Search

The spin relaxation time in solids is determined by several competing energy scales and processes, and distinct methods are called for to analyze the various regimes. We present a stochastic model for the spin dynamics in solids which is equivalent to solving the spin Boltzmann equation and takes the relevant processes into account on an equal footing. The calculations reveal yet unknown parts of the spin-relaxation phase diagram, where strong reversible spin dephasing occurs in addition to spin relaxation. Spin-relaxation times are obtained for this regime by introducing the numerical Loschmidt echo. This allows us to construct a generic approximate formula for the spin-relaxation time, τ_{s}, for the entire phase diagram, involving the quasiparticle scattering rate, Γ, spin-orbit coupling strength, L, and a magnetic term, Δ_{Z} due to the Zeeman effect. The generic expression reads as ℏ/τ_{s}≈ΓL^{2}/(Γ^{2}+L^{2}+Δ_{Z}^{2}).

Economic experiments have shown that punishment can increase public goods game contributions over time. However, the effectiveness of punishment is challenged by second-order free-riding and antisocial punishment. The latter implies that noncooperators punish cooperators, while the former implies unwillingness to shoulder the cost of punishment. Here, we extend the theory of cooperation in the spatial public goods game by considering four competing strategies, which are traditional cooperators and defectors, as well as cooperators who punish defectors and defectors who punish cooperators. We show that if the synergistic effects are high enough to sustain cooperation based on network reciprocity alone, antisocial punishment does not deter public cooperation. Conversely, if synergistic effects are low and punishment is actively needed to sustain cooperation, antisocial punishment does is viable, but only if the cost-to-fine ratio is low. If the costs are relatively high, cooperation again dominates as a result of spatial pattern formation. Counterintuitively, defectors who do not punish cooperators, and are thus effectively second-order free-riding on antisocial punishment, form an active layer around punishing cooperators, which protects them against defectors that punish cooperators. A stable three-strategy phase that is sustained by the spontaneous emergence of cyclic dominance is also possible via the same route. The microscopic mechanism behind the reported evolutionary outcomes can be explained by the comparison of invasion rates that determine the stability of subsystem solutions. Our results reveal an unlikely evolutionary escape from adverse effects of antisocial punishment, and they provide a rationale for why second-order free-riding is not always an impediment to the evolutionary stability of punishment.

Economic experiments reveal that humans value cooperation and fairness. Punishing unfair behavior is therefore common, and according to the theory of strong reciprocity, it is also directly related to rewarding cooperative behavior. However, empirical data fail to confirm that positive and negative reciprocity are correlated. Inspired by this disagreement, we determine whether the combined application of reward and punishment is evolutionarily advantageous. We study a spatial public goods game, where in addition to the three elementary strategies of defection, rewarding, and punishment, a fourth strategy that combines the latter two competes for space. We find rich dynamical behavior that gives rise to intricate phase diagrams where continuous and discontinuous phase transitions occur in succession. Indirect territorial competition, spontaneous emergence of cyclic dominance, as well as divergent fluctuations of oscillations that terminate in an absorbing phase are observed. Yet, despite the high complexity of solutions, the combined strategy can survive only in very narrow and unrealistic parameter regions. Elementary strategies, either in pure or mixed phases, are much more common and likely to prevail. Our results highlight the importance of patterns and structure in human cooperation, which should be considered in future experiments.

Leaving the joint enterprise when defection is unveiled is always a viable option to avoid being exploited. Although loner strategy helps the population not to be trapped into the tragedy of the commons state, it could offer only a modest income for nonparticipants. In this paper we demonstrate that showing some tolerance toward defectors could not only save cooperation in harsh environments but in fact results in a surprisingly high average payoff for group members in public goods games. Phase diagrams and the underlying spatial patterns reveal the high complexity of evolving states where cyclic dominant strategies or two-strategy alliances can characterize the final state of evolution. We identify microscopic mechanisms which are responsible for the superiority of global solutions containing tolerant players. This phenomenon is robust and can be observed both in well-mixed and in structured populations highlighting the importance of tolerance in our everyday life. [ABSTRACT FROM AUTHOR]

Inspired by the fact that people have diverse propensities to punish wrongdoers, we study a spatial public goods game with defectors and different types of punishing cooperators. During the game, cooperators punish defectors with class-specific probabilities and subsequently share the associated costs of sanctioning. We show that in the presence of different punishing cooperators the highest level of public cooperation is always attainable through a selection mechanism. Interestingly, the selection does not necessarily favor the evolution of punishers who would be able to prevail on their own against the defectors, nor does it always hinder the evolution of punishers who would be unable to prevail on their own. Instead, the evolutionary success of punishing strategies depends sensitively on their invasion velocities, which in turn reveals fascinating examples of both competition and cooperation among them. Furthermore, we show that under favorable conditions, when punishment is not strictly necessary for the maintenance of public cooperation, the less aggressive, mild form of sanctioning is the sole victor of the selection process. Our work reveals that natural strategy selection cannot only promote, but sometimes also hinders competition among prosocial strategies. [ABSTRACT FROM AUTHOR]

