.
We have found that the level of cooperation can be both higher and lower in the AS game compared with the synchronous game. Actually, the area of the parameter space where full cooperation ({C, C}) is the winning strategy is larger for the synchronous game (compare Figures 3 and 4). On the other hand, if players decide simultaneously, then unconditional defection ({D, D}) will become a possible outcome. Accordingly, in some cases, a synchronization of players' move would render an elevated level of cooperation possible. In other cases, the same would be achieved if one of the payers would clearly show its intention before the other decides. Synchronization of decisions is very difficult and is presumably only possible in experimental settings or in human interactions. However, it is still very interesting that changing from synchrony to asynchrony or vice versa might have such a profound effect on the attainable level of cooperation. We have the impression that such a change in behavior is unattainable in animal societies; therefore, the proposed asynchronous game is a better representation of the problem of animal cooperation.
Below, we review observations, experiments, and theoretical studies that fit the snowdrift game and where asynchrony in the timing of the behavior and/or synergistic effect of cooperation can be found. We exclude studies and models that focus on aggression and competition, which are traditionally modeled with the help of the hawk–dove game (e.g., Maynard Smith 1982; Sirot 2000; Dubois and Giraldeau 2003; Dubois et al. 2003).
Territorial defense and predator guarding
Territorial defense and guarding against predators are examples to which the snowdrift game might apply. It is trivial that lack of cooperative behavior in all members of an animal group that is involved in such an activity (i.e., none of them defend the territory or none of them watch for predators) can have dire consequences (P < S). Reports on meerkats (Suricata suricatta) indicate that they take turns of guarding while other members dig in search for invertebrates and small vertebrates (Clutton-Brock et al. 1999). It is highly unlikely that 2 individuals would choose to go on raised guard at the same time; thus, it can be considered an asynchronous game. Furthermore, there is no synergistic effect of multiple guardians ( = 0), and thus, only one individual is on the watch (see also Bednekoff 2001). Similar findings were reported, for example, for Arabian babblers (Turdoides squamiceps) (Wright et al. 2001) and Florida scrub jays (Aphelocoma coerulescens) (Bednekoff and Woolfenden 2003). However, in the case of territorial defense, synergistic effects might indeed occur. Depending, for example, on the number of intruders, more individuals are needed for successful defense. Lions readily approach intruders (Heinsohn and Packer 1995); thus, full cooperation is attainable in these animal groups. It has to be noted that the behavior of different individuals can vary, and there are individuals that are less inclined to cooperate (Heinsohn and Packer 1995).
Kleptoparasitism
Group foraging is another behavior that can be characterized as a snowdrift game. Here, a group of animals forage for patchy food resources, and when one individual finds a food patch (the cooperator), the others (the defectors) flock there and join the exploitation of the patch. If all animals are actively foraging and all of them have roughly similar chances of finding a food patch, then the situation can be characterized by an AS game (termed the "information-sharing model," see review in Giraldeau and Beauchamp 1999). Note that unless at least one individual forages, no one is able to find a food patch, thus P < S. In a different but related approach, it is assumed that individuals either forage (producers) or exploit the findings of others (scrounger) ("producer–scrounger" model, e.g., Vickery et al. 1991). This model is well suited to the study of inter- and intraspecific kleptoparasitism (e.g., Broom and Ruxton 1998, 2003; Hamilton 2002); both types are widespread in nature (e.g., Brockmann and Barnard 1979; Koops and Giraldeau 1996; Gorman et al. 1998; Grant et al. 2002; Hamilton and Dill 2003a, 2003b). None of the studies reported synergistic effects of joint foraging. The general result is that scroungers and/or opportunists (individuals that forage or scrounge depending on the conditions) coexist with producers, which translates to a mixed strategy of cooperation and defection either on the population or on the individual level.
