The basin of attraction of the local stable equilibrium

Evolutionary stability in the asymmetric volunteer's dilemma. He JZ, Wang RW, Li YT - PLoS ONE (2014) Bottom Line: A further analysis revealed that the strong player adopts a pure strategy but mixed strategies for the weak players to produce the public good, and that the probability of volunteering by weak players decreases with increasing group size or decreasing cost-benefit ratio.Our model shows that the defection probability of a "strong" player is greater than the "weak" players in the model of Diekmann (1993).Our public good model with ESS has thus extended previous interpretations that the public good can only be produced by strong players in an asymmetric game. View Article: PubMed Central - PubMed Affiliation: Statistics and Mathematics College, Yunnan University of Finance and Economics, Kunming, Yunnan, China; State Key Laboratory of Genetic Resources and Evolution, Kunming Institute of Zoology, Chinese Academy of Science, Kunming, Yunnan, China. ABSTRACT It is often assumed that in public goods games, contributors are either strong or weak players and each individual has an equal probability of exhibiting cooperation. It is difficult to explain why the public good is produced by strong individuals in some cooperation systems, and by weak individuals in others. Viewing the asymmetric volunteer's dilemma game as an evolutionary game, we find that whether the strong or the weak players produce the public good depends on the initial condition (i.e., phenotype or initial strategy of individuals). These different evolutionarily stable strategies (ESS) associated with different initial conditions, can be interpreted as the production modes of public goods of different cooperation systems. A further analysis revealed that the strong player adopts a pure strategy but mixed strategies for the weak players to produce the public good, and that the probability of volunteering by weak players decreases with increasing group size or decreasing cost-benefit ratio. Our model shows that the defection probability of a "strong" player is greater than the "weak" players in the model of Diekmann (1993). This contradicts Selten's (1980) model that public goods can only be produced by a strong player, is not an evolutionarily stable strategy, and will therefore disappear over evolutionary time. Our public good model with ESS has thus extended previous interpretations that the public good can only be produced by strong players in an asymmetric game. Show MeSH Major Biological Evolution* Game Theory* Models, Theoretical* Volunteers/psychology* Minor Cooperative Behavior Humans Probability Related in: MedlinePlus	© Copyright Policy Related In: Results - Collection License Show All Figures getmorefigures.php?uid=PMC4128801&req=5 pone-0103931-g002: The basin of attraction of the local stable equilibrium points A (“shadow region”) and E (“blank region”).The parameters are fixed at and . Mentions: By analyzing the property of the matrices' eigenvalues, produced by the linearization of the replicator dynamics at these six equilibrium points (see Appendix S1, Figure 1 and Figure 2), we find that and are sinks (stable), and are sources (unstable), while and are saddles (unstable) [41], [48]. It is also noteworthy that the stability of equilibrium point depends on the special condition, that is, the equilibrium point is only stable if and only if the cost-benefit ratio of the “strong/weak” player and the group size satisfy inequality (see Figure 3 and Appendix S1). Furthermore, the rest point exists in the interior of the unit square if and only if this same inequality holds and the group size N is not too large (i.e. ). Thus, the replicator dynamics of the game are illustrated in two evolutionary stability points and (Figure 1).

Evolutionary stability in the asymmetric volunteer's dilemma.

He JZ, Wang RW, Li YT - PLoS ONE (2014)

Bottom Line: A further analysis revealed that the strong player adopts a pure strategy but mixed strategies for the weak players to produce the public good, and that the probability of volunteering by weak players decreases with increasing group size or decreasing cost-benefit ratio.Our model shows that the defection probability of a "strong" player is greater than the "weak" players in the model of Diekmann (1993).Our public good model with ESS has thus extended previous interpretations that the public good can only be produced by strong players in an asymmetric game.

View Article: PubMed Central - PubMed

Affiliation: Statistics and Mathematics College, Yunnan University of Finance and Economics, Kunming, Yunnan, China; State Key Laboratory of Genetic Resources and Evolution, Kunming Institute of Zoology, Chinese Academy of Science, Kunming, Yunnan, China.

ABSTRACT

It is often assumed that in public goods games, contributors are either strong or weak players and each individual has an equal probability of exhibiting cooperation. It is difficult to explain why the public good is produced by strong individuals in some cooperation systems, and by weak individuals in others. Viewing the asymmetric volunteer's dilemma game as an evolutionary game, we find that whether the strong or the weak players produce the public good depends on the initial condition (i.e., phenotype or initial strategy of individuals). These different evolutionarily stable strategies (ESS) associated with different initial conditions, can be interpreted as the production modes of public goods of different cooperation systems. A further analysis revealed that the strong player adopts a pure strategy but mixed strategies for the weak players to produce the public good, and that the probability of volunteering by weak players decreases with increasing group size or decreasing cost-benefit ratio. Our model shows that the defection probability of a "strong" player is greater than the "weak" players in the model of Diekmann (1993). This contradicts Selten's (1980) model that public goods can only be produced by a strong player, is not an evolutionarily stable strategy, and will therefore disappear over evolutionary time. Our public good model with ESS has thus extended previous interpretations that the public good can only be produced by strong players in an asymmetric game.

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Related in: MedlinePlus

The basin of attraction of the local stable equilibrium points A (“shadow region”) and E (“blank region”).The parameters are fixed at and .

Related In: Results - Collection

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pone-0103931-g002: The basin of attraction of the local stable equilibrium points A (“shadow region”) and E (“blank region”).The parameters are fixed at and .

Mentions: By analyzing the property of the matrices' eigenvalues, produced by the linearization of the replicator dynamics at these six equilibrium points (see Appendix S1, Figure 1 and Figure 2), we find that and are sinks (stable), and are sources (unstable), while and are saddles (unstable) [41], [48]. It is also noteworthy that the stability of equilibrium point depends on the special condition, that is, the equilibrium point is only stable if and only if the cost-benefit ratio of the “strong/weak” player and the group size satisfy inequality (see Figure 3 and Appendix S1). Furthermore, the rest point exists in the interior of the unit square if and only if this same inequality holds and the group size N is not too large (i.e. ). Thus, the replicator dynamics of the game are illustrated in two evolutionary stability points and (Figure 1).