Evolutionary social dilemma games are extended by an additional matching-pennies game that modifies the collected payoffs. In a spatial version players are distributed on a square lattice and interact with their neighbors. First, we show that the matching-pennies game can be considered as the microscopic force of the Red Queen effect that breaks the detailed balance and induces eddies in the microscopic probability currents if the strategy update is analogous to the Glauber dynamics for the kinetic Ising models. The resulting loops in probability current breaks symmetry between the chessboardlike arrangements of strategies via a bottleneck effect occurring along the four-edge loops in the microscopic states. The impact of this congestion is analogous to the application of a staggered magnetic field in the Ising model; that is, the order-disorder critical transition is wiped out by noise. It is illustrated that the congestion induced symmetry breaking can be beneficial for the whole community within a certain region of parameters. [ABSTRACT FROM AUTHOR]

We study the rock-paper-scissors game in structured populations, where the invasion rates determine individual payoffs that govern the process of strategy change. The traditional version of the game is recovered if the payoffs for each potential invasion stem from a single pairwise interaction. However, the transformation of invasion rates to payoffs also allows the usage of larger interaction ranges. In addition to the traditional pairwise interaction, we therefore consider simultaneous interactions with all nearest neighbors, as well as with all nearest and next-nearest neighbors, thus effectively going from single pair to group interactions in games of cyclic dominance. We show that differences in the interaction range affect not only the stationary fractions of strategies but also their relations of dominance. The transition from pairwise to group interactions can thus decelerate and even revert the direction of the invasion between the competing strategies. Like in evolutionary social dilemmas, in games of cyclic dominance, too, the indirect multipoint interactions that are due to group interactions hence play a pivotal role. Our results indicate that, in addition to the invasion rates, the interaction range is at least as important for the maintenance of biodiversity among cyclically competing strategies [ABSTRACT FROM AUTHOR]

Reciprocity is firmly established as an important mechanism that promotes cooperation. An efficient information exchange is likewise important, especially on structured populations, where interactions between players are limited. Motivated by these two facts, we explore the role of facilitators in social dilemmas on networks. Facilitators are here mirrors to their neighbors--they cooperate with cooperators and defect with defectors--but they do not participate in the exchange of strategies. As such, in addition to introducing direct reciprocity, they also obstruct information exchange. In well-mixed populations, facilitators favor the replacement and invasion of defection by cooperation as long as their number exceeds a critical value. In structured populations, on the other hand, there exists a delicate balance between the benefits of reciprocity and the deterioration of information exchange. Extensive Monte Carlo simulations of social dilemmas on various interaction networks reveal that there exists an optimal interplay between reciprocity and information exchange, which sets in only when a small number of facilitators occupy the main hubs of the scale-free network. The drawbacks of missing cooperative hubs are more than compensated for by reciprocity and, at the same time, the compromised information exchange is routed via the auxiliary hubs with only marginal losses in effectivity. These results indicate that it is not always optimal for the main hubs to become leaders of the masses, but rather to exploit their highly connected state to promote tit-for-tat-like behavior. [ABSTRACT FROM AUTHOR]

Monod and Beuneu [P. Monod and F. Beuneu, Phys. Rev. B 19, 911 (1979)] established the validity of the Elliott-Yafet theory for elemental metals through correlating the experimental electron spin resonance linewidth with the so-called spin-orbit admixture coefficients and the momentum-relaxation theory. The spin-orbit admixture coefficients data were based on atomic spin-orbit splitting. We highlight two shortcomings of the previous description: (i) the momentum-relaxation involves the Debye temperature and the electron-phonon coupling whose variation among the elemental metals was neglected, (ii) the Elliott-Yafet theory involves matrix elements of the spin-orbit coupling (SOC), which are however not identical to the SOC induced energy splitting of the atomic levels, even though the two have similar magnitudes. We obtain the empirical spin-orbit admixture parameters for the alkali metals by considering the proper description of the momentum relaxation theory. In addition we present a model calculation, which highlights the difference between the SOC matrix element and energy splitting. [ABSTRACT FROM AUTHOR]

Extortion strategies can dominate any opponent in an iterated prisoner's dilemma game. But if players are able to adopt the strategies performing better, extortion becomes widespread and evolutionary unstable. It may sometimes act as a catalyst for the evolution of cooperation, and it can also emerge in interactions between two populations, yet it is not the evolutionarily stable outcome. Here we revisit these results in the realm of spatial games. We find that pairwise imitation and birth-death dynamics return known evolutionary outcomes. Myopic best response strategy updating, on the other hand, reveals counterintuitive solutions. Defectors and extortioners coarsen spontaneously, which allows cooperators to prevail even at prohibitively high temptations to defect. Here extortion strategies play the role of a Trojan horse. They may emerge among defectors by chance, and once they do, cooperators become viable as well. These results are independent of the interaction topology, and they highlight the importance of coarsening, checkerboard ordering, and best response updating in evolutionary games. [ABSTRACT FROM AUTHOR]

