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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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Zero-determinant strategies for the iterated snowdrift game (with , , and , , , ).(a) The grey shaded area gives the space of feasible payoff pairs in the snowdrift game. The three colored lines give three examples of possible payoff combinations if the focal player uses a strategy that enforces a linear relation between payoffs. Unlike in the iterated prisoner’s dilemma, the slope of  is positive, . (b) The grey-shaded area depicts the space of possible combinations of baseline payoff  and slopes  that are enforceable in the snowdrift game. A comparison with Fig. 2 shows that the state space differs considerably from the state space of a prisoner’s dilemma game. However, the qualitative dynamics within the state space remains unchanged.
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pone-0077886-g007: Zero-determinant strategies for the iterated snowdrift game (with , , and , , , ).(a) The grey shaded area gives the space of feasible payoff pairs in the snowdrift game. The three colored lines give three examples of possible payoff combinations if the focal player uses a strategy that enforces a linear relation between payoffs. Unlike in the iterated prisoner’s dilemma, the slope of is positive, . (b) The grey-shaded area depicts the space of possible combinations of baseline payoff and slopes that are enforceable in the snowdrift game. A comparison with Fig. 2 shows that the state space differs considerably from the state space of a prisoner’s dilemma game. However, the qualitative dynamics within the state space remains unchanged.

Mentions: Throughout this manuscript, we have focused on the dynamics of an iterated prisoner’s dilemma. However, only a few of our results actually depend on the characteristic order of payoffs, . In fact, the only result specific to the prisoner’s dilemma concerns the characterization of enforceable () pairs in Eq. (6). For games that are different from the prisoner’s dilemma, the geometry of the state space may thus be different, but the dynamics on the respective state space remains unchanged. In Figure 7, we illustrate this observation by considering the dynamics of an iterated snowdrift game (which is defined by the payoff relations , , , with such that , see [39], [40]). For snowdrift games we observe that only a subset of extortionate strategies is feasible [41]: extortionate strategies with need to fulfill the requirement (i.e. the maximum extortion factor is ). Moreover, only strategies that yield a baseline payoff higher than can enforce a payoff relation with negative slope, . As a consequence, any sufficiently large initial population that yields a payoff less than against itself can be replaced by more cooperative mutant strategies with higher baseline payoffs. As in the prisoner’s dilemma, this dynamics leads to the edge of compliers, which can only be left by neutral invasion of altruists.


Adaptive dynamics of extortion and compliance.

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

Zero-determinant strategies for the iterated snowdrift game (with , , and , , , ).(a) The grey shaded area gives the space of feasible payoff pairs in the snowdrift game. The three colored lines give three examples of possible payoff combinations if the focal player uses a strategy that enforces a linear relation between payoffs. Unlike in the iterated prisoner’s dilemma, the slope of  is positive, . (b) The grey-shaded area depicts the space of possible combinations of baseline payoff  and slopes  that are enforceable in the snowdrift game. A comparison with Fig. 2 shows that the state space differs considerably from the state space of a prisoner’s dilemma game. However, the qualitative dynamics within the state space remains unchanged.
© Copyright Policy
Related In: Results  -  Collection

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

pone-0077886-g007: Zero-determinant strategies for the iterated snowdrift game (with , , and , , , ).(a) The grey shaded area gives the space of feasible payoff pairs in the snowdrift game. The three colored lines give three examples of possible payoff combinations if the focal player uses a strategy that enforces a linear relation between payoffs. Unlike in the iterated prisoner’s dilemma, the slope of is positive, . (b) The grey-shaded area depicts the space of possible combinations of baseline payoff and slopes that are enforceable in the snowdrift game. A comparison with Fig. 2 shows that the state space differs considerably from the state space of a prisoner’s dilemma game. However, the qualitative dynamics within the state space remains unchanged.
Mentions: Throughout this manuscript, we have focused on the dynamics of an iterated prisoner’s dilemma. However, only a few of our results actually depend on the characteristic order of payoffs, . In fact, the only result specific to the prisoner’s dilemma concerns the characterization of enforceable () pairs in Eq. (6). For games that are different from the prisoner’s dilemma, the geometry of the state space may thus be different, but the dynamics on the respective state space remains unchanged. In Figure 7, we illustrate this observation by considering the dynamics of an iterated snowdrift game (which is defined by the payoff relations , , , with such that , see [39], [40]). For snowdrift games we observe that only a subset of extortionate strategies is feasible [41]: extortionate strategies with need to fulfill the requirement (i.e. the maximum extortion factor is ). Moreover, only strategies that yield a baseline payoff higher than can enforce a payoff relation with negative slope, . As a consequence, any sufficiently large initial population that yields a payoff less than against itself can be replaced by more cooperative mutant strategies with higher baseline payoffs. As in the prisoner’s dilemma, this dynamics leads to the edge of compliers, which can only be left by neutral invasion of altruists.

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