Parental care
In animals that require the care of at least one of the parents for raising the offspring, there is a conflict between the parents: which should raise them. In essence, either of the parents can desert, as long as the other remains [so the desertion of both parents is the worst result (P < S)]. In most situations, it is the male who can choose first whether to desert or stay, creating an asymmetry in the conflict. Simple models predict that the male should desert, and it is often observed in nature (see Maynard Smith 1982, Chapter 10, and references therein). However, other factors might change the situation in such a way that it becomes advantageous for the male to stay and cooperate with the female (see, e.g., in McNamara et al. 2000; Barta et al. 2002). Synergy can be assumed, albeit there are empirical results suggesting that sexual conflicts between the parents can reduce the fitness of offspring that were raised together (Royle et al. 2002 and see the model of McNamara et al. 2003). In some other cases, for example, in the mouth-brooding cichlid, Galilee St Peter's fish (Sarotherodon galilaeus), both sexes can choose to be the first to take a portion of the eggs. Here, cooperation has a synergistic effect, and full cooperation can be an ESS in some conditions (Yaniv and Motro 2004).
Cooperative hunting
As mentioned in the introduction, cooperative hunting presents a case to which our model fits nicely. During cooperative hunting, a group coordinates its activity in order to enhance their hunting efficiency, which usually means that they can bring down large prey that no individual hunter can handle alone. Thus, in this case, a synergistic effect of cooperation is present as demonstrated for a range of species (e.g., wolf [Canis lupus]: Mech 1970; spotted hyena [Crocuta crocuta]: Kruuk 1972; African wild dog [Lycaon pictus]: Creel S and Creel NM 1995; Creel 1997; chimpanzee [Pan troglodytes]: Boesch 1994, 2002; lion [Panthera leo]: Scheel and Packer 1991; Packer and Pusey 1997; bottlenose dolphin [Tursiops truncatus]: Gazda et al. 2005, Florida scrub jay [A. coerulescens]: Bowman 2003; Lanner falcon [Falco biarmicus]: Leonardi 1999; Saker falcon [Falco cherrug]: Eakle et al. 2004; Aplomodo falcon [Falco femoralis]: Hector 1986; Taita falcon [Falco faschiinucha]: Hartley et al. 1993; loggerhead shrike [Lanius ludovicianus]: Frye and Gerhardt 2001; Harris' hawk [Parabuteo unicinctus]: Bednarz 1988). On the other hand, individuals have the choice to join the hunt or lag behind but take share from the captured prey (Scheel and Packer 1991; Boesch 1994). Cooperation strongly depends on the cost to benefit ratio of such a behavior. For example, lions hunt in groups for larger prey (e.g., buffalos) but do not join group members in hunts for smaller animals (e.g., warthogs) (Scheel and Packer 1991). Synergism seems to be a rather important component of cooperative hunting because without the added benefits of it, group hunting is no longer observable. Chimpanzees that are not very good hunters individually, and thus benefit more from cooperative hunting, often hunt in groups (Boesch 1994). In contrast, chimpanzees that hunt quite successfully even alone, and could therefore experience low or no synergism, have rarely been observed to cooperate (Boesch 1994).
The above examples describe situations that conform to the AS game. These demonstrate on the one hand that the snowdrift game is widespread in nature and underline on the other hand the importance of the sequence in which the players act. The player acting first can usually force the decision of the second player (see, e.g., the experiments of Bornstein et al. 1997, p. 400; Andreoni et al. 2002), thus creating a situation drastically different from the simultaneous case, which is usually assumed in the game theoretical models and experiments (e.g., in sociological experiments employing the chicken game: Bornstein 2003; Bornstein and Gilgula 2003).