A five-species predator-prey model is studied on a square lattice where each species has two prey and two predators on the analogy to the rock-paper-scissors-lizard-Spock game. The evolution of the spatial distribution of species is governed by site exchange and invasion between the neighboring predator-prey pairs, where the cyclic symmetry can be characterized by two different invasion rates. The mean-field analysis has indicated periodic oscillations in the species densities with a frequency becoming zero for a specific ratio of invasion rates. When varying the ratio of invasion rates, the appearance of this zero-eigenvalue mode is accompanied by neutrality between the species associations. Monte Carlo simulations of the spatial system reveal diverging fluctuations at a specific invasion rate, which can be related to the vanishing dominance between all pairs of species associations. [ABSTRACT FROM AUTHOR]

We study the evolution of cooperation in the spatial public goods game, focusing on the effects that are brought about by the delayed distribution of goods that accumulate in groups due to the continuous investments of cooperators. We find that intermediate delays enhance network reciprocity because of a decelerated invasion of defectors, who are unable to reap the same high short-term benefits as they do in the absence of delayed distribution. Long delays, however, introduce a risk because the large accumulated wealth might fall into the wrong hands. Indeed, as soon as the curvature of a cooperative cluster turns negative, the engulfed defectors can collect the heritage of many generations of cooperators and by doing so start a waning-moon pattern that nullifies the benefits of decelerated invasion. Accidental meeting points of growing cooperative clusters may also act as triggers for the waning-moon effect, thus linking the success of cooperators with their propensity to fail in a rather bizarre way. Our results highlight that "investing in the future" is a good idea only if that future is sufficiently near and not likely to be burdened by inflation. [ABSTRACT FROM AUTHOR]

We show that the resolution of social dilemmas in random graphs and scale-free networks is facilitated by imitating not the strategy of better-performing players but, rather, their emotions. We assume sympathy and envy to be the two emotions that determine the strategy of each player in any given interaction, and we define them as the probabilities of cooperating with players having a lower and a higher payoff, respectively. Starting with a population where all possible combinations of the two emotions are available, the evolutionary process leads to a spontaneous fixation to a single emotional profile that is eventually adopted by all players. However, this emotional profile depends not only on the payoffs but also on the heterogeneity of the interaction network. Homogeneous networks, such as lattices and regular random graphs, lead to fixations that are characterized by high sympathy and high envy, while heterogeneous networks lead to low or modest sympathy but also low envy. Our results thus suggest that public emotions and the propensity to cooperate at large depend, and are in fact determined by, the properties of the interaction network. [ABSTRACT FROM AUTHOR]

A collective-risk social dilemma implies that personal endowments will be lost if contributions to the common pool within a group are too small. Failure to reach the collective target thus has dire consequences for all group members, independently of their strategies. Wanting to move away from unfavorable locations is therefore anything but surprising. Inspired by these observations, we here propose and study a collective-risk social dilemma where players are allowed to move if the collective failure becomes too probable. More precisely, this so-called risk-driven migration is launched depending on the difference between the actual contributions and the declared target. Mobility therefore becomes an inherent property that is utilized in an entirely self-organizing manner. We show that under these assumptions cooperation is promoted much more effectively than under the action of manually determined migration rates. For the latter, we in fact identify parameter regions where the evolution of cooperation is greatly inhibited. Moreover, we find unexpected spatial patterns where cooperators that do not form compact clusters outperform those that do, and where defectors are able to utilize strikingly different ways of invasion. The presented results support the recently revealed importance of percolation for the successful evolution of public cooperation, while at the same time revealing surprisingly simple methods of self-organization towards socially desirable states. [ABSTRACT FROM AUTHOR]

Experiments on the ultimatum game have revealed that humans are remarkably fond of fair play. When asked to share an amount of money, unfair offers are rare and their acceptance rate small. While empathy and spatiality may lead to the evolution of fairness, thus far considered continuous strategies have precluded the observation of solutions that would be driven by pattern formation. Here we introduce a spatial ultimatum game with discrete strategies, and we show that this simple alteration opens the gate to fascinatingly rich dynamical behavior. In addition to mixed stationary states, we report the occurrence of traveling waves and cyclic dominance, where one strategy in the cycle can be an alliance of two strategies. The highly webbed phase diagram, entailing continuous and discontinuous phase transitions, reveals hidden complexity in the pursuit of human fair play. [ABSTRACT FROM AUTHOR]

Cooperation and defection may be considered to be two extreme responses to a social dilemma. Yet the reality is much less clear-cut. Between the two extremes lies an interval of ambivalent choices, which may be captured theoretically by means of continuous strategies defining the extent of the contributions of each individual player to the common pool. If strategies are chosen from the unit interval, where 0 corresponds to pure defection and 1 corresponds to the maximal contribution, the question is what is the characteristic level of individual investments to the common pool that emerges if the evolution is guided by different benefit functions. Here we consider the steepness and the threshold as two parameters defining an array of generalized benefit functions, and we show that in a structured population there exist intermediate values of both at which the collective contributions are maximal. However, as the cost-to-benefit ratio of cooperation increases, the characteristic threshold decreases while the corresponding steepness increases. Our observations remain valid if more complex sigmoid functions are used, thus reenforcing the importance of carefully adjusted benefits for high levels of public cooperation. [ABSTRACT FROM AUTHOR]

