Self-organization in a distributed coordination game through heuristic rules

In this paper, we consider a distributed coordination game played by a large number ofagents with finite information sets, which characterizes emergence of a single dominantattribute out of a large number of competitors. Formally, N agents play acoordination game repeatedly, which has exactly N pure strategy Nash equilibria, and all of theequilibria are equally preferred by the agents. The problem is to select one equilibriumout of Npossible equilibria in the least number of attempts. We propose a number of heuristicrules based on reinforcement learning to solve the coordination problem. We see that theagents self-organize into clusters with varying intensities depending on the heuristicrule applied, although all clusters but one are transitory in most cases. Finally, wecharacterize a trade-off in terms of the time requirement to achieve a degree of stabilityin strategies versus the efficiency of such a solution.     

Evolution of ethnocentrism on undirected and directed Barabási-Albert networks

Using Monte Carlo simulations, we study the evolution of contingent cooperation and ethnocentrism in the one-shot game. Interactions and reproduction among computational agents are simulated on undirected and directed Barabási-Albert (BA) networks. We first replicate the Hammond-Axelrod model of in-group favoritism on a square lattice and then generalize this model on undirected and directed BA networks for both asexual and sexual reproduction cases. Our simulations demonstrate that irrespective of the mode of reproduction, the ethnocentric strategy becomes common even though cooperation is individually costly and mechanisms such as reciprocity or conformity are absent. Moreover, our results indicate that the spread of favoritism towards similar others highly depends on the network topology and the associated heterogeneity of the studied population.     

Quantum-classical correspondence and nonclassical state generation in dissipative quantum optical systems

We develop a semiclassical method to determine the nonlinear dynamics of dissipative quantum optical systems in the limit of large number of photons N; it is based on the 1/N-expansion and the quantum-classical correspondence. The method is used to tackle two problems: the study of the dynamics of nonclassical state generation in higher order anharmonic dissipative oscillators and the establishment of the difference between the quantum and classical dynamics of the second-harmonic generation in a self-pulsing regime. In addressing the first problem, we obtain an explicit time dependence of the squeezing and the Fano factor for an arbitrary degree of anharmonism in the short-time approximation. For the second problem, we analytically find a characteristic time scale at which the quantum dynamics differs insignificantly from the classical one.     

Emergent cooperation amongst competing agents in minority games

We study a variation of the minority game. There are N agents. Each has to choose between one of two alternatives every day, and there is a reward to each member of the smaller group. The agents cannot communicate with each other, but try to guess the choice others will make, based only on the past history of the number of people choosing the two alternatives. We describe a simple probabilistic strategy using which the agents, acting independently, and trying to maximize their individual expected payoff, still achieve a very efficient overall utilization of resources, and the average deviation of the number of happy agents per day from the maximum possible can be made O(N^?), for any ?>0. We also show that a single agent does not expect to gain by not following the strategy.     

Why only few are so successful?

In many professions employees are rewarded according to their relative performance. Corresponding economy can be modeled by taking N independent agents who gain from the market with a rate which depends on their current gain. We argue that this simple realistic rate generates a scale-free distribution even though intrinsic ability of agents are marginally different from each other. As an evidence we provide distribution of scores for two different systems (a) the global stock game where players invest in real stock market and (b) the international cricket.     

Geometrical aspects and quantum brachistochrone problem for a collection of N spin-s system with long-range Ising-type interaction

We consider a physical system of N interacting qudits consisting of N spin-s particles coupled via the long-range interaction of Ising-type. We investigate the corresponding dynamics, define the associated quantum state manifold and we give the related Fubini-Study metric. We derive the Gaussian curvature and using the Gauss-Bonnet theorem, we show that the dynamics happen on a two-parametric manifold of spherical topology. We examine the geometrical phase acquired by the system under arbitrary and cyclic evolutions. Further, we study the quantum brachistochrone problem concerning the determination of the smallest possible time to realize a time-optimal evolution. By restricting our study to a two-qubit system under the Ising interaction, a detailed analysis is performed for the Fubini-Study metric, the Gaussian curvature, the geometrical phase and the optimal time in relation with the entanglement of the two qubits.     

Evolution of cooperation in well-mixed N-person snowdrift games

We study the time evolution of cooperation in a recently proposed N-person evolutionary snowdrift game, by focusing on the details of the evolutionary dynamics. It is found that the analytic solution for the equilibrium fraction of cooperators as given previously by the replicator dynamics stems from a balance between the terms: the cost to contribute to a common task and the risk in refusing to participate in a common task. Analytic expressions for these two terms are given, and their magnitudes are studied over the whole range of parameter space. Away from equilibrium, it is the imbalance between these terms that drives the system to equilibrium. A continuous time first-order differential equation for the degree of cooperation is derived, for arbitrary interacting group size N and cost-to-benefit ratio. Analytic solutions to the time evolution of cooperation for the cases of N=2 and N=3 are obtained, with results in good agreement with those obtained by numerical simulations. For arbitrary N, numerical solutions to the equation give the time evolution of cooperation, with the long time limit giving the equilibrium fraction of cooperators.     