The examples further underline the importance of both the magnitude of synergism of cooperation and the population structure and/or social networks. Fully cooperative behavior ({C, C}), when all players cooperate (note that in the snowdrift game at least one player cooperates) can only be found when there is both synergy and some population structure. Without both, as demonstrated, for example, by the cases of kleptoparasitism, cheating behavior can be observed. This corresponds to the AS game in well-mixed population with = 0, where {C, D} or {D, C} strategies were found to be the ESS. Moreover, synergy or some kind of population structure in itself is not enough to establish {C, C} as the dominant strategy (Figure 3). In predator guarding, the group might be a close-knit community (e.g., in meerkat groups Clutton-Brock et al. 1999), but as there is no synergistic effect from having numerous sentinels, then usually only one of them watches for predators. Similarly, in the example of the yeast cells on a sucrose plate where the population is structured in space, but again there is no synergistic effect from cooperative production of the invertase enzyme, ample examples of defection are observed (Greig and Travisano 2004). However, in social groups where cooperation can have a synergistic effect, full cooperation can be easily realized, as observed in the case of cooperative hunting and in some cases of territorial defense and also demonstrated in our study.
We can conclude that the AS game describes many of the situations where cooperation among nonkin animals has been observed. Population structure coupled with synergistic effects of cooperation can induce full cooperation in group-forming animals.
A new paradigm of viewing nonkin cooperation in animals is emerging, which tries to unify the theoretical models of the last decades and reconcile them with observations and empirical findings. The main characteristic of this paradigm shift is that the PD is no longer viewed as the only way of looking at animal cooperation (Noë 1990; Dugatkin et al. 1992). On the contrary, depending on parameters, many situations should be treated as potentially resembling the PD, the snowdrift, or the assurance game.
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APPENDIX |
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TOPABSTRACTINTRODUCTIONAS GAME WITH SYNERGISTIC...AS GAME IN INFINITE,...AS GAME IN FINITE...RESULTSDISCUSSIONAPPENDIXREFERENCES |
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The replicator dynamics of the AS game
First, the 2 x 2 bimatrix game is transformed into a 4 x 4 symmetric normal form game. We can define 4 pure strategies based on the bimatrix game: {C, C}, {D, C}, {C, D}, and {D, D}, where the first letter in the braces denotes the strategy of the player if it initiates the interaction and the second letter denotes the strategy adopted if it acts second. Each player finds itself in the role of the initiator of the interaction or in the role of the reactor with the same probability. Accordingly, the payoffs of the strategy pairs can be computed by taking these two roles into account with the same weight. This leads to the following 4 x 4 matrix (Table A1).
Table A1 The payoff matrix of the asymmetric snowdrift game as a 4 x 4 symmetric normal form game
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Let us denote the frequency of the different strategies by p, q, r, and s, respectively, where p + q + r + s = 1. It can be seen from the matrix that strategies {C, D} and {D, C} are identical. The strategy {C, D} obtains a fitness value of WCD = p(1 – /4 + /2) + q(1/2 – /4 + /2) + r(1 – /2) + s(1/2 – /2) in an arbitrary population. Similarly, the strategy {D, C} obtains a fitness given by WDC = p(1 – /4 + /2) + q(1 – /2) + r(1/2 – /4 + /2) + s(1/2 – /2). By comparing the 2 fitnesses, it is evident that WDC > WCD 
q > r; thus, from these 2 strategies, the one having greater densities would always have greater fitness and would outcompete the other. Thus, if initially q > r, then r 0 (this does not rule out q 0), and thus, we can simplify the replicator dynamics by including either the strategy {D, C} or the strategy {C, D} depending on the initial frequencies. Let us assume that initially q > r, and thus, only the strategy {D, C} will be considered. By leaving out the strategy {C, D} from the payoff matrix presented in Table A1, we arrive at the following payoff matrix (Table A2).
Table A2 The payoff matrix of the simplified asymmetric snowdrift game as a 3 x 3 symmetric normal form game
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It can be seen from the matrix that the strategy {D, C} in an ESS if < /2, and the strategy {C, C} is an ESS if > /2. In this setting, {D, D} cannot be an ESS, unless > 1 (Maynard Smith 1982).