While worldwide census data provide statistical evidence that firmly link the population density with several indicators of social welfare, the precise mechanisms underlying these observations are largely unknown. Here we study the impact of population density on the evolution of public cooperation in structured populations and find that the optimal density is uniquely related to the percolation threshold of the host graph irrespective of its topological details. We explain our observations by showing that spatial reciprocity peaks in the vicinity of the percolation threshold, when the emergence of a giant cooperative cluster is hindered neither by vacancy nor by invading defectors, thus discovering an intuitive yet universal law that links the population density with social prosperity. [ABSTRACT FROM AUTHOR]

The fact that individuals will most likely behave differently in different situations begets the introduction of conditional strategies. Inspired by this, we study the evolution of cooperation in the spatial public goods game, where, besides unconditional cooperators and defectors, also different types of conditional cooperators compete for space. Conditional cooperators will contribute to the public good only if other players within the group are likely to cooperate as well but will withhold their contribution otherwise. Depending on the number of other cooperators that are required to elicit cooperation of a conditional cooperator, the latter can be classified in as many types as there are players within each group. We find that the most cautious cooperators, who require all other players within a group to be conditional cooperators, are the undisputed victors of the evolutionary process, even at very low synergy factors. We show that the remarkable promotion of cooperation is due primarily to the spontaneous emergence of quarantining of defectors, who become surrounded by conditional cooperators and are forced into isolated convex "bubbles" from which they are unable to exploit the public good. This phenomenon can be observed only in structured populations, thus adding to the relevance of pattern formation for the successful evolution of cooperation. [ABSTRACT FROM AUTHOR]

We study the evolution of cooperation in public goods games on the square lattice, focusing on the effects that are brought about by different sizes of groups where individuals collect their payoffs and search for potential strategy donors. We find that increasing the group size does not necessarily lead to mean-field behavior, as is traditionally observed for games governed by pairwise interactions, but rather that public cooperation may be additionally promoted by means of enhanced spatial reciprocity that sets in for very large groups. Our results highlight that the promotion of cooperation due to spatial interactions is not rooted solely in having restricted connections among players, but also in individuals having the opportunity to collect payoffs separately from their direct opponents. Moreover, in large groups the presence of a small number of defectors is bearable, which makes the mixed-phase region expand with increasing group size. Having a chance of exploiting distant players, however, offers defectors a different way to break the phalanx of cooperators and even to resurrect from small numbers to eventually completely invade the population. [ABSTRACT FROM AUTHOR]

We have studied the evolution of strategies in spatial public goods games where both individual (peer) and institutional (pool) punishments are present in addition to unconditional defector and cooperator strategies. The evolution of strategy distribution is governed by imitation based on the random sequential comparison of neighbors' payoff for a fixed level of noise. Using numerical simulations, we evaluate the strategy frequencies and phase diagrams when varying the synergy factor, punishment cost, and fine. Our attention is focused on two extreme cases describing all the relevant behaviors in such a complex system. According to our numerical data peer punishers prevail and control the system behavior in a large segments of parameters while pool punishers can only survive in the limit of weak peer punishment when a rich variety of solutions is observed. Paradoxically, the two types of punishment may extinguish each other's impact, resulting in the triumph of defectors. The technical difficulties and suggested methods are briefly discussed. [ABSTRACT FROM AUTHOR]

The efficiency of institutionalized punishment is studied by evaluating the stationary states in the spatial public goods game comprising unconditional defectors, cooperators, and cooperating pool punishers as the three competing strategies. Fines and costs of pool punishment are considered as the two main parameters determining the stationary distributions of strategies on the square lattice. Each player collects a payoff from five five-person public goods games, and the evolution of strategies is subsequently governed by imitation based on pairwise comparisons at a low level of noise. The impact of pool punishment on the evolution of cooperation in structured populations is significantly different from that reported previously for peer punishment. Representative phase diagrams reveal remarkably rich behavior, depending also on the value of the synergy factor that characterizes the efficiency of investments payed into the common pool. Besides traditional single- and two-strategy stationary states, a rock-paper-scissors type of cyclic dominance can emerge in strikingly different ways. [ABSTRACT FROM AUTHOR]

According to the evolutionary game theory principle, a strategy representing a higher payoff can spread among competitors. But there are cases when a player consistently overestimates or underestimates her own payoff, which undermines proper comparison. Interestingly, both underconfident and overconfident individuals are capable of elevating the cooperation level significantly. While former players stimulate a local coordination of strategies, the presence of overconfident individuals enhances the spatial reciprocity mechanism. In both cases the propagations of competing strategies are influenced in a biased way resulting in a cooperation supporting environment. These effects are strongly related to the nonlinear character of invasion probabilities which is a common and frequently observed feature of microscopic dynamics. [ABSTRACT FROM AUTHOR]

