Time averages, recurrence and transience in the stochastic replicator dynamics

We investigate the long-run behavior of a stochastic replicator process, which describes game dynamics for a symmetric two-player game under aggregate shocks. We establish an averaging principle that relates time averages of the process and Nash equilibria of a suitably modified game. Furthermore, a sufficient condition for transience is given in terms of mixed equilibria and definiteness of the payoff matrix. We also present necessary and sufficient conditions for stochastic stability of pure equilibria.Comment: Published in at http://dx.doi.org/10.1214/08-AAP577 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Mobility and asymmetry effects in one-dimensional rock-paper-scissors games

As the behavior of a system composed of cyclically competing species is strongly influenced by the presence of fluctuations, it is of interest to study cyclic dominance in low dimensions where these effects are the most prominent. We here discuss rock-paper-scissors games on a one-dimensional lattice where the interaction rates and the mobility can be species dependent. Allowing only single site occupation, we realize mobility by exchanging individuals of different species. When the interaction and swapping rates are symmetric, a strongly enhanced swapping rate yields an increased mixing of the species, leading to a mean-field like coexistence even in one-dimensional systems. This coexistence is transient when the rates are asymmetric, and eventually only one species will survive. Interestingly, in our spatial games the dominating species can differ from the species that would dominate in the corresponding nonspatial model. We identify different regimes in the parameter space and construct the corresponding dynamical phase diagram.Comment: 6 pages, 5 figures, to appear in Physical Review

The Dynamics of Asymmetric Games

A game dynamical analysis of a simple asymmetric game (two roles with two alternatives each) shows that an interesting class of "semi-stable" heteroclinic cycles leading to a highly unpredictable behavior can occur in a robust way. Biological examples related to conflicts over ownership and parental investment are analyzed

Enhancement of cooperation in highly clustered scale-free networks

We study the effect of clustering on the organization of cooperation, by analyzing the evolutionary dynamics of the Prisoner's Dilemma on scale-free networks with a tunable value of clustering. We find that a high value of the clustering coefficient produces an overall enhancement of cooperation in the network, even for a very high temptation to defect. On the other hand, high clustering homogeneizes the process of invasion of degree classes by defectors, decreasing the chances of survival of low densities of cooperator strategists in the network.Comment: 4 pages, 3 figure

Stochastic gain in population dynamics

We introduce an extension of the usual replicator dynamics to adaptive learning rates. We show that a population with a dynamic learning rate can gain an increased average payoff in transient phases and can also exploit external noise, leading the system away from the Nash equilibrium, in a reasonance-like fashion. The payoff versus noise curve resembles the signal to noise ratio curve in stochastic resonance. Seen in this broad context, we introduce another mechanism that exploits fluctuations in order to improve properties of the system. Such a mechanism could be of particular interest in economic systems.Comment: accepted for publication in Phys. Rev. Let

Impact of generalized benefit functions on the evolution of cooperation in spatial public goods games with continuous strategies

Cooperation and defection may be considered as 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.Comment: 8 two-column pages, 8 figures; accepted for publication in Physical Review

Dephasing of quantum dot exciton polaritons in electrically tunable nanocavities

We experimentally and theoretically investigate dephasing of zero dimensional microcavity polaritons in electrically tunable single dot photonic crystal nanocavities. Such devices allow us to alter the dot-cavity detuning in-situ and to directly probe the influence on the emission spectrum of varying the incoherent excitation level and the lattice temperature. By comparing our results with theory we obtain the polariton dephasing rate and clarify its dependence on optical excitation power and lattice temperature. For low excitation levels we observe a linear temperature dependence, indicative of phonon mediated polariton dephasing. At higher excitation levels, excitation induced dephasing is observed due to coupling to the solid-state environment. The results provide new information on coherence properties of quantum dot microcavity polaritons.Comment: Figure 2, panel (b) changed to logarithmic + linear scal

Random replicators with high-order interactions

We use tools of the equilibrium statistical mechanics of disordered systems to study analytically the statistical properties of an ecosystem composed of N species interacting via random, Gaussian interactions of order p >= 2, and deterministic self-interactions u <= 0. We show that for nonzero u the effect of increasing the order of the interactions is to make the system more cooperative, in the sense that the fraction of extinct species is greatly reduced. Furthermore, we find that for p > 2 there is a threshold value which gives a lower bound to the concentration of the surviving species, preventing then the existence of rare species and, consequently, increasing the robustness of the ecosystem to external perturbations.Comment: 7 pages, 4 Postscript figure

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

Random Topologies and the emergence of cooperation: the role of short-cuts

We study in detail the role of short-cuts in promoting the emergence of cooperation in a network of agents playing the Prisoner's Dilemma Game (PDG). We introduce a model whose topology interpolates between the one-dimensional euclidean lattice (a ring) and the complete graph by changing the value of one parameter (the probability p to add a link between two nodes not already connected in the euclidean configuration). We show that there is a region of values of p in which cooperation is largely enhanced, whilst for smaller values of p only a few cooperators are present in the final state, and for p \rightarrow 1- cooperation is totally suppressed. We present analytical arguments that provide a very plausible interpretation of the simulation results, thus unveiling the mechanism by which short-cuts contribute to promote (or suppress) cooperation