It is known that if a strategy is an ESS then it is a local asymptotic fixed point in the corresponding replicator dynamics (Hofbauer and Sigmund 1998, p. 70). However, it is possible that the replicator dynamics have stable fixed points that are not ESS (Hofbauer and Sigmund 1998).
Let us now investigate the replicator dynamics corresponding to the above game. Using the same notations as above, we get the following system of differential equations:
 | (A1) |
is the average fitness in the population, and p + q + s = 1. Based on the payoff matrix of the game (Table A2), we have
 | (A2) |
The fixed points of Equations A1 are then
 | (A3) |
To study the local asymptotic stability of the fixed points, we follow the standard techniques by forming the right-hand side Jacobian matrix of Equations A1, and substituting the corresponding value of the 
fixed points to determine the eigenvalues. After the necessary operations, we obtain the following 
eigenvalues for the fixed points:
 | (A4) |
Because 2 < < 1, the 
fixed points are unstable (each has at least one positive eigenvalue), and only the fixed point 
is stable with 3 negative eigenvalues.
For 
the situation is special because it has 2 negative eigenvalues and 1 zero eigenvalue. So, this fixed point is locally marginally stable, and their stability can be analyzed if we would consider higher order terms around the fixed point. However, this higher order analysis seems to be analytically untreatable. Thus, we integrated Equations A1 numerically around 
to study the stability of this fixed point. After extensive numerical simulations, we conclude that is nonlinearly unstable (Figure A1). We can conclude that the only locally asymptotically stable fixed point of the dynamics corresponds with the ESS of the game, namely, the strategy {D, C}. (The calculations were done with Mathematica 3.0.)
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We have to note that if the synergistic effect is so strong as
> /2, then 
will become the only stable fixed point. That is, {C, C} is the ESS and the only stable fixed point in the replicator dynamics. Numerical simulations indicate that besides this fixed point there are no other attractors (i.e., limit cycle, chaotic attractor) in the system. This fixed point is globally stable.
Naturally, we can get the same result by using the strategy {C, D} instead of {D, C}. In a general system, where both {C, D} and {D, C} are present, the strategy that was initially more abundant will emerge as the winner (see Discussion). This result is also supported by numerical simulations.
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ACKNOWLEDGEMENTS |
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We are grateful to János Podani, Viktor Müller, Katalin Csepi, and the 3 anonymous referees for their helpful comments on an earlier draft of the manuscript. This work was supported by grant from the Hungarian National Research Fund (OTKA T035009, T049692, T037726). Á.K. is a postdoctoral fellow of OTKA (D048406).
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REFERENCES |
|---|
|
TOPABSTRACTINTRODUCTIONAS GAME WITH SYNERGISTIC...AS GAME IN INFINITE,...AS GAME IN FINITE...RESULTSDISCUSSIONAPPENDIXREFERENCES |
|---|
Andreoni J, Brown PM, Vesterlund L. 2002. What makes an allocation fair? Some experimental evidence. Games Econ Behav 40:1–24.
Axelrod R, Hamilton WD. 1981. The evolution of cooperation. Science 211:1390–6.
Barta Z, Houston AI, McNamara JM, Székely T. 2002. Sexual conflict about parental care: the role of reserves. Am Nat 159:687–705.[CrossRef][Web of Science][Medline]
Bednarz JC. 1988. Cooperative hunting in Harris' Hawks (Parabuteo unicinctus). Science 239:1525–7.
Bednekoff PA. 2001. Coordination of safe, selfish sentinels based on mutual benefits. Ann Zool Fenn 38:5–14.
Bednekoff PA, Woolfenden GE. 2003. Florida scrub-jays (Aphelocoma coerulescens) are sentinels more when well-fed (even with no kin nearby). Ethology 109:895–903.[CrossRef][Web of Science]
Boesch C. 1994. Cooperative hunting in chimpanzees. Anim Behav 48:653–67.[CrossRef]
Boesch C. 2002. Cooperative hunting roles among Taï chimpanzees. Hum Nat 13:27–46.