Spin relaxation is conventionally discussed using two different approaches for materials with and without inversion symmetry. The former is known as the Elliott-Yafet (EY) theory and for the latter the D'yakonov-Perel' (DP) theory applies. We discuss herein a simple and intuitive approach to demonstrate that the two seemingly disparate mechanisms are closely related. A compelling analogy between the respective Hamiltonians is presented, and that the usual derivation of spin-relaxation times, in the respective frameworks of the two theories, can be performed. The result also allows us to obtain less canonical spin-relaxation regimes, i.e. the generalization of the EY when the material has a large quasiparticle broadening, and the DP mechanism in ultrapure semiconductors. The method also allows a practical and intuitive numerical implementation of the spin-relaxation calculation, which is demonstrated for MgB2, which has anomalous spin-relaxation properties. [ABSTRACT FROM AUTHOR]

Detecting defection and alarming partners about the possible danger could be essential to avoid being exploited. This act, however, may require a huge individual effort from those who take this job, hence such a strategy seems to be unfavorable. But structured populations can provide an opportunity where a largely unselfish excluder strategy can form an effective alliance with other cooperative strategies, hence they can sweep out defection. Interestingly, this alliance is functioning even at the extremely high cost of exclusion where the sole application of an exclusion strategy would be harmful otherwise. These results may explain why the emergence of extreme selfless behavior is not necessarily against individual selection but could be the result of an evolutionary process. [ABSTRACT FROM AUTHOR]

In the traditional setup of the public goods game all players are involved in every available group and the mutual benefit is shared among competing cooperator and defector strategies. However, in real life situations the group formation of players could be more sophisticated because not all players are attractive enough for others to participate in a joint venture. What if only those players who are successful enough to the neighbors can initiate a group formation and establish a game? To elaborate this idea we employ a modified protocol and demonstrate that a carefully chosen threshold to establish a joint venture could efficiently improve the cooperation level even if the synergy factor would suggest a full defector state otherwise. The microscopic mechanism that is responsible for this effect is based on the asymmetric consequences of competing strategies: while the success of a cooperator provides a long-time well-being for the neighborhood, the temporary advantage of defection cannot be maintained if the protocol is based on the success of leaders. [ABSTRACT FROM AUTHOR]

The rock-paper-scissors game is a paradigmatic model for biodiversity, with applications ranging from microbial populations to human societies. Research has shown, however, that mobility jeopardizes biodiversity by promoting the formation of spiral waves, especially if there is no conservation law in place for the total number of competing players. First, we show that even if such a conservation law applies, mobility still jeopardizes biodiversity in the spatial rock-paper-scissors game if only a small fraction of links of the square lattice is randomly rewired. Secondly, we show that zealots are very effective in taming the amplitude of oscillations that emerge due to mobility and/or interaction randomness, and this regardless of whether the later is quenched or annealed. While even a tiny fraction of zealots brings significant benefits, at 5% occupancy zealots practically destroy all oscillations regardless of the intensity of mobility, and regardless of the type and strength of randomness in the interaction structure. Interestingly, by annealed randomness the impact of zealots is qualitatively the same as by mobility, which highlights that fast diffusion does not necessarily destroy the coexistence of species, and that zealotry thus helps to recover the stable mean-field solution. Our results strengthen the important role of zealots in models of cyclic dominance, and they reveal fascinating evolutionary outcomes in structured populations that are a unique consequence of such uncompromising behavior. [ABSTRACT FROM AUTHOR]

Rock-scissors-paper game, as the simplest model of intransitive relation between competing agents, is a frequently quoted model to explain the stable diversity of competitors in the race of surviving. When increasing the number of competitors we may face a novel situation because beside the mentioned unidirectional predator-prey-like dominance a balanced or peer relation can emerge between some competitors. By utilizing this possibility in the present work we generalize a four-state predator-prey-type model where we establish two groups of species labeled by even and odd numbers. In particular, we introduce different invasion probabilities between and within these groups, which results in a tunable intensity of bidirectional invasion among peer species. Our study reveals an exceptional richness of pattern formations where five quantitatively different phases are observed by varying solely the strength of the mentioned inner invasion. The related transition points can be identified with the help of appropriate order parameters based on the spatial autocorrelation decay, on the fraction of empty sites, and on the variance of the species density. Furthermore, the application of diverse, alliance-specific inner invasion rates for different groups may result in the extinction of the pair of species where this inner invasion is moderate. These observations highlight that beyond the well-known and intensively studied cyclic dominance there is an additional source of complexity of pattern formation that has not been explored earlier. [ABSTRACT FROM AUTHOR]

Temperature-dependent electron spin resonance (ESR) measurements are reported on stage-1 potassium-doped graphite, a model system of biased graphene. The ESR linewidth is nearly isotropic, and although the g factor has a sizable anisotropy, its majority is shown to arise due to macroscopic magnetization. Even though the homogeneous ESR Unewidth shows an unusual, nonlinear temperature dependence, it appears to be proportional to the resistivity which is a quadratic function of the temperature. These observations suggests the validity of the Elliott-Yafet relaxation mechanism in KC8 and allows us to place KC8 on the empirical Beuneu-Monod plot among ordinary elemental metals. [ABSTRACT FROM AUTHOR]

