Dynamical Organization of Cooperation in Complex Topologies

In this Letter, we study how cooperation is organized in complex topologies by analyzing the evolutionary (replicator) dynamics of the Prisoner's Dilemma, a two-players game with two available strategies, defection and cooperation, whose payoff matrix favors defection. We show that, asymptotically, the population is partitioned into three subsets: individuals that always cooperate ({\em pure cooperators}), always defect ({\em pure defectors}) and those that intermittently change their strategy. In fact the size of the latter set is the biggest for a wide range of the "stimulus to defect" parameter. While in homogeneous random graphs pure cooperators are grouped into several clusters, in heterogeneous scale-free (SF) networks they always form a single cluster containing the most connected individuals (hubs). Our results give further insights into why cooperation in SF networks is favored.Comment: 4 pages and 4 figures. Final version as published in Physical Review Letter

Evolutionary Prisoner's Dilemma game on the Newman-Watts networks

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 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.Comment: 4 figure

Restricted connections among distinguished players support cooperation

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 with 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. 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.Comment: 6 two-column pages, 6 figures; accepted for publication in Physical Review

Evolutionary Dynamics of Populations with Conflicting Interactions: Classification and Analytical Treatment Considering Asymmetry and Power

Evolutionary game theory has been successfully used to investigate the dynamics of systems, in which many entities have competitive interactions. From a physics point of view, it is interesting to study conditions under which a coordination or cooperation of interacting entities will occur, be it spins, particles, bacteria, animals, or humans. Here, we analyze the case, where the entities are heterogeneous, particularly the case of two populations with conflicting interactions and two possible states. For such systems, explicit mathematical formulas will be determined for the stationary solutions and the associated eigenvalues, which determine their stability. In this way, four different types of system dynamics can be classified, and the various kinds of phase transitions between them will be discussed. While these results are interesting from a physics point of view, they are also relevant for social, economic, and biological systems, as they allow one to understand conditions for (1) the breakdown of cooperation, (2) the coexistence of different behaviors ("subcultures"), (2) the evolution of commonly shared behaviors ("norms"), and (4) the occurrence of polarization or conflict. We point out that norms have a similar function in social systems that forces have in physics

Impact of aging on the evolution of cooperation in the spatial prisoner's dilemma game

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.Comment: 7 two-column pages, 5 figures; accepted for publication in Physical Review

Adaptive Investment Strategies For Periodic Environments

In this paper, we present an adaptive investment strategy for environments with periodic returns on investment. In our approach, we consider an investment model where the agent decides at every time step the proportion of wealth to invest in a risky asset, keeping the rest of the budget in a risk-free asset. Every investment is evaluated in the market via a stylized return on investment function (RoI), which is modeled by a stochastic process with unknown periodicities and levels of noise. For comparison reasons, we present two reference strategies which represent the case of agents with zero-knowledge and complete-knowledge of the dynamics of the returns. We consider also an investment strategy based on technical analysis to forecast the next return by fitting a trend line to previous received returns. To account for the performance of the different strategies, we perform some computer experiments to calculate the average budget that can be obtained with them over a certain number of time steps. To assure for fair comparisons, we first tune the parameters of each strategy. Afterwards, we compare the performance of these strategies for RoIs with different periodicities and levels of noise.Comment: Paper submitted to Advances in Complex Systems (November, 2007) 22 pages, 9 figure

Prisoner's dilemma in structured scale-free networks

The conventional wisdom is that scale-free networks are prone to cooperation spreading. In this paper we investigate the cooperative behaviors on the structured scale-free network. On the contrary of the conventional wisdom that scale-free networks are prone to cooperation spreading, the evolution of cooperation is inhibited on the structured scale-free network while performing the prisoner's dilemma (PD) game. Firstly, we demonstrate that neither the scale-free property nor the high clustering coefficient is responsible for the inhibition of cooperation spreading on the structured scale-free network. Then we provide one heuristic method to argue that the lack of age correlations and its associated `large-world' behavior in the structured scale-free network inhibit the spread of cooperation. The findings may help enlighten further studies on evolutionary dynamics of the PD game in scale-free networks.Comment: Definitive version accepted for publication in Journal of Physics

Similarity based cooperation and spatial segregation

We analyze a cooperative game, where the cooperative act is not based on the previous behaviour of the co-player, but on the similarity between the players. This system has been studied in a mean-field description recently [A. Traulsen and H. G. Schuster, Phys. Rev. E 68, 046129 (2003)]. Here, the spatial extension to a two-dimensional lattice is studied, where each player interacts with eight players in a Moore neighborhood. The system shows a strong segregation independent on parameters. The introduction of a local conversion mechanism towards tolerance allows for four-state cycles and the emergence of spiral waves in the spatial game. In the case of asymmetric costs of cooperation a rich variety of complex behavior is observed depending on both cooperation costs. Finally, we study the stabilization of a cooperative fixed point of a forecast rule in the symmetric game, which corresponds to cooperation across segregation borders. This fixed point becomes unstable for high cooperation costs, but can be stabilized by a linear feedback mechanism.Comment: 7 pages, 9 figure

Ordering in spatial evolutionary games for pairwise collective strategy updates

Evolutionary $2 \times 2$ 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 state assuming quenched strategies in their neighborhood. In the case of the anti-coordination game this system behaves alike an anti-ferromagnetic 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.Comment: 9 pages, 6 figures; accepted for publication in Physical Review

Stochasticity and evolutionary stability

In stochastic dynamical systems, different concepts of stability can be obtained in different limits. A particularly interesting example is evolutionary game theory, which is traditionally based on infinite populations, where strict Nash equilibria correspond to stable fixed points that are always evolutionarily stable. However, in finite populations stochastic effects can drive the system away from strict Nash equilibria, which gives rise to a new concept for evolutionary stability. The conventional and the new stability concepts may apparently contradict each other leading to conflicting predictions in large yet finite populations. We show that the two concepts can be derived from the frequency dependent Moran process in different limits. Our results help to determine the appropriate stability concept in large finite populations. The general validity of our findings is demonstrated showing that the same results are valid employing vastly different co-evolutionary processes