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Evolution of all-or-none strategies in repeated public goods dilemmas.

Pinheiro FL, Vasconcelos VV, Santos FC, Pacheco JM - PLoS Comput. Biol. (2014)

Bottom Line: We study both analytically and by computer simulations the evolutionary dynamics encompassing this extensive strategy space, witnessing the emergence of a surprisingly simple strategy that we call All-Or-None (AoN).AoN consists in cooperating only after a round of unanimous group behavior (cooperation or defection), and proves robust in the presence of errors, thus fostering cooperation in a wide range of group sizes.The principles encapsulated in this strategy share a level of complexity reminiscent of that found already in 2-person games under direct and indirect reciprocity, reducing, in fact, to the well-known Win-Stay-Lose-Shift strategy in the limit of the repeated 2-person Prisoner's Dilemma.

View Article: PubMed Central - PubMed

Affiliation: Centro de Biologia Molecular e Ambiental da Universidade do Minho, Braga, Portugal; INESC-ID & Instituto Superior Técnico, Universidade de Lisboa, Taguspark, Porto Salvo, Portugal; Centro de Física da Universidade do Minho, Braga, Portugal; ATP-group, CMAF, Instituto para a Investigação Interdisciplinar, Lisboa, Portugal.

ABSTRACT
Many problems of cooperation involve repeated interactions among the same groups of individuals. When collective action is at stake, groups often engage in Public Goods Games (PGG), where individuals contribute (or not) to a common pool, subsequently sharing the resources. Such scenarios of repeated group interactions materialize situations in which direct reciprocation to groups may be at work. Here we study direct group reciprocity considering the complete set of reactive strategies, where individuals behave conditionally on what they observed in the previous round. We study both analytically and by computer simulations the evolutionary dynamics encompassing this extensive strategy space, witnessing the emergence of a surprisingly simple strategy that we call All-Or-None (AoN). AoN consists in cooperating only after a round of unanimous group behavior (cooperation or defection), and proves robust in the presence of errors, thus fostering cooperation in a wide range of group sizes. The principles encapsulated in this strategy share a level of complexity reminiscent of that found already in 2-person games under direct and indirect reciprocity, reducing, in fact, to the well-known Win-Stay-Lose-Shift strategy in the limit of the repeated 2-person Prisoner's Dilemma.

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Stationary bit distribution as a function of the error rate.We plot (log-linear scale) the fraction of time the population spends in a strategy with bq = 1 for a broad range of error probabilities ε. Circles on the left indicate the values obtained for ε = 0.0, gray areas show the range of values for which bits were defined to have a dominant behavior. Note that for ε = 0.5 all strategies behave randomly. The bar plot on the right shows the results for ε = 0.06 (vertical dashed line). Other model parameters: Z = 100, β = 1.0, N = 5, F/N = 0.85, w = 0.96 and μ≪1/Z.
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pcbi-1003945-g002: Stationary bit distribution as a function of the error rate.We plot (log-linear scale) the fraction of time the population spends in a strategy with bq = 1 for a broad range of error probabilities ε. Circles on the left indicate the values obtained for ε = 0.0, gray areas show the range of values for which bits were defined to have a dominant behavior. Note that for ε = 0.5 all strategies behave randomly. The bar plot on the right shows the results for ε = 0.06 (vertical dashed line). Other model parameters: Z = 100, β = 1.0, N = 5, F/N = 0.85, w = 0.96 and μ≪1/Z.

Mentions: To investigate the robustness of AoN we show, in Figure 2, the effect of execution errors on the stationary bit distribution () for a fixed group size (here N = 5): Clearly, both b0 and bN remain associated with cooperation for a wide range of error probabilities (ε≤0.2). The internal bits, in turn, remain qualitatively close to the AoN profile (i.e. bq = 0 for 0<q<N), undergoing changes as the error rate increases, allowing an efficient resume into full cooperation, after (at least) one behavioral error. In particular, for 0.01<ε<0.1, evolution selects for defection in bits b1 to bN−1, with particular incidence to adjacent bits of b0 and bN, allowing a fast error recovery. This feature gets enhanced with increasing ε. For larger values of ε (ε>0.1), unanimity becomes less likely and we witness an adaptation of the predominant strategy that acts to reduce the interval of bits in which defection prevails. In other words, it is as if execution errors redefine the notion of unanimity itself or, alternatively, individuals become more tolerant as execution errors become more likely. It is also noteworthy that the non-monotonous response to errors shown in Figure 2 has been previously observed in other evolutionary models of cooperation [48] where intermediate degrees of stochasticity emerge as maximizers of cooperation. We confirmed that the results remain qualitatively equivalent for different group sizes.