The Effects of Sacred Value Networks Within an Evolutionary, Adversarial Game

The effects of personal relationships and shared ideologies on levels of crime and the formation of criminal coalitions are studied within the context of an adversarial, evolutionary game first introduced in Short et al. (Phys. Rev. E 82:066114, 2010). Here, we interpret these relationships as connections on a graph of N players. These connections are then used in a variety of ways to define each player’s “sacred value network”—groups of individuals that are subject to special consideration or treatment by that player. We explore the effects on the dynamics of the system that these networks introduce, through various forms of protection from both victimization and punishment. Under local protection, these networks introduce a new fixed point within the game dynamics, which we find through a continuum approximation of the discrete game. Under more complicated, extended protection, we numerically observe the emergence of criminal coalitions, or “gangs”. We also find that a high-crime steady state is much more frequent in the context of extended protection networks, in both the case of Erd?s-Rényi and small world random graphs.     

General linear response formula for non integrable systems obeying the Vlasov equation

Long-range interacting N-particle systems get trapped into long-living out-of-equilibrium stationary states called quasi-stationary states (QSS). We study here the response to a small external perturbation when such systems are settled into a QSS. In the N → ∞ limit the system is described by the Vlasov equation and QSS are mapped into stable stationary solutions of such equation. We consider this problem in the context of a model that has recently attracted considerable attention, the Hamiltonian mean field (HMF) model. For such a model, stationary inhomogeneous and homogeneous states determine an integrable dynamics in the mean-field effective potential and an action-angle transformation allows one to derive an exact linear response formula. However, such a result would be of limited interest if restricted to the integrable case. In this paper, we show how to derive a general linear response formula which does not use integrability as a requirement. The presence of conservation laws (mass, energy, momentum, etc.) and of further Casimir invariants can be imposed a posteriori. We perform an analysis of the infinite time asymptotics of the response formula for a specific observable, the magnetization in the HMF model, as a result of the application of an external magnetic field, for two stationary stable distributions: the Boltzmann-Gibbs equilibrium distribution and the Fermi-Dirac one. When compared with numerical simulations the predictions of the theory are very good away from the transition energy from inhomogeneous to homogeneous states.     

A myopic random walk on a finite chain

We solve analytically the problem of a biased random walk on a finite chain of ‘sites’ (1,2,…,N) in discrete time, with ‘myopic boundary conditions’—a walker at 1 (orN) at timen moves to 2 (orN − 1) with probability one at time (n + 1). The Markov chain has period two; there is no unique stationary distribution, and the moments of the displacement of the walker oscillate about certain mean values asn → ∞, with amplitudes proportional to 1/N. In the continuous-time limit, the oscillating behaviour of the probability distribution disappears, but the stationary distribution is depleted at the terminal sites owing to the boundary conditions. In the limit of continuous space as well, the problem becomes identical to that of diffusion on a line segment with the standard reflecting boundary conditions. The first passage time problem is also solved, and the differences between the walks with myopic and reflecting boundaries are brought out.     

On the dynamical structure of the Dicke maser model

For suitable states of the Dicke maser model we study the time evolution of the mean photon number n(t) in the limit N → ∞. Here N is the number of maser active atoms. Our starting point is a well-tuned cavity with only one mode of the radiation field excited. Introducing new dynamical variables we are able to exploit fully the conservation laws so as to get a simple but completely rigorous solution. We find that n(t) is periodic, and given by a Weierstrassian elliptic function, provided a net polarization is present in the cavity. No approximation is involved.     

One-Shot Decoupling

If a quantum system A, which is initially correlated to another system, E, undergoes an evolution separated from E, then the correlation to E generally decreases. Here, we study the conditions under which the correlation disappears (almost) completely, resulting in a decoupling of A from E. We give a criterion for decoupling in terms of two smooth entropies, one quantifying the amount of initial correlation between A and E, and the other characterizing the mapping that describes the evolution of A. The criterion applies to arbitrary such mappings in the general one-shot setting. Furthermore, the criterion is tight for mappings that satisfy certain natural conditions. One-shot decoupling has a number of applications both in physics and information theory, e.g., as a building block for quantum information processing protocols. As an example, we give a one-shot state merging protocol and show that it is essentially optimal in terms of its entanglement consumption/production.     