Bornstein G. 2003. Intergroup conflict: individual, group, and collective interests. Pers Soc Psychol Rev 7:129–45.
Bornstein G, Budescu D, Zamir S. 1997. Cooperation in intergroup, N-person, and two-person games of chicken. J Confl Resolut 31:384–406.
Bornstein G, Gilgula Z. 2003. Between-group communication and conflict resolution in assurance and chicken games. J Confl Resolut 47:326–39.[CrossRef]
Bowman R. 2003. Apparent cooperative hunting in Florida scrub-jays. Wilson Bull 115:197–9.[CrossRef]
Brembs B. 1996. Chaos, cheating and cooperation: potential solutions to the prisoner's dilemma. Oikos 76:14–24.
Brockmann J, Barnard CJ. 1979. Kleptoparasitism in birds. Anim Behav 27:487–514.
Broom M, Ruxton GD. 1998. Evolutionary stable stealing: game theory applied to kleptoparasitism. Behav Ecol 9:397–403.
Broom M, Ruxton GD. 2003. Evolutionarily stable kleptoparasitism: consequences of different prey types. Behav Ecol 14:23–33.
Clutton-Brock TH, O'Riain MJ, Brotherton PNM, Gaynor D, Kansky R, Griffin AS, Manser M. 1999. Selfish sentinels in cooperative mammals. Science 284:1640–4.
Creel S. 1997. Cooperative hunting and group size: assumption and currencies. Anim Behav 54:1319–24.[CrossRef][Web of Science][Medline]
Creel S, Creel NM. 1995. Communal hunting and pack size in African wild dogs, Lycan pictus. Anim Behav 50:1325–39.
Dawes RM, Messick DM. 2000. Social dilemmas. Int J Psychol 35:111–6.[CrossRef]
Doebeli M, Hauert C. 2005. Models of cooperation based on the prisoner's dilemma and the snowdrift game. Ecol Lett 8:748–66.[CrossRef][Web of Science]
Doebeli M, Hauert C, Killingback T. 2004. The evolutionary origin of cooperators and defectors. Science 306:859–62.
Dubois F, Giraldeau L-A. 2003. The forager's dilemma: food sharing and food defense as risk-sensitive foraging options. Am Nat 162:768–79.[CrossRef][Medline]
Dubois F, Giraldeau L-A, Grant JWA. 2003. Resource defense in a group-foraging context. Behav Ecol 14:2–9.
Dugatkin LA. 1991. Dynamics of the tit-for-tat strategy during predator inspection in the guppy (Poecilia reticulata). Behav Ecol Sociobiol 29:127–32.[CrossRef][Web of Science]
Dugatkin LA. 1997. Cooperation among animals: an evolutionary perspective. New York: Oxford University Press.[CrossRef][Web of Science]
Dugatkin LA. 2002. Animal cooperation among unrelated individuals. Naturwissenschaften 89:533–41.[Web of Science][Medline]
Dugatkin LA, Mesterson-Gibbons M, Houston AI. 1992. Beyond the prisoner's dilemma: toward models to discriminate among mechanism of cooperation in nature. Trends Evol Ecol 7:202–5.
Eakle WL, Millier C, Mineau P, Világosi J. 2004. An example of cooperative hunting by Saker Falcons in Hungary. J Raptor Res 38:292–3.
Frean MR. 1994. The prisoner's dilemma without synchrony. Proc R Soc Lond B Biol Sci 257:75–9.[Medline]
Frye GG, Gerhardt RP. 2001. Apparent cooperative hunting in loggerhead shrikes. Wilson Bull 113:462–4.
Gazda S, Connor RC, Edgar RK, Cox F. 2005. A division of labour with role specialization in group-hunting bottlenose dolphins (Tursiops truncatus) off Cedar Key, Florida. Proc R Soc Lond B Biol Sci 272:135–40.