Evolution of all-or-none strategies in repeated public goods dilemmas.

Pinheiro FL, Vasconcelos VV, Santos FC, Pacheco JM - PLoS Comput. Biol. (2014)

Stationary bit distribution as a function of the error rate.We plot (log-linear scale) the fraction of time the population spends in a strategy with bq = 1 for a broad range of error probabilities ε. Circles on the left indicate the values obtained for ε = 0.0, gray areas show the range of values for which bits were defined to have a dominant behavior. Note that for ε = 0.5 all strategies behave randomly. The bar plot on the right shows the results for ε = 0.06 (vertical dashed line). Other model parameters: Z = 100, β = 1.0, N = 5, F/N = 0.85, w = 0.96 and μ≪1/Z.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4230726&req=5

pcbi-1003945-g002: Stationary bit distribution as a function of the error rate.We plot (log-linear scale) the fraction of time the population spends in a strategy with bq = 1 for a broad range of error probabilities ε. Circles on the left indicate the values obtained for ε = 0.0, gray areas show the range of values for which bits were defined to have a dominant behavior. Note that for ε = 0.5 all strategies behave randomly. The bar plot on the right shows the results for ε = 0.06 (vertical dashed line). Other model parameters: Z = 100, β = 1.0, N = 5, F/N = 0.85, w = 0.96 and μ≪1/Z.
Mentions: To investigate the robustness of AoN we show, in Figure 2, the effect of execution errors on the stationary bit distribution () for a fixed group size (here N = 5): Clearly, both b0 and bN remain associated with cooperation for a wide range of error probabilities (ε≤0.2). The internal bits, in turn, remain qualitatively close to the AoN profile (i.e. bq = 0 for 0<q<N), undergoing changes as the error rate increases, allowing an efficient resume into full cooperation, after (at least) one behavioral error. In particular, for 0.01<ε<0.1, evolution selects for defection in bits b1 to bN−1, with particular incidence to adjacent bits of b0 and bN, allowing a fast error recovery. This feature gets enhanced with increasing ε. For larger values of ε (ε>0.1), unanimity becomes less likely and we witness an adaptation of the predominant strategy that acts to reduce the interval of bits in which defection prevails. In other words, it is as if execution errors redefine the notion of unanimity itself or, alternatively, individuals become more tolerant as execution errors become more likely. It is also noteworthy that the non-monotonous response to errors shown in Figure 2 has been previously observed in other evolutionary models of cooperation [48] where intermediate degrees of stochasticity emerge as maximizers of cooperation. We confirmed that the results remain qualitatively equivalent for different group sizes.

Bottom Line: We study both analytically and by computer simulations the evolutionary dynamics encompassing this extensive strategy space, witnessing the emergence of a surprisingly simple strategy that we call All-Or-None (AoN).AoN consists in cooperating only after a round of unanimous group behavior (cooperation or defection), and proves robust in the presence of errors, thus fostering cooperation in a wide range of group sizes.The principles encapsulated in this strategy share a level of complexity reminiscent of that found already in 2-person games under direct and indirect reciprocity, reducing, in fact, to the well-known Win-Stay-Lose-Shift strategy in the limit of the repeated 2-person Prisoner's Dilemma.

View Article: PubMed Central - PubMed

Affiliation: Centro de Biologia Molecular e Ambiental da Universidade do Minho, Braga, Portugal; INESC-ID & Instituto Superior Técnico, Universidade de Lisboa, Taguspark, Porto Salvo, Portugal; Centro de Física da Universidade do Minho, Braga, Portugal; ATP-group, CMAF, Instituto para a Investigação Interdisciplinar, Lisboa, Portugal.

ABSTRACT
Many problems of cooperation involve repeated interactions among the same groups of individuals. When collective action is at stake, groups often engage in Public Goods Games (PGG), where individuals contribute (or not) to a common pool, subsequently sharing the resources. Such scenarios of repeated group interactions materialize situations in which direct reciprocation to groups may be at work. Here we study direct group reciprocity considering the complete set of reactive strategies, where individuals behave conditionally on what they observed in the previous round. We study both analytically and by computer simulations the evolutionary dynamics encompassing this extensive strategy space, witnessing the emergence of a surprisingly simple strategy that we call All-Or-None (AoN). AoN consists in cooperating only after a round of unanimous group behavior (cooperation or defection), and proves robust in the presence of errors, thus fostering cooperation in a wide range of group sizes. The principles encapsulated in this strategy share a level of complexity reminiscent of that found already in 2-person games under direct and indirect reciprocity, reducing, in fact, to the well-known Win-Stay-Lose-Shift strategy in the limit of the repeated 2-person Prisoner's Dilemma.

Show MeSH