Flatness Is a Criterion for Selection of Maximizing Measures

For the one-dimensional classical spin system, each spin being able to get Np+1 values, and for a non-positive potential, locally proportional to the distance to one of N disjoint configurations set {(j?1)p+1,??,jp}^?, we prove that the equilibrium state converges as the temperature goes to 0. The main result is that the limit is a convex combination of the two ergodic measures with maximal entropy among maximizing measures and whose supports are the two shifts where the potential is the flattest. In particular, this is a hint to solve the open problem of selection, and this indicates that flatness is probably a/the criterion for selection as it was conjectured by A.O. Lopes. As a by product we get convergence of the eigenfunction at the log-scale to a unique calibrated subaction.     

Two-dimensional convection in a finite box dynamics of a convective pattern with variable number of rolls

We propose a model in which a convection dynamics results from the interaction of three hydrodynamic modes. At small enough Prandtl number, we show the existence of a transition from an N-roll pattern to an (N?1) one. This transition may be time dependent, giving rise to relaxation oscillations.     

GLAD: a simple adaptive strategy that yields cooperation in dilemma games

We introduce a simple adaptive rule where agents choose a cooperative effort on a grid. Agents can adjust this effort step by step and G ains and L osses A djust D irections. We show that this process converges to the cooperative outcome in a two-person Prisoners’ Dilemma game, and we provide simulations showing that the results also holds with a larger number of agents.     

Stochastic analysis of an agent-based model

We analyze the dynamics of a forecasting game that exhibits the phenomenon of information cascades. Each agent aims at correctly predicting a binary variable and he/she can either look for independent information or herd on the choice of others. We show that dynamics can be analytically described in terms of a Langevin equation and its collective behavior is described by the solution of a Kramers’ problem. This provides very accurate results in the region where the vast majority of agents herd, that corresponds to the most interesting one from a game theoretic point of view.     

Swarming on Random Graphs

We consider a compromise model in one dimension in which pairs of agents interact through first-order dynamics that involve both attraction and repulsion. In the case of all-to-all coupling of agents, this system has a lowest energy state in which half of the agents agree upon one value and the other half agree upon a different value. The purpose of this paper is to study the behavior of this compromise model when the interaction between the N agents occurs according to an Erd?s-Rényi random graph $\mathcal{G}(N,p)$ . We study the effect of changing p on the stability of the compromised state, and derive both rigorous and asymptotic results suggesting that the stability is preserved for probabilities greater than $p_{c}=O(\frac{\log N}{N})$ . In other words, relatively few interactions are needed to preserve stability of the state. The results rely on basic probability arguments and the theory of eigenvalues of random matrices.     

Boson-like quantum dynamics of association in ultracold Fermi gases

We study the collective association dynamics of a cold Fermi gas of 2N atoms in M atomic modes into a single molecular bosonic mode. When the atomic translational motion is slow compared to the atom-molecule conversion rate, the many-body fermionic problem for 2^M amplitudes is effectively reduced to a dynamical system of min{N, M} + 1 amplitudes, making the solution no more complex than the solution of a two-mode Bose-Einstein condensate and allowing realistic calculations with up to 10⁴ particles. The many-body dynamics is shown to be formally similar to the dynamics of the bosonic system under the mapping of boson particles to fermion holes, producing collective enhancement effects due to many-particle constructive interference.     

From Particle Systems to the Landau Equation: A Consistency Result

We consider a system of N classical particles, interacting via a smooth, short-range potential, in a weak-coupling regime. This means that N tends to infinity when the interaction is suitably rescaled. The j-particle marginals, which obey the usual BBGKY hierarchy, are decomposed into two contributions: one small but strongly oscillating, the other hopefully smooth. Eliminating the first, we arrive to establish the dynamical problem in terms of a new hierarchy (for the smooth part) involving a memory term. We show that the first order correction to the free flow converges, as N →∞, to the corresponding term associated to the Landau equation. We also show the related propagation of chaos.     

Evolutionary minority game on complex networks

In this work we investigate the dynamics of networked evolutionary minority game (NEMG) wherein each agent is allowed to evolve its strategy according to the information obtained from its neighbors in the network. We investigate four kinds of networks, including star network, regular network, random network and scale-free network. Simulation results indicate that the dynamics of the system depends crucially on the structure of the underlying network. The strategy distribution in a star network is sensitive to the precise value of the mutation magnitude L, in contrast to the strategy distribution in regular, random and scale-free networks, which is easily affected by the value of the prize-to-fine ratio R. Under a simple evolutionary scheme, the networked system with suitable parameters evolves to a high level of global coordination among its agents. In particular, the performance of the system is correlated to the clustering property of the network, where larger clustering coefficient leads to better performance.