On the coexistence of cooperators, defectors and conditional cooperators in the multiplayer iterated Prisoner's Dilemma

Recent experimental evidence [Gruji\'c et al., PLoS ONE 5, e13749 (2010)] on the spatial Prisoner's Dilemma suggests that players choosing to cooperate or not on the basis of their previous action and the actions of their neighbors coexist with steady defectors and cooperators. We here study the coexistence of these three strategies in the multiplayer iterated Prisoner's Dilemma by means of the replicator dynamics. We consider groups with n = 2, 3, 4 and 5 players and compute the payoffs to every type of player as the limit of a Markov chain where the transition probabilities between actions are found from the corresponding strategies. We show that for group sizes up to n = 4 there exists an interior point in which the three strategies coexist, the corresponding basin of attraction decreasing with increasing number of players, whereas we have not been able to locate such a point for n = 5. We analytically show that in the infinite n limit no interior points can arise. We conclude by discussing the implications of this theoretical approach on the behavior observed in experiments.Comment: 12 pages, 10 figures, uses elsart.cl

Micromagnetic Simulation of Nanoscale Films with Perpendicular Anisotropy

A model is studied for the theoretical description of nanoscale magnetic films with high perpendicular anisotropy. In the model the magnetic film is described in terms of single domain magnetic grains with Ising-like behavior, interacting via exchange as well as via dipolar forces. Additionally, the model contains an energy barrier and a coupling to an external magnetic field. Disorder is taken into account in order to describe realistic domain and domain wall structures. The influence of a finite temperature as well as the dynamics can be modeled by a Monte Carlo simulation. Many of the experimental findings can be investigated and at least partly understood by the model introduced above. For thin films the magnetisation reversal is driven by domain wall motion. The results for the field and temperature dependence of the domain wall velocity suggest that for thin films hysteresis can be described as a depinning transition of the domain walls rounded by thermal activation for finite temperatures.Comment: Revtex, Postscript Figures, to be published in J. Appl.Phy

Quasiorders, Tolerance Relations and Corresponding “Partitions”

The paper deals with a generalization of the notion of partition for wider classes of binary relations than equivalences: for quasiorders and tolerance relations. The counterpart of partition for the quasiorders is based on a generalization of the notion of equivalence class while it is shown that such a generalization does not work in case of tolerances. Some results from [5] are proved in a much more simple way. The third kind of “partition” corresponding to tolerances, not occurring in [5], is introduced