Sample Path Large Deviations for Stochastic Evolutionary Game Dynamics

Sample path Large Deviations and optimal importance sampling for stochastic volatility models

  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.   Key words: Fitness, fixation probability, invasion, game theory, economics, hierarchy, evolutionary game.

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

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.

Gaussian Stochastic Volatility Models: Scaling Regimes, Large Deviations, and Moment Explosions

Stochastic evolutionary game dynamics: foundations, deterministic approximation, and equilibrium selection

Large deviation principles for sequences of logarithmically weighted means

By proving the continuity of multi-dimensional Skorokhod maps in a quasi-linearly discounted uniform norm on the doubly infinite time interval R, and strengthening know sample path large deviation principles for fractional Brownian motion to this topology, we obtain large deviation principles for the image of multi-dimensional fractional Brownian motions under Skorokhod maps as an immediate consequence of the contraction principle. As an application, we explicitly calculate large deviation decay rates for steady-state tail probabilities of certain queueing systems in multi-dimensional heavy traffic models driven by fractional Brownian motions.

The voluntary prisoner’s dilemma (VPD) game has sparked interest from various fields since it was proposed as an effective mechanism to incentivize cooperative behavior. Current studies show that the inherent cyclic dominance of the strategies of the VPD game results in periodic oscillations in population. This paper investigated the influence of the level of individual rationality and the size of a population on the evolutionary dynamics of the VPD game. Different deterministic dynamics, such as the replicator dynamic, the Smith dynamic, the Brown-von Neumann-Nash (BNN) dynamic and the best response (BR) dynamic, for the evolutionary VPD game were modeled and simulated. The stochastic evolutionary dynamics based on quasi birth and death (QBD) process was proposed for the evolutionary VPD game and compared with deterministic dynamics. The results indicated that with the increase of the loners’ fixed payoff, the loner is more likely to remain in the stable state of a VPD game under any of the dynamics mentioned above. However, the different speeds of motion under the dynamics in the cycle dominance proved to be diverse under different evolutionary dynamics and also highly sensitive to the rationality of individuals in a population. Furthermore, in QBD stochastic dynamics, the size of the population has a remarkable effect on the possibility distribution. When the population size increases, the limited distribution of the QBD process will be in accordance with the results in the deterministic dynamics.

This paper introduces a new approach for the local reconstruction of coupled map lattice (CML) models of stochastic spatio-temporal dynamics from measured data. The nonlinear functionals describing the evolution of the spatio-temporal patterns are constructed using B-spline wavelet and scaling functions. This provides a multiresolution approximation for the underlying spatio-temporal dynamics. An orthogonal least squares algorithm is used to determine significant terms from wavelet functions to form an accurate representation of the nonlinear spatio-temporal dynamics. Two examples are used to demonstrate the application of the proposed new approach.

Mutation–selection equilibrium in games with mixed strategies

In physics, there is a growing interest in studying stochastic processes described by evolution equations such as nonlinear master equations and nonlinear Fokker–Planck equations that define the so-called nonlinear Markov processes and are nonlinear with respect to probability densities. In this context, however, relatively little is known about nonlinear Markov processes defined by nonlinear Markov chains. In the present work, we demonstrate explicitly how the nonlinear Markov chain approach can be carried out by addressing a model for evolutionary population dynamics. In line with the nonlinear Markov chain approach, we derive a measure that tells us how attractive it is for a biological entity to evolve towards a particular biological type. Likewise, a measure for the noise level of the evolutionary process is obtained. Both measures are found to be implicitly time dependent. Finally, a simulation scheme for the many-body system corresponding to the Markov chain model is discussed.

Matrix Models for Population, Disease, and Evolutionary Dynamics

In this paper, we use evolutionary dynamics to understand the origin of social norms and dominant behavioral and cultural trends in social networks that lead to emergence of "hubs" such as few companies, politicians, celebrities, and fashion trends with overwhelming popular support. We define the notion of "behavior networks" and introduce a novel social choice model (i.e. mutation model) for evolutionary dynamics of behavior in social networks that exhibits a rich set of emergent phases of evolution. For behavioral networks with Watts-Strogatz small-world topology covering a broad range from highly clustered nearest-neighbor graphs to random graphs, we demonstrate the existence of S-shape phase transition curves relating diversity of behaviors to the mutation rate. This curve suggests the emergence of four key phases of behavior in the society ranging from one or a few dominant behaviors to collapse (or lack of dominance). Preliminary stability analysis and characterization of properties of equilibria are provided that suggest the proposed social choice model is plausible.

Strong seasonality at high latitudes represents a major challenge for many endotherms as they must balance survival and reproduction in an environment that varies widely in food availability and temperature. To avoid energetic mismatches caused by limited foraging time and stochastic weather conditions, bats employ the energy-saving state of torpor during summer to save accumulated energy reserves. However, at high-latitude small-bats-in-summer face a particular challenge: as nocturnal foragers, they rely on the darkness at night to avoid predators and/or interspecific competition, but live in an environment with short, light summer nights, and even a lack of true night at the northernmost distributions of some bat species. To predict optimal behaviour in relation to latitudinal variation in diurnal cycles, we constructed a stochastic dynamic programming model of bats living at high latitudes. Using a stochastic dynamic programming framework with values that are representative for our study system, we show that individual energetic reserves are a strong driver of daytime use of torpor and night-time foraging behaviour alike, with these linked effects being both temperature- and photoperiod-dependent. We further used the model to predict survival probabilities at five locations across a latitudinal gradient (60.1° N to 70.9° N), finding that combinations of photoperiod and temperature conditions limited population distributions in the model. To verify our model results, we compared predictions for optimal decisions with our own empirical data collected on northern bats (Eptesicus nilssonii) from two latitudes in Norway. The similarities between our predictions and observations provide strong evidence that this model framework incorporates the most important drivers of diurnal decision-making in bat physiology and behaviour. Comparing empirical data and model predictions also revealed that bats facing lighter night conditions further north restrict their mass gain, which strengthens the hypothesis that predation threat is a main driver of bat nocturnality. Our model findings regarding state-dependent decisions in bats should contribute to the understanding of how bats cope with the summer challenges at high latitudes.