Limits...
Beyond pairwise strategy updating in the prisoner's dilemma game.

Wang X, Perc M, Liu Y, Chen X, Wang L - Sci Rep (2012)

Bottom Line: In spatial games players typically alter their strategy by imitating the most successful or one randomly selected neighbor.This leads to phase diagrams that differ significantly from those obtained by means of pairwise strategy updating.In particular, the survivability of cooperators is possible even by high temptations to defect and over a much wider uncertainty range.

View Article: PubMed Central - PubMed

Affiliation: Center for Complex Systems, Xidian University, Xi'an 710071, China.

ABSTRACT
In spatial games players typically alter their strategy by imitating the most successful or one randomly selected neighbor. Since a single neighbor is taken as reference, the information stemming from other neighbors is neglected, which begets the consideration of alternative, possibly more realistic approaches. Here we show that strategy changes inspired not only by the performance of individual neighbors but rather by entire neighborhoods introduce a qualitatively different evolutionary dynamics that is able to support the stable existence of very small cooperative clusters. This leads to phase diagrams that differ significantly from those obtained by means of pairwise strategy updating. In particular, the survivability of cooperators is possible even by high temptations to defect and over a much wider uncertainty range. We support the simulation results by means of pair approximations and analysis of spatial patterns, which jointly highlight the importance of local information for the resolution of social dilemmas.

Show MeSH

Related in: MedlinePlus

Schematic presentation of two representative cooperative (blue) clusters surrounded by defectors (red).The cluster depicted left has no chances of survival under pairwise or locally influenced strategy updating. The cluster on the right, however, cannot prevail under pairwise imitation, but can do so under locally influenced strategy updating. This is because the core of the cooperative cluster (C1 in the figure) is quarantined from defectors in case imitation proceeds according to local influence (see main text for details).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC3472391&req=5

f5: Schematic presentation of two representative cooperative (blue) clusters surrounded by defectors (red).The cluster depicted left has no chances of survival under pairwise or locally influenced strategy updating. The cluster on the right, however, cannot prevail under pairwise imitation, but can do so under locally influenced strategy updating. This is because the core of the cooperative cluster (C1 in the figure) is quarantined from defectors in case imitation proceeds according to local influence (see main text for details).

Mentions: To confirm these conjectures, we present in Fig. 5 two typical C-cluster configurations and analyze the survivability of cooperators separately for each particular case. For the sake of simplicity but without loss of generality, we consider for the following analysis only the K → 0 limit. Then if the payoff of each cooperator along the boundary is larger than that of each defector in its neighborhood, we are allowed to conclude that such a C-cluster will survive. For the left C-cluster pattern in Fig. 5 under pairwise updating, the payoffs of a cooperator C (PC) and defector D (PD) along the boundary are respectively. For locally influenced updating, however, the average payoff of cooperators () and the average payoff of defectors () along the boundary are given by respectively. Thus for such a C-cluster pattern to survive, both update rules lead to r < –0.25. Indeed, neither locally influenced nor pairwise strategy updating support the survivability of such a pattern. Performing the same analysis for the configuration on the right, however, yields a different outcome. The payoff of a cooperator C2 () on the boundary and that of the two types of defectors D1 and D2 ( and ) are respectively. For locally influenced updating the corresponding payoffs are Accordingly, we find that under pairwise updating the condition for survivability is r < –0.25, while under locally influenced updating it is only . Hence, locally influenced strategy updating can warrant the survivability of cooperators when grouped in this way, while pairwise updating can not. Note also that the C-cluster configuration on the right of Fig. 5 is the smallest one which can persist in the population under the most hostile conditions under locally influenced strategy updating. Based on this analysis, we can in fact estimate the extinction threshold in the limit K → 0, and indeed we find excellent agreement between this analytical approximation and the simulation results presented in Fig. 2(c).


Beyond pairwise strategy updating in the prisoner's dilemma game.

Wang X, Perc M, Liu Y, Chen X, Wang L - Sci Rep (2012)

Schematic presentation of two representative cooperative (blue) clusters surrounded by defectors (red).The cluster depicted left has no chances of survival under pairwise or locally influenced strategy updating. The cluster on the right, however, cannot prevail under pairwise imitation, but can do so under locally influenced strategy updating. This is because the core of the cooperative cluster (C1 in the figure) is quarantined from defectors in case imitation proceeds according to local influence (see main text for details).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3472391&req=5

f5: Schematic presentation of two representative cooperative (blue) clusters surrounded by defectors (red).The cluster depicted left has no chances of survival under pairwise or locally influenced strategy updating. The cluster on the right, however, cannot prevail under pairwise imitation, but can do so under locally influenced strategy updating. This is because the core of the cooperative cluster (C1 in the figure) is quarantined from defectors in case imitation proceeds according to local influence (see main text for details).
Mentions: To confirm these conjectures, we present in Fig. 5 two typical C-cluster configurations and analyze the survivability of cooperators separately for each particular case. For the sake of simplicity but without loss of generality, we consider for the following analysis only the K → 0 limit. Then if the payoff of each cooperator along the boundary is larger than that of each defector in its neighborhood, we are allowed to conclude that such a C-cluster will survive. For the left C-cluster pattern in Fig. 5 under pairwise updating, the payoffs of a cooperator C (PC) and defector D (PD) along the boundary are respectively. For locally influenced updating, however, the average payoff of cooperators () and the average payoff of defectors () along the boundary are given by respectively. Thus for such a C-cluster pattern to survive, both update rules lead to r < –0.25. Indeed, neither locally influenced nor pairwise strategy updating support the survivability of such a pattern. Performing the same analysis for the configuration on the right, however, yields a different outcome. The payoff of a cooperator C2 () on the boundary and that of the two types of defectors D1 and D2 ( and ) are respectively. For locally influenced updating the corresponding payoffs are Accordingly, we find that under pairwise updating the condition for survivability is r < –0.25, while under locally influenced updating it is only . Hence, locally influenced strategy updating can warrant the survivability of cooperators when grouped in this way, while pairwise updating can not. Note also that the C-cluster configuration on the right of Fig. 5 is the smallest one which can persist in the population under the most hostile conditions under locally influenced strategy updating. Based on this analysis, we can in fact estimate the extinction threshold in the limit K → 0, and indeed we find excellent agreement between this analytical approximation and the simulation results presented in Fig. 2(c).

Bottom Line: In spatial games players typically alter their strategy by imitating the most successful or one randomly selected neighbor.This leads to phase diagrams that differ significantly from those obtained by means of pairwise strategy updating.In particular, the survivability of cooperators is possible even by high temptations to defect and over a much wider uncertainty range.

View Article: PubMed Central - PubMed

Affiliation: Center for Complex Systems, Xidian University, Xi'an 710071, China.

ABSTRACT
In spatial games players typically alter their strategy by imitating the most successful or one randomly selected neighbor. Since a single neighbor is taken as reference, the information stemming from other neighbors is neglected, which begets the consideration of alternative, possibly more realistic approaches. Here we show that strategy changes inspired not only by the performance of individual neighbors but rather by entire neighborhoods introduce a qualitatively different evolutionary dynamics that is able to support the stable existence of very small cooperative clusters. This leads to phase diagrams that differ significantly from those obtained by means of pairwise strategy updating. In particular, the survivability of cooperators is possible even by high temptations to defect and over a much wider uncertainty range. We support the simulation results by means of pair approximations and analysis of spatial patterns, which jointly highlight the importance of local information for the resolution of social dilemmas.

Show MeSH
Related in: MedlinePlus