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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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Related in: MedlinePlus

Average payoffs for the four different mutation treatments.In rare-mutation treatments, the mutation rate is set to , whereas in frequent-mutation treatments the mutation rate is . Local mutations are randomly drawn from a Gaussian distribution around the parent strategy, non-local mutations are randomly drawn from the entire state space. The rare local mutations correspond to the previous simulations in Figs. 3 and 4. All other parameters are the same as before.
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pone-0077886-g005: Average payoffs for the four different mutation treatments.In rare-mutation treatments, the mutation rate is set to , whereas in frequent-mutation treatments the mutation rate is . Local mutations are randomly drawn from a Gaussian distribution around the parent strategy, non-local mutations are randomly drawn from the entire state space. The rare local mutations correspond to the previous simulations in Figs. 3 and 4. All other parameters are the same as before.

Mentions: So far, we have assumed that a mutant’s strategy is close to the parent’s strategy (which allowed us to use derivatives to approximate the dynamics), and that mutations are rare (which allowed us to focus on games between a resident and one mutant strategy). Let us now weaken these assumptions and numerically explore the impact of non-local mutations, and of different mutation rates, respectively. In Fig. 5, we distinguish four simulations, according to whether the mutation rate is high or low ( vs. ), and whether mutations occur on a local or on a global level (mutant strategies are drawn from a normal distribution around the parent’s strategy, vs. mutant strategies are uniformly distributed over the set of enforceable pairs). These simulations indicate that all treatments follow the same pattern: average payoffs are close to the minimum in small populations, and they increase with population size. However, there is a clear difference between treatments with local mutations and treatments with non-local mutations. If mutations are local, populations can be trapped in regions with a low payoff for a considerable time, although distant mutant strategies would offer an immediate escape. For example, we have seen that any strategy of the form forms a stable fixed point of the adaptive dynamics. However, once we allow mutants to adopt any strategy of the state space, mutants with close to one and can easily invade (in fact, in Stewart and Plotkin [25] it is shown that in sufficiently large populations, compliant strategies with can replace any noncooperative zero-determinant strategy). Overall, non-local mutations thus lead to a shift of the invariant distribution towards more cooperative strategies.


Adaptive dynamics of extortion and compliance.

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

Average payoffs for the four different mutation treatments.In rare-mutation treatments, the mutation rate is set to , whereas in frequent-mutation treatments the mutation rate is . Local mutations are randomly drawn from a Gaussian distribution around the parent strategy, non-local mutations are randomly drawn from the entire state space. The rare local mutations correspond to the previous simulations in Figs. 3 and 4. All other parameters are the same as before.
© Copyright Policy
Related In: Results  -  Collection

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

pone-0077886-g005: Average payoffs for the four different mutation treatments.In rare-mutation treatments, the mutation rate is set to , whereas in frequent-mutation treatments the mutation rate is . Local mutations are randomly drawn from a Gaussian distribution around the parent strategy, non-local mutations are randomly drawn from the entire state space. The rare local mutations correspond to the previous simulations in Figs. 3 and 4. All other parameters are the same as before.
Mentions: So far, we have assumed that a mutant’s strategy is close to the parent’s strategy (which allowed us to use derivatives to approximate the dynamics), and that mutations are rare (which allowed us to focus on games between a resident and one mutant strategy). Let us now weaken these assumptions and numerically explore the impact of non-local mutations, and of different mutation rates, respectively. In Fig. 5, we distinguish four simulations, according to whether the mutation rate is high or low ( vs. ), and whether mutations occur on a local or on a global level (mutant strategies are drawn from a normal distribution around the parent’s strategy, vs. mutant strategies are uniformly distributed over the set of enforceable pairs). These simulations indicate that all treatments follow the same pattern: average payoffs are close to the minimum in small populations, and they increase with population size. However, there is a clear difference between treatments with local mutations and treatments with non-local mutations. If mutations are local, populations can be trapped in regions with a low payoff for a considerable time, although distant mutant strategies would offer an immediate escape. For example, we have seen that any strategy of the form forms a stable fixed point of the adaptive dynamics. However, once we allow mutants to adopt any strategy of the state space, mutants with close to one and can easily invade (in fact, in Stewart and Plotkin [25] it is shown that in sufficiently large populations, compliant strategies with can replace any noncooperative zero-determinant strategy). Overall, non-local mutations thus lead to a shift of the invariant distribution towards more cooperative strategies.

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