Evolutionary Voluntary Prisoner’s Dilemma Game under Deterministic and Stochastic Dynamics

We have studied whether living things investigated under the same measuring conditions can generate signals with different types of dynamics. We also wanted to detect the possible effects of low-intensity microwaves using parameters of deterministic chaos. For this purpose, two sets of electroretinograms were analysed by methods aimed at recognizing different types of dynamics. Both sets included the time series recorded from objects exposed to low-intensity microwaves and those that were not exposed. The analytical methods are based on nonlinear forecasting and a “surrogate data” technique. Although the experimental conditions were identical for the two sets, we have shown that both have time series with deterministic and stochastic dynamics. We also found that the use of parameters of deterministic dynamics is insufficient to distinguish between the sets.

Motivated by important geophysical applications we study a non-linear model of geyser dynamics under the influence of external stochastic forcing. It is shown that the deterministic dynamics is substantially dependent on system parameters leading to the following evolutionary scenaria: (i) oscillations near a stable equilibrium and a transient tendency of the phase trajectories to a spiral sink or a stable node (pre-eruption regime), and (ii) fast escape from equilibrium (eruption regime). Even a small noise changes the system dynamics drastically. Namely, a low-intensity noise generates the small amplitude stochastic oscillations in the regions adjoining to the stable equilibrium point. A small buildup of noise intensity throws the system over its separatrix and leads to eruption. The role of the friction coefficient and relative pressure in the deterministic and stochastic dynamics is studied by direct numerical simulations and stochastic sensitivity functions technique.

The longstanding question of how stochastic behaviour arises from deterministic Hamiltonian dynamics is of great importance, and any truly holistic theory must be capable of describing this transition. In this review, we introduce the influence functional formalism in both the quantum and classical regimes. Using this technique, we demonstrate how irreversible behaviour arises generically from the reduced microscopic dynamics of a system-environment amalgam. The influence functional is then used to rigorously derive stochastic equations of motion from a microscopic Hamiltonian. In this method stochastic terms are not identified heuristically, but instead arise from an exact mapping only available in the path-integral formalism. The interpretability of the individual stochastic trajectories arising from the mapping is also discussed. As a consequence of these results, we are also able to show that the proper classical limit of stochastic quantum dynamics corresponds non-trivially to a generalised Langevin equation derived with the classical influence functional. This provides a further unifying link between open quantum systems and their classical equivalent, highlighting the utility of influence functionals and their potential as a tool in both fundamental and applied research.

Ordinary differential equation (ODE) solvers with randomly sampled timestep sizes appear in the context of chaotic dynamical systems, differential equations with low regularity and, implicitly, in stochastic optimization. In this work, we propose and study the stochastic Euler dynamics—a continuous-time Markov process that is equivalent to a linear spline interpolation of a random timestep (forward) Euler method. We understand the stochastic Euler dynamics as a path-valued ansatz for the ODE solution that shall be approximated. We first obtain qualitative insights by studying deterministic Euler dynamics, which we derive through a first-order approximation to the infinitesimal generator of the stochastic Euler dynamics. In the context of linear ODEs, these deterministic Euler dynamics describe the dynamics of the expectation of the stochastic Euler dynamics. Then we show convergence of the stochastic Euler dynamics to the ODE solution by studying the associated infinitesimal generators and by a novel local truncation error analysis. Next, we prove stability by an immediate analysis of the random timestep Euler method and by deriving Foster–Lyapunov criteria for the stochastic Euler dynamics; the latter also yield bounds on the speed of convergence to stationarity. The paper ends with a discussion of second-order stochastic Euler dynamics and a series of numerical experiments that appear to verify our analytical results.

Although the long-term behavior of stochastic evolutionary game dynamics in finite populations has been fully investigated, its evolutionary characteristics in a limited period of time is still unclear. In order to answer this question, we introduce the formulation of the path integral approach for evolutionary game theory. In this framework, the transition probability is the sum of all the evolutionary paths. The path integral formula of the transition probability is expected to be a new mathematical tool to explore the stochastic game evolutionary dynamics. As an example, we derive the transition probability for stochastic evolutionary game dynamics by the path integral in a limited period of time with the updating rule of the Wright-Fisher process.

The impact of initial cooperation fraction on the evolutionary fate in a spatial prisoner's dilemma game

This paper studies the evolutionary prisoner's dilemma game (PDG) in finite dynamic scale-free networks, where the dynamic property is fulfilled by considering an epidemic process in networks. When a person is in infected state, he will not play PDG. Only healthy persons play PDG with their healthy neighbors. Our simulations show that (i) the ratio of healthy persons, R_s, depends not only on the spreading rate &lambda;, but also on the recovery rate &delta;; (ii) The relationship between cooperation behaviors and the spreading rate &lambda; depends on the value of &delta;; (iii) Given the same value of R_s and payoff parameter <i>b</i>, the cooperation frequency <i>f</i> changes with &delta;; (iv) Some curves of <i>f</i> against R_s are monotonic while others are non-monotonic. We have qualitatively explained results (ii)-(iv) through competition mechanism of cooperation enhancement effect and cooperation suppression effect. Our work sheds some lights on the important effect of dynamic topology on evolutionary game.

Chaos, oscillation and the evolution of indirect reciprocity in n-person games

Selection intensity and risk-dominant strategy: A two-strategy stochastic evolutionary game dynamics in finite population

Evolutionary game dynamics in finite size populations can be described by a fitness-dependent Wright-Fisher process. We consider symmetric 2×2 games in a well-mixed population. In our model, two parameters to describe the level of player's rationality and noise intensity in environment are introduced. In contrast with the fixation probability method that used in a noiseless case, the introducing of the noise intensity parameter makes the process an ergodic Markov process and based on the limit distribution of the process, we can analysis the evolutionary stable strategy (ESS) of the games. We illustrate the effects of the two parameters on the ESS of games using the Prisoner's dilemma games (PDG) and the snowdrift games (SG). We also compare the ESS of our model with that of the replicator dynamics in infinite size populations. The results are determined by simulation experiments.

Biological populations obeying difference equations: The effects of stochastic perturbation

Linear quadratic regulation of systems with stochastic parameter uncertainties

Economic MPC of Markov Decision Processes: Dissipativity in undiscounted infinite-horizon optimal control

&nbsp; Stochastic evolutionary game dynamics in hierarchy are very popular in economics and in management sciences. This study extends the model of stochastic evolutionary game dynamics. This paper highlights hierarchy selection model and we show that the hierarchy selection games are more stable without structures in this paper. Furthermore, an example in industrial organization is given to rationally explain the theory in this work. &nbsp; Key words:&nbsp;Fitness, fixation probability, invasion, game theory, economics, hierarchy, evolutionary game.

Effects of inertia on the evolution of cooperation in the voluntary prisoner’s dilemma game