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Adaptive dynamics of extortion and compliance.

Hilbe C, Nowak MA, Traulsen A - PLoS ONE (2013)

Bottom Line: Our results are not restricted to the case of the prisoners dilemma, but can be extended to other social dilemmas, such as the snowdrift game.Iterated social dilemmas in large populations do not lead to the evolution of strategies that aim to dominate their co-player.Instead, generosity succeeds.

View Article: PubMed Central - PubMed

Affiliation: Evolutionary Theory Group, Max-Planck Institute for Evolutionary Biology, Plön, Germany.

ABSTRACT
Direct reciprocity is a mechanism for the evolution of cooperation. For the iterated prisoner's dilemma, a new class of strategies has recently been described, the so-called zero-determinant strategies. Using such a strategy, a player can unilaterally enforce a linear relationship between his own payoff and the co-player's payoff. In particular the player may act in such a way that it becomes optimal for the co-player to cooperate unconditionally. In this way, a player can manipulate and extort his co-player, thereby ensuring that the own payoff never falls below the co-player's payoff. However, using a compliant strategy instead, a player can also ensure that his own payoff never exceeds the co-player's payoff. Here, we use adaptive dynamics to study when evolution leads to extortion and when it leads to compliance. We find a remarkable cyclic dynamics: in sufficiently large populations, extortioners play a transient role, helping the population to move from selfish strategies to compliance. Compliant strategies, however, can be subverted by altruists, which in turn give rise to selfish strategies. Whether cooperative strategies are favored in the long run critically depends on the size of the population; we show that cooperation is most abundant in large populations, in which case average payoffs approach the social optimum. Our results are not restricted to the case of the prisoners dilemma, but can be extended to other social dilemmas, such as the snowdrift game. Iterated social dilemmas in large populations do not lead to the evolution of strategies that aim to dominate their co-player. Instead, generosity succeeds.

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Stochastic dynamics for different population sizes.We consider a homogeneous population of size . Once a mutation occurs, the mutant strategy either takes over the whole population (with probability ), or goes extinct before the next mutation arises. This leads to a sequence of residents in the state space, which is shown in the upper three graphs (the dashed line corresponds to the threshold ). The lower three graphs give the distribution of the resulting payoffs in the population. (a) In the extreme case of , most players enforce a strategy with baseline payoff . In particular, extortion strategies can persist. (b) As population size increases, a bistable situation emerges: the population clusters along the edges with  and . (c) For large population sizes, this implies that the edge of compliers is (neutrally) stable, whereas the edge of extortioners is unstable. As a consequence, mean payoffs increase with population size. The figure shows simulation runs for  residents for a prisoner’s dilemma with , , , . New mutant strategies are randomly drawn from a Gaussian distribution around the parent strategy (). The invasion probability  of a mutant is calculated as , where  and  are the respective payoffs of mutants and residents, and where  is the strength of selection.
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pone-0077886-g003: Stochastic dynamics for different population sizes.We consider a homogeneous population of size . Once a mutation occurs, the mutant strategy either takes over the whole population (with probability ), or goes extinct before the next mutation arises. This leads to a sequence of residents in the state space, which is shown in the upper three graphs (the dashed line corresponds to the threshold ). The lower three graphs give the distribution of the resulting payoffs in the population. (a) In the extreme case of , most players enforce a strategy with baseline payoff . In particular, extortion strategies can persist. (b) As population size increases, a bistable situation emerges: the population clusters along the edges with and . (c) For large population sizes, this implies that the edge of compliers is (neutrally) stable, whereas the edge of extortioners is unstable. As a consequence, mean payoffs increase with population size. The figure shows simulation runs for residents for a prisoner’s dilemma with , , , . New mutant strategies are randomly drawn from a Gaussian distribution around the parent strategy (). The invasion probability of a mutant is calculated as , where and are the respective payoffs of mutants and residents, and where is the strength of selection.

Mentions: In order to confirm these predictions, we have simulated the dynamics in finite populations for a pairwise comparison process, where the probability to switch to the role model’s strategy is given by a Fermi function [37], [38]. We assume that mutations follow Gaussian distributions around and and focus on the distribution of strategies and on the distribution of payoffs. For we find that the population clusters around the edge of low population payoffs (see Fig. 3a), and the density function for the payoffs has a single peak at . Increasing the population size has a two-fold effect (Fig. 3b and 3c). First, compliant strategies with become stable, such that the density function of the population payoffs has a second peak at . Second, increasing the population size reduces the stochastic noise; as a consequence almost all the mass is concentrated around the two peaks and . As predicted by adaptive dynamics, and in line with previous results [23], larger populations exhibit larger payoffs. For example, payoffs for a population size exceed the payoffs for by more than a factor of six.


Adaptive dynamics of extortion and compliance.