"Three is a crowd" is an old proverb that applies as much to social interactions as it does to frustrated configurations in statistical physics models. Accordingly, social relations within a triangle deserve special attention. With this motivation, we explore the impact of topological frustration on the evolutionary dynamics of the snowdrift game on a triangular lattice. This topology provides an irreconcilable frustration, which prevents anticoordination of competing strategies that would be needed for an optimal outcome of the game. By using different strategy updating protocols, we observe complex spatial patterns in dependence on payoff values that are reminiscent to a honeycomb-like organization, which helps to minimize the negative consequence of the topological frustration. We relate the emergence of these patterns to the microscopic dynamics of the evolutionary process, both by means of mean-field approximations and Monte Carlo simulations. For comparison, we also consider the same evolutionary dynamics on the square lattice, where of course the topological frustration is absent. However, with the deletion of diagonal links of the triangular lattice, we can gradually bridge the gap to the square lattice. Interestingly, in this case the level of cooperation in the system is a direct indicator of the level of topological frustration, thus providing a method to determine frustration levels in an arbitrary interaction network. [ABSTRACT FROM AUTHOR]

According to the evolutionary death-birth protocol, a player is chosen randomly to die and neighbors compete for the available position proportional to their fitness. Hence, the status of the focal player is completely ignored and has no impact on the strategy update. In this paper, we revisit and generalize this rule by introducing a weight factor to compare the payoff values of the focal and invading neighbors. By means of evolutionary graph theory, we analyze the model on joint transitive graphs to explore the possible consequences of the presence of a weight factor. We find that focal weight always hinders cooperation under weak selection strength. Surprisingly, the results show a nontrivial tipping point of the weight factor where the threshold of cooperation success shifts from positive to negative infinity. Once focal weight exceeds this tipping point, cooperation becomes unreachable. Our theoretical predictions are confirmed by Monte Carlo simulations on a square lattice of different sizes. We also verify the robustness of the conclusions to arbitrary two-player prisoner's dilemmas, to dispersal graphs with arbitrary edge weights, and to interaction and dispersal graphs overlapping arbitrarily.

Inspired by the fact that people have diverse propensities to punish wrongdoers, we study a spatial public goods game with defectors and different types of punishing cooperators. During the game, cooperators punish defectors with class-specific probabilities and subsequently share the associated costs of sanctioning. We show that in the presence of different punishing cooperators the highest level of public cooperation is always attainable through a selection mechanism. Interestingly, the selection does not necessarily favor the evolution of punishers who would be able to prevail on their own against the defectors, nor does it always hinder the evolution of punishers who would be unable to prevail on their own. Instead, the evolutionary success of punishing strategies depends sensitively on their invasion velocities, which in turn reveals fascinating examples of both competition and cooperation among them. Furthermore, we show that under favorable conditions, when punishment is not strictly necessary for the maintenance of public cooperation, the less aggressive, mild form of sanctioning is the sole victor of the selection process. Our work reveals that natural strategy selection cannot only promote, but sometimes also hinders competition among prosocial strategies.

Motivated by the fact that the same social dilemma can be perceived differently by different players, we here study evolutionary multigames in structured populations. While the core game is the weak prisoner's dilemma, a fraction of the population adopts either a positive or a negative value of the sucker's payoff, thus playing either the traditional prisoner's dilemma or the snowdrift game. We show that the higher the fraction of the population adopting a different payoff matrix the more the evolution of cooperation is promoted. The microscopic mechanism responsible for this outcome is unique to structured populations, and it is due to the payoff heterogeneity, which spontaneously introduces strong cooperative leaders that give rise to an asymmetric strategy imitation flow in favor of cooperation. We demonstrate that the reported evolutionary outcomes are robust against variations of the interaction network, and they also remain valid if players are allowed to vary which game they play over time. These results corroborate existing evidence in favor of heterogeneity-enhanced network reciprocity, and they reveal how different perceptions of social dilemmas may contribute to their resolution.

While worldwide census data provide statistical evidence that firmly link the population density with several indicators of social welfare, the precise mechanisms underlying these observations are largely unknown. Here we study the impact of population density on the evolution of public cooperation in structured populations and find that the optimal density is uniquely related to the percolation threshold of the host graph irrespective of its topological details. We explain our observations by showing that spatial reciprocity peaks in the vicinity of the percolation threshold, when the emergence of a giant cooperative cluster is hindered neither by vacancy nor by invading defectors, thus discovering an intuitive yet universal law that links the population density with social prosperity.

We have studied the impact of time-dependent learning capacities of players in the framework of spatial prisoner's dilemma game. In our model, this capacity of players may decrease or increase in time after strategy adoption according to a steplike function. We investigated both possibilities separately and observed significantly different mechanisms that form the stationary pattern of the system. The time decreasing learning activity helps cooperator domains to recover the possible intrude of defectors hence supports cooperation. In the other case the temporary restrained learning activity generates a cyclic dominance between defector and cooperator strategies, which helps to maintain the diversity of strategies via propagating waves. The results are robust and remain valid by changing payoff values, interaction graphs or functions characterizing time dependence of learning activity. Our observations suggest that dynamically generated mechanisms may offer alternative ways to keep cooperators alive even at very larger temptation to defect.