Giraldeau L-A, Beauchamp G. 1999. Food exploitation: searching for the optimal joining policy. Trends Evol Ecol 14:102–6.
Gorman ML, Mills MG, Raath JP, Speakman JR. 1998. High hunting costs make African wild dogs vulnerable to kleptoparasitism by hyaenas. Nature 391:479–81.[CrossRef]
Grant JWA, Girard IL, Breau C, Weir LK. 2002. Influence of food abundance on competitive aggression in juvenile convict cichlids. Anim Behav 63:323–30.
Greig D, Travisano M. 2004. The prisoner's dilemma and polymorphism in yeast SUC genes. Proc R Soc Lond B Biol Sci 271:S25–6.
Hamilton IM. 2002. Kleptoparasitism and the distribution of unequal competitors. Behav Ecol 13:260–7.
Hamilton IM, Dill LM. 2003a. Group foraging by a kleptoparasitic fish: a strong inference test of social foraging models. Ecology 12:3349–59.
Hamilton IM, Dill LM. 2003b. The use of territorial gardening versus kleptoparasitism by a subtropical reef fish (Kyphosus cornelii) is influenced by territory defendability. Behav Ecol 14:561–8.
Hammerstein P. 1981. The role of asymmetries in animal contests. Anim Behav 29:193–205.
Hartley RR, Bodington G, Dunkley AS, Groenewald A. 1993. Notes on the breeding biology, hunting behaviour, and ecology of the Taita Falcon in Zimbabwe. J Raptor Res 27:133–42.
Hauert C. 2002. Effects of space in 2x2 games. Int J Bifurcat Chaos 12:1531–48.[CrossRef][Web of Science]
Hauert C. 2005. Spatial effects in social dilemmas. J Theor Biol. 10.1016/j.jtbi.2005.10.024.
Hauert C, Doebeli M. 2004. Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature 428:643–6.[CrossRef][Medline]
Hauert C, Schuster HG. 1998. Extending the iterated prisoner's dilemma without synchrony. J Theor Biol 192:155–66.[CrossRef][Web of Science][Medline]
Hector DP. 1986. Cooperative hunting and its relationship to foraging success and prey size in an avian predator. Ethology 73:247–57.
Heinsohn R, Packer C. 1995. Complex cooperative strategies in group-territorial African lions. Science 269:1260–2.
Hofbauer J, Sigmund K. 1998. Evolutionary games and population dynamics. Cambridge, UK: Cambridge University Press.
Killingback T, Doebeli M. 2002. The continuous prisoner's dilemma and the evolution of cooperation through reciprocal altruism with variable investment. Am Nat 160:421–38.
Killingback T, Doebeli M, Knowlton N. 1999. Variable investment, the continuous prisoner's dilemma, and the origin of cooperation. Proc R Soc Lond B Biol Sci 266:1723–8.
Kollock P. 1998. Social dilemmas: the anatomy of cooperation. Annu Rev Sociol 24:183–214.[CrossRef][Web of Science]
Koops MA, Giraldeau L-A. 1996. Producer–scrounger foraging games in starlings: a test of rate-maximizing and risk-sensitive models. Anim Behav 51:773–83.[CrossRef]
Kruuk H. 1972. The spotted hyaena. Chicago: University of Chicago Press.
Leonardi G. 1999. Cooperative hunting of jackdaws by the Lanner Falcon (Falco biarmicus). J Raptor Res 33:123–7.
Maynard Smith J. 1982. Evolution and the theory of games. Cambridge, UK: Cambridge University Press.
McNamara JM, Houston AI, Barta Z, Osorno J-L. 2003. Should young ever be better off with one parent than with two? Behav Ecol 14:301–10.
McNamara JM, Székely T, Webb JN, Houston AI. 2000. A dynamic game-theoretic model of parental care. J Theor Biol 205:605–23.[CrossRef][Web of Science][Medline]
Mech DL. 1970. The wolf. New York: Natural History Press.
Milinski M. 1987. Tit for tat in sticklebacks and the evolution of cooperation. Nature 325:433–5.[CrossRef][Medline]
Neill DB. 2001. Optimality under noise: higher memory strategies for the alternating prisoner's dilemma. J Theor Biol 201:159–80.
Noë R. 1990. A veto game played by baboons: a challenge to the use of the prisoner's dilemma as a paradigm for reciprocity and cooperation. Anim Behav 39:78–90.[CrossRef]
Nowak MA, Bonhoeffer S, May RM. 1994a. More spatial games. Int J Bifurcat Chaos 4:33–56.[CrossRef][Web of Science]
Nowak MA, Bonhoeffer S, May RM. 1994b. Spatial games and the maintenance of cooperation. Proc Natl Acad Sci USA 91:4877–81.
Nowak MA, May RM. 1992. Evolutionary games and spatial chaos. Nature 359:826–9.[CrossRef]
Nowak MA, Sasaki A, Taylor C, Fudenberg D. 2004. Emergence of cooperation and evolutionary stability in finite populations. Nature 428:646–50.[CrossRef][Medline]
Nowak MA, Sigmund K. 1992. Tit for tat in heterogeneous population. Nature 355:250–2.[CrossRef][Web of Science]
Nowak MA, Sigmund K. 1993. A strategy of win-stay, lose-shift that outperforms tit-for-tat in the prisoner's dilemma game. Nature 364:56–8.[CrossRef][Medline]
Nowak MA, Sigmund K. 1994. The alternating prisoner's dilemma. J Theor Biol 168:219–26.[CrossRef]
Nunn CL, Lewis RJ. 2001. Cooperation and collective action in animal behaviour. In: Noë R, van Hooff JARAM, Hammerstein P, editors. Economics in nature. Cambridge, UK: Cambridge University Press. p 42–66.[CrossRef][Web of Science]
Packer C, Pusey AE. 1997. Divided we fall: cooperation among lions. Sci Am 276:32–9.
Roberts G, Sherratt TN. 1998. Development of cooperative relationship through increasing investment. Nature 394:175–9.[CrossRef][Medline]
Royle NJ, Hartley IR, Parker GA. 2002. Sexual conflict reduces offspring fitness in zebra finches. Nature 416:733–6.[CrossRef][Medline]
Scheel D, Packer C. 1991. Group hunting behaviour of lions: a search for cooperation. Anim Behav 41:697–709.
Sigmund K, Hauert C. 2002. Altruism. Curr Biol 12:R270–2.[CrossRef][Web of Science][Medline]
Sirot E. 2000. Evolutionarily stable kleptoparasitism: consequences of different prey types. Behav Ecol 11:351–6.
Traulsen A, Claussen JC, Hauert C. 2005. Coevolutionary dynamics: from finite to infinite populations. Phys Rev Lett 95:238701.[CrossRef][Medline]
Trivers R. 1971. The evolution of reciprocal altruism. Q Rev Biol 46:35–57.[CrossRef]
van Baalen M, Rand DA. 1998. The unit of selection in viscous populations and the evolution of altruism. J Theor Biol 193:631–48.[CrossRef][Web of Science][Medline]
Vickery WL, Giraldeau L-A, Templeton JJ, Kramper DL, Chapman CA. 1991. Producers, scroungers, and group foraging. Am Nat 137:847–63.
Wright J, Berg E, de Kort SR, Khazin V, Maklakov AA. 2001. Safe selfish sentinels in a cooperative bird. J Anim Ecol 70:1070–9.[CrossRef]
Yaniv O, Motro U. 2004. The parental investment conflict in continuous time: St. Peter's fish as an example. J Theor Biol 228:377–88.[CrossRef][Web of Science][Medline]
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