Hilbe C, Nowak MA, Traulsen A - PLoS ONE (2013)

Stochastic dynamics for different population sizes.We consider a homogeneous population of size . Once a mutation occurs, the mutant strategy either takes over the whole population (with probability ), or goes extinct before the next mutation arises. This leads to a sequence of residents in the state space, which is shown in the upper three graphs (the dashed line corresponds to the threshold ). The lower three graphs give the distribution of the resulting payoffs in the population. (a) In the extreme case of , most players enforce a strategy with baseline payoff . In particular, extortion strategies can persist. (b) As population size increases, a bistable situation emerges: the population clusters along the edges with  and . (c) For large population sizes, this implies that the edge of compliers is (neutrally) stable, whereas the edge of extortioners is unstable. As a consequence, mean payoffs increase with population size. The figure shows simulation runs for  residents for a prisoner’s dilemma with , , , . New mutant strategies are randomly drawn from a Gaussian distribution around the parent strategy (). The invasion probability  of a mutant is calculated as , where  and  are the respective payoffs of mutants and residents, and where  is the strength of selection.
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Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3815207&req=5

pone-0077886-g003: Stochastic dynamics for different population sizes.We consider a homogeneous population of size . Once a mutation occurs, the mutant strategy either takes over the whole population (with probability ), or goes extinct before the next mutation arises. This leads to a sequence of residents in the state space, which is shown in the upper three graphs (the dashed line corresponds to the threshold ). The lower three graphs give the distribution of the resulting payoffs in the population. (a) In the extreme case of , most players enforce a strategy with baseline payoff . In particular, extortion strategies can persist. (b) As population size increases, a bistable situation emerges: the population clusters along the edges with and . (c) For large population sizes, this implies that the edge of compliers is (neutrally) stable, whereas the edge of extortioners is unstable. As a consequence, mean payoffs increase with population size. The figure shows simulation runs for residents for a prisoner’s dilemma with , , , . New mutant strategies are randomly drawn from a Gaussian distribution around the parent strategy (). The invasion probability of a mutant is calculated as , where and are the respective payoffs of mutants and residents, and where is the strength of selection.
Mentions: In order to confirm these predictions, we have simulated the dynamics in finite populations for a pairwise comparison process, where the probability to switch to the role model’s strategy is given by a Fermi function [37], [38]. We assume that mutations follow Gaussian distributions around and and focus on the distribution of strategies and on the distribution of payoffs. For we find that the population clusters around the edge of low population payoffs (see Fig. 3a), and the density function for the payoffs has a single peak at . Increasing the population size has a two-fold effect (Fig. 3b and 3c). First, compliant strategies with become stable, such that the density function of the population payoffs has a second peak at . Second, increasing the population size reduces the stochastic noise; as a consequence almost all the mass is concentrated around the two peaks and . As predicted by adaptive dynamics, and in line with previous results [23], larger populations exhibit larger payoffs. For example, payoffs for a population size exceed the payoffs for by more than a factor of six.

Bottom Line: Our results are not restricted to the case of the prisoners dilemma, but can be extended to other social dilemmas, such as the snowdrift game.Iterated social dilemmas in large populations do not lead to the evolution of strategies that aim to dominate their co-player.Instead, generosity succeeds.

View Article: PubMed Central - PubMed

Affiliation: Evolutionary Theory Group, Max-Planck Institute for Evolutionary Biology, Plön, Germany.

ABSTRACT
Direct reciprocity is a mechanism for the evolution of cooperation. For the iterated prisoner's dilemma, a new class of strategies has recently been described, the so-called zero-determinant strategies. Using such a strategy, a player can unilaterally enforce a linear relationship between his own payoff and the co-player's payoff. In particular the player may act in such a way that it becomes optimal for the co-player to cooperate unconditionally. In this way, a player can manipulate and extort his co-player, thereby ensuring that the own payoff never falls below the co-player's payoff. However, using a compliant strategy instead, a player can also ensure that his own payoff never exceeds the co-player's payoff. Here, we use adaptive dynamics to study when evolution leads to extortion and when it leads to compliance. We find a remarkable cyclic dynamics: in sufficiently large populations, extortioners play a transient role, helping the population to move from selfish strategies to compliance. Compliant strategies, however, can be subverted by altruists, which in turn give rise to selfish strategies. Whether cooperative strategies are favored in the long run critically depends on the size of the population; we show that cooperation is most abundant in large populations, in which case average payoffs approach the social optimum. Our results are not restricted to the case of the prisoners dilemma, but can be extended to other social dilemmas, such as the snowdrift game. Iterated social dilemmas in large populations do not lead to the evolution of strategies that aim to dominate their co-player. Instead, generosity succeeds.

Show MeSH
Related in: MedlinePlus