Evolutionary 2×2 games are studied with players located on a square lattice. During the evolution the randomly chosen neighboring players try to maximize their collective income by adopting a random strategy pair with a probability dependent on the difference of their summed payoffs between the final and initial states assuming quenched strategies in their neighborhood. In the case of the anticoordination game this system behaves like an antiferromagnetic kinetic Ising model. Within a wide region of social dilemmas this dynamical rule supports the formation of similar spatial arrangement of the cooperators and defectors ensuring the optimum total payoff if the temptation to choose defection exceeds a threshold value dependent on the sucker's payoff. The comparison of the results with those achieved for pairwise imitation and myopic strategy updates has indicated the relevant advantage of pairwise collective strategy update in the maintenance of cooperation.

We study the evolution of cooperation in spatial public goods games with four competing strategies: cooperators, defectors, punishing cooperators, and punishing defectors. To explore the robustness of the cooperation-promoting effect of costly punishment, besides the usual strategy adoption dynamics we also apply strategy mutations. As expected, frequent mutations create kind of well-mixed conditions, which support the spreading of defectors. However, when the mutation rate is small, the final stationary state does not significantly differ from the state of the mutation-free model, independently of the values of the punishment fine and cost. Nevertheless, the mutation rate affects the relaxation dynamics. Rare mutations can largely accelerate the spreading of costly punishment. This is due to the fact that the presence of defectors breaks the balance of power between both cooperative strategies, which leads to a different kind of dynamics.

We study the evolution of cooperation under the assumption that the collective benefits of group membership can only be harvested if the fraction of cooperators within the group, i.e., their critical mass, exceeds a threshold value. Considering structured populations, we show that a moderate fraction of cooperators can prevail even at very low multiplication factors if the critical mass is minimal. For larger multiplication factors, however, the level of cooperation is highest at an intermediate value of the critical mass. The latter is robust to variations of the group size and the interaction network topology. Applying the optimal critical mass threshold, we show that the fraction of cooperators in public goods games is significantly larger than in the traditional linear model, where the produced public good is proportional to the fraction of cooperators within the group.

We studied spatial Prisoner's Dilemma and Stag Hunt games where both the strategy distribution and the players' individual noise level could evolve to reach higher individual payoff. Players are located on the sites of different two-dimensional lattices and gain their payoff from games with their neighbors by choosing unconditional cooperation or defection. The way of strategy adoption can be characterized by a single K (temperaturelike) parameter describing how strongly adoptions depend on the payoff difference. If we start the system from a random strategy distribution with many different player specific K parameters, the simultaneous evolution of strategies and K parameters drives the system to a final stationary state where only one K value remains. In the coexistence phase of cooperator and defector strategies the surviving K parameter is in good agreement with the noise level that ensures the highest cooperation level if uniform K is supposed for all players. In this paper we give a thorough overview about the properties of this evolutionary process.

Evolutionary prisoner's dilemma games are studied with players located on square lattice and random regular graph defining four neighbors for each one. The players follow one of the three strategies: tit-for-tat, unconditional cooperation, and defection. The simplified payoff matrix is characterized by two parameters: the temptation b to choose defection and the cost c of inspection reducing the income of tit-for-tat. The strategy imitation from one of the neighbors is controlled by pairwise comparison at a fixed level of noise. Using Monte Carlo simulations and the extended versions of pair approximation we have evaluated the b-c phase diagrams indicating a rich plethora of phase transitions between stationary coexistence, absorbing, and oscillatory states, including continuous and discontinuous phase transitions. By reasonable costs the tit-for-tat strategy prevents extinction of cooperators across the whole span of b determining the prisoner's dilemma game, irrespective of the connectivity structure. We also demonstrate that the system can exhibit a repetitive succession of oscillatory and stationary states upon changing a single payoff value, which highlights the remarkable sensitivity of cyclical interactions on the parameters that define the strength of dominance.

We study the evolution of cooperation in public goods games on different regular graphs as a function of the noise level underlying strategy adoptions. We focus on the effects that are brought about by different group sizes of public goods games in which individuals participate, revealing that larger groups of players may induce qualitatively different behavior when approaching the deterministic limit of strategy adoption. While by pairwise interactions an intermediate uncertainty by strategy adoptions may ensure optimal conditions for the survival of cooperators at a specific graph topology, larger groups warrant this only in the vicinity of the deterministic limit independently from the underlying graph. These discrepancies are attributed to the indirect linkage of otherwise not directly connected players, which is brought about by joint memberships within the larger groups. Thus, we show that increasing the group size may introduce an effective transition of the interaction topology, and that the latter shapes the noise dependence of the evolution of cooperation in case of pairwise interactions only.

Aging is always present, tailoring our interactions with others, and postulating a finite lifespan during which we are able to exercise them. We consider the prisoner's dilemma game on a square lattice and examine how quenched age distributions and different aging protocols influence the evolution of cooperation when taking the life experience and knowledge accumulation into account as time passes. In agreement with previous studies, we find that a quenched assignment of age to players, introducing heterogeneity to the game, substantially promotes cooperative behavior. Introduction of aging and subsequent death as a coevolutionary process may act detrimental on cooperation but enhances it efficiently if the offspring of individuals that have successfully passed their strategy is considered newborn. We study resulting age distributions of players and show that the heterogeneity is vital-yet insufficient-for explaining the observed differences in cooperator abundance on the spatial grid. The unexpected increment of cooperation levels can be explained by a dynamical effect that has a highly selective impact on the propagation of cooperator and defector states.

We study a spatial two-strategy (cooperation and defection) prisoner's dilemma game with two types ( A and B ) of players located on the sites of a square lattice. The evolution of strategy distribution is governed by iterated strategy adoption from a randomly selected neighbor with a probability depending on the payoff difference and also on the type of the neighbor. The strategy adoption probability is reduced by a prefactor (w<1) from the players of type B . We consider the competition between two opposite effects when increasing the number of neighbors ( k=4 , 8, and 24). Within a range of the portion of influential players (type A ) the inhomogeneous activity in strategy transfer yields a relevant increase (dependent on w ) in the density of cooperators. The noise dependence of this phenomenon is also discussed by evaluating phase diagrams.

We study the evolution of cooperation within the spatial prisoner's dilemma game on a square lattice where a fraction of players mu can spread their strategy more easily than the rest due to a predetermined larger teaching capability. In addition, players characterized by the larger teaching capability are allowed to temporarily link with distant opponents of the same kind with probability p , thus introducing shortcut connections among the distinguished players. We show that these additional temporary connections are able to sustain cooperation throughout the whole range of the temptation to defect. Remarkably, we observe that, as the temptation to defect increases the optimal mu decreases, and moreover only minute values of p warrant the best promotion of cooperation. Our study thus indicates that influential individuals must be few and sparsely connected in order for cooperation to thrive in a defection-prone environment.

Formation and competition of associations are studied in a six-species ecological model where each species has two predators and two prey. Each site of a square lattice is occupied by an individual belonging to one of the six species. The evolution of the spatial distribution of species is governed by iterated invasions between the neighboring predator-prey pairs with species specific rates and by site exchange between the neutral pairs with a probability X . This dynamical rule yields the formation of five associations composed of two or three species with proper spatiotemporal patterns. For large X a cyclic dominance can occur between the three two-species associations whereas one of the two three-species associations prevails in the whole system for low values of X in the final state. Within an intermediate range of X all the five associations coexist due to the fact that cyclic invasions between the two-species associations reduce their resistance temporarily against the invasion of three-species associations.

Maintenance of cooperation was studied for a two-strategy evolutionary prisoner's dilemma game where the players are located on a one-dimensional chain and their payoff comes from games with the nearest- and next-nearest-neighbor interactions. The applied host geometry makes it possible to study the impacts of two conflicting topological features. The evolutionary rule involves some noise affecting the strategy adoptions between the interacting players. Using Monte Carlo simulations and the extended versions of dynamical mean-field theory we determined the phase diagram as a function of noise level and a payoff parameter. The peculiar feature of the diagram is changed significantly when the connectivity structure is extended by extra links as suggested by Newman and Watts.

The diversity in wealth and social status is present not only among humans, but throughout the animal world. We account for this observation by generating random variables that determine the social diversity of players engaging in the prisoner's dilemma game. Here the term social diversity is used to address extrinsic factors that determine the mapping of game payoffs to individual fitness. These factors may increase or decrease the fitness of a player depending on its location on the spatial grid. We consider different distributions of extrinsic factors that determine the social diversity of players, and find that the power-law distribution enables the best promotion of cooperation. The facilitation of the cooperative strategy relies mostly on the inhomogeneous social state of players, resulting in the formation of cooperative clusters which are ruled by socially high-ranking players that are able to prevail against the defectors even when there is a large temptation to defect. To confirm this, we also study the impact of spatially correlated social diversity and find that cooperation deteriorates as the spatial correlation length increases. Our results suggest that the distribution of wealth and social status might have played a crucial role by the evolution of cooperation amongst egoistic individuals.

Cyclic predator-prey models with four or six species are studied on a square lattice when the invasion rates are varied. It is found that the cyclic invasions maintain a self-organizing pattern as long as the deviation of the invasion rate(s) from a uniform value does not exceed a threshold value. For larger deviations, the system exhibits a continuous phase transition into a frozen distribution of odd (or even) label species.

We study a spatial cyclic predator-prey model with an even number of species (for n=4, 6, and 8) that allows the formation of two defensive alliances consisting of the even and odd label species. The species are distributed on the sites of a square lattice. The evolution of spatial distribution is governed by iteration of two elementary processes on neighboring sites chosen randomly: if the sites are occupied by a predator-prey pair then the predator invades the prey's site; otherwise the species exchange their sites with a probability X . For low X values, a self-organizing pattern is maintained by cyclic invasions. If X exceeds a threshold value, then two types of domain grow up that are formed by the odd and even label species, respectively. Monte Carlo simulations indicate the blocking of this segregation process within a range of